ABSTRACT
Statistical Monitoring of a Process with Autocorrelated Output and Observable Autocorrelated Measurement Error
Jesús Cuéllar Fuentes, Ph.D.
Co Mentors: John W. Seaman, Jr., Ph.D., Jack D. Tubbs, Ph.D.
Our objective in this work is to monitor a production process yielding output that is correlated and contaminated with autocorrelated measurement error. Often, the elimination of the causes of the autocorrelation of the measurement error and the reduction of the measurement error to a negligible level is not feasible because of regulatory restrictions, technological limitations, or the expense of requisite modifications. In this process, reference material is measured to verify the performance of the measurement process, before the product material is measured.
We propose the use of a transfer function to account for measurement error in the
product measurements. We obtain the base production signal and use a modified version of the common cause (CC) chart and the special cause control (SCC) chart, originally proposed by Roberts and Alwan (1988), to monitor the base production process. We incorporate control limits in the CC chart as suggested by Alwan (1991) and
Montgomery and Mastrangelo (1991) and add MR-chart to the original SCC chart.
The common cause control (CCC) chart and SCC charts comprise a flexible monitoring scheme capable of detecting not only changes in the process mean, but also shifts in the mean and the variance of the random shocks that generate the base process.
Statistical Monitoring of a Process with Autocorrelated Output and Observable Autocorrelated Measurement Error
by
Jesús Cuéllar Fuentes, M.S.
A Dissertation
Approved by the Department of Statistical Science
______Jack D. Tubbs, Ph.D., Chairperson
Submitted to the Graduate Faculty of Baylor University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Approved by the Dissertation Committee
______John W. Seaman, Jr., Ph.D., Co-Chairperson
______Jack D. Tubbs, Ph.D., Co-Chairperson
______Jane L. Harvill, Ph.D.
______Elisabeth M. Umble, Ph.D.
______Dean M. Young, Ph.D.
Accepted by the Graduate School May 2008
______J. Larry Lyon, Ph.D., Dean
Page bearing signatures is kept on file in the Graduate School. Copyright © 2008 by Jesús Cuéllar Fuentes
All rights reserved
TABLE OF CONTENTS
List of Figures viii
List of Tables xii
List of Abbreviations xv
Acknowledgments xvi
Dedication xvii
CHAPTER 1 – Introduction 1
1.1 Control Charts and Autocorrelated Observations 4
1.2 Schemes to Monitor Processes with Autocorrelated Output with Unobserved Measurement Error 8
1.2.1 Methods that Reduce or Remove Autocorrelation 9
1.2.2 Methods that Account for Autocorrelation 11
1.2.2.1 Observation-based methods. 11
1.2.2.2 Residual-based methods. 13
1.2.2.3 Specialized monitoring schemes. 17
1.3 Schemes to Monitor Processes with Autocorrelated Output and Observed Correlated Measurement Error 18
1.4 Summary and Discussion 29
CHAPTER 2 – Monitoring Schemes 30
2.1 Characteristics of a Successful Monitoring Scheme 31
2.2 Issues to Consider when Monitoring a Process with an Autocorrelated Output and no Observed Measurement Error 32
2.2.1 Special Causes and Types of Disturbances 33
iii 2.2.2 Dynamic Behavior of the Residuals after a Change in the Process Mean 34
2.2.3 Assessing Performance of a Control Scheme 34
2.2.4 Calculation of Control Limits: Moving Range vs. Standard Deviation 36
2.3 Selection of a Monitoring Scheme for a Process with Autocorrelated Output and Unobserved Measurement Error 37
2.3.1 Construction of the Common-Cause Control Chart 39
2.3.2 The Special-Cause Charts and Control Limits. 40
2.3.3 Procedure to Implement the Selected Monitoring Scheme for a Process with Autocorrelated Output and Unobserved Measurement Error 42
2.4 Monitoring Scheme for a Process with Autocorrelated Output and Observed Autocorrelated Measurement Error 44
2.4.1 Procedure to Implement the CCC and SCC Charts to Monitor a Process with Autocorrelated Output and Observed Autocorrelated Measurement Error 47
2.5 Summary and Discussion 49
CHAPTER 3 – Behavior of Observations, Forecasts, and Residuals after Changes in the Mean and Variance 51
3.1 Behavior of the Lead-One Forecasts and Residuals for Processes with Correlated Output and Unobserved Measurement Error 51
3.1.1 Behavior of the Residuals after a Step Shift in the Process Mean 53
3.1.2 Behavior of the Residuals after a Step Change in the Process Variance 60
3.1.3 Behavior of the Observations, Lead-One Forecasts, and Residuals after a Step Change in the Variance of the Random Shocks 62
3.1.4 Behavior of the Observations, Lead-One Forecasts, and Residuals after a Step Change in the Mean of the Random Shocks 67
3.2 Behavior of the Lead-One Forecasts and Residuals for Processes with Correlated Output and with Observed Correlated Measurement Error 69
iv 3.2.1 Behavior of the Residuals after a Step Shift in the Mean of the Base Process 70
3.2.2 Behavior of the Residuals after a Step Shift in the Variance of the Base Process 77
3.2.3 Behavior of the Residuals after a Step Shift in the Mean of the Random Shocks of the Base Process 78
3.2.4 Behavior of the Residuals after a Step Shift in the Variance of the Random Shocks of the Base Process 83
3.3 Summary and Discussion 88
CHAPTER 4 – Illustration of the Monitoring of a Process with Autocorrelated Output and Unobservable Measurement Error 90
4.1 Phase I Implementation 91
4.1.1 Identification of the ARIMA Model and Estimation of Its Parameters 92
4.1.2 Obtaining Lead-One Forecasts 107
4.1.3 Construction of the Common Cause Control Chart 111
4.1.4 Construction of the Special Cause Control Charts 111
4.2 Phase II Implementation – Process Monitoring 116
4.3 Detection of Out-of-Control Conditions 119
4.3.1 Procedure to Generate Data with Out-of-Control Conditions 120
4.3.2 Effect of a Step Shift in the Mean of the Process 123
4.3.3 Effect of a Step Shift in the Mean of the Random Shocks 131
4.3.4 Effect of a Step Shift in the Variance of the Random Shocks 138
4.4 Summary and Discussion 145
v CHAPTER 5 – Illustration of the Monitoring of a Process with Autocorrelated Output and with an Observable Autocorrelated Measurement Error 147
5.1 Phase I of the Implementation of the CCC and SCC Charts 148
5.1.1 Identification of the Transfer Function and Estimation of its Parameters 152
5.1.2 Identification of the Base Process Model and Estimation of its Parameters 158
5.1.3 Construction of the CCC and SCC Charts 168
5.2 Phase II – Implementation of the Monitoring Scheme for the Base Process 173
5.3 Detection of Out-of-Control Conditions 176
5.3.1 Detection of Step Shifts in the Mean of the Base Process 178
5.3.2 Detection of Step Shifts in the Mean of the Random Shocks of the Base Process 185
5.3.3 Detection of Step Shifts in the Variance of the Random Shocks of the Base Process 190
5.4 Summary and Discussion 198
CHAPTER 6 – Performance of the CCC and SCC Charts when the Mean or the Variance of the Base Process or the Random Shock Shift 199
6.1 Description of the Simulation Study 200
6.1.1 Simulation Procedure to Obtain Run Lengths 204
6.2 Determination of the Control Limits for the I-Chart and the MR-Chart 206
6.3 Performance of the CCC and SCC Charts for a Step Shift in the Mean of the Base Process 209
6.4 Performance of the CCC and SCC Charts for a Step Shift in the Mean of the Random Shocks of the Base Process 216
6.5 Performance of the CCC and SCC Charts for a Step Shift in the Variance of the Random Shocks of the Base Process 222
6.6 Summary and Discussion 225
vi CHAPTER 7 – Conclusions and Future Research 227
APPENDICES 232
Appendix A – Literature Summary 233
Appendix B – Variance of the Base Process 240
Appendix C – SAS Code 242
REFERENCES 263
vii
LIST OF FIGURES
Figure 1. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process (P), the measurement process (X), and overall process (Y). 2
( prod ) Figure 2. Aggregation of the measurement process signal X t and the intrinsic production process signal Pt into the output signal Yt. 18
Figure 3. Change in the measurement process from measuring the reference materials. 24
Figure 4. Diagram representing the use of the transfer function as a filter to obtain the base process signal. 28
Figure 5. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process and the measurement process. 30
Figure 6. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process and the measurement process. 90
Figure 7. Time plot of Phase I measurement process data. 93
Figure 8. ACF of the measurement process series. 95
Figure 9. PACF of the measurement process series. 95
Figure 10. ACF and PACF of the residuals from model (4.10). 105
Figure 11. ACF and PACF of the residuals from model (4.11). 105
Figure 12. Histogram and normal probability plot of the residuals from model (411). 106
Figure 13. Observed versus forecasted measurement process series. 110
Figure 14. CCC chart for the Phase I measurement process data. 112
Figure 15. I-chart and MR-chart of the residuals from model (4.11). 114
Figure 16. Distribution gamma(1, 4) fitted to the observed MR values. 116
viii Figure 17. CCC Chart of Phase II Data 119
Figure 18. I-chart and MR-chart of Phase II data. 120
Figure 19. CCC and SCC charts with a 3σW step shift in mean of the process 125
Figure 20. Expected behavior of the residuals after a shift in the mean of the measurement process. 128
Figure 21. CCC and SCC Charts with a 2σW Step Shift in Mean of the Process. 129
Figure 22. CCC and SCC charts with a1.5σW step shift in mean of the process. 130
Figure 23. CCC and SCC Charts with a 3σa step shift in the mean of the random shocks. 132
Figure 24. CCC and SCC charts with a 2σa step shift in the mean of the random shocks. 136
Figure 25. CCC and SCC charts with a 1.5σa step shift in the mean of the random shocks. 137
Figure 26. CCC and SCC charts with a 3σa step shift in the variance of the random shocks. 139
Figure 27. CCC and SCC charts with a 2σa step shift in the variance of the random shocks. 143
Figure 28. CCC and SCC charts with a 0.5σa step shift in the variance of the random shocks. 144
Figure 29. Time plot of the measurement process and overall process observations. 153
Figure 30. ACF and PACF of the overall process series. 154
Figure 31. CCF of the prewhitened Xt and Yt series. 155
Figure 32. ACF and PACF of the residuals after model (5.8) is fitted. 158
Figure 33. ACF and PACF of the residuals ut from model (3.35). 164
Figure 34. Time plot of the overall process observations and their lead-one forecasts. 166
Figure 35. Histogram and normal probability plot of the residuals from model (5.11). 167
ix Figure 36. CCC chart for the overall process observations. 170
Figure 37. SCC charts of the residuals of the overall process. 172
Figure 38. Phase II CCC chart of the overall process observations. 175
Figure 39. Phase II I-chart and MR-chart of residuals. 175
Figure 40. CCC and SCC charts of the overall process with a 3σu shift in the mean of the base process. 179
Figure 41. Expected behavior of the overall process observations, lead-one forecasts, and residuals after a 3σu shift in the mean of the base process. 182
Figure 42. CCC and SCC charts of the overall process with a 2σu shift in the mean of the base process. 183
Figure 43. CCC and SCC charts of the overall process with a 1.5σu shift in the mean of the base process. 184
Figure 44. CCC and SCC charts of the overall process with a 3σu shift in the mean of the random shocks of the base process. 186
Figure 45. CCC and SCC charts of the overall process with a 2σu shift in the mean of the random shocks of the base process. 189
Figure 46. CCC and SCC charts of the overall process with a 1.5σu shift in the mean of the random shocks of the base process. 190
Figure 47. CCC and SCC charts of the overall process with a 3σu shift in the variance of the random shocks of the base process. 191
Figure 48. CCC and SCC charts of the overall process with a 2σu shift in the variance of the random shocks of the base process. 192
Figure 49. CCC and SCC charts of the overall process with a 0.5σu shift in the variance of the random shocks of the base process. 193
Figure 50. Probability and cumulative probability distribution functions of the run length for all charts combined for φbp = – 0.95 and θbp = – 0.90 a step shift in the mean of the base process. 210
Figure 51. Behavior of the observed, lead-one forecast, and residuals values after a shift in the mean of the random shocks at observation 100 and for combinations of φbp and θbp. 221
x Figure 52. CDF plots of the run length for all charts combined for different shifts in the mean of the random shocks of the base process. 221
Figure 53. CDF plots of the run length for all charts combined for different shifts in the variance of the random shocks of the base process. 225
xi
LIST OF TABLES
Table 1. Description of the Special-Cause Rules 6
Table 2. Classification of Methods to Monitor Processes with Autocorrelated Data 9
Table 3. Decisions when the Overall and the Measurement Processes are monitored separately 21
Table 4. Transient Behavior and Steady State Level of Residuals for Various Process Models 59
Table 5. Guidelines to identify a model based on the ACF and PACF 96
Table 6. Conditional Least Squares Estimates of the Coefficient in Model with Components at Lags 1, 2, 3, 4, 10, 11, 12, and 13 102
Table 7. Conditional Least Squares Estimates of the Coefficients in Model (4.10) 102
Table 8. Conditional Least Squares Estimates of the Coefficients in Model (4.11) 103
Table 9. Portmanteaus Test of Autocorrelation of the Residuals Model (4.10) 103
Table 10. Portmanteaus Test of Autocorrelation of the Residuals Model (4.11) 104
Table 11. Comparison of Model Selection Criteria 106
Table 12. Tests of Normality of the Residuals from Model (4.11) 106
Table 13. Out-of-Control Conditions 121
Table 14. Portmanteau Test of the Cross-Correlation of the Residuals,
For the Transfer function vBˆ( ) = ωˆ 0 156
Table 15. Portmanteau Test of the Cross-Correlation of the Residuals For the Transfer function (3.31) 157
Table 16. Conditional Least Squares Estimates of the parameters of Model (5.10). 159
Table 17. Portmanteau Lack of Fit Test that the Autocorrelation of the Residuals of Model (5.7) are Equal to Zero 160
xii Table 18. Portmanteau Lack of Fit Test that the Cross-Correlation of the Residuals of Model (5.10) and the Xt series are Equal to Zero 161
Table 19. Conditional Least Squares Estimates of the parameters of Model (5.11) 162
Table 20. Portmanteau Lack of Fit Test that the Autocorrelation of the Residuals of Model (5.11) are Equal to Zero 162
Table 21. Portmanteau Lack of Fit Test that the Cross-Correlation of the Residuals of Model (5.11) and the Xt series are Equal to Zero 163
Table 22. Comparison of Variance Estimates and Information Criteria for Models (5.10) and (5.11) 165
Table 23. Correlation between Parameter Estimates of Model (5.11) 165
Table 24. Tests for Normality of the Residuals from Model (3.35) 168
Table 25. Types of Step Shifts and their Magnitudes 176
Table 26. Expected values of the lead-one forecasts and the Residuals of the Overall Process 182
Table 27. Types of Step Shifts and their Magnitude 201
Table 28. Simulation Design Points for the Parameters φbp, and θbp each at 3 Levels 202
Table 29. Designed Experiment of the Parameters φbp, and θbp each at 3 Levels 208
Table 30. Run Length Summary Statistics for step shifts in the mean of the base process 211
Table 31. Run Length Summary Statistics for Step Shifts in the Mean of the Random Shocks 217
Table 32. Run Length Summary Statistics for Step Shifts in the Variance of the Random Shocks 223
xiii
LIST OF ABBREVIATIONS
Page where First Acronym Meaning appears ACF Autocorrelation function 17 AR Autoregressive 17 ARL Average run length 16 ARMA Autoregressive moving average 10 ARIMA Autoregressive integrated moving average 9 CC chart Common cause chart 37 CCC chart Common cause control chart 37 CCF Cross-correlation function 149 CDF Cumulative distribution function 202 CL Center line 6 CSE Combined Shewhart and EWMA chart 14 CUSUM Cumulative sum 8 EWMA Exponentially weighted moving average 10 EWMV Exponentially waited moving variance 12 EWMV Exponentially weighted root mean square 12 GLRT Generalized likelihood ratio test 14 I-chart Individual observation control chart 7 IID Individually and identically distributed 6 IMA Integrated moving average 14 IQRRL Run length interquartile range 202 LCL Lower control limit 5 MCAP Max-CUSUM for autocorrelated processes 15 MR-chart Moving range control chart 7 MRL Median run length 202 OGLF Optimal generalized linear filter 16 OSLF Optimal second-order linear filter 16 PACF Partial autocorrelation function 92 PID Proportional-Integral-Derivative 14 RL Run length 34 RMA Reverse moving average 15 SCC chart Special cause control chart 37 SCR Special cause rule 4 SDRL Standard deviation of the run length 34 SPC Statistical process control 2 UCL Upper control limit 5
xiv
ACKNOWLEDGMENTS
My wife and my daughters deserve my most gratitude for their encouragement, support, and patience. Dr. Seaman, Dr. Tubbs, and Dr. Young thank you for your continuous encouragement and valuable conversations. Also, I want to thank the faculty and staff of the Department of Statistical Science for their teachings and help.
xv
A Yolanda, Victoria, Amanda y Julieta. ¡La esencia de mi vida!
A mis papas. ¡Mi cimiento!
xvi
CHAPTER ONE
Introduction
The subject of this work was motivated by the need to monitor a pharmaceutical
process. An assay is performed to assess the properties of the active pharmaceutical ingredient and the final drug product that the patient will use. In this assay the pharmaceutical product is dissolved in a buffered solution and placed into a transparent holder (called a cuvette). A beam of light of a specific wavelength is shone onto the sample and the amount of light absorbed is measured. The performance of the measurement process is assessed by measuring a sample of a reference or control material with an absorbance that is known to be within a specified tolerance. The measurement process is affected by variations in the batches of buffered solutions, by the deterioration of the instrument’s lamp and optical detectors, by the analyst’s technique, by the instrument setup, and by the variation of the physical characteristics of the cuvettes.
We will use Yt to represent the absorbance of the product solution, Xt to represent
the absorbance of the reference material, and Pt to represent the intrinsic absorbance of
the product generated by the base production process. The subscript t represents the time
period at which the absorbance value is determined. The quantity Pt cannot be measured
unless it goes through the measurement process. The measurement process
( prod ) “contaminates” the actual absorbance value, Pt, with a measurement error, X t , producing the measurement of the product material, Yt. We will refer to Yt as the output
1 of the overall process because it contains the intrinsic absorbance value and the product measurement error. Also, we will refer to Xt as the output of the measurement process and it represents the measurement error of the reference material. Figure 1 depicts this pharmaceutical process.
Base Process
Product P Y
Measurement Process
X Reference
Figure 1. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process (P), the measurement process (X), and overall process (Y).
The overall process and the measurement process generate values that are naturally autocorrelated. In the real-world pharmaceutical problem that motivated this research, the removal of the cause or causes of the autocorrelation of the measurement process and the reduction of its contribution to the overall process measurements to a negligible level, is not feasible because of regulatory constraints and expensive requisite modifications.
The objective of the monitoring of the pharmaceutical process is to detect out-of- control conditions of the base production process and eliminate their causes, so that the base process is maintained in control. This requires that we remove the measurement error from the overall process observation.
2 Contamination of a base process by a measurement process is certainly not unique
to pharmaceutical production. For example, another process where the product material
and a reference material are measured is in the production of metallic pigments. Thin
metal plates are ground while bathed in oil to provide cooling and a suspension for the
fine metal particles, which are the final product. The key characteristic of the product is assessed by first measuring the particle size of a reference suspension with metal particles of known size, Xt, and then measuring a sample of the suspension generated by the base
process, Yt. The objective of measuring the reference suspension first is to verify the
performance of the measurement process. In both examples the overall process and the
measurement process observations are autocorrelated. Another motivation for this work
is to present a practical monitoring scheme that can be used in an industrial setting. In
fact, we believe that the most important contribution of this work is to provide a practical
perspective for a user of statistical process control (SPC) in a typical manufacturing
operation with autocorrelated output; that is, we provide a usable methodology for
someone with knowledge of basic process improvement principles, SPC control charts,
and regression analysis such as those holding Six Sigma Green or Black Belts.
This chapter is organized as follows. In Section 1.1 we discuss control charts in
general and how autocorrelation affects their performance. In Section 1.2 we summarize
the different methods that have been proposed to monitor processes with an autocorrelated output, but with an unobserved measurement error. Also, in this section we discuss possible ways to monitor the base process. In Section 1.3 we summarize and discuss our findings. The specific contributions made in this chapter are:
3 • A synthesis of the methods that have been proposed to monitor processes with an
autocorrelated output over the last 25+ years.
• A summary and a discussion of the issues that affect the performance of a control
charts when data are correlated.
• A discussion of different approaches to monitor processes with autocorrelated output
and an observable autocorrelated measurement error.
The remainder of the dissertation is organized as follows. In Chapter 2 we
discuss the characteristics of a successful monitoring scheme and, based on these, we
propose a monitoring scheme for the measurement process and the base process. In
Chapter 3 we develop the expressions that describe the behavior of the observations, forecasts, and residuals after the process is affected by different types of disturbances.
The implementation of the monitoring of the measurement process is illustrated in
Chapter 4 and in Chapter 5 we illustrate the implementation of the monitoring of the base
process. In Chapter 6 we present a simulation study to investigate the detection
capability of the proposed monitoring scheme. Finally, in Chapter 7 we summarize our
findings, establish conclusions, and suggest future research.
1.1 Control Charts and Autocorrelated Observations
A desired property of a process is that it is in a state of control. A process is said
to be in a state of control, or stable, when based on past observations the process output
characteristic can be expected to vary within specified limits (Alwan and Roberts, 1988).
That is, the location, variation, and shape of the distribution of the observations of a
4 process output characteristic remains constant over time. The measurement of the output
characteristic at time t, Xt, of a stable process can be represented as
X t= μ+εt, (1.1)
where μ represents the mean of the process output and εi is a sequence of independently
2 and identically distributed random variables with E[εt] = 0 and Var[εt] = σε (Harris and
Ross, 1991, Montgomery and Mastrangelo, 1991, Box and Luceño, 1997; p. 13).
However, in most cases only Xt is observed, therefore a stable process will have E[Xt] = μ
2 2 and Var[Xt] = σX = σε .
Statistical Process Control (SPC) is the methodology used to achieve process
stability. SPC is generally implemented in two phases. In Phase I data are collected
when the process is believed to be stable. These data are used to calculate control limits
and construct control charts to monitor the location and the dispersion of the distribution
of the output characteristic.
In Phase II, the process is monitored using the control charts with the control
limits calculated in Phase I. If an observation violates a Special-Cause Rule (SCR), the
process is said to be out of control. Table 1 summarizes the SCRs (Nelson, 1984). SCR
1 is always applied. The other rules are applied according to the effect of potential
special causes on the quality characteristic Y. For example, SCR 3 may be implemented
to accelerate the detection of a change in the mean of the process, μ.
If an observation is larger than the upper control limit (UCL), smaller than the
lower control limit, (LCL), or violates a specified SCR, the process is said to be out of
control. Generally, when an out-of-control condition is detected the process is stopped;
the cause of the out-of-control condition is identified, and permanently eliminated. Then,
5 the process operation is resumed assuming that the process has been returned to its original state of control (Woodall and Montgomery 1999, Montgomery 1996, Jensen et
al. 2006).
Table 1. Description of the Special-Cause Rules
Special-Cause Description Rule No.
1 One point outside the control limits
2 Nine points in a row above or below the center line
3 Six points in a row steadily increasing or decreasing
4 Fourteen points in a row alternating up and down
Two out of three points in the region between 2 and 3 5 standard deviations above or below the center line. Four out of five points in a row in the region beyond 1 6 standard deviation above or below the center line Fifteen points in a row within ±1 standard deviation of the 7 center line Eight points in a row on both sides of the center line with 8 none within the ±1 standard deviation of the center line
A control chart is a time-ordered plot of a statistic of the observed data (means, a
single observation, standard deviations, ranges, etc.) with a fixed upper control limit,
lower control limit, and center line (CL). The UCL and LCL are calculated such that 99 to 100% of the observations fall between these limits when the process is stable. Control
charts are used in pairs. One control chart monitors the location of the process (e.g. the
mean) and the second chart monitors the variation of the process (e.g. the standard
deviation). The CL is just the mean of the plotted statistic. Therefore, the control limits
6 are traditionally placed 3 standard deviations above and below the mean. The generic
definition of the control limits of a control chart are:
LCL = μ−3 σε CL =μ (1.2)
UCL = μ+3 σε
Where μ is the mean and σε is the standard deviation of the plotted statistic. For example, if the plotted statistic is normally distributed, then about 99.73% of the values will be within the UCL and LCL calculated using (1.2).
This specification of the control limits assumes that the values of the statistic
plotted on a control chart are randomly selected from a common distribution with a mean
μ and a standard deviation σε. That is, the plotted values of the statistic are independent
and identically distributed (IID). The data collected during Phase I are used to obtain
estimates of μ and σε. In Phase II the estimates of μ and σε are traditionally used as if
they were fixed and equal to the actual parameters of the distribution of the plotted
statistic; i.e. the estimates of μ and σε are not updated using Phase II observations.
If the values of the plotted statistic are not independent, a higher number of false
out-of-control signals are observed (Faltin, Mastrangelo, Runger, Ryan, 1997). For
example, Maragah (1989) showed that for an autoregressive process of order 1, AR(1),
the control limits are narrower with positive autocorrelation and wider for negative
autocorrelation than the control limits for independent data, leading to a higher or lower number of false out-of-control signals than for a process with independent observations.
There are processes where a single measurement of the quality characteristic (n =
1) is sufficient to assess the performance of the process. The pharmaceutical process that motivated our research is an example of this. In this case the statistic to plot on the
7 control chart to monitor the mean is the individual observation (the Individuals control
chart, or the I-chart), and the moving range on a second chart to monitor the process
variation (the moving range chart or the MR-chart). There are other processes where
more than one measurement (n ≥ 1) is needed to assess their performance, for example
measuring five consecutive pieces of product or measuring the product on five different
locations (n = 5). In these cases, the mean of the n observations is plotted on the control
chart to monitor the process location (X-bar control chart), and the standard deviation of
the n observations on the control chart to monitor the process variation (S control chart).
If the individual observations or the means were autocorrelated and the control limits were calculated using (1.2), then the I-chart or the X-bar chart will either generate more false alarms than expected or will not detect real out-of-control situations.
1.2 Schemes to Monitor Processes with Autocorrelated Output with Unobserved Measurement Error
A plethora of approaches have been proposed to deal with processes that naturally generate serially dependent observations, where the measurement error is not considered and where the removal of the sources of autocorrelation is not practical or economically
feasible. Frequent sampling of continuous processes, for example water treatment
(Berthouex, Hunter, and Pallesen, 1978), nuclear reactions (Ermer, Chow, Wu, 1979),
and clean room air purification (Ramiréz, 1998), generate serially correlated data. Also, batch processes with carryover effects generate serially correlated observations, like
biochemical processes (Winkel and Zhang, 2004), or measurement processes such as the
one in our motivating pharmaceutical process.
There have been a large number of methods proposed to monitor processes with
autocorrelated observations. These methods have been classified as residual-based
8 control charts or observation-based control charts with modified control limits (Lu and
Reynolds, 1999a; Lee, 2004). A more discriminating classification of methods proposed for this problem seems to be in order. Specifically, we distinguish between methods that reduce, remove, or account for autocorrelation. We now summarize these methods. In the following, refer to the taxonomy presented in Table 2.
1.2.1 Methods that Reduce or Remove Autocorrelation
In our discussion of the methods summarized in Table 2, we shall refer to the
author and date of publication, as usual, as well as the reference number used in the table.
The references are organized alphabetically at the end of the dissertation, and
chronologically in Appendix A. Yaschin (1993, 21) proposed monitoring a sequence of
score transforms with a cumulative sum (CUSUM) chart. These scores are the logarithm
of the ratio of the on- and off-target density functions, which substantially reduce the
magnitude of the serial correlation.
Runger and Willemain (1995, 27) proposed two monitoring schemes for
processes that generate a large amount of serial data. The first scheme constructs
weighted averages of consecutive data that are uncorrelated. The weights are derived
based on the underlying autoregressive integrated moving average (ARIMA) process and
the batch size is selected to detect a specified shift in the process mean. In the second
scheme, called un-weighted batch means, consecutive data is grouped into batches. The
batch size is selected to reduce the lag 1 autocorrelation of the means to 0.1 or less. The
advantage of this approach is that a time series model is not needed. Similarly,
Willemain and Runger (1995, 27) introduced a monitoring scheme based on the run
lengths of observations above and below a target value. Since these run lengths are IID
9 any of the standard control charts for IID data (Shewhart, CUSUM, exponentially weighted moving average (EWMA)) can be used to monitor the mean of the process.
Table 2. Classification of Methods to Monitor Processes with Autocorrelated Data (number in parenthesis is the reference number in Appendix A; for an explanation of the acronyms used in the table, see the accompanying text).
Transformation (21) Run Sums (42) Reduce/Remove Run Lengths (42) Autocorrelation Weighted batch means (28) Un-weighted batch means (28) Adjust X, X-Bar, S chart limits (5, 15, 26, 49) Adjust CUSUM (2, 13, 27, 33, 41, 53, 56) Observation-based Adjust EWMA (26, 33, 39, 45) Methods ARMA chart (48) T2 control chart (55) Approach EWMV and EWMS (22) Fit ARIMA - Residual & forecast (7, 14) Fit EWMA - Residual forecast & Observed (16) Fit ARIMA - CSE (29) Fit ARIMA - Generalized likelihood ratio test chart (44) Residual-based Fit ARIMA - log res2, res2charts (46) Account for Methods Fit PID model - PID control chart (58) autocorrelation Fit ARIMA- Combined X-S2 X and EWMA charts (59) Fit ARIMA - Worst-case EWMA (62, 65) Fit ARIMA - Reverse moving average chart (63) Fit ARIMA - MCAP for mean and variance (66) Fit ARIMA - OSLF chart (70) Fit ARIMA - OGLF chart (71)
Spectral control chart (8) Specialized Detect outliers or level shifts (40) Applications Detect changes in autocorrelation structure (57) Adaptive EWMA (61)
10 Also, Willemain and Runger (1998, 41) proposed the use of run sums to monitor the mean of a process with autocorrelated data. The performance of control charts of run sums is equivalent to that of control charts of IID data.
The proponents of the methods to reduce or eliminate autocorrelation claim that their schemes perform better than the schemes that use a Shewhart chart of individual forecast errors from an autoregressive moving average (ARMA) model. However,
Yashin’s (1993, 21) approach is cumbersome and requires simulation to tune the parameters of his CUSUM chart. Some of the methods proposed by Willemain and
Runger (1995, 27) have the advantage of not requiring the fit of an ARIMA model, but the batching of consecutive observations, counting or summing run lengths, hides the actual dynamic behavior of the process. Furthermore, the statistic plotted on the control chart cannot be interpreted directly to identify the potential cause of out-of control conditions.
1.2.2 Methods that Account for Autocorrelation
These methods can be further classified as observation-based, residual-based, or specialized monitoring schemes.
1.2.2.1 Observation-based methods. These methods plot statistics of the observed data on control charts with control limits adjusted for the autocorrelation of the data.
Modifications of the individuals (X), mean ( X ), and standard deviation (S) Shewhart charts have been proposed by Vasilopoulos and Stamboulis (1978, 5), English,
Krishnamurti, and Sastri (1991, 15), and Kramer and Schmid (2000, 48). Also, Wardell,
Moskowitz, and Plante (1994b, 25) suggest adding a likelihood ratio statistic to an X
11 chart to aid in deciding if a point that violates the SCR 1 represents an actual out of
control condition.
Procedures to design observation-based CUSUM charts to detect specific changes
in the mean for specific models have been proposed by Johnson and Bagshaw (1975,2),
Harris and Ross (1991, 13), Runger, Willemain, and Prabhu (1995, 26), VanBrackle and
Reynolds (1997, 32), Timmer, Pignatiello, and Longnecker (1998, 40), Lou and Reynolds
(2001, 52), and Atienza, Tang, and Ang (2002a, 55).
Similarly, modifications of the standard, observation-based EWMA chart to
account for specific autocorrelation were proposed by Wardell, Moskowitz, and Plante
(1994b, 25), VanBrackle and Reynolds (1997, 32), Zhang (1998, 38), and Lou and
Reynolds (1999a, 44).
Jiang, Tsui, and Woodall (2000, 47) introduced an observation-based ARMA
chart that plots a statistic based on the first-order autoregressive moving average or
ARMA (1, 1) model. The variance of the autocorrelated process at different lags is used
to compute the control limits. Also, Apley and Tsung (2002, 54) proposed the
autoregressive T2 chart. This control chart monitors the Hotelling’s T2 statistic
constructed using a vectors of specified size of subsequent observations that are updated
as new observations are obtained. The upper control limit is the 1 – α percentile of a χ2 with degrees of freedom equal to the dimension of the vectors.
McGregor and Harris (1993, 22) presented the exponentially weighted moving variance (EWMV) and the exponentially weighted root mean square (EWRMS) charts to monitor the variation of a process that generates individual autocorrelated observations.
These charts use statistics based on the squared deviation of an observation from the
12 mean of the process. The EWRMS statistic uses the squared deviation of the
observations from the known process mean or from a specified target value. The EWMV
statistic uses the squared deviations of the observations from an estimate of the process
mean.
The main advantage of the observation-based methods is that there is no need to
fit a model to represent the autocorrelation of the data. Another advantage is that the
actual observations or statistics based on the observations are displayed on the control chart, facilitating the interpretation of out of control signals. However, as Alwan (1991,
14) points out, the adjustment of the control limits are devised for specific autocorrelation and such adjustments cannot be generalized. Also, the modified control limits of
Shewhart-type charts consider only violations of SCR 1, but because of the nonrandom behavior of the data plotted on the control chart, using the other SCR’s would likely generate false alarms.
1.2.2.2 Residual-based methods. The common characteristic of these methods is
the use an ARIMA or other time-dependent models to generate a sequence of IID lead-
one or one-step ahead forecast errors or residuals (from here on the term residual will
refer to the lead-one forecast error). These residuals are plotted on control charts for IID
data. Several authors have reported the use of this approach (Chow, Wu, and Ermer,
1979; Berthouex, Hunter, and Pallesen, 1978, 4; Notohardjono, Ermer, 1986, 6), however
most of the recent papers refer to the approach proposed by Alwan and Roberts (1988, 7).
The method proposed by these authors consists of the following steps: 1) fitting an
ARIMA model (Box, Jenkins, and Reinsel, 1994); 2) Construct the common-cause (CC)
ˆ chart which is a time-ordered plot of the lead-one forecasts, X t−1 ()1 ; and 3) Construct
13 the special-cause control (SCC) chart which is an individuals Shewhart control chart of
ˆ the residuals, eXXttt=−−1 ()1 , with control limits calculated using (2.1).
Alwan (1991, 14) presented an example of the construction of the CCC and SCC
and proposed the use of a single control chart, instead of two, of lead-one forecasts with
ˆ control limits calculated as in (2.2), but with μ taken to be equal to X t−1 ()1 and σε is estimated by the standard deviation of the residuals.
Montgomery and Mastrangelo (1991, 16) suggested a similar approach as Alwan
and Roberts (1988, 7), but they proposed optimizing the parameter of a EWMA forecast
to approximate the lead-one forecasts of the actual ARIMA model that describes the
process. This approximation is adequate for processes with positive autocorrelation and a
slow moving mean. They suggested constructing two control charts. The first is a time-
ordered plot of the observations, their corresponding EWMA forecasts, Zt, and control
limits constructed around the forecasted value Zt. These control limits are the same as
those proposed by Alwan (1991, 14). The second chart is the same as the SCC chart
proposed by Alwan and Roberts (1988, 7).
Several control charts of residuals have been proposed in an effort to detect small
shifts of the process mean and take advantage of its effect on the residuals (this is
discussed in detail in Chapter 3) or changes in the variance for different ARMA or IMA
(integrated moving average) models.
• CSE Chart – Lin and Adams (1996, 28) proposed a combined residual-based
Shewhart and EWMA (CSE) chart. The Shewhart chart detects large changes in the
residual that occur immediately after a shift in the process mean and the EWMA
14 detects smaller changes in the residuals that occur several observations after the
mean shift.
• GLRT Chart – Apley and Shi (1999, 43) suggested the use of the residual-based
generalized likelihood ratio test (GLRT) control chart. The GLRT statistic plotted
on the chart is computed from the fault signature that accounts for the transient
behavior of the residuals after a change in the process mean has occurred.
• log(e2) EWMA Chart – Lu and Reynolds (1999b, 45) used the EWMA chart of the
logarithm of the squared of the residuals as well as individuals chart of the residuals
squared to monitor the variance of a process with autocorrelated output.
• PID Chart – Jiang. Wu, Tsung, Nair, and Tsui (2002, 57) introduced the
Proportional-Integral-Derivative (PID) chart based on the three-term controller
equation. The PID statistic consists of the proportional (P) term that considers the
direct effect of the forecast error, the integral (I) term considers the additive effect of
current and previous forecast errors, and the derivative (D) term accounts for the
effect due to the difference between the current and the previous forecast error (see
Box, Jenkings, and Reinsel; 1994 p.493). The authors “tune” the PID statistic to
detect a specific change in the mean and to achieve a particular in-control average
run length.
• X - S2 EWMA Chart – This chart was introduced by Knoth and Schmid (2002, 58)
to monitor the mean and the variance of a process that generates autocorrelated
observations. This approach plots the EWMA statistic of the mean of the residuals
and the EWMA of the sample variance of the residuals on the same chart. Two-
sided control limits are used for the EWMA chart on the mean and an upper control
15 limit is used for the EWMA on the variance. If either EWMA statistic exceeds the
corresponding control limits, then the process is out of control.
• Worst-Case EWMA Chart – Apley and Lee (2003, 61) and Lee (2004, 64) presented
a modified residuals-based EWMA chart to account for the uncertainty of the
estimated values of the ARMA model fitted to the data. This worst-case EWMA
chart has wider limits than the EWMA of residuals based on the assumption that
there is no estimation error.
• RMA Chart – Dyer, Adams, and Conerly (2003, 62) proposed the reverse moving
average (RMA) control chart. The RMA is an average of previous residuals. The
number of residuals to average is determined through simulations of the observations
from a specified ARMA and for specific changes in the process mean.
• MCAP Chart – The Max-CUSUM for autocorrelated processes or MCAP chart was
introduced by Cheng and Thaga (2005, 65) to simultaneously monitor the mean and
the standard deviation. They calculate a statistic, Zi, for the mean that is equal to the
Z-score based on the mean and the standard deviation of the stable process. Also,
they calculate a statistic, Yi, for the standard deviation. This statistic is equal to the
inverse function of the normal cumulative distribution of the ratio of the mean-
squared of the residuals and the standard deviation of the stable process. Then,
CUSUM statistics for Zi and Yi are computed. Since Zi and Yi are independent and
have the same distribution, a single statistic, defined as the maximum of the Zi and Yi
CUSUM statistics, is plotted on the CUSUM chart that is used to monitor the
process.
16 • OSLF Chart – Chin and Apley (2006, 69) proposed the optimal second-order linear
filter (OSLF) control chart. The statistic plotted on the chart corresponds to an
ARMA(2, 1) filter of the residuals from an ARMA model. The parameters of the
filter are obtained by a numerical minimization of the out-of-control ARL for a
specified in-control ARL and for a specific change in the process mean (shifts,
spikes, and sinusoidal changes).
• OGLF Charts – The optimal generalized linear filter (OGLF) control chart was also
introduced by Apley and Chin (2007, 70). The statistic plotted on the control chart is
a truncated sum of the weighted residuals. The weights are found by the numerical
minimization of the out-of-control average run length (ARL) for a specified in-
control ARL and for a specific shift in the process mean.
1.2.2.3 Specialized monitoring schemes. These are methods that monitor specific characteristics of a process.
• Spectral Chart – This chart was introduced by Beneke, Leemis, Schiegel, and Foote
(1988, 8) and it is based on the periodogram and its objective is to detect cyclical
patterns in the process data.
• λLS Chart – Atienza, Tang, and Ang (1998, 39) proposed plotting test statistics,
computed using the estimated parameters of an ARMA model, which detect additive
outliers, innovative outliers, or level shifts.
• ACF and Q Charts – The autocorrelation function (ACF) chart and the Box-Ljung-
Pierce statistic (Q) chart were conceived by Atienza, Tang, and Ang (2002b, 56) with
the purpose to detect changes in the stochastic model that describes the
autocorrelation of the process
17 • AEWMA Chart – The adaptive EWMA chart devised by Nembhard and Kao (2003,
60) monitors the output of a process during a process transition (e.g. a warm-up
period). The EWMA parameter and the first-order autoregressive or AR(1) process
parameter are adjusted dynamically to represent the signal of the process during the
transition period. At the end of the transition the AR(1) parameter is restarted so that
the EWMA continuous to monitor the output of the process.
At this point one is confronted with the difficult decision to select an appropriate
monitoring scheme. We will discuss this in Chapter 2.
1.3 Schemes to Monitor Processes with Autocorrelated Output and Observed Correlated Measurement Error
As mentioned above, the overall process observations, Yt, contain information
about the base process that generates the product, Pt, and the measurement process that
( prod ) generates the measurement, X t (see Figure 2). In processes like this the objective of
the monitoring scheme is to detect out-of-control condition in the base process, as well as to maintain the measurement process under control.
Yt Pt Base Measurement X ( prod ) Process Product Process t
( prod ) Figure 2. Aggregation of the measurement process signal X t and the intrinsic production process signal Pt into the output signal Yt.
18 It is important to note that Xt represents the signal of the measurement process
( prod ) obtained from measuring the reference material and X t represents the unobservable
contribution of the measurement process contained in the measurement of the product
material Yt.
The measurement of the reference material, Xt, is used to verify the performance
of the measurement process and it is assumed to represent the measurement error
contained in the measurement of the product.
The construction of an appropriate monitoring system requires that either the
( prod ) measurement process signal X t be removed from the overall output measurement or
be accounted for so that the behavior of the base process can be observed.
MacGregor and Harris (1993) and Lu and Reynolds (1999a) considered the
situation where the process output can be described as
X tt= μ+εt. (1.3)
The mean of the random process, μt, is represented by the first-order autoregressive or
AR(1) model
(1− φμ−μ=XtXB)( ) bt,
where E[Xt] = μX is the process mean or process level, φX is the first-order autoregressive
coefficient, B is the backwards shift operator, defined as BXXtt= −1 . The bt’s are called
the random shocks and form a white noise series, that is, a sequence of independent
2 random variables with mean zero and variance σb . The εt are also independent normal
2 random variables with mean zero and variance σε that represent the variation introduced
by the measurement system. It has been shown that an AR(1) process with added white
19 noise corresponds to a first-order autoregressive moving average, or ARMA(1, 1) process
(Box, Jenkins, and Reinsel, 1994; p. 174). Therefore, Xt in (1.3) can be represented by the model
(11−φXtB)( X −η) =( −θ XB)at, (1.4) where θX is the first-order moving average coefficient, at is a white noise series of are
2 independent random shocks with mean zero and varianceσa . Lu and Reynolds (1999a) used the standard deviation of the residuals from model (1.4) to calculate the control limits for the I-chart of residuals. The variance of the residuals from (1.4) is a function of
2 2 the variance of the random shocks, σb , and of the measurement error,σε . However, to monitor the stability of the underlying AR(1) process, i.e. excluding the variation introduced by the measurement system, the I-chart should be constructed using the residuals from the AR(1) model with control limits calculated using the standard deviation of these residuals, σb. The I-chart used by Lu and Reynolds (1999a) would be appropriate when the measurement error is small compared to the process variance.
In the processes that we are considering the overall process observations are autocorrelated as well as the observations from the measurement process. The results of
Lu and Reynolds (1999a) are not applicable in this case. However, we will follow their approach to study the behavior of the residuals after special causes affect the base process
(see Chapter 3).
As far as we know there are no other methods proposed to monitor processes with autocorrelated output and with an observed autocorrelated measurement error that must be accommodated. Therefore, we will discuss three possible approaches to monitor the process described in Figure 1 and select one to monitor the base process.
20 The first approach consists of monitoring the measurement process to ensure that it is maintained in control. Then, an ARIMA model is fitted to the overall process
ˆ ˆ observations, Yt. The lead-one forecastsYt−1 (1) and the residuals YYtt− −1 ()1 can be used to monitor the overall process output. This monitoring scheme will lead us to the situations described in Table 1 below.
( prod ) If Xt is unrelated to X t , this scheme is not appropriate. The decisions in the
( prod ) body of Table 3 are based on the fact that Xt is at least proportional to X t . Also, from
Table 3 we see that when the control charts of the measurement and the overall processes detect an out-of-control condition at the same time, we cannot be entirely sure that the out-of-control condition detected in the overall process control chart is solely due to an out-of-control condition in the measurement process. This out-of-control condition could be due to both the base and to the measurement process being out of control. In this situation, both processes need to be investigated to identify the process that is out of control. This monitoring scheme is not very practical and will not be considered further.
Table 3. Decisions when the Overall and the Measurement Processes are monitored separately
Yt Xt In control Out of control
In control Base process in control Base process is out of control Base process may be in control or Out of control Base process in control may not be in control
21 A second approach is to obtain the signal from the base process by subtracting the
measurements of the reference material from the measurements of the product material, that is
PYXtt= − t. (1.5)
This approach requires the following two assumptions. First, the behavior of the
measurement process when the product material is measured must be the same as when
( prod ) the reference material is measured; that is X t= X t with probability one. Second, the
base process must be independent of the measurement process.
If the foregoing assumptions hold, then an ARIMA model is fitted to the values
ˆ ˆ obtained from (1.5). The lead-one forecasts, Pt−1 (1) , and the residuals PPtt− −1 ()1 can be used to monitor the base process. If an out-of-control condition is detected by these control charts, then we know that the base process is out of control.
( prod ) This approach is not appropriate if X t is not equal to X t with probability one since use of equation (3.3) implicitly assumes that
( prod ) PYXtt= −=− t YX t t.
A third approach to monitor the base process is to provide for a more general
( prod ) relationship between X t and X t . Ideally we would determine the bias and precision
when the product and reference material are measured. This allows us to model the
relationship as
prod X t=κ12 +κ X t, (1.6)
where κ1 corrects for the difference in bias and κ2 for the difference in precision. This
expression then can be used to obtain the base process signal
22 PYtt= − ζ +κ Xt. (1.7)
In many cases (1.6) cannot be established because tests are destructive or because they are expensive. In this work we will assume that relationship (1.6) cannot be established and that the only information available is the autocorrelated data from the measurements of the product and the reference material.
This situation forces us to view the problem differently. A relationship similar to
(1.7) can be established by recognizing that the measurements of the reference and product material are not equally affected by the transitory and non-transitory behavior of the measurement process, and that the measurement of the reference material can influence the measurement of the product material (or vice versa if the order of measurement is inverted). The consecutive measurement of a reference material and a product material causes a dependency between the two values. This dependency may be caused, for example, when the depth of an amorphous metal connector in a semiconductor is measured by running a needle over the surface and measuring the drop of the needle. However, as the needle moves over the surface it accumulates a minute amount of metal, then the following measurement will be slightly biased because the size of the tip of the needle is not the same, and as more measurements are taken the bias increases. Other examples of the dependency between measurements occur when the raw material needed in the measurement is slowly depleted, or when taking a measurement causes an unavoidable physical change (e.g. warm-up effects caused by the deterioration of an optical system, etc.), or when a chemical or a biochemical contamination is introduced by measuring the reference material sample (e.g. when the same buffered solution is used to generate the reference and product samples), etc. Furthermore, it may
23 be impractical or expensive to restore the measurement process to its original state after a
measurement is performed. Figure 3 depicts this view of the measurement process where
the shaded box indicates that the measurement of the reference material affects the
behavior of the measurement process when the product material is measured.
The third approach is to assume that the effect of the measurement of the reference material, Xt, on the measurement of the product, Yt, is linear, so that their
dependency can be represented as
YvBXt= ( ) t, (1.8)
2 where vB()=+ v01 vBvB + 2 +… is a discrete linear transfer function. The vj’s are called
∞ the impulse response weights and v < ∞ . It is possible that the dependency ∑ j=0 j between the reference and product measurements is nonlinear, but we do not consider this situation.
Reference Product Material Material
(Changed Measurement Measurement measurement Process Process process)
X Y t t
Figure 3. Change in the measurement process from measuring the reference materials.
Assuming that the reference material is always measured before the product
materials, then (1.8) is a causal transfer function, meaning that Xt affects Yt (or vice
versa, if the product material is measured before the reference material), and that if Xt is
24 stationary, Yt is also stationary (Zhang, 1997). If Xt is not stationary, it can be
transformed into a stationary process.
The transfer function in (1.8) can be parsimoniously parameterized as (Box,
Jenkins, Reinsel, 1994; p. 415)
ω(BB) b Y = X , (1.9) tδ()B t
2 r where X tt=−XEX[ t], the polynomial δ(B) = 1 – δ1B – δ2B – … – δrB represents the
2 s autoregressive behavior of Yt, and the polynomial ω(B) = ω0 – ω1B – ω2B – … – δsB represents the dependency of Yt on current and previous values of Yt. The exponent b corresponds to the delay, in time lags, of the effect of measuring the reference material on the measurement of the product. This delay may occur, for example, because after the reference material is measured b times enough contamination accumulates to cause a
change in the product measurements.
The transfer function is identified using the cross-correlation of simultaneous
pairs of observations (X1, Y1), (X2, Y2), …, (Xn, Yn) obtained at equal time intervals 1, 2,
…, n.
The measurement process does not physically affect the behavior of the base
process and the output of the overall process is an aggregation of the signals of the base process and the measurement process, hence the observations of the overall process can be described as
YvBXPtt= ( ) + t, (1.10)
prod where X t= vBX() t and Xt and Pt are independent. Equation (1.10) implies that the
prod transfer function describes the relationship between X t and Xt. There are situations
25 when the error of measurement depends on the magnitude of the characteristics being
measured resulting in a nonlinear aggregation of the intrinsic characteristics and the
measurement error. We will not consider this nonlinear aggregation in this work.
prod If v(B) = ω0 =1, then X t= X tand model (1.10) reduces to the relationship
described by model (1.5). Then, the second monitoring scheme discussed above is a
special case of this monitoring scheme.
If we assume that the measurement process is described by model (1.4), then
θx (B) X tX−μ = at. φx ()B
Furthermore, if the base process signal is autocorrelated, then it can be represented using the model
θbp (B) Putbp−μ = t. φbp ()B
Under these conditions, expression (1.10) can be written as
b ωθ()BB⎡⎤x ( B) θbp (B) YatXt=μ++μ+⎢⎥bput, (1.11) δφ()BB⎣⎦⎢⎥xb() φp()B
where ω(B), δ(B), φx(B), θx(B), φbp(B), and θbp(B) are finite-order polynomials, and the roots of ω(B) = 0, δ(B) = 0, φx(B) = 0, θx(B) = 0, φbp(B) = 0, and θbp(B) = 0 are all outside
the unit circle (Wei, 2006, p.323). Also, at and ut are independent, with common zero
2 2 mean and variances σa and σu , respectively. Note that at and ut are mutually independent because the base process and the measurement process are independent.
26 Once the transfer function is identified and its parameters estimated, the
ARIMA(pBP, dBP, qBP) model that represents the unobservable base process can be identified and its parameters estimated. We will illustrate this in Chapter 5.
Having identified the ARIMA model and estimated its parameters, the lead-one
forecasts of Yt can be computed by first writing (1.11) as (Wei, 2006, p. 332-333)
cBY( ) tt=+ dBX( ) −b gBu( ) t, where
22r π cB()=δ () B φbp () B =()11 −δ12 B −δ B −…… −δr B( −φbp1 B −φ bp 2 B − −φbpπ B) 22s π dB()=ω () B φbp () B =() ω01 −ω B −ω 2 B −…… −ωs B(1 −φbp1 B −φ bp 2 B − −φbpπ B) 22rq gB()=δ () B θbp () B =()11 −δ12 B −δ B −…… −δr B( −θbp1 B −θ bp 2 B − −θbpq B).
Then,
ˆ YYt−−1111211223211233()1 =δ+φ()(bp t +δ−δφbp +φ bp ) Y t − +δ−δφ( bp −δφ bp +φ bp)Y t− +… −() δrbprbprbptr φπ−2112 +δ − φ π− +δ − φ πYY − −π+ 2 −() δ rbprbptr φπ− 11 +δ − φ π − −π+ 1 +δφrbptrπ−−πYX +ω0101121102tb − −()( ω +ωφbp X tb−− − ω −ωφbp +ωφ bp )X tb−−2 −ω−()3 ωφ21bp −ωφ 12 bp +ωφ 03 bpX t−− b 3 +… (1.12) +ωφ()sbpπ−2112 +ωφ s − bp π− +ω s − φ bp πXX tbs − − −π+2 +ωφ( sbpπ− 11 +ωφ s − bp π) tbs − − −π+1 +δsbp φπ−−−πXr tbs −()( δ11121122 + θbpt− − δ −δ θbp + θ bp ) r t−
−() δ32112 −δ θbp −δθ bp +θ bp 33r t−− +… +() δr θ bpq2 +δ r− 1 θ bpq− 1 +δ r− 2 θ bpqr t−−+ r q 2
+δθ()r bpq−−11 +δθ r bpqrr t −−+ r q 1 +δθ r bpq t −− r q.
ˆ The lead-one forecasts from (1.12) and their corresponding residuals, rYYttt=−−1 (1) , can be used to monitor the base process.
The monitoring of an unobservable base process by using a transfer function to separate the signal of the measurement process from the overall process signal (see
Figure 4) is, to our knowledge, a new application of the transfer function.
27
Yt
Xt Transfer Function
(Pt)
Figure 4. Diagram representing the use of the transfer function as a filter to obtain the base process signal.
Transfer function models are traditionally used to describe the dynamic behavior of a process output caused by a change in an input variable (Box, Jenkins, Reinsel, 1994; p. 373. Wei, 2006; p. 322). For example, Zhang (1997) used the transfer function to model the dependency between input and outputs of chemical processes. This model is then used to study data reconciliation and gross error detection in the output based on the measured amounts of materials entering the process.
The transfer function has also been used in process monitoring. In these cases the transfer function is used to model the dependency of the process output on the process input. For example, West, Dellana, and Jarrett (2002) used a transfer function to model the dynamic effect of changes in the input biochemical oxygen demand (BOD) on the output BOD of a wastewater treatment plant. Tests are performed to detect additive and innovative outliers. When outliers are detected the residuals from the initial fit of the
ARIMA model are adjusted and the standard deviation of the adjusted residuals,σˆ a , is
recalculated. The new σˆ a is used to test for more outliers. This procedure continued until
28 all outliers are identified. Then, an ARIMA intervention model is constructed. The σˆ a calculated from the residuals of this model are used to calculate the control limits of the I- chart of residuals. Nugent, Baykal-Gürsoy, and Gürsoy (2005) used a transfer function to relate the input to the output of a process and to introduce various types of disturbances.
The process is controlled with a minimum variance controller. The actions of a minimum variance controller are based on the transfer function that relates the input and the output of the process. The controlled output is autocorrelated and is described by an MA process of order k–1, where k is the delay in the system. The residuals of the MA model are used to construct a residual I-chart, a residual CUSUM chart, and a residual EWMA chart.
These charts are used to determine if the disturbances are detected.
In Chapter 2 we will present the desirable characteristics of a monitoring system and discuss and propose procedures to monitor processes with autocorrelated outputs and with and without an observed autocorrelated measurement error.
1.4 Summary and Discussion
We have seen that standard control charts cannot be employed to monitor processes with autocorrelated output. Therefore, we have identified and summarized the different methods that have been proposed to date to monitor processes with autocorrelated output, but where the measurement error is not observed, nor considered.
We are now confronted with the decision to select a particular method. We will consider the practically of the methods to select a monitoring approach in Chapter 2.
As far as we know, no methods have been proposed to monitor a base production processes with autocorrelated output and with an observed autocorrelated measurement error. We discussed three possible approaches to monitor the base production process
29 and found that a transfer function is a plausible approach to model the relationship between the product measurements and the reference material measurements, which indirectly allows us to obtain the base process signal. This signal can then be used to model the base process. In Chapter 2 we propose an approach to construct control charts for monitoring the base process.
30
CHAPTER TWO
Monitoring Schemes
The problem of monitoring a production process that generates an autocorrelated output with an added autocorrelated measurement noise was introduced in Chapter 1. A particular example of such a process is the pharmaceutical process depicted in Figure 5.
The detection of out-of-control conditions in the base production process requires monitoring and controlling the performance of the measurement process.
Base Process
Product P Y
Measurement Process
X Reference
Figure 5. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process and the measurement process.
The measurement process is assessed by measuring the quality characteristic of a reference material every time a product material is measured. The measurement process generates autocorrelated observations. This autocorrelation is natural to many production processes and its removal is often not economically feasible. In such cases, standard control charts for independent observations are not appropriate. In this chapter we
30
propose a practical scheme to monitor the location (mean) and the variation (standard deviation) of a process with autocorrelated output. In Chapter 4 we use this scheme and the measurements of the reference material to monitor the measurement process shown in
Figure 1.
Also, in this chapter we propose an approach to monitor the base production process using a transfer function. This function exploits the cross-correlation between the product material measurements and the reference material measurements, to implicitly establish the relationship between the error of measurement when the product material is measured and the error of measurement when the reference material is measured. In this way, the measurement error can be taken into account, allowing us to monitor the unobserved base production process.
In Section 2.1 we discuss the characteristics of a monitoring scheme that can be successfully implemented in a typical industrial setting. The issues that arise in monitoring a process with autocorrelated output are discussed in section 2.2. In Section
2.3, we use the criteria established in Section 2.1, the issues identified in Section 2.2, and the practical merits of the methods discussed in Chapter 1 to select a monitoring scheme for a process with correlated output and no observed measurement error. In Section 2.4 we discuss the use of the proposed scheme to monitor the base process and how it can be implemented. The results are summarized and discussed in Section 2.5.
2.1 Characteristics of a Successful Monitoring Scheme
Specific criteria have been established for the successful implementation of SPC in industry (Montgomery, 1996, p. 159; Rungasamy, Antony, Ghosh, 2002). One of the key criteria is the use of appropriate monitoring schemes. An appropriate monitoring
31
scheme detects actual out-of-control conditions and provides information that expedites the identification and elimination of special causes (Gruska and Kimal, 2006).
Based on author’s experience as a process engineer, the criteria that will ensure the acceptance and use of a monitoring scheme are:
• Method is logical, versatile, and easy to understand.
• Implementation is easy and inexpensive.
• Control charts provide useful information to identify special causes.
• Reliable detection of out-of-control conditions.
The versatility of the scheme refers to the flexibility of models than can be used and the variety of effects of special causes than can be reliably detected.
2.2 Issues to Consider when Monitoring a Process with an Autocorrelated Output and no Observed Measurement Error
The autocorrelation in a process output can be modeled by identifying the dependency of a current observation, Xt, on previous observations, Xt–k, k = 1, 2, …, and
on random shocks both present, at, and previous, at–k with k = 1, 2, … . These
dependencies introduce the problem of detecting disturbances that have a transitory effect
on the process output. Furthermore, process disturbances can affect the mean or the
variance of the observations, Xt, or the mean or the variance of the random shocks, at
(Runger, 2002). In this section we discuss these and other issues that must be considered when one is trying to identify an appropriate monitoring scheme.
32
2.2.1 Special Causes and Types of Disturbances
Every process is affected by two sources of variation:
• Common variation due to uncontrollable common causes, intrinsic to the process,
and can only be altered by modifying the process.
• Special-cause variation due to unusual disruptions or special causes that can be
economically discovered and removed.
The special causes are also called assignable causes (Woodall, 2000; Juran and Gryna, p.
24.3, 1988). Wheeler and Chambers (1986, p. 9-10), describe a special cause as being
abnormal and not being part of the overall system.
Practically all the monitoring schemes for processes with autocorrelated output and no observed measurement error discussed in Chapter 1 only consider abrupt changes in the process mean, due to special causes such as the use of wrong raw materials, setting a process parameter incorrectly, deviations from the standard operating procedure, etc.
Runger (2002), on the other hand, argues that special causes can also affect the mean or the variance of the white noise that generates the process output. This can happen when the output slowly drifts to a new level or its variation slowly increases due, for example, to a damaged component in a machine, or to the accidental contamination of a raw material, or to the failure of a controller, etc. These changes in the mean or the standard deviation of the white noise are likely to be detected by residual-based control charts.
Therefore, the monitoring scheme that we select must be able to not only detect shifts in the mean of the process, but also shifts in the variance of the process as well as shifts in the mean and the variance of the random shocks.
33
2.2.2 Dynamic Behavior of the Residuals after a Change in the Process Mean
Changes in the mean of the process output will cause the residuals to have a transient behavior and eventually reach a steady state that may or may not differ from the original in-control level. This final level, and how fast the residuals reach it, depends on the characteristics of the autocorrelation structure of the data (this is discussed in detail in
Chapter 3). Many of the more elaborate methods described in this chapter are tailored to detect signals in the residuals that disappear very quickly. However, the detection of these signals depends on the magnitude of shift in the process mean.
2.2.3 Assessing Performance of a Control Scheme
In most cases authors have evaluated the performance of control charts using the out-of-control average run length for specific out-of-control situations; e.g. a step change in the mean of magnitude δσa, while ensuring that the in-control average run
length is maintained at some specified level. The ARL is the average number of points
that are plotted on the control chart before an out-of-control condition is detected. It is desirable for an in-control ARL to be large when the process is stable and the out-of control ARL to be small when an out-of-control condition occurs. The run length (RL)
follows a geometric distribution and the ARL is equal to the inverse of the probability of
a false alarm (Montgomery, 1996; p. 141). For example, the ARL of an I-chart of IID
normally distributed observations when the process is stable is 370. Wardell, Moskowitz,
and Plante (1994a) point out that, in the case of the I-chart, the standard deviation of the
run length (SDRL) is also about 370. Therefore, it is possible to have a control chart
signal an out-of-control condition much sooner than expected when there is a shift in the
34
process mean. Similarly, if there is not a shift in the process mean, the control chart may signal much later than expected.
Furthermore, comparisons using the ARL performance assume that the fitted model represents the actual process output exactly and that model parameters are
ˆ perfectly estimated, e.g. θ=θ11. Vander Wiel, (1996) shows that the ARL of a CUSUM chart varies from –40% to +35% when the parameter of a EWMA forecast varies by ±
0.1. Adams and Tseng (1998) study the effect of the uncertainty of the estimated parameters concluding that large samples are necessary to obtain accurate estimates and suggested constant updating of these estimates.
In almost every study of the performance of residual control charts only SCR #1
is used. An exception is Adams and Tseng (1998) where they included SCR #5 (see
Table 1 of Chapter 1) in the evaluation of overestimation and underestimation of the
values of the parameters in an AR(1) and in an IMA (1, 1) model. They found that the
performance of the CUSUM and EWMA control charts is seriously affected by the
underestimation or overestimation of the model parameter, however the performance of
the individuals control chart with SCR #1 and the individuals control chart with SCR #1
and #5 are not impacted as severely by incorrectly estimating the parameter. Although
the I-chart with SCR #1 and #5 performs worse than the I-chart with SCR #1.
Claims of superiority made by various authors of the methods discussed in
Chapter 1 are always based on the ARL alone and for specific changes in the process
mean. In general, there is not a perfect control chart for detecting changes in the process
mean for every possible autocorrelation structure (Wardell, Moskowitz, Plante, 1992),
35
however the I-control chart of residuals performs well in many situations and it appears to be robust to the error of estimation of the model parameters (Adams and Tseng, 1998).
2.2.4 Calculation of Control Limits: Moving Range vs. Standard Deviation
Cryer and Ryan (1990) compare the moving range estimate, MR /d2, and the standard deviation estimate, S/c4, of the process standard deviation, σ, of the IID
observations plotted on an I-control chart. Here, S is the standard deviation of the
observations, c4 is a correction constant whose value depends on the number of
observations used to calculate S, MR is the average moving range of the data, and d2 is a
correction constant equal to 1.128. These authors show that the estimate MR/d2 is between 61 and 64% larger than the estimate S/c4 for sample size between 20 and 100.
Also, they point out that, if the lag 1 autocorrelation is positive, MR/d2 underestimates σ,
and if the lag 1 autocorrelation is negative it overestimates σ. However, the bias of S/c4 caused by the autocorrelation of the observations decreases as the number of observations used to compute S increases. Given these facts the authors suggest using the MR/d2 estimator during the retrospective analysis of the process (i.e. during Phase I of the implementation of the monitoring system) and using the S/c4 estimator once it has been verified that indeed the process is stable and that the observations are IID. That is, recalculate the control limits using S/c4 and use them for the Phase II of the
implementation of the monitoring system.
Another approach is to estimate the variance using the most recent observations.
Montgomery and Mastrangelo (1991) suggest the use of a mean absolute deviation
estimate or a smoothed variance estimate. These estimates seem more appropriate for a
36
process that has been stable for some time so that the updating of the variance produces a reliable estimate.
2.3 Selection of a Monitoring Scheme for a Process with Autocorrelated Output and Unobserved Measurement Error
In this work we only consider processes where a single observation is obtained at
each sampling interval, i.e. m = 1. Amin, Schmid, and Frank (1997) provide three cases
of autocorrelation for processes where m ≥ 1 and Lu and Reynolds (1996b) discuss the
monitoring of mean and the variance when m ≥ 1.
From all the methods proposed to monitor a process with autocorrelated output,
the method proposed by Alwan and Roberts (1988) (CC/SCC charts) seems the more
plausible for a practical application because of its simplicity. Wardell, Moskowitz, and
Plante (1992) noted the following advantages of their monitoring scheme:
• The use of the correlation of the process output to forecast the output for the next
time interval.
2 • The SCC chart is constructed with residuals that are approximately IID N(0, σa ),
hence special cause rules can be applied,
• The construction of the CC and the SCC chart is straight forward and user-friendly
software packages facilitate the fitting of ARIMA models.
• This method is suitable for any time-series model and it is not restricted to particular
models such as the MA(1) or IMA(1, 1) models, for example.
In this work we propose using the Roberts and Alwan (1988) approach to monitor
a process with a correlated output with the following modifications:
37
• The Common Cause Control (CCC) Chart: Plot control limits based on 99.7%
confidence interval of the forecasts and the actual observations in addition to plotting
the lead-one forecasts (Alwan, 1991, and Montgomery and Mastrangelo, 1991,
suggested these modifications).
• The Special Cause Control (SCC) charts: Include the MR-chart of the residuals in
addition to the I-chart chart of residuals.
Alwan (1991) suggested that only the CCC chart is needed to monitor the stability
of the process. However, the CCC and the SCC charts facilitate detection of special
causes and visualize the dynamic behavior of the process output (Montgomery and
Mastrangelo, 1991). Ramírez (1998) has illustrated the use of this modified control
scheme (with a 95% confidence interval of the forecasts and without the moving range
chart) to monitor the dew point of a make-up air handler system in a clean room.
The Modified CCC/SCC chart scheme conserves the advantages noted by
Wardell, Moskowitz, and Plante (1992) and meets the success criteria listed in Section
2.1 for the following reasons:
• Personnel responsible for implementing and managing process monitoring systems
are likely to understand the logic behind the construction and interpretation of the
CCC and SCC charts and accept this approach more readily because of their
familiarity with fitting regression models and performing residual diagnostics;
• There is a well established procedure to identify ARIMA models, estimate their
parameters, and generate forecasts (Box, Jenkins, and Reinsel, 1994, or Wei, 2006) as
well as a large variety of software packages that can be used to do this (R, SAS,
JMP, Minitab, etc.);
38
• Simple recursive algorithms can be coded into existing SPC systems or time series
packages can be linked to the on-line SPC system to generate the lead-one forecasts,
the residuals, and the CCC and SCC charts;
• The CCC/SCC chart scheme is a reliable approach to detect out-of control conditions
that affect the mean and variance of the process output and/or the underlying random
shocks that drive the process;
• The use of special-cause tests, which are available and easily turned on or off in an
existing on-line SPC system will allow faster and reliable detection of smaller
changes in the mean and variance (Nelson, 1984, 1985; Montgomery, 1996);
• The CCC and SCC charts provide useful clues that may expedite the identification of
the causes of the out-of-control condition.
In the next sections the construction of the modified SCC and SCC charts is presented, and a general procedure for their implementation is outlined.
2.3.1 Construction of the Common-Cause Control Chart
The data collected in the Phase I of the implementation are used to identify a suitable ARIMA model, obtain estimates of the model parameters, and of the residual
ˆ standard deviation, σa. Using the fitted model the lead-one forecast, X t ()1 , is generated.
Following the traditional SPC approach to chart construction and monitoring, the model parameter estimates, the estimated process level, and the estimated standard deviation are treated as if they were the actual population parameters. Then, the control limits for this chart are calculated as
39
ˆ UCL=+σ X ta(13) ˆ ˆ CL= X t ()1 (2.1) ˆ LCL=−σ X ta()1 3ˆ .
The CCC charts is constructed by plotting the observations, Xt, the lead-one
ˆ forecast, X t ()1 , and the control limits (2.1). If an observation falls outside the control limits and this event is confirmed by the SCC charts, then the process is considered to be out of control.
2.3.2 The Special-Cause Charts and Control Limits.
The construction of these control charts requires computing the residuals
ˆ eXXttt=−(1) , (2.2)
and the residual moving ranges
MRett= +1 − et. (2.3)
The control limits of the I-control charts are calculated as
MR UCL=+ X 3 d2 CL= X (2.4) MR LCL=− X 3 . d2
Where MR is the average of the n–1 individual moving ranges and the correction
constant d2 is 1.128, because the moving range is computed from two consecutive
observations. See Montgomery (1996), Appendix A-15 for tabulated values of this and
other correction constants. The I-chart is constructed by plotting the time ordered
residuals, et, and the control limits (2.4).
40
The control limits of the MR- chart are computed as
UCL= D4 MR CL= MR (2.5)
LCL= D3 MR
The constants D3 and D4 are
Ddd432=+1 3( /) = 3.268 (2.6) Ddd33= max{} 1−= 3() /2 ,0 0
The MR-chart is constructed by plotting the time ordered moving ranges, MRt, and the control limits (2.5).
Out-of-control conditions due to changes in the mean or variance of the process or the underlying noise can be detected by the I-chart.
Montgomery (1996, p. 224) notes that some authors recommend not using the
MR-chart because the control limits are not accurate. The inaccuracy results from the fact that the constants D3 and D4 , in (2.6), are based on independent observations, but the
moving ranges are not independent because they are calculated using consecutive
observations (Ryan, 1989; p. 160).
The advantage of the MR-chart is that large changes in the mean appear as large
peaks in the MR-chart and changes in the variance appear as multiple oscillating peaks.
When an out-of-control situation is detected by the MR-chart it should be investigated,
especially when the I-chart and the CCC chart also detect it.
Another important point is that there may be situations when the CCC chart may
show a point outside the control limits, but the same point may be within the control
limits of the I-chart. This is because the CCC chart uses the standard deviation of the
residuals to compute the control limits, whereas the I-chart uses control limits based on
41
the estimator MR /d2. This discrepancy can be corrected by computing the control limits
of the I-chart and MR-chart using the residual standard deviation, S, in Phase II. If a
point is out of control on either chart it should be investigated to verify if the process is
out of control.
2.3.3 Procedure to Implement the Selected Monitoring Scheme for a Process with Autocorrelated Output and Unobserved Measurement Error
The procedure to implement the modified CCC and SCC chart scheme is as
follows:
Phase I
1. Identify the quality characteristic of interest and create standard operating procedures
for the process and to measure the quality of characteristic (if necessary);
2. Ensure that the process is stable (calibrate gauges, verify capability of measurement
devices, follow the standard operating procedure);
3. Collect data following standard operating procedure;
4. Verify independence of the data using the autocorrelation function (ACF) graph (see
Box, Jenkins, and Reinsel, 1996, or Wei, 2006);
5. If the data is not autocorrelated, use standard control charts and techniques for
independent data (see Montgomery, 1996);
6. If the data is autocorrelated and the removal of the source of autocorrelation is not
feasible, then
• identify an ARIMA model that fits the data,
• estimate the model coefficients,
• obtain estimate of the residual variance,
42
• compute lead-one forecasts and residuals,
• compute control limits for the CCC and SCC charts;
7. Determine if there are any out-of-control conditions. Investigate observations that
correspond to out-of control conditions to determine if a special cause exists;
• If a special cause is identified, then take action to remove it permanently, delete
the observation and repeat step 6;
• If a special cause cannot be found, do not remove the observation from the dataset
and continue with Step 8;
Phase II
8. Implement control charts and include instructions for how to use them and how to
proceed if an out-of-control condition is detected;
9. Update the estimate of the residual variance based on the most recent w observations
using the smoothed variance estimate (Montgomery and Mastrangelo, 1991)
22 2 σˆˆat(te) =α +(11 −α) σ a( t − ) ,
where α = 2 /(w + 1). For example if w = 100, then α ≈ 0.02;
10. Recalculate control limits for the I and MR charts using σˆ a from Step 10;
11. Periodically repeat Step 6 to revise the ARIMA model, to update the parameter
estimates, and to update the control limits of the CCC and SCC charts.
The construction of the CCC and SCC to monitor a process with autocorrelated output and unobserved measurement error will be illustrated using the measurement process of the pharmaceutical process in Chapter 4.
43
2.4 Monitoring Scheme for a Process with Autocorrelated Output and Observed Autocorrelated Measurement Error
In this section we consider the monitoring of a process where the autocorrelated
overall process output, Yt, and the autocorrelated measurement process output, Xt, are
observed, and where it is not feasible to modify the measurement process to reduce its
contribution to the overall process measurements to a negligible level.
In Section 1.3 of Chapter 1 we propose the use of a discrete linear transfer
function
YvBXt= ( ) t, (2.7)
∞ where vB=+ v vBvB +2 +… s and v < ∞ , to model the effect of the () 01 2 ∑ j=0 j
measurement of the reference material, Xt, on the measurement of the product, Yt. This
transfer function in (2.7) can be parsimoniously parameterized as (Box, Jenkins, Reinsel,
1994; p. 415)
ω(BB) b Y= X, (2.8) tδ()B t
2 r where the polynomial δ(B) = 1 – δ1B – δ2B – … – δrB represents the autoregressive
2 s behavior of Yt, and the polynomial ω(B) = ω0 – ω1B – ω2B – … – δsB represents the
dependency of Yt on current and previous values of Yt. The exponent b corresponds to the
delay, in time lags, of the effect of measuring the reference material on the measurement of the product.
The measurement process does not physically affect the behavior of the base
process and the output of the overall process is an aggregation of the signals of the base
44
process and the measurement process, hence the observations of the overall process can be described as
YvBXPtt= ( ) + t, (2.9)
prod where X t= vBX() t, and Xt and Pt are independent. We noted in Chapter 1 that we
will not consider nonlinear relationships between Yt and Xt, nor nonlinear aggregation of
the intrinsic characteristic, Pt, and the measurement error.
The impulse response weights, vj, in the transfer function v(B) in (2.8) are
obtained by first computing the the cross-covariance of the simultaneous pairs of
observations (X1, Y1), (X2, Y2), …, (Xn, Yn) obtained at equal time intervals 1, 2,, …, n, as
γ=kEX⎡ −μ−μ Y ⎤ xy ()⎣ (t x )( t y )⎦
for k = 0, ±1, ±2, …, and then computing the cross-correlation function (CCF) as
γ xy (k ) ρ=xy ()k . (2.10) σxσ y
Then, (Wei, 2006; p. 329)
σ y vkx=ρy()k. (2.11) σx
The estimates of the CCF (2.10) are used to estimate the weights vk using (2.11). Based on the pattern of the vk’s we identify the delay b, and the order r and s of the polynomials in the transfer function (2.8). Then, the coefficients ω0, ω1, …, ωs and δ1, δ2, …, δr are
obtained in the same way as the estimates of the vk’s, since vB( )()(=ω B δ B).
We have assumed that Yt and Xt are stationary processes. If the measurement
processes series, Xt, is not stationary a variance-stabilization or differencing
transformation is used to transform it to a stationary process (Box, Jenkins, Reinsel,
45
1994; p. 92-96). Since, we are assuming that Xt causes Yt, then the same transformation is applied to the overall process series, Yt, to make it stationary (Zhang, 1997).
Once the transfer function is identified and its parameters estimated, the base process series is calculated as
ω(B) PY=− BXb . ttδ()B t
These values are used to identify the ARMA model,
θbp (B) Putbp−μ = t, φbp ()B that best describes the base process and to estimate its parameters (Wei, 2006; p. 331).
Then expression (2.9) can be written as
b ω(BB) θbp (B) YXtt=+μ+bput, δφ()BBbp () and the minimum mean squared error lead-one forecast is computed as
ˆ YYt−−1111211223211233()1 =δ+φ()(bp t +δ−δφbp +φ bp ) Y t − +δ−δφ( bp −δφ bp +φ bp)Y t− +…
−() δrbprbprbptr φπ−2112 +δ − φ π− +δ − φ πYY − −π+ 2 −() δ rbprbptr φπ− 11 +δ − φ π − −π+ 1
+δφrbptrπ−−πYX +ω0101121102tb − −()( ω +ωφbp X tb−− − ω −ωφbp +ωφ bp )X tb−−2
−ω−ωφ()3211bp −ωφ+ωφbp203 bpX t−− b 3 +… (2.12) +ωφ()sbpπ−2112 +ωφ s − bp π− +ω s − φ bp πXX tbs − − −π+2 +ωφ( sbpπ− 11 +ωφ s − bp π) tbs − − −π+1
+δsbp φπ−−−πXr tbs −()( δ11121122 + θbpt− − δ −δ θbp + θ bp ) r t−
−() δ32112 −δ θbp −δθ bp +θ bp 33rr t− +… +( δr θ bpq−−−−2 +δ r 1 θ bpq 1 +δ r 2 θ bpq) t −−+ r q 2
+δ()rθ+δθbpq−−11 r bpqrr t −−+ r q 1 +δθ r bpq t −− r q . and the corresponding residuals as
ˆ rYYttt=−−1 (1) . (2.13)
46
The transfer function depends exclusively on the cross-correlation between Yt and
Xt and the base process has no effect on its determination. In practice the transfer
function is established from data collected when both the overall production process and
the measurement processes are in control (Phase I of implementation of the monitoring
scheme). In the monitoring phase of the base process (Phase II of implementation of the
monitoring scheme), the measurement process is kept in control and the transfer function,
the parameters of the models that describe the base process and the measurement process are assumed to be fixed Therefore, its seems reasonable to monitor the base process using the CCC and SCC charts, constructed using the observations, Yt, the lead-one
forecasts from (2.12) and the residuals from (2.13). Since the lead-one forecasts from
(2.12) and the residuals from (2.13) do account for the amount of error introduce in the
product measurements, any out-of-control condition detected by these charts will be due
to the base process being out of control. In Chapter 3 we will see these charts share the
same advantages and disadvantages of the SCC and SCC charts used to monitor a process
with autocorrelated output with unobserved measurement error.
2.4.1 Procedure to Implement the CCC and SCC Charts to Monitor a Process with Autocorrelated Output and Observed Autocorrelated Measurement Error
The procedure to construct the CCC and SCC and monitor a process with
autocorrelated output and account for the autocorrelated measurement error is:
Phase I
1. Identify the quality characteristic of interest and create standard operating procedures
for the process and to measure the quality of characteristic (if necessary);
2. Ensure that the measurement process is stable and remains stable during Phase I data
collection;
47
3. Ensure that the base production process is operating under standard conditions
(calibrate gauges, verify process setting, follow the standard operating procedure)
4. Collect data following standard operating procedure;
5. Identify the form of the transfer function and estimate its parameters;
6. Identify the base process model and estimate its parameters;
7. Compute the lead-one forecasts, the residuals, and the moving range of the residuals;
8. Compute the control limits of the CCC charts as
ˆ UCL= Ytf(13) +σˆ , ˆ CL= Yt ()1, (2.14) ˆ LCL= Ytf()13−σˆ ,
where σˆ f is the estimated standard deviation of the lead-one forecast. The control
limits (2.14) are the limits of the 99.87% confidence interval of the lead-one forecast;
9. Compute the control limits of the of the I-chart of the residuals as
MR UCL=+ r 3,r 1.128 CL= r , (2.15) MR LCL=− r 3;r 1.128
10. Compute the control limits of the moving ranges of the residuals, MRr=−t+1 rt using
the equations
UCL= 3.268 MRr ,
CL= MRr , (2.16) LCL = 0;
11. Construct the CCC and SCC charts;
48
12. Investigate any point outside the control limits. Remove or retain such points using
the following criteria:
o If there is a special cause that made the observations associated with the point
outside the control limits, then remove it from the data set and repeat the entire
process;
o If a special cause cannot be found, then the observations associated with the point
out of control are kept in the data and models. Parameter estimates, and control
limits are not revised;
Phase II
13. Monitor the base process using the established transfer function, base process model
and control limits of the CCC and SCC charts;
12. Periodically repeat Steps 5 through 12 to revise models, to update the parameter
estimates, and to update the control limits of the CCC and SCC charts.
The construction of the CCC and SCC to monitor a process with autocorrelated output and with an observed autocorrelated measurement error will be illustrated using the pharmaceutical process in Chapter 5.
2.5 Summary and Discussion
In this chapter we presented the criteria that will ensure that a monitoring scheme is successfully implemented. We found that a successful monitoring scheme must be able to detect not only changes in the mean of the process, but also changes in the process variance and changes in the mean and variance of the random shocks. Also, a successful monitoring scheme should be able to detect changes in the process mean despite the transient behavior of the residuals. Many of the methods proposed to monitor processes
49
with autocorrelated output, presented in Chapter 1, have been tailored to achieve a high level of detection of specific changes in the process mean. However, their capability is based ARL which is highly variable and not appropriate, because the distribution of the run length is highly skewed. Given all of these considerations we decided to adopt the common cause (CC) chart and special cause control (SCC) chart suggested by Alwan and
Roberts (1988), but with the following modifications. We converted the CC chart into a control chart by adding the control limits suggested by Alwan (1991) and Montgomery and Mastrangelo (1991), therefore becoming the common cause control (CCC) chart. We also added the MR-chart to the original SCC chart that only consisted of the I-chart. In this Chapter we also proposed the use of the CCC chart and the SCC charts scheme to monitor the base process. These charts are constructed using the product measurements, the lead-one forecasts, that rely on a transfer function to account for the error of measurement, and the residuals. In Chapter 3 we will see that the advantages and disadvantages of the CCC and SCC charts are the same when they are used to monitor a process with autocorrelated output with an unobserved measurement error or with an observed autocorrelated measurement error.
50
CHAPTER THREE
Behavior of Observations, Forecasts, and Residuals after Changes in the Mean and Variance
Several authors have documented the transient behavior of the residuals caused by a shift in the mean of a process with autocorrelated output (see for example Harris and
Ross, 1991), and several of them have documented different ways of obtaining these expressions (Harris and Ross, 1991; Wardell, Moskowitz, and Plante, 1992; Runger and
Willemain, 1995, among others). However, expressions for the behavior of the residuals when there is a shift in the variance of the process or shifts in the mean or the variance of the random shocks have not been published.
In this chapter we derive the expressions that describe the behavior of the residuals, the lead-one forecasts, and, when necessary, the behavior of the observations for step shifts in the mean or variance of the process and for step shifts in the mean or variance of the random shocks. In Section 3.1 we derive the expressions for processes with correlated output, but no observed measurement error. In section 3.2 we derive the expressions for processes with correlated output and observed correlated measurement error. In Section 3.3 we summarize and discuss our results.
3.1 Behavior of the Lead-One Forecasts and Residuals for Processes with Correlated Output and Unobserved Measurement Error
In the following sections, as well as in Section 3.2, we will use specific models from the general class of autoregressive integrated moving average or ARIMA (p, d, q)
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models (Box, Jenkins, and Reinsel; 1994) to represent the autocorrelated series of observations, Xt. The general ARIMA (p, d, q) model is
d φ∇()B ( XtX −μ=θ) (Ba) t, (3.1) where:
• μX is the level of the process; i.e., EX[ tX] = μ .
2 p • φ()B =1 −φ12BB −φ − −φp B is a polynomial of order p, that represents the
autoregressive component of the series, AR(p).
• B is the backwards shift operator, defined as BXXtt= −1 .
• ∇d is the backward difference operator of order d, which is defined as
d d ∇=−X t()1 BXt. For example, ∇X ttt=−XX−1 , and
2 ∇=∇∇=∇−X tttttt( XXXXXX) ( −11) =−2 −− +t2.
2 q • θ()B =1 −θ12BB −θ − −θq B is a polynomial of order q, that describes the
moving average part of the series, MA(q).
• at is the random shock at time t. Here {at} is a white noise series, that is a sequence
2 of IID values with mean 0 and variance σa .
For example, when sequence of observations, Xt, is described by the specific first- order autoregressive, or AR(1) or ARIMA (1, 0, 0) model, then the general model (3.1) reduces to
(1− φ−μ=1B)( XatX) t, or
X tXt= (1−φ111) μ +φXa− + t.
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Then, we see that the current value of the series depends on the last observed value and the current random shock. If the series Xt is describe by a first-order moving average, or
MA(1), or ARIMA(0, 0, 1) model, then (3.1) reduces to
X tX−μ =(1 −θ1Ba) t, or
X tXt= μ+aa −θ11 t− .
In this case the current observation is generated by the current and previous random shocks. Another common model found in practice is the mixed first order autoregressive moving average model, or ARMA(1, 1), or ARIMA (1, 0, 1). In this case (3.1) becomes
(11−φ11B)( XBtX −μ) =( −θ )at, or
X tXtt=−φμ+φ(1 1111) Xaa− +−θt−1
Here, the current observation depends on the previous observation, the current, as well as the previous random shocks.
3.1.1 Behavior of the Residuals after a Step Shift in the Process Mean
Abrupt changes in the level (or mean) of the process output due to a special cause
(e.g. use of wrong raw materials, setting a process parameter incorrectly, failure of a part, deviations from the standard operating procedure, etc.) cause a dynamic behavior that may or may not allow the detection of out-of-control conditions. The dynamic behavior of the residuals is obtained by following their evolution after a shift in the mean occurs
(Runger and Willemain, 1995; Lu and Reynolds, 1999a).
Example 3.1. To illustrate this consider a process described by an ARMA(1, 1) model with a time dependent mean (Lu and Reynolds, 1999a)
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X tt= μ−φμ11 t− +φ 1Xaa t− 1 + t −θ 11 t−, (3.2) which is affected by a special cause that shifts the process mean at t = τ, as follows:
⎧μ0 t =τ−1, τ− 2,… μ=t ⎨ (3.3) ⎩μ=ττ+1 t ,1,… .
The lead-one minimum mean squared error forecast of an ARMA(1, 1) process is
(see Chapter 5 of Box, Jenkins, and Reinsel, 1994)
ˆ XXtt−101101(1) =μ +φ( − −μ) −θet−1. (3.4)
Subtracting (3.4) from (3.2) yields the recursive expression to compute the residuals
eXt=−μ−φ−μ+θttt01( X− 10) 1 e−1.
The expected value of the residuals, E[et], is equal to 0 for t = τ–1, τ–2, …, and
⎧μ0 t =τ−1, τ− 2,… EX[]t = ⎨ ⎩μ=ττ+1 t ,1,… .
Then the expected values of the residuals, starting at τ–1, are
Ee[ τ−1101201] =−μ−φ−μ+θ= E⎣⎦⎡⎤ Xτ− () Xτ− eτ−20
Ee[]τ =−μ−φ−μ+θ=δ E⎣⎦⎡⎤ Xτ 01() Xτ− 10 11 eτ−
Ee[]τ+1=−μ−φ−μ+θ=δ−φδ+θδ E⎣⎦⎡⎤ Xτ+ 101() Xτ 01 eτ 11
=−φ+θδ[]1 11
Ee[]τ+2201101=−μ−φ−μ+θ=δ−φδ+θ−φ+θδ E⎣⎦⎡⎤ Xτ+ () Xτ+ eτ+11111[]1 (3.5) ⎡⎤2 =θ+⎣⎦111()()11 −φ +θ δ ⎡⎤⎡ 2 ⎤ Ee[]τ+3 = E⎣⎦ Xτ+301−μ −φ()Xeτ+ 20 −μ +θ 12τ+ =δ−φδ+θ 1 11⎣ θ +()(11 −φ 1 +θ 1)⎦ δ =θ+⎡⎤3211 −φ +θ+θ δ ⎣⎦1111()()
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Proceeding in this way and noting that we are only interested in a finite number of observations after the shift of the mean, we obtain the general expression
k Ee = ⎡θ+kj()1 −φ θ−1 ⎤ δ. []τ+k ⎣ 111∑ j=1 ⎦
Since,
k k 1− θ θjk−−121 =1 +θ +θ +… +θ = 1 , ∑ j=1 1111 1− θ1 then
kk θ11(111 −θ) +( −φ 1)( −θ 1) Ee[]τ+k = δ 1−θ1 θkk−θ θ +1 −θ k −φ +φθ k = 111 1111δ 1−θ1 θφ−θ+−φk ()1 =δ11 1 1. 1−θ1
The change in the expected value of the residuals is summarized as
⎧01t =τ−, τ−2,… ⎪ ⎪δ=t τ Ee[]t = ⎨ θφ−θ−φ+k ()1 ⎪ 11 1 1 δ=τ+=tkk; 1,2,… . ⎪ ⎩ 1−θ1
This expression indicates that the residuals reach their highest value immediately after the shift in the mean occurs and that, as time progresses, the residuals approach the steady-state level (Lu and Reynolds, 1999a)
1− φ1 lim Ee[]τ+k = . k→∞ 1− θ1
2 These residuals are independent and have variance σa . Wardell, Moskowitz and Plante
(1992) use z transforms to obtain an alternative non-recursive expression to calculate the lead-one forecast errors after a shift in the mean.
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Example 3.2. When a process can be represented by the AR(1) or ARIMA(1, 0,
0) model
X tt= μ+φ11( Xa t− −μ+ t) t, and it is affected by the special cause defined in (3.3), it can be shown, using the recursive approach (3.5), that the expected values of the residuals are (Runger and
Willemain, 1995; Hu and Roan, 1996)
⎧01t =τ−, τ−2,… ⎪ Ee[]t = ⎨δ=τt ⎪ ⎩()11−φ1 δt =τ+ ,… .
The expected value of the residuals of an AR(1) process will equal the magnitude of the change in the mean immediately after the shift occurs and will drop to a level determined by the value of (1 – φ1). If φ1 (the autocorrelation at lag 1) is positive and close to 1 the expected value of the lead-one forecast errors will drop close to the original level. In this situation if the shift in the mean is not large enough (i.e. δ larger than 1.5 or
2σa), it may not be detected by a control chart of residuals. On the other hand, if φ1 is negative and close to –1, then the expected values of the lead-one forecast errors will increase to a level higher than the magnitude of the shift in the mean. In this situation a control chart of residuals may be able to detect changes smaller than 1.5σa. These facts have been reported extensively, for example see Longnecker and Ryan (1990) and Harris and Ross (1991).
Example 3.3. When a process is represented by an MA(1) or ARIMA(0, 0, 1) model,
X ttt= μ+aa −θ11 t− ,
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and it is affected by the special cause defined in (2.5), the expected value of the residuals are
⎧01t =τ−,,1… ⎪ ⎪δ=τt Ee[]t = ⎨ ⎛⎞1−θk ⎪ 1 δ=τ+=tkk; 1,2,… . ⎪⎜⎟ ⎩⎝⎠1−θ1
The residuals of an MA(1) process increase immediately after a shift in the mean to a level equal to the magnitude of the change in the mean and decrease at a rate given by the value of θ1, to the steady state
1 lim Ee[]τ+k = . k→∞ 1− θ1
Example 3.4. Consider a process with output that can be described by an IMA(1,
1), or ARIMA(0, 1, 1) model,
X tt− Xaa−11=−θ t t−1.
This process does not have a constant mean. To account for special causes that shift the level of the process output, Vander Wiel (1996) suggested the following variable change
⎧Xtt =1, 2,… ,τ− 1 Zt = ⎨ ⎩Xtt +δ =τ,1,, τ+ … such that
⎧⎪ EX[ t ] = 01,2,, t=τ… −1 EZ[]t = ⎨ ⎩⎪EX[]t +δ =δ t =τ,1, τ+ … .
Then, the IMA(1, 1) model in terms of the new variable is
Ztt= Zaa−11+−θ t t−1, and the lead-one forecast is
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ˆ Ztt−111(1) =−θZe−−t1.
Then, the residuals and expected residuals, starting at t = τ are
ˆ eZZτττ−ττ−τ−=−11(1) =− ZZ +θ1 e1
Ee[]τ = EZ [τ ]−+θ=δ EZ [τ−11 ] Ee [τ− 1 ]
Ee[]τ+11=−+θ=θδ EZ [τ+ ] EZ []τ Ee []τ
2 Ee[]τ+2211= EZ [τ+ ][]−+θ= EZτ+ Ee []τ+ θδ k Ee[]τ+k =θ δ.
2 These residuals continue to be independent with variance σa . The summary of the expected value of the residuals is (Vander Wiel, 1996)
⎧01,,1t =τ−… ⎪ Ee[]t = ⎨δ=τt ⎪ k ⎩θδ1 tkk =τ+; = 1,2,… .
Since | θ1| < 1, the lead-one forecast errors decrease at a rate given by the value of the parameter θ1, to the steady-state level
limEe[ τ+k ] = 0 . k→∞
Example 3.5. Finally, we consider a process that can be represented by an
ARIMA(1, 1, 1) model. The expected value of the residuals after the process undergoes a step shift in the mean can be obtained using the same change in variable as for the
2 IMA(1, 1) case. The residuals after the shift are independent with variance σa and mean
⎧01t =τ−,…,1 ⎪ Ee[]t = ⎨δ=t τ ⎪ l−1 ⎩θθ−φδ=τ+=1 ()tll; 1,2,… .
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This result has not been explicitly published, but Vander Wiel (1996) gives a general expression to obtain the residuals for a general ARIMA (p, d, q) model.
Since, | θ1| < 1, the expected values of these errors decrease at a rate that depends on the value of θ1 to the steady-state
limEe[ τ+k ] = 0 . k→∞
Vander Wiel (1996) points out that the residuals for non-stationary processes (i.e. d = 0) will tend exponentially to zero, whereas the residuals of a stationary processes (i.e. d ≥ 1) will tend to a nonzero level.
Table 4 summarizes the behavior of the residuals after a step shift in the process mean. It can be seen that the residuals for stationary processes decrease to a non-zero level and the residuals of non-stationary processes decrease towards zero. Another important observation is that the residuals have their largest expected value immediately after the shift in the mean occurs.
Table 4. Transient Behavior and Steady State Level of Residuals for Various Process Models
Transient Behavior Steady-State Model t < τ t = τ t ≥ τ+1, ..., τ+k, … Level
AR(1) 0 δ (1− φδ1 ) ()1−φ1 δ ⎛⎞1−θk δ MA(1) 0 δ 1 δ ⎜⎟ 1−θ ⎝⎠1−θ1 1 k θ (φ−θ) −φ+1 1−φ1 ARMA(1, 1) 0 δ 11 1 1 δ δ 1−θ 1−θ1 1 k IMA(1, 1) 0 δ θ1 δ 0 k −1 ARIMA(1, 1, 1) 0 δ θ1 (θ−φ) δ 0
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The transient behavior of the lead-one forecast errors needs to be considered when designing a monitoring scheme. If the initial jump in the error exceeds 1.5σa, then the I- chart of the forecast errors using only SCR #1 is likely to detect the change in the process mean (Montgomery, 1996; p. 314). Smaller changes may be detected if other SCR’s are used, however the EWMA or CUSUM control chart of residuals are alternatives to detect smaller shifts without the need of SCR’s (Montgomery, 1996; p. 150).
3.1.2 Behavior of the Residuals after a Step Change in the Process Variance
2 The detection of special causes that affect the variance of the process, σ X , or the
2 variance of the noise series, σa , has received considerably less attention in the literature than the detection of changes in the mean. Cryer and Ryan (1990) considered the effect of a lag 1 autocorrelation on the estimators of the process variance and its impact on a moving range and a standard deviation control charts. MacGregor and Harris (1993) evaluated the exponentially weighted moving variance and the exponentially weighted mean square control charts for an AR(1) process with added noise. Amin, Schmid, and
Frank (1997) compared the Range chart and the S2 chart for the cases when there is autocorrelation within samples but not between samples, and when there is autocorrelation between samples but not within samples. Lu and Reynolds (1999b) studied control charts of the logarithm of the square of the residuals (loge2) and the squared residuals (e2).
Insight in the behavior of the observed values, the lead-one forecasts, and the residuals after a shift in the process variance can be obtained by noting that the variance
2 2 of the process output, σ X , is a function of the variance of the noise series, σa . This
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dependency comes from the characteristic that a stochastic process can be expressed as a linear function of the random shocks (Box, Jenkins, Resinsel, 1994; p. 46),
∞ X =+ψ+ψ+=+aa a… a ψ a, tt11 t− 2 t−− 2 t∑ j=1 jtj such that ∞ σ22=σ ψ2. Xa∑ j=0 j
For example, the variance of a process that can be represented with an AR(1) is
2 2 σa σ=X 2 . 1− φ1
2 Thus, it seems reasonable to assume that a shift in the process variance σ X is
2 due to a change in the variance of the random shocks σa . This has the important implication that a special cause that changes the variance of the random shocks will also change the variance of the process. Based on this assumption Lu and Reynolds (1999) studied the behavior of the residuals after a shift in the variance for a process where the
observations are described as X tt= μ+εt, where μt follows an AR(1) process with a
2 white noise at (mean 0 and variance σa ) and the εt are observable random errors with
2 mean 0 and variance σε . Therefore, Xt is described by ARMA (1, 1) process with two
222 sources of variation, that is σ=σ+σX μ ε .
Here we also assume that the change in the process variance is due to a change in the variance of the random shocks and use the same approach as Lu and Reynolds (1999) to determine the behavior of the forecasts and the residuals after a shift in the variance of
2 the random shocks, at. We do not include random noise, σε , in our model because it is not observable and we have no means to estimate its variability. Therefore the
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observations Xt are directly represented by an ARIMA model. As far as we know the following results have not been published before.
3.1.3 Behavior of the Observations, Lead-One Forecasts, and Residuals after a Step Change in the Variance of the Random Shocks
Example 3.6. Here we assume that the autocorrelation of the observations can be represented with an AR(1) model,
X tt= φ+11Xa− t,
where XXtt=−μX, and that the variance of the process changes from its stable level,
2 2 σa , to the out of control level, σd , between time intervals τ – 1 and τ.
The change in the variance of the process is represented by superimposing a disturbance, dτ, onto the random shocks, at. The disturbances, dτ, dτ+1,…, are normally
222 distributed with mean zero and variance σda=δσ −σa and are independent of the random variable shocks at. The observed value, at t = τ, is
X = φ++Xad τ 11τ− τ τ 0 =+Xdττ,
0 where the part of the model that represents the stable process is designated as X τ ; i.e. without the disturbance. Expressions for subsequent observations are obtained by recursive substitution of the expression of previous observations:
XXadτ+11=φ τ + τ+ 1 + τ+ 1 =φ11() φX τ−τττ+τ+ 1 +ad + + a1 + d 1
2 =φ111X τ−ττ+ττ +φaa + 11 +φ dd + + 1 0 =+φ+Xddτ+11 τ τ+ 1.
Proceeding in the same way we obtain
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32 2 X τ+2111=φXaaadddτ− +φ τ +φ 1121 τ+ +t + +φ τ +φ 11 τ+ + τ+2 02 =+φ+φ+Xdddτ+21 τ 11 τ+ τ+ 2 kk XX =φkk + φ−−jk a + φ j d τ+kj11τ− ∑∑jj==00 1τ+ 1τ+ j k =+Xd 0 φkj− . τ+kj∑ j=0 1 τ+
Thus, the variance of the observations is
k Var⎡⎤ X =σ222 + δσ −σ φ2()kj− . ⎣⎦τ+kX( aa)∑ j=0 1
Therefore, variation of the the observed values have a dynamic behavior; their variation will begin to increase from their in-control level, immediately after the shift, to a new
22 level determined by the value φ1 and the magnitude of the shift (δσ−σaa) .
ˆ The minimum mean squared lead-one forecasts are given by X t()1 =φX t−1 . The first lead-one forecast after the shift is
ˆˆ 0 X τ−τ11()1 =φXX− = τ,
ˆ 0 where, again, the quantity X τ corresponds to the lead-one forecast of the stable process.
This first forecast is not affected by the shift in the process variance. Lead-one forecasts for periods after t = τ are also obtained by recursive substitution of the expression of previous observations and forecasted values. These forecasts are
ˆ 2 XXadττ()1 =φ−1 +φt +φ τ ˆ 0 =+φXdτ+1 τ ˆ 32 2 X τ+11()1 =φXaaddτ− +φ τ +φ τ+ 1 +φ τ +φ τ+1 ˆ 02 =+φ+φXddτ+21 τ τ+
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kk−−11 X ˆ 1 =φ+φkk+−1 Xaj +φk−jd τ+kj −11() τ− ∑∑jj==00τ+ τ+ j
k −1 =+Xd ˆ 0 φkj− . τ+kj∑ j=0 τ+
These expression indicate that the lead-one forecasts also exhibit a dynamic behavior; they will exhibit larger variability, starting one time period after the shift, determined by the value φ1 and the variation of the dτ+k, k = 0, 1, 2, … .
After the shift in the variance the residuals, et, are calculated by directly
ˆ subtracting the expressions for X t−1 ()1 from the expression for X t . These residuals are
00ˆ eXdXt =+−ττ τ 0 =+edττ 00ˆ eXτ+11=+φ+−−φ τ+ ddX τ τ+ 11 τ+ d τ 0 =+edτ+11 τ+ 02 ˆ 02 eXτ+22=+φ+φ+−−φ−φ τ+ dddX τ τ+ 122 τ+ τ+ dd τ τ+1 0 =+edτ+22 τ+
kk−1 eX=+ 00 φkj−− dX −−ˆ φkj d τ+kk τ+ ∑∑jj==00τ+jk τ+ τ+ j 0 =+edτ+kk τ+ .
We have denoted the residuals that result from the difference between the observations
0 and forecasts from the stable process by eτ+k . The variance of the residuals is
2 Var[ eτ+ka] = δσ .
Therefore, a shift in the process variance will have an immediate and persistent effect on the variance of the residuals.
Example 3.7. Now let us assume that the autocorrelation of a process can be represented with an ARMA(1, 1) model,
X ttt= φ+−θXaa−11t− ,
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2 and that variance of the process increases from its stable level, σa , to the out of control
222 level σ=δσ−σdaa between time intervals τ–1 and τ. The first observation after the shift in the process variance is
0 X ττ−τττ−τ= φ++−θ=+XadaXd11τ,
0 where the term X τ represents the observation not affected by the shift in the variance.
Expressions for subsequent observations are obtained by recursive substitution of the expressions of previous observations. Hence,
0 XXdτ+111=++φ−θ τ+ τ+ ( ) dτ 0 XXdτ+222=++φ−θ+φ τ+ τ+ ()() ddτ+ 1 τ
k XXd =++φ−θφ01j− d. τ+kkk τ+ τ+ ()∑ j=1 τ+k − j
The variance of the observations is
2 k Var⎡⎤ X =σ222 + δσ −σ⎡1. +() φ−θ φ21()j− ⎤ ⎣⎦τ+kX( aa) ⎣ ∑ j=1 ⎦
As in the previous example, the variation of the observations will increase, immediately after the shift, to a new level determined by the values φ1 and θ1, and the magnitude of the shift. The lead-one forecasts obtained in a similar manner as the observations are
ˆˆ0 XXτ−1 ()1 = τ ˆˆ0 XXττ()1 =+φ−θ+1 ( ) d τ ˆˆ0 XXτ+12()1 =+φ−θ+φτ+ ( )( ddτ+ 1 τ ) ˆˆ02 XXτ+23()1 =+φ−θ+φ+φτ+ ( )() dddτ+ 21 τ+ τ ˆˆ02⎡ 3⎤ X τ+34()1 =+φ−θ+φ+φ+φXdddτ+ ( )⎣ τ+ 321 τ+ τ+ dt ⎦
k XX ˆˆ1.=+φ−θφ01j− d τ+kk −1 ()τ+ ( )∑ j=1 τ+k − j
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Lead-one forecasts will exhibit an increase variation, one time interval after the shift, determined by the values φ1 and θ1, and the magnitude of the shift.
ˆ The residuals, obtained by subtracting the expression for X t−1 ()1 from the
expression for X t , are
0 eedττ=+ τ 0 eedτ+111=+ τ+ τ+ 0 eedτ+τ+τ+22=+2 0 eedτ+τ+τ+kk=+k.
The variance of the residuals is
2 Var[ eτ+ka] = δσ .
As in the case where the observations are represented with an AR(1) model, the shift in the process variance results in an immediate and constant change in the variance of the residuals. The same behavior is observed for processes that can be represented by an IMA(1, 1) model.
The assumption that the variance of the process is dependent on variance of the random shocks may be unreasonable when there is not a physical relationship between them; i.e. when the ARIMA model is just an empirical representation of the observed autocorrelation. This could happen, for example, given an accidental shift to a different process setting that allows larger fluctuations of temperature. In these cases, the variance of the residuals will not be constant. This is an opportunity for future investigation.
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3.1.4 Behavior of the Observations, Lead-One Forecasts, and Residuals after a Step Change in the Mean of the Random Shocks
Another type of special cause is the one that shifts the mean of the random shocks. The shift of the mean of the random shocks occurs between the time intervals τ –
1 and τ, that is
⎧01,2t =τ− τ− ,… Ea[]t = ⎨ ⎩δ=ττ+t ,1,… .
The effect of the change in the mean of the at can be modeled as in Section 3.1.3, that is the by adding the disturbance dt. However, dt is treated as a constant equal to the magnitude of the shift in the mean, δ.
Example 3.8. Suppose the autocorrelation of the observations can be represented with an AR(1) model,
X tt= φ+11Xa− t,
where XXtt=−μX. The observations, starting at t = τ, are obtained by recursively substituting a representation of the previous observations into that for the current observations. Thus, we have
0 XXaττ−τ= φ++δ=+11( ) X τδ 0 XXτ+11=++φδ τ+ ()1 1 02 XXτ+22= τ+ +()1 +φ 11 +φ δ 023 XXτ+33= τ+ ++φ+φ+φδ()1 111 k XX =+δφ0 j τ+kk τ+ ∑ j=0 1 k 0 1−φ1 =+δX τ+k , 1−φ1 where
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k k 1− φ φ=δj 1 , ∑ j=0 1 1− φ1
0 and X τ+k , k = 0, 1, 2…, represent the undisturbed value of the observation. The expected values of the observations are given by
k 0 1− φ1 EX[]τ =μX +δ , 1− φ1
⎡⎤00 where EX⎣⎦τ+k= μx is the level of the stable process. It follows that, after the shift in the mean of the random shocks, the observations will drift to a new level determined by
0 1 lim EX[]τ =μX +δ . k→∞ 1− φ1
The corresponding lead-one forecasts are also obtained by recursive substitution of previous observation and forecasted values starting at t = τ. Thus
ˆˆ0 XXXτ−11()1 =φτ− = τ ˆˆ0 XXττ()1 =+φδ+11 ˆ ˆ 02 XXτ+1211()1 =+φ+φδτ+ ()
k XXˆ 11= ˆ 0 +δ φj − , τ+kk −11() τ+ ()∑ j=0
ˆ 0 where the terms X τ+k , k = 0, 1, 2…, represent the lead-one forecast based on observations and random shocks of the stable process. Since,
k k 1− φ φ=δj 1 , ∑ j=0 1 1− φ1 then
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⎛⎞k ˆ ˆ 0 1−φ1 XXτ+kk −1 ()11= τ+ +δ⎜⎟ − ⎝⎠1−φ1
⎛⎞k ˆ 0 φ−φ11 =+δX τ+k ⎜⎟. ⎝⎠1−φ1 Hence, the lead-one forecasts will begin to increase one time interval after the shift,
k because the term δ()φ11− φ (1− φ 1) is being added onto the forecasts from the stable process.
The residuals result from subtracting the lead-one forecast from the corresponding observed value and are given by
0 eeτ+kk=+δ= τ+ , k 0,1,2,…,
00ˆ 0 where eXXτ+kk=− τ+ τ+k. The expected values of the residuals are
Ee[ τ+k ] =δ, k = 0,1,2,… .
Therefore, the residuals will exhibit an increase in level immediately after the shift in the mean of the random shocks.
3.2 Behavior of the Lead-One Forecasts and Residuals for Processes with Correlated Output and with Observed Correlated Measurement Error
In this section we investigate the behavior of the residuals when there is a step shift in the mean of the base process, μbp, or the mean or variance of the random shocks of the base process , ut. We do not specifically investigate the effect of a step shift in the variance of the base process because we assume that it depends on the variance of the
2 random shock, and therefore we focus on the effect of shifts in the variance σu .
In Chapter 2, Section 2.4, we propose using the model
YvBXPt= 0 ( ) t+ t, (3.6)
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where vB0 () is a discrete linear transfer function, to account for the contribution of the error introduced by the measurement process and to monitor the signal of the base process, Pt. We assume that the measurement process is in control and that the base process signal is autocorrelated and that it can be represented by a general ARIMA(pbp, dbp, qbp). However, we will only consider stationary models (i.e. dbp = 0), since nonstationary models can be transformed to stationary models. Therefore, in the following development we will use specific cases of the general ARMA(pbp, qbp) model
φ−μ=θbp(B)(P t bp) bp(Bu) t , (3.7) where
• μbp is the level of the base process; i.e., EP[ tb] = μ p.
2 p • φbp ()B =1 −φbp12BB −φ bp − −φbpp B represents the autoregressive component of
the series.
• B is the backwards shift operator, defined as BXXtt= −1 .
2 q • θ=−θ−θ−−θbp ()B 1 bp12BB bp bpq B describes the moving average part of the
series.
• ut is the random shock of the base process at time t. The {ut} is a white noise series
2 with mean 0 and variance σa .
3.2.1 Behavior of the Residuals after a Step Shift in the Mean of the Base Process
In this section we investigate the behavior of the residuals when there is a step shift in the mean of the base process. The cause of a step shift in the mean of the base process may be due to the accidental change of the set point of a process variable, or to
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the use of the wrong raw material, or to the incorrect use of the standard operational procedure, etc.
Example 3.9. We first assume that the unobservable output of the base process can be described by the AR(1) model. In that case, (3.7) reduces to
PPttbptt= μ+φ( −1 −μ+) ut, where the level of the base process, E[Pt] = μt, is made time dependent to account for the shift of the mean. Then, equation (3.6) can be written as
YvBXtttb=+μ+φ−μ01( ) p( Ptt− ) + ut. (3.8)
Here, v0(B) represents the transfer function established during Phase I; i.e. when the overall process, the base process, and the measurement process were in control.
The lead-one forecasts of Yt can be obtained by first writing (3.8) as
YvB= ⎡⎤μ + φ X− μ ++ aμ + φ P− μ + u. (3.9) tx()0 ⎣⎦x( txtb−101) p( t−0) t where all values in (3.9) are realizations of random quantities or calculated parameters, except for as yet unobserved values of at and ut. Therefore, the conditional expectation of
(3.9) with respect to previous observations of the output yields
ˆ YvBXtx−−10()1 = ( )⎣⎦⎡⎤μ + φx( t1− μxb) + μ 0+ φ p(Pt− 1− μ0)
ˆ =+vBX010()tb−− ()1.μ+φ−p (Pt 10μ )
The conditional expectations of at and ut are equal to zero (Box, Jenkins, Reinsel, 1994; p. 445).
Assume that a special cause shifts the in-control level of the base process, μ0, to a new level, μ1, between time periods τ – 1 and τ; that is
⎧μ0 t =τ−1, τ− 2,… μ=t ⎨ ⎩μ=ττ+1 t ,1,,…
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hence,
⎧μ=τ−τ−0 t 1, 2,… EP[]t = ⎨ ⎩μ=ττ+1 t ,1,… .
Then, the first residual after the shift at t = τ is
rYY=−ˆ (1) τττ−1 ˆ =+−−vBX001()τ P τ vBX ()τ− ()1,μ01− φbp (Pτ− − μ0 ) and its expected value is
⎡ ⎤ Er[ τττ] =+− Ev⎣ 001( Be) tb Pμ − φ p( P−− μ0)⎦
=μ10 −μ −φbp () μ 00 −μ
=μ10 −μ =δ,
ˆ where eXXτττ−=−1 ()1 and, because the measurement process is in control, E[et] = 0.
Subsequent residuals and their expected values are obtained in the same manner:
⎡⎤ Er[ τ+10110] = Ev⎣⎦() Beτ++ P τ+ −μ −φbp ( Pτ −μ 0) =δ(1 −φbp )
Er[]τ+2022010= Ev⎡⎤() Beτ++ P τ+ −μ −φbp () Pτ+ −μ =δ1 −φbp ⎣⎦() ⎡⎤ Er[]τ+kk= Ev⎣⎦000() Beτ++ P τ+kb −μ −φp() Pτ+k −μ =δ()1. −φbp
These expressions indicate that, on average, the first residual value after the shift in the mean of the base process will jump to a level equal to the magnitude of the shift, δ, and subsequent residual values will be at a new level determined by the magnitude and sign of φbp. If φbp is near +1, then the new level will decrease to almost its original level, whereas if φbp is close to –1, the new level will be close to twice the magnitude of the shift in the mean. This behavior of the residuals is the same as that of the residuals after a shift in the mean of a process with autocorrelated output and no observed measurement error (see Table 1 in Section 3.1.1).
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Assuming that the base process and the measurement process are stationary or that they can be transformed into stationary processes, the variance of these residuals is obtained by first noting that the general ARMA model that describes the bases process can be written as
θbp (B) Putbp−μ = t, φbp ()B or it can be written in the moving average form (Box, Jenkins, Reinsel, 1994; p. 46)
PBtbpbp−μ =ψ ( )ut.
Then ,
θbp (B) 2 ψ=bp ()BB =+ψ+ψ+1 12B…. (3.10) φbp ()B
The general ARMA model that describes the measurement process is
θx (B) X tx−μ = at. φx ()B
Hence, vBX0 ( ) t can also be expressed in terms of the random shocks, at, as
b ωθ(BB) x ( B) 2 ζ ()B==ζ01+ ζ B+ ζ 2B+…. (3.11) δφ()BBx ()
Therefore, equation (3.6) can be written as a function of the white noise processes
YBaBtt=ζ( ) +ψ( )ut, where ω(B), δ(B), φx(B), θx(B), φbp(B), and θbp(B) are finite-order polynomials in B, and the roots of ω(B) = 0, δ(B) = 0, φx(B) = 0, θx(B) = 0, φbp(B) = 0, and θbp(B) = 0, are all outside the unit circle. Also, at and ut are mutually independent and each of them are
2 2 independent random variables with zero means and variances σa and σu , respectively.
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The variance of the lead-k forecast error (the lead-k residual) is given by (Wei,
2006; p. 343. Box, Jenkins, Reinsel, 1994; p. 446)
2 kk−−11 Var⎡⎤ r k=− E⎡⎤ Y Yˆ k =σ222ζ +σ ψ2. ⎣⎦() ⎣⎦tt−1 () a∑∑jj==00 j u j
Hence, the variance of the residuals (i.e. the lead-one forecast error) is
22 2 Var[ rtr] = σ=ζσ+ψσ00 a u.
Here, the coefficients ζ0 and ψ0 are the first coefficient in equations (3.10) and (3.11), respectively. Also, from theses equations it can be found that ψ0 = 1 and that ζ0 = 0, if b
≠ 0, or ζ0 = ω0 if b = 0.
Therefore, the residuals rt, for t = τ, τ + 1, τ + 2, …, are independent normal variates with mean
⎧01t =τ−, τ−2,… ⎪δ=τt Er[]t = ⎨ ⎪δ−φ1;tkk =τ+ =1,2,…, ⎩ ()bp
2222 and variance σ=ζσ+σra0 u, where
⎧ω0 if b = 0 ζ=0 ⎨ ⎩ 0if b ≠ 0 .
Example 3.10. Next, suppose that the unobservable output of the base process can be described by the ARMA(1, 1) model. Then (3.7) reduces to
PPuttbptt=μ +φ( −11 −μ) + tbpt −θ u− .
As before, the level of the base process is made time dependent to account for the shift of the mean. The overall process is now described by the expression
YvBXtttb=+μ+φ−μ+−θ01() p( Pttt− ) ubp ut−1.
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Suppose the mean of the base process shifts between time periods τ – 1 and τ, such that
⎧μ0 t =τ−1, τ− 2,… μ=t ⎨ ⎩μ=ττ+1 t ,1,,… and,
⎧μ=τ−τ−0 t 1, 2,… EP[]t = ⎨ ⎩μ=ττ+1 t ,1,… .
The minimum mean squared error forecast of the in-control processes is
ˆˆ YvBXttb−−10()11=+( ) 10( ) μ+φ−p( Pt − 10μ−) θbprt−1.
The residuals are given by the expression
rvBePtttb=+−μ−φ−μ+θ001( ) p( Pt− 0) bp rt−1, where E[et] = 0. Also, E[rt] = 0 for t = τ – 1, τ – 2, ... . Starting at t = τ the expected values of the residuals are
Er[ τ ] =δ Er =θ⎡⎤ +1 −φ δ []τ+1 ⎣⎦bp ()bp Er =θ⎡⎤2 +11 −φ +θ δ []τ+2 ⎣⎦bp ()()bp bp Er =θ⎡⎤32 +11 −φ +θ+θ δ []τ+3 ⎣⎦bp ()bp() bp bp Er = ⎡θ42 +11 −φ +θ +θ +θ3⎤ δ []τ+4 ⎣ bp ()bp() bp bp bp ⎦
k Er =θ⎡⎤kj +1 −φ θ−1 δ . []τ+kbpbpbp⎣⎦()∑ j=1
Since,
k k 1− θ θjk−−121 =1 +θ +θ +… +θ = bp , ∑ j=1 bp bp bp bp 1− θbp then the general expression of the mean of the residuals is
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k (φ−θθ+−φbp bp) bp1 bp Er[]τ+k = δ . 1−θbp
Assuming that the shift of the mean of the bases process remains unchanged, the residuals approach the steady-state level
1− φbp lim Er[]τ+k = δ , k→∞ 1−θbp at a rate determined by the magnitude of the coefficient θbp. Therefore, the residuals after the shift in the mean of the process are independent normal variates with mean
⎧01t =τ−, τ−2,… ⎪ ⎪δ=t τ Er[]t = ⎨ k ()φ−θθ+−φbp bp bp1 bp ⎪ δ=τ+=tkk; 1,2,… , ⎪ ⎩ 1−θbp
2222 and variance σ=ζσ+σra0 u, where
⎧ω0 if b = 0 ζ=0 ⎨ ⎩ 0if b ≠ 0.
This is the same behavior of the residuals when there is a step shift in the mean of a process with autocorrelated observations (see Table 1 in Section 3.1.1). That is, the first residual after the shift in the mean of the base process is expected to jump to a new level given by the magnitude of the shift, δ. Subsequent residuals decrease steadily at a rate given by the magnitude of the coefficient θBP until it reaches the level
−1 ()()11−φbp −θ bp δ.
From these two examples, we see that the behavior of the residuals
ˆ rYYttt=−−1 ()1 after a step shift in the mean of the bases process is determined by the
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ARMA (or ARIMA) model that describes the autocorrelation of the base process.
Therefore, the behavior of the residuals from model (3.6) after a step shift in the mean of the bases process is also summarized by Table 1 in Section 3.1.1.
3.2.2 Behavior of the Residuals after a Step Shift in the Variance of the Base Process
The moving average representation of the base process is
∞ Pa =+ψ+ψ+= a a … ψa, tt11 t− 2 t−− 2 ∑ j=0 jtj
where PPttb=−μp. Hence,
∞ Var⎡⎤ P = σ=σ22 ψ2. (3.12) ⎣⎦tPu∑ j=0 j
The variance of the base process is a function of the variance of the random shocks ut. Thus, it seems reasonable to assume that changes in the variance of the output of the base process are due to special causes that change the variance of the random shocks. In other words, the variation of a process output is caused by changes in the internal variation (random shocks) of the process. For example, an incorrectly installed power supply may cause an erratic voltage output that in turn causes the process to generate a product with highly variable characteristics.
The variance of the overall process is a function of the variance of the random shocks of the measurement process and the base process, that is
∞∞ σ22=σ ζ 22+σ ψ2. (3.13) Ya∑ jj=0 ju∑ =0j
Where this last expression was obtained from the moving average representation of the overall process (Wei, 2006; p. 342),
∞∞ Ya=ζ+ψu. tj∑ jj==00t− jj∑ t− j
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Therefore, changes in the overall process variance, given that the measurement process is in control, are caused by changes in the variance of the random shocks of the base process.
In the following two subsections we are going to investigate the behavior of the observations, lead-one forecasts, and the residuals of the overall process when there is a step shift in the mean or in the variance of the random shocks.
3.2.3 Behavior of the Residuals after a Step Shift in the Mean of the Random Shocks of the Base Process
Example 3.11. We first consider the simple case where the base process can be described by the AR(1) model
PPtbpt= φ+−1 ut,
where PPttb=−μp.
The step shift in the mean of the random shocks of the base process can be modeled by superimposing a step disturbance onto the random shocks as
⎧utt =τ−1, τ− 2,… ut = ⎨ ⎩utt +δ =τ,1,, τ+ … such that,
⎧01,2t =τ− τ− ,… Eu[]t = ⎨ ⎩δ=ττ+t ,1,… .
The base process values evolve as follows after the shift in the variance:
0 PPuττ−ττ=φbp 1 +( +δ) = P +δ 0 PPuPτ+111=φbp τ +() τ+ +δ =τ+ +()1 +φbp δ 02 PPuPτ+2122=φbp τ+ +() τ+ +δ =τ+ +()1 +φbp +φ bp δ
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k PP = 0 +δ φj , τ+kk τ+ ∑ j=0 bp
0 where Pτ+k , k =0, 1, 2, …, represents the in-control part of the base process. Then, based on (3.6), the overall process observations after the shift in the variance are described as
YvBXPvBXτττττ=+=+φ++δ001() ( ) bp P−τ( u ) 00 =++δ=+δvBX0 () ττ P Y τ 0 Yτ+10=+=+++φδ v() BXτ+ 1 P τ+ 10 v() BXτ+ 1 P τ+ 1()1 bp 0 =++φδYτ+1 ()1 bp (3.14) 02 YvBXPvBXPτ+20=+=+++φ+φ() τ+ 2 τ+ 20() τ+ 2 τ+ 2()1 bp bp δ 02 =Yτ+2 +()1 +φbp +φ bp δ k YY=+δφ0 j , τ+kk τ+ ∑ j=0 bp
0 0 where the YvBXPτ+kk=0 () τ++ τ+k, k = 0, 1, 2, …, represents the in-control part of the
overall process, and vB0 ()is the transfer function determined in Phase I from the in- control measurement and overall process signals.
The minimum mean squared lead-one forecast of the base process is given by
ˆ PPtbp()1 =φ t−1 ,
ˆ ˆ where PPtt()11=−μ () bp. Hence, the lead-one forecast of the overall process can be written as
ˆˆ YvBXtt−10(1) =+φ( ) bpPt−1.
The lead-one forecasts starting at t = τ are also obtained by recursive substitution of the expression of previous observations and forecasted values of the base process.
These forecasts are
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ˆˆ ˆ0 YvBXPYτ−10()11=+( ) τ− 1( ) φ=bp τ− 1 τ ˆˆ ˆ0 YvBXPYτττ()11=+01 ( ) () φ=+bp τ+φδbp ˆˆ ˆ02 YvBXPYτ+10()11=+ ( )τ+ 1 () φ=+bp τ+ 12 τ+ (φ+bpφδ bp ) (3.15) k YYˆˆ1.=+δφ0 j τ+kk −1 () τ+ ∑ j=1 bp
ˆ Subtracting the expressions for Yt−1 (1) from the expression for Yt yields the residuals
0 rrτ+kk= τ+ +δ, k =0, 1, 2, … .
0 0 The residuals rτ+k , k =0, 1, 2, …, result from the difference between the observations Yτ
ˆ 0 and forecasts Yτ from the stable process . The expected values of the residuals are
⎧01,2t =τ− τ− ,… Er[]t = ⎨ ⎩δ=ττ+t ,1,,
2222 their variance isσ=ζσ+σra0 u, where
⎧ω0 if b = 0 ζ=0 ⎨ ⎩ 0if b ≠ 0.
Therefore, a shift in the mean of the random shocks of the base process causes an immediate and persistent change in the mean of the residuals.
Example 3.12. Another case is when the base process can be described by the
ARMA(1, 1) model
PPuut=φ bpt−11 + t −θ bpt− .
The values of the base process after the shift in the mean of the random shocks ut are
0 PPuττ−τ= φ++δ−θ=+δbp 11( ) bp uP τ−τ 0 ⎡⎤ PPτ+11=++φ−θδ τ+ ⎣⎦1 bp bp PP = 02+⎡11 +φ+φ−θ +φ⎤ δ τ+22 τ+ ⎣ bp bp bp() bp ⎦
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PP =02 +⎡11 +φ +φ +φ3 −θ +φ +φ2⎤ δ τ+33 τ+ ⎣ bp bp bp bp( bp bp )⎦
kk−1 PP =+0 ⎡⎤ φ−θφδjj. τ+kk τ+ ⎣⎦∑∑jj==00 bpbpbp
Substituting the expressions for he base process values into equation (3.6) yields the overall process values
0 YvBXPYττττ=+=+δ0 () 0 ⎡⎤ YvBXPYτ+10=+=++φ−θδ() τ+ 1 τ+ 1 τ+ 1⎣⎦1 bp bp YvBXPY= + =02 +⎡⎤11 +φ +φ −θ +φ δ τ+20() τ+ 2 τ+ 2 τ+ 2⎣⎦bp bp bp() bp (3.16) YvBXPY= + =02 +⎡11 +φ +φ +φ3 −θ +φ +φ2⎤ δ τ+30() τ+ 3 τ+ 3 τ+ 3⎣ bp bp bp bp() bp bp ⎦
k k −1 YvBXPY=+=+φ() 0 ⎡⎤j −θ φj δ. τ+kkkkb0 τ+ τ+ τ+ ⎣⎦∑ j=0 pbp∑ j=0 bp
The lead-one forecast of the overall process value at t = τ is
ˆˆ ˆ0 YvBXPrτ−10()11= ( ) τ− 1( ) +φbpτ− 1 −θ bp τ− 1 =Y τ ,
ˆ 0 where Yτ is the lead-one forecast of the in-control overall process and vB0 () is the transfer function determined in Phase I from the in control measurement and overall process signals. The lead-one forecast at t = τ + 1 is
ˆˆ YvBXPτ (11) = 0 ( ) ττ( ) +φbp −θ bprτ and the residual rτ is ˆˆ000 rYYτττ−τ= −1 (1) = Y +δ− Y τ = r τ +δ.
Hence,
YvBXPˆˆ()11=+( ) ( ) φ+ 00δ−θ r+δ τττ0 bp ( )τ bp ( τ) ˆ ˆ 0 =++φ−vBX01()ττ+ ()1 P ()bpθδ bp ˆ 0 =+φ−θδYτ+1 ()bp bp ,
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ˆ ˆ 0 ˆ 0 where X τ ()1 , Pτ+1 , and Yτ+1 are the lead-one forecasts of the in-control measurement process, the base process, and the overall process, respectively.
Proceeding in the same way, the expression of subsequent lead-one forecast and their corresponding residuals are
0 rrτ+11=+δ τ+ ˆˆ YvBXPrτ+10()11=+ ( )τ+ 1 () φ−bpτ+ 1θ bp τ+ 1 =+φ+φ−θ+φδYˆ 02⎡⎤1 τ+2 ⎣⎦bp bp bp() bp 0 rrτ+22=+δ τ+ ˆˆ YvBXPrτ+20()11=+ ( )τ+ 2 () φ−bp τ+ 2θbp τ+ 2 (3.17) =Yˆ 0232 +φ+φ+φ−θ⎡⎤1 +φ+φ δ τ+3 ⎣⎦bp bp bp bp() bp bp 0 rrτ+kk −11=+δ τ+ − kk−1 YvBXˆˆ()11=+ ( ) () φPrY − θ=+φ−θφˆ 0 ⎡ jj⎤ δ τ+kk −10τ+ bpτ+kbpkk −11 τ+ − τ+ ⎣∑∑jj==10 bpbpbp⎦ 0 rrτ+kk=+δ τ+ .
0 ˆ 0 ˆ 0 0 In these equations Pτ+k , X τ+k , X τ+k −1 (1) , Yτ+k , Yτ+k , and rτ+k , k = 0, 1, 2, …, represent the quantities that correspond to the in-control part of each of the processes.
Therefore, the residuals of the overall process are independent normal variates with mean
⎧01,2t =τ− τ− ,… Er[]t = ⎨ ⎩δ=ττ+t ,1,,…
2 and variance σr .
Expressions (3.14) and (3.16) show that overall process observations will begin to deviate from their in-control level immediately after the shift in the mean of the random shocks of the base process until they reach a stable level that is determined by the
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magnitude and sign of the model coefficients. Expressions (3.15) and (3.17) indicate that the lead-one forecasts of the overall process will also increase, but their increment will start one time period after the shift in the mean of the random shocks and they will stay below their corresponding observations by the magnitude of the shift, i.e. δ.
3.2.4 Behavior of the Residuals after a Step Shift in the Variance of the Random Shocks of the Base Process
In this subsection we study the behavior of the residuals after a shift in the
2 variance of the random shocks of the base process from its stable level, σu , to the out of
2 control level σd between time intervals τ – 1 and τ.
Example 3.13. Let’s assume that the base process can be described by the AR(1) model
PPtbpt= φ+−1 ut,
where PPttb=−μp. The change in the variance of the process is represented by superimposing a disturbance d onto the random shocks, ut, starting at t = τ. The disturbances dτ, dτ+1,… are normally distributed with mean zero and variance
222 σ=δσ−σduu. The disturbance series is independent of the random shocks, ut, and of the random shocks, at. This implies that the base process represented as
⎧φ+Pu t =τ−τ−1, 2,… ⎪ bp t−1 t Pt = ⎨ … ⎩⎪φ++=ττ+bpPudt t−1 t t ,1, .
The base process values after the shift in the variance are
0 PPudPdττ−ττττ=φbp 1 +( +) = +
0 PPudPddτ+11=φbp τ +() τ+ + τ+11 = τ+ +φbp τ + τ+1
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02 PPudPdddτ+21222=φbp τ+ +( τ+ + τ+) = τ+ +φbpτ +φ bp τ+ 1 + τ+2 k PP =+0 φkj− d, τ+kk τ+ ∑ j=0 bpjτ+
0 where Pτ+k , k =0, 1, 2, …, represents the in-control part of the base process. The variance of the base process after the shift is
k Var⎡⎤ P =σ222 + δσ −σ φ2()kj− . ⎣⎦τ+kbpuu( )∑ j=0 bp
The expressions that describe the overall process output after the shift in the
variance, obtained from substituting the expressions for Pτ+k , k = 0, 1, 2 …, into equation
(3.8), are
00 YvBXPdYτ = 0 ( ) ττ++=+ ττ dτ 0 YYτ+11=+φ+ τ+bp dd τ τ+1 02 YYτ+22=+φ+φ+ τ+ bp dτ bp dd τ+12 τ+ (3.18) k YY=+0 φkj− d. τ+kk τ+ ∑ j=0 bpjτ+
where vB0 () is the transfer function determine in Phase I from the in-control
0 0 measurement and overall processes, and YvBXPτ+τkk=0 ( ) ++τ+k, k = 0, 1, 2, …, is the in- control part of the overall process. The variance of the overall process observations is given by
k Var Y =σ222 + δσ −σ φ2()kj− . [ τ+kY] ( uu)∑ j=0 bp
The minimum mean squared lead-one forecast of the base process is given by
ˆ PPtbp()1 =φ t−1 .
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Hence, the lead-one forecast of the overall process can be expressed as
ˆˆ YvBXtt−10(11) =+( ) −− 1( ) φbpPt1.
The lead-one forecasts starting at t = τ, obtained by recursive substitution of the expressions of previous observations and lead-one forecasts are
ˆˆ ˆ0 YvBXPYτ−10()11=+ ( ) τ− 1( ) φ=bp τ− 1 τ ˆˆ 00 ˆ YvBXPdYdτττ()11=+01 ( ) () φ+=+bp ()ττ+τφbp ˆˆ 00ˆ2 YvBXPddYddτ+10()11=+ ( )τ+ 1 () φ+bp ()τ+ 1φ+=+bp τ τ+ 12 τ+ φ+bpτφ bp τ+1 (3.19) k−1 YYˆˆ1.=+0 φkj− d τ+kk() τ+ ∑ j=0 bpjτ+
ˆ Subtracting the expressions for Yt−1 (1) from the expression for Yt yields the residuals
0 rrdτ+τ+τ+kk=+K, k =0, 1, 2, … .
0 0 The residuals, rτ+k , k =0, 1, 2, …, result from the difference between the observations Yτ
ˆ 0 and forecasts Yτ from the stable process. The variance of the residuals is
2 Var[ rτ+ku] =δσ, k = 0,1,2,… ,
2 where σu is the variance of the residuals from the in-control process. Therefore, a shift in the variance of the random shocks of the base process will have an immediate and persistent effect on the variance of the residuals; their mean, however, does not change.
Example 3.14. Now we consider the case when the ARMA(1, 1) model
PPuut=φ bpt−11 + t −θ bpt− describes the behavior of the base process. As in the previous case, the change in the variance of the random shocks of the base process is represented by superimposing a
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disturbance d onto the random shocks ut starting at t = τ. These disturbance values are
222 normally distributed with mean zero and variance σdu=δσ −σu. The disturbance series is independent of the random shocks ut and of the random shocks at. The base process is represented as
⎧φ+−θPu u t =τ−τ−1, 2,… ⎪ bp t−−11 t bp t Pt = ⎨ … ⎩⎪φ++−θ=ττ+bpPudut t−−11() t t bp t ,1,, and the expressions that represent it after the shift in the variance are
0 PPuduPdττ−τττ−ττ=φbp 11 +( +) −θbp = + PPududτ+11=φbp τ +()() τ+ + τ+1 −θbp τ + τ 0 =++φ−θPdτ+11 τ+ ()bp bp dτ PPududτ+212211=φbp τ+ +()() τ+ + τ+ −θbp τ+ + τ+ 0 =++φ−θφ+Pdτ+22 τ+ ()()bp bp bpddτ τ+1 PPududτ+323322=φbp τ+ +()() τ+ + τ+ −θbp τ+ + τ+ 0 2 =+Pdτ+3 τ+31+φ()bp −θ bp() φ bpdddτ +φ bp τ+ + τ+2 k −1 PPd =++φ−θφ01kj−− d, τ+kkkbpbpbpj τ+ τ+ ()∑ j=0 τ+
0 where Pτ+k , k =0, 1, 2, …, represents the in-control part of the base process. The variance of the base process after the shift is
k−1 Var⎡⎤ P =σ222 + δσ −σ⎡1 + φ −θ φkj−−1⎤ . ⎣⎦τ+kbpuu( ) ⎣ ( bpbpbp)∑ j=0 ⎦
The expressions that describe the overall process output after the shift in the
variance, obtained from substituting the expressions for Pτ+k , k = 0, 1, 2 …, into equation
(3.8), are
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00 YvBXPdYττττττ=++=+0 () d 0 YYdτ+111=++φ−θ τ+ τ+ ()bp bp dτ 0 YYdτ+222=++φ−θφ+ τ+ τ+ ()()bp bp bpddτ τ+1 (3.20) 02 YYdτ+333=++φ−θφ+φ+ τ+ τ+ ()bp bp() bpdτ bp dd τ+12 τ+ k −1 YYd=++φ−θφ01kj−− d, τ+kkkbpbpbpj τ+ τ+ ()∑ j=0 τ+
where vB0 ()is the transfer function determined in Phase I from the in-control overall and
0 0 measurement processes, and YvBXPτ+τkk=0 ( ) ++τ+k, k = 0, 1, 2, …, is the in-control part of the overall process. The variance of the overall process is given by
k −1 Var Y =σ222 + δσ −σ⎡1 + φ −θ φkj−−1⎤ . []τ+k Y( u u ) ⎣ ( bp bp )∑ j=0 bp ⎦
The minimum mean squared lead-one forecast of the base process is given by
ˆ PPtbp(1) =φ t−1 .
Hence, the lead-one forecast of the overall process can be expressed as
ˆˆ YvBXPttb−−−10(11) =+( ) 1( ) φ−pt 1θbprt−1.
The lead-one forecasts at t = τ is
ˆˆ ˆ0 YvBXPrτ−10()11= ( ) τ− 1( ) +φbpτ− 1 −θ bp τ− 1 =Y τ ,
ˆ 0 where Yτ is the lead-one forecast of the in-control overall process. The next forecast is
ˆˆ YvBXPrτττ()11=+0 ( ) ( ) φ−bpθ bp τ ˆ 00 =+vBX0 ()τ ()1 φ+−bp ()( Pττ dθbp r ττ+ d), ˆ 0 =+φ−θYdτ+1 ()bp bp τ, where the residual rτ is given by
ˆˆ000 rYYτττ−τ=−1 (1) =+−=+ Y dY τττ r dτ.
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The following residuals and lead-one forecasts are obtained in the same manner:
0 rrdτ+11=+ τ+ τ+ 1 ˆˆ YvBXPrτ+10()11=+ ( )τ+ 1 () φ−bpτ+ 1θ bp τ+ 1 ˆ 0 =+φ−θφ+Ydτ+21()()bp bp bp τd τ+ 0 rrdτ+22=+ τ+ τ+ 2 YvBXPrˆˆ()11=+ ( ) () φ− θ τ+20τ+ 1bp τ+ 2bp τ+ 2 (3.21) ˆ 02 =+φ−θφ+φ+Ydτ+31()bp bp() bpτ bpd τ+d τ+2 0 rrdτ+kk −11=+ τ+ − τ+ k − 1 k −1 YYˆˆ1 =+φ−θ0 φkj−−1 d τ+kkb −1 () τ+ ()pbp∑ j=0 bpτ+ j 0 rrdτ+kk=+ τ+ τ+ k.
ˆ 0 0 In these equationsYτ+k , and rτ+k , k = 0, 1, 2, …, represent the lead-one forecasts and the residuals of the in-control processes.
The variance of the residuals is
2 Var[ rτ+ku] =δσ, k = 0,1,2,… ,
2 where σu is the variance of the residuals from the in-control process.
The variance of the residuals of the overall process will increase immediately after the shift in the variance of the random shocks of the base process, but their mean will remain unchanged. Also, the variance of the observed and lead-one forecast values of the overall process will continue to increase until it reaches some constant level, as shown by equations (3.18), (3.19), (3.20), and (3.21).
3.3 Summary and Discussion
We derived the expressions that describe the behavior of the residuals, the lead- one forecasts, and, when necessary, the behavior of the observations for step shifts in the
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mean or variance of the process and for step shifts in the mean or variance of the random shocks. These expressions can be extended to describe the effect of other types of changes, such as shifts that have a limited duration, in the process observations, lead-one forecasts, and the residuals.
The importance of these expressions is that they helps us recognize patterns that indicate the presence of out-of-control conditions and how the special causes that generate them are affecting the process. This will facilitate the identification of the special cause and its permanent elimination. For example, suppose that the CCC chart displays observed values that start to drift upwards followed by the forecasts, which lag behind one time period, with constant difference between the two series. At the same time that the observations begin to drift we see an upwards step change in the I-chart.
This behavior of the observations, forecasts on the CCC chart and residuals on the I-chart is a clear indication that the special cause is affecting the mean of the random shocks.
Perhaps, the culprit of this behavior is an internal component of the process that has suffered some damage or deterioration. Furthermore, these characteristic patterns suggest the use of specific SCR’s (see Table 1 in Chapter 1) to detect smaller shifts faster.
In Chapters 4 and 5 we illustrate Phase I and Phase II of the implementation of the monitoring schemes proposed in Chapter 2, as well as the effect of the shifts in the mean and the variance of the process and random shocks on the CCC chart and the SCC charts.
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CHAPTER FOUR
Illustration of the Monitoring of a Process with Autocorrelated Output and Unobservable Measurement Error
In this chapter we will illustrate the monitoring scheme proposed in Chapter 2 for a process with autocorrelated output where the measurement error is not observed. The measurement process (outlined in Figure 6) of the pharmaceutical process introduced in
Chapter 1, is an example of a process that generates autocorrelated observations with no observable error of measurement (this error could be obtained if the same reference material were measured several times).
Base Process
Product Pt Yt
Measurement Process
Xt Reference
Figure 6. Diagram of the measurement of the product and reference materials and corresponding signals of the base pharmaceutical process and the measurement process.
The measurement process is assessed by measuring the absorbance of a reference material every time a product material is measured. These measurements are naturally autocorrelated and the removal of the causes of the autocorrelations is not economically
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feasible. Therefore, we will use the proposed CCC/SCC chart scheme to verify that the measurement process is in control and to maintain it in control.
The implementation of the monitoring scheme for the measurement process will be performed using the two-phase procedure detailed in Section 1.1 of Chapter 1. In
Phase I of the implementation, 983 observations were obtained from the actual pharmaceutical process described in Chapter 1. It is believed that the measurement process was stable (i.e. not affected by out-of-control conditions) when these data were collected. Because of the lack of Phase II data to illustrate the monitoring of the stability of the process, the 983 observations are split into two sets. The first set consists of the first 900 observations and will be used for Phase I; i.e., used to fit the ARIMA model and construct the CCC and SCC charts. The remaining 83 observations will be used to illustrate Phase II of the implementation; i.e. the actual monitoring of the measurement process using the CCC and SCC from Phase I.
4.1 Phase I Implementation
The procedure to implement the proposed CCC and SCC chart scheme is described in Section 2.3.3 of Chapter 2. In our case the quality characteristic of interest has been determined (i.e. the absorbance of the reference buffered solution). Also, we believe that the measurement process remained stable during data collection and used standard procedures to obtain the reference material measurements. We will follow the procedure in Section 2.3.3 to verify that the data is autocorrelated, use the autocorrelation structure to identify an appropriate ARIMA model, estimate the model parameters, and construct the CCC and SCC charts.
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4.1.1 Identification of the ARIMA Model and Estimation of Its Parameters
The objective in this part of Phase I is to select (i.e. identify) a model or models that are likely to fit the measurement process data from the family of ARIMA(p, d, q) models (see Section 3.1 in Chapter 3),
d φ∇=θ(B) XBt( )at. (4.1)
The first task of the identification step is to determine if the sequence of reference
material measurements (hereafter called the measurement process series) generated by
the measurement process is stationary. If the series is not stationary, then the level of
difference, d, needed to transform the original series into a stationary series needs to be determined. Stationarity ensures that the variance of the autocorrelated process is finite
(Box, Jenkins, and Reinsel, 1994; p. 49-50. Wei, 2006; p. 6-10).
We use PROC ARIMA in SAS 9.1.3 to identify an appropriate ARIMA model, to
estimate model parameters and the residual variance, and to generate lead-one forecasts.
The measurement process series plotted against the observation number, a time
index, is shown in Figure 7. The series exhibits a non-constant overall mean, but with
parts of the series having similar patterns. This may be an indication of a homogeneous
non-stationary process, and that the first-order difference of the series may yield a
stationary process (Box, Jenkins, and Reinsel, 1994; p. 95-96).
Also, the autocorrelation function (ACF) and the partial autocorrelation function
(PACF) of the data can be used to confirm that a process is non-stationary. The ACF and
PACF are generated using PROC ARIMA by including the IDENTIFY statement (See
SAS Code 1 in Appendix C, where the measurement process series data is specified as
X).
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I measurement process data. process I measurement Observation No. Observation Figure 7. Time plot of Phase Figure 7. Time plot of Phase 0 100 200 300 400 500 600 700 800 900
40 30 20 10 Data
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Figure 8 shows the ACF plot of the measurement process series. The height of the vertical line represents the magnitude of the autocorrelation coefficient, ρk, from time lag k = 0 (autocorrelation of the data with itself) to time lag k = 24. The correlation coefficient is defined as:
Cov( X, X ) γ ρ= ttk+ = k , k γ Var() Xtt Var() X +k0
where γ=ktEX()( −μ X tk+ −μ), ρ0 = 1, and if the process is stationary the variance,
2 γ=σ0 X , at time t is the same at time t + k.
The horizontal curves, above and below the zero autocorrelation point, represent the 95% confidence interval of ρk, which is approximately ± 2 n (Box, Jenkins, and
Reinsel, 1994; p. 188). If a bar protrudes beyond the 95% confidence interval curve, then the null hypothesis that ρk = 0 is rejected. The ρk’s that are not zero may correspond to
MA components that need to be included in model.
Figure 9 shows the PACF plot of the measurement process data. The partial autocorrelation is defined as
Cov⎡ X−− Xˆˆ, X X ⎤ ⎣( t t) ( tk++ tk)⎦ Pk = , ˆˆ Var() Xtt−− X() X tktk++ X
ˆ where X tk++−+−−=+bX1122 tk bX tk ++ bk 1 X t+1, and the bj (1≤ j ≤k–1) are mean-squared
ˆ 2 linear regression coefficients calculated from minimizing EX( tk++− X tk) (Wei, 2006, p.
11-14). The height of the vertical lines in the PACF corresponds to the magnitude of the
Pk, from time lags 0 to 24. The horizontal curves above and below Pk = 0 correspond to
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the 95% confidence interval, given by ± 2 n (Box, Jenkins, and Reinsel, 1994; p. 188).
If a bar protrudes beyond the 95% confidence interval curve, then the null hypothesis that
Pk = 0 is rejected. The Pk’s that are not zero may correspond to AR components that need to be included in the model.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Autocorrelation 0.0 -0.1 -0.2 -0.3
0123456789101112131415161718192021222324 Lag Figure 8. ACF of the measurement process series.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Partial Autocorrelation Partial 0.1
0.0
-0.1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Lag Figure 9. PACF of the measurement process series.
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The guidelines to identify a possible model to represent the behavior of the data based on the ACF and PACF are given in Table 5 (Box, Jenkins, and Reinsel, 1994; p.
186. Wei, 2006; p. 54-59).
Table 5. Guidelines to identify a model based on the ACF and PACF
Process ACF PACF
AR(p) Decrease towards zero (“tails off”) Cuts off at lag p
MA(q) Cuts off at lag q Decrease towards zero (“tails off”)
ARMA (p, q) Decrease towards zero (“tails off”) Decrease towards zero (“tails off”)
Decrease slowly, almost constantly Nonstationary Decrease towards zero (“tails off”) towards zero
The ACF and PACF in Figures 8 and 9 suggest that the measurement process series follows an AR process. On the other hand, the slow decreasing autocorrelation pattern of the ACF in Figure 8 may be an indication that the process is nonstationary.
The ARMA(p, q) model φ=θ(B) XBt( )at, where XXtt= −μ, will define a stationary process if all roots of the characteristic equation φ(B) = 0 are outside the unit circle. If the ARMA process is stationary it will have an infinite moving average representation (Box, Jenkins, and Reinsel, 1994; p. 77-78),
∞ X =ψBa = ψ a , tt() ∑ j=0 jt− j
∞ with ψ = 1, and ψ <∞, and will have a finite variance given by 0 ∑0 j
∞ Var X = σ2ψ2. (4.2) ( ta) ∑ j=0 j
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If any of the roots of the characteristic equation are inside the unit circle, the X t values described by an ARMA process exhibit exponential growth. If d of the roots lie on the unit circle, the process is still not stationary, but it can be made stationary by using
th d d the d difference of X t ; i.e. ∇=−X tt()1 BX.
The autocorrelation function of an ARMA(p, q) satisfies the difference equation
th φρ=()B k 0 , where ρk is the k autocorrelation and k > q. The solution of the difference equationφ=()B 0 is
p φ=B 1 −GB , ( ) ∏i=1( i ) where 1/Gi are the roots of the characteristic equation. Assuming that these roots are not equal, the autocorrelation function is (Box, Jenkins, and Reinsel, 1994; p. 184-185)
kk k ρ=kA11GAG + 2 2 +… + AGpp, (4.3) for k > q – p. The stationarity requirement is that the 1/Gi lie outside the unit circle, in which case the Gi values lie inside the unit circle; i.e. | Gi | < 1. Depending on the value of the roots of (4.3) we have three situations:
1. If the values Gi are less than 1, equation (4.3) shows that autocorrelation values will
tend towards zero as k increases. That is the ACF graph will show a pattern where
the ρk decrease towards zero;
2. If one or more of the values of Gi are equal to 1, then equation (4.3) reduces to a
constant for moderate and large k. For example, if one root is equal to 1, say G2,
then ρk = A2G2. In this case the ACF will show the ρk decreasing slowly to a
constant level and does not decrease to zero for large k;
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3. If one of the Gi values is close to 1, the ACF will show a pattern similar to the case
when one of the roots is equal to one; i.e. the ACF decreases slowly, and if k is large
enough, the autocorrelations will decay to zero.
For moderately large values of k it is difficult to distinguish between the ACF of a stationary and the ACF of a nonstationary process. Therefore, it is important to interpret the ACF and the PACF together. If the PACF indicates that the series can be represented with an AR(p) model, even though it is not clear from the ACF if the process
2 is nonstationary, we would proceed to fit it. If the residuals are IID N(0, σa ) then we know that the AR(p) model can be used to represent the data.
When identifying a model to be used for process control and forecasting it is important to keep in mind that, should a root of φ(B) = 0 be close to 1, the process will wander away from is mean and will take several periods before it returns to its overall level; i.e. the process behaves more like a nonstationary model. Therefore, if the ACF seems to indicate that a root of φ(B) = 0 is 1 or close to 1, and the purpose of the model is to forecast and monitor a process, it may be better to use a nonstationary model because it does not include the mean; i.e. ()1− B d μ=0(Box, Jenkins, and Reinsel, 1994; p. 207-208).
The ACF in Figure 8 and the PACF in Figure 9 seem to indicate that a subset of an AR(13) process may be appropriate to represent the MP data because autocorrelations at lags 1, 2, 3, 4, 10, 11, and 13 are larger than the upper 95% confidence interval limit.
Then, a tentative model to represent the MP data might be
234 101113 ()1−φ1B −φ 2BBB −φ 3 −φ 4 −φ 10 B −φ 11 B −φ 13 BX( t −μ) =at. (4.4)
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It is likely that a model simpler than (4.4) can represent the data well. Therefore model (4.4) and simpler variations of it where were fitted using PROC ARIMA in SAS
9.1.3. In the IDENTIFY statement the MP series data is designated as X. In the
ESTIMATE statement in the PROC ARIMA we specify the model to be fitted by setting the values of Q and P equal to the order of the AR and MA polynomials, respectively.
For example, to fit model (4.4) we specify P = (1, 2, 3, 4, 10, 11) (see SAS Code 1 in
Appendix C).
Also, in the ESTIMATE statement no method of estimation is specified. The default estimation method is conditional least squares (conditional on the series of values already observed). The conditional least squares estimates are obtained by fitting an
ARMA(p, q), an ARMA(p, 0), or an ARMA(0, q) to the {X t } data. The general
ARMA(p, q) model can be expressed as
aXtt= −φ11 X t−− −φ 2 X t 2 − −φ ptpt X −−− +θ1122 a +θ a t + +θ qt a−q (4.5)
with XXtt=−μ. The joint probability density of all random shocks, aa12,,,… an ,
2 assuming that they are IID N (0, σa ) , is
−n 2 ⎧ 1 n ⎫ Paa,,,… a |,,φθσ=πσ22 2 exp − a2, ()12 na( a)⎨ 2 ∑t=1 t⎬ ⎩⎭2σa
with φ =φφ()12,,,… φp , and θ = (θθ12,,,… θq ) . Then, the log-likelihood function of the
2 parameters ()φθ,,σa is
S φθ, 22 n ( ) l ()φθ,,σ=−πσ−aa |X ln2() 2 , (4.6) 22σa where, X is the vector of observed values and
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n Saφθ,= 2 φθ,|X . (4.7) ( ) ∑t=1 t ( )
The values φˆ and θˆ that maximize function (4.6) are the conditional maximum likelihood estimators. Since the data, X , influences the log-likelihood function only through the conditional sum of squares function, then the values φˆ and θˆ can be obtained by minimizing the sum of squares in (4.7). That is why these estimators are called the least squares estimators.
The minimization of S (φθ, ) requires the calculation of the aa12,,,… an using
(4.5). However, to use this equation it is necessary to specify p starting values of X ,
represented by X* , and q starting values of a, represented by a* . A less computationally intensive minimization of (4.7) can be achieved by conditioning S (φθ, ) on the
assumption that the unobserved X* and a* are equal to zero. The conditional estimates adequately approximate the unconditional estimates when the number of observations, n, is moderate or large (Box, Jenkins, and Reinsel, 1994; p. 227). In the particular case of the measurement process series, the conditional estimates are appropriate because the sample size of n = 900 is relatively large.
The conditional log-likelihood function is
22n S* (φθ, ) l* ()φθ,,σ=−aa ln2() πσ− 2 . 22σa
Where
n Saφθ,,= 2 φθ|XaX ,, . (4.8) *( ) ∑t=1 t ( *** )
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ˆ Therefore, the values φˆ and θ that minimize (4.8) are found by computing S* (φθ, ) for different values of φ and θ , starting with the preliminary estimates computed form the autocorrelations used to construct the ACF (Wei, 2006; p.136-138). Finally, the estimate
2 of σa is computed as
S (φθ, ) σ=ˆ 2 * , (4.9) a df where df=−−+()( n p p q), n – p is the number of terms in the sum in (4.9) and p + q is the number of estimated parameters (Wei, 2006; p.139-140). Further details about the different methods used to estimate the parameters of an ARMA(p, q) are given in the
SAS 9.1.3 On-Line Help and Documentation (ETS-Estimation Details), Box, Jenkins, and Reinsel (1994, p. 226-228), and Wei (2006, p. 138-140).
The first model fitted to the measurement process data is (4.4). Table 6 shows the conditional least square estimates of the coefficients in model (4.4) and the p-value associated with the null hypothesis that the coefficients in the model are equal to zero.
The p-values for the coefficients φ3 and φ3 are larger than the p-values of other coefficients and are likely not to be significant.
Various subsets of model (4.4) were fitted in order to find the simplest model that can represent the behavior of the measurement process series. Two simpler models seem to properly represent the MP data. The first model is an AR with components at lags 1,
2, 4, 11, and 13,
24 1113 (1−φ12B −φBB −φ 4 −φ 1113 B −φ BX)( t −μ) =at. (4.10)
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Table 6. Conditional Least Squares Estimates of the Coefficient in Model with Components at Lags 1, 2, 3, 4, 10, 11, 12, and 13
Standard Approx Parameter Estimate Error t Value Pr > |t| Lag μ 26.03540 1.32470 19.65 <0.0001 0
φ1 0.34225 0.03263 10.49 <0.0001 1
φ2 0.06059 0.03508 1.73 0.0845 2
φ3 0.05190 0.03469 1.50 0.1350 3
φ4 0.11330 0.03235 3.50 0.0005 4
φ10 0.09430 0.03234 2.92 0.0036 10
φ11 0.16493 0.03322 4.97 <0.0001 11
φ13 0.08796 0.03145 2.80 0.0053 13
The second model is also an AR with components at lags 1, 2, 4, 10, and 11,
24 1011 (1−φ12B −φBB −φ 4 −φ 1011 B −φ BX)( t −μ) =at. (4.11)
Table 7 and Table 8 show the conditional least squares estimates and corresponding p- values for models (4.10) and (4.11), respectively. The coefficients of both models are non-zero; however the p-value for φ2 in model (4.11) is smaller than for model (4.10).
Table 7. Conditional Least Squares Estimates of the Coefficients in Model (4.10)
Standard Approx Parameter Estimate Error t Value Pr > |t| Lag μ 25.64172 1.06915 23.98 <0.0001 0
φ1 0.37553 0.03244 11.58 <0.0001 1
φ2 0.07061 0.03384 2.09 0.0372 2
φ4 0.14631 0.03029 4.83 <0.0001 4
φ11 0.16572 0.02961 5.60 <0.0001 10
φ13 0.12582 0.03098 4.06 <0.0001 13
Table 9 and 10 show the results of the standard portmanteau lack of fit test (Box,
Jenkings, Reinsel, 1994; p. 314) of models (4.10) and (4.11), respectively.
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Table 8. Conditional Least Squares Estimates of the Coefficients in Model (4.11)
Standard Approx Parameter Estimate Error t Value Pr > |t| Lag μ 25.64675 1.10421 23.23 <0.0001 0
φ1 0.35533 0.03254 10.92 <0.0001 1
φ2 0.10104 0.03249 3.11 0.0019 2
φ4 0.13490 0.03015 4.47 <0.0001 4
φ10 0.11157 0.03206 3.48 0.0005 10
φ11 0.18687 0.03259 5.73 <0.0001 11
The portmanteau lack of fit test uses the residual autocorrelation in groups of 6 to
test the null hypothesis H:01ρ=ρii++ = =ρ i 5 =0, where the default values are i =1, 7,
13, 19, 25, 31, 37, and 43.
Table 9. Portmanteaus Test of Autocorrelation of the Residuals Model (4.10)
To Pr > Lag Chi-Square DF ChiSq Autocorrelations
6 3.11 1 0.0779 –0.016 –0.032 0.023 –0.040 0.001 0.002 12 12.72 7 0.0791 0.017 0.005 –0.057 0.068 –0.029 –0.039 18 17.29 13 0.1862 –0.062 0.005 0.014 –0.010 –0.020 0.021 24 24.02 19 0.1954 –0.054 0.018 0.051 0.004 –0.033 0.017 30 28.96 25 0.2657 0.014 0.030 0.023 –0.009 0.050 0.032 36 34.19 31 0.3168 0.018 –0.031 0.008 0.043 0.047 0.015 42 37.00 37 0.4691 –0.015 –0.016 0.038 0.015 0.025 –0.015 48 39.37 43 0.6294 –0.035 0.004 0.010 0.030 0.017 0.004
The p-values of the portmanteaus test of model (4.10) seem to indicate that the residuals are independent because the p-values are relative large; i.e. larger than a risk of
0.05 of falsely deciding that the residuals are still autocorrelated.
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Table 10. Portmanteaus Test of Autocorrelation of the Residuals from Model (4.11)
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 4.07 1 0.0436 –0.013 –0.046 0.019 –0.043 0.000 0.001 12 12.74 7 0.0787 0.019 0.010 –0.052 –0.039 –0.046 –0.052 18 15.38 13 0.2841 0.034 0.023 0.020 –0.004 –0.010 0.026 24 21.88 19 0.2905 –0.054 0.014 0.042 0.003 –0.043 0.017 30 27.71 25 0.3212 0.004 0.037 0.031 –0.002 0.055 0.030 36 33.82 31 0.3328 0.014 –0.041 0.014 0.046 0.049 0.001 42 37.68 37 0.4378 –0.026 –0.017 0.041 0.025 0.028 –0.007 48 40.63 43 0.5745 –0.037 –0.001 0.005 0.034 0.021 0.011
The p-values of the portmanteaus test of residuals of model (4.11) show that perhaps the first 6 residuals are still autocorrelated. However, the ACF and PACF of the residuals of both models depicted in Figures 10 and 11, respectively, show that none of the autocorrelations or partial autocorrelations exceeds the 95% confidence interval.
Therefore, we can conclude that the residuals from both models are not correlated.
Table 11 compares the Akaike’s information criterion (AIC), the Schwarz’s
Bayesian criterion (SBC), as well as the standard error of models (2.22) and (2.23) fitted to the MP series.
The AIC is computed as
2 AIC=σ+ nln( ˆ a ) 2 M , where n is the number of observations in the series, M is the number of estimated
2 2 parameters, p + q, and σˆ a is the maximum likelihood estimate of σa (Wei, 2006, p. 156-
157). The objective is to identify the model that achieves the smallest AIC, because then
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the model will have the smallest number of parameters and the smallest estimated
2 variance, σˆ a .
1.0 1.0
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
Autocorrelation 0.2 0.2
0.1 Autocorrelation Partial 0.1
0.0 0.0
-0.1 -0.1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Lag Lag Figure 10. ACF and PACF of the residuals from model (4.10).
1.0 1.0
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
Autocorrelation 0.2 0.2
0.1 Autocorrelation Partial 0.1
0.0 0.0
-0.1 -0.1
0123456789101112131415161718192021222324 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Lag Lag Figure 11. ACF and PACF of the residuals from model (4.11).
The SBC is similar to the AIC and it is computed as (Wei, 2006, p. 157)
2 SBC=σ+ nln( ˆ a ) M ln ( n) .
The objective, as with the AIC, is to find a model that minimizes the SBC.
Based on the AIC and SBC criteria model (4.11) fits the MP data better than model (4.10). For this reason model (4.11) will be used to obtain the lead-one forecasts and residuals to construct the CCC and SCC charts.
Figure 12 shows the histogram and the normal probability plot of the residuals from model (4.10). Both graphs seem to indicate that the residuals are approximately normal.
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Table 11. Comparison of Model Selection Criteria
Statistic Model (4.10) Model (4.11)
2 σˆ a 16.895 16.596 AIC 5098.703 5082.645 SBC 5127.511 5111.453
25 R 15 e s i 20 d 10 u a l P : 5 e 15 r A c c t 0 e u n 10 a t l - -5 F o 5 r e -10 c a s t 0 -15 -13-11-9-7-5-3-1135791113 0.01 0.1 1 5 10 25 50 75 90 95 99 99.9 99.99 Residual: Actual-Forecast Normal Percentiles Figure 12. Histogram, and normal probability plot of the residuals from model (411).
Table 12 shows the Shapiro-Wilk and the Anderson-Darling test of the null hypothesis that the data is sampled from a normal distribution. The p-values of these tests are relatively large indicating that the residuals come from a normal distribution.
The graphs in Figures 11 and 12 and the results in Table 12 confirm that model (4.11) appropriately represents the measurement process series and that the residuals are IID
2 N(0,σa ) . Therefore, these residuals can be used to construct the SCC I-chart and MR- chart, and the Special Cause Rules can be applied.
Table 12. Tests of Normality of the Residuals from Model (4.11)
Test Statistic p Value Shapiro-Wilk W 0.997387 Pr < W 0.1605 Anderson-Darling A-Sq 0.687258 Pr > A-Sq 0.0765
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It is important to note that other ARMA and ARIMA model were explored and two of them could have been used to describe the autocorrelation of the MP process. The first one is the ARMA model
410 ()11−φ410B −φBX( tt −μ) =( −θ1 Ba) .
A possible problem with this model is that the AR and MA coefficients are highly autocorrelated. The second model is the nonstationary ARIMA model
11 ()111−φ111B −φBBXB( −) tt =( −θ1)a.
This model is not considered because we have no evidence that the measurement process naturally will be nonstationary. Another important point is to note that the autoregressive component φ11 at lag 11 coincides with a natural cycle of the process (personal communication with process expert); i.e. Xt is related to Xt–11 because of a systematic change in the process, such as change of a raw material every 12 runs.
4.1.2 Obtaining Lead-One Forecasts
The next step in this part of Phase I is to generate the lead-1 forecast using the selected model, in this case, that given by (4.11).
Forecasts are obtained by expressing the invertible ARIMA(p, d, q) as a function of previous observations, that is
X tl+ = π+π++1122XX tl+− tl +− atl+ , (4.12) where Xt+l is an observation l lags beyond the last observed value, Xt, and the invertibility
∞ condition implies that π forms a convergent series. The minimum mean square ∑ j=1 j
ˆ error forecast, X t ()l , of Xt+l, made at time t, is given by the conditional expectation of
(4.12), (Box, Jenkins, Reinsel, 1994, p. 136)
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Xˆ () l== EX ( |, X X ,……) π EX( |, X X ,) +π EX( |, X X ,…) + tt+−lttt111 +l−−ttt 122 +l−tt − 1 + Ea()tl+− | X t , X t1 ,… , where
EX()tj−−| X t , X t1 ,… == Xtj − ; j 0,1,2, ˆ EX()tj+−| X t , X t1 ,… == Xt() j ; j 1,2,
ˆ Ea()tj−−| X t , X t11 ,… == atj −−− X tj − X tj−()1 ; j = 0,1, 2,
Ea()tj+−| X t , X t1 ,… = 0; j = 1,2, .
Hence,
ˆˆ ˆ Xltt()=π111 Xl ( −11 ) +… +π ltl−+ X( ) +π Xtltlt +π X−1 +π+2 X−2.
Since this is a convergent series, only a finite number of terms are needed to calculate a forecast to a sufficient degree of accuracy (Box, Jenkins, Reinsel, 1994; p. 136). Then, to calculate a forecast it is necessary to compute estimates of the weights, {πi}, as follows.
Write expression (4.12) as
π(B) Xat= t, (4.13)
2t−1 where π=−π−π−−π()B 1 12BB t− 1 B is a finite sum with the number of terms determined by the number of observations. Note that the ARIMA(p, d, q) model (4.1) is
d φ−(B)(1 BX) tt =θ( Ba) .
Then, from (4.13) and (4.1), the weights πi, i = 1, 2, ..., t − 1, are given by the expression
φ−()(B 1 B )d π=()B . (4.14) θ()B
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The estimates of the πi, i = 1, 2, ..., t − 1, are computed by equating terms in (4.14) and using the least squares estimates of (φ1, φ2, …, φp) and (θ1, θ2, …, θq). The forecast l- steps ahead is calculated with the expression
ˆ X tt()lX=πˆˆ1122+−lt +π X +−lt + +π ˆ+−l 1X1.
Finally, the one-step ahead or the lead-one forecast is calculated as
ˆ X ttt(1) = π+π++πˆˆ121XX− ˆtX1
For the particular case of model (4.11), the estimates πˆ ’s are given by
ˆˆ21 ˆt− ˆˆ 241 ˆ ˆ0 ˆ11 11−π1B −π 2 BBBBBB − −πt−1 =( −φ1 −φ 2 −φ 4 −φ 10 −φ 11B) .
j The coefficients πi, i = 1, 2, ..., t − 1, are determined by equating the coefficients of B , j
ˆ ˆ ˆ = 0, 1, 2, ..., t − 1, on both sides of the equation to obtain πˆ 11=φ , π=φˆ 22 , πˆ 44=φ ,
ˆ ˆ π=φˆ 10 10 , and πˆ 11=φ 11 . All other weights are equal to zero. The lead-one forecast equation is
ˆ ˆˆˆˆ ˆ X ttttt()1 =φ01122441010111 +φXXX− +φ−− +φ +φ X − +φ Xt−1 (4.15)
ˆˆˆˆ ˆ where φ01241 =()1 −φ −φ −φ −φ01 −φ1 μ.
ˆ The lead-one forecasts, X t−1 (1) , are obtained using the FORECAST statement in the PROC ARIMA with LEAD=1 (see SAS Code 1 in Appendix C).
The time plot of the observed measurement process series and its corresponding lead-one forecasted values are shown in Figure 13. This plot shows that there is a good agreement between the forecasted and the actual measurement process series.
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Measurement Process Series Measurement Process
○ Series Overall Process ted measurement process series. process ted measurement Observation No. Observation Figure 13. Observed versus forecas 0 100 200 300 400 500 600 700 800 900
40 30 20 10 Output
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Once the lead-one forecasts have been obtained, the residuals are calculated as
ˆ eXXttt=−−1 (1) . (4.16)
We are now ready to construct the CCC and SCC charts.
4.1.3 Construction of the Common Cause Control Chart
This chart is constructed by plotting the observed values, the lead-one forecasted values, and the upper and lower control limits on the Y-axis and the corresponding time or observation number on the X-axis. The control limits of this chart are calculated using
(2.1), in Section 2.3.1 in Chapter 2, and the estimate of the standard deviation of the residuals (square root of the variance estimate for model (4.11), shown in Table 11),
σ=ˆ a 4.063 . These limits are:
ˆ UCL=+ X t (1) 12.188 ˆ CL= X t ()1 ˆ LCL=− X t ()1 12.188 .
The CCC chart, shown in Figure 14, has no points outside the control limits and the series of lead-one forecasted values matches the trend of the observed values.
4.1.4 Construction of the Special Cause Control Charts
The SCC charts monitors the mean and the standard deviation of the distribution of the residuals. The I-chart is constructed by plotting the residual values together with the control limits (2.4), (see Section 2.3.2), on the Y-axis and the corresponding time or observation number on the X-axis. The residuals are calculated using (4.16) and the moving ranges of the residuals are computed using
MRett= +1 − et (4.17)
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Measurement Process Series Process Measurement
○ Series Overall Process Observation No. Observation Figure 14. CCC chart for the Phase I measurement process data. UCL LCL 0 100 200 300 400 500 600 700 800 900 0
50 40 30 20 10 Output
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The mean of the residuals is –0.05, the average moving range of the residuals is 4.371
and MR/ d2 = 3.875. The control limits of the I-chart are
UCL =11.57 CL =−0.05 LCL =−11.67
The control limits of the MR-charts are calculated using (2.5) and the constants
D3 and D4 are given by (2.6), in Section 2.3.2. These control limits are
UCL =14.28 CL = 4.371 LCL = 0.
The I-chart and MR-chart were constructed using JMP 6.03 and are shown in
Figure 10. In the I-chart the points that violate an SCR are circled and the number of the
SRC that they violated appears next to the circle (only SCR #1 was used). The moving range values that exceed the control limits are encircled and an asterisk is displayed next to them.
Figure 15 shows that 8 points failed SCR #1 in the I-Chart and 19 in the MR-
Chart. The out of control points in the MR-chart are verified by inspecting the I-chart and the CCC charts. For example, the moving range 144 is associated with the difference between residuals 143, which is close to the upper control limit of the I-chart, and residual 144. Even though observations 143 and 144 are not outside the control limits of
I-chart or the CCC chart, it would be appropriate to investigate these observations to determine if the process is out of control. There are other moving ranges, like moving ranges, like number 379, that are outside the upper control limit, but residuals 378 and
379 are not outside the control limits of the I-chart and observations 378 and 379 are not even near the upper control limit of the CCC chart. This is likely be a false alarm
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because the MR-chart with upper control limit defined by (2.6) and with D4 = 3.267 has a higher proportion of moving ranges that naturally fall outside the upper control limit than the I-chart (see discussion below). In fact, the constant D4 needs to be between 4 and 5 to have the same performance as the I-Chart (Montgomery, 1996, p. 225). In what follows we will calculate the MR-chart upper control limit using (2.6) with D4 = 3.267 and will say that a moving range outside the upper control limit indicates an out-of-control condition only if the residuals or observations associated with it are outside the control limits of the CCC chart or the I-chart.
Obs. # 379 1 Obs. # 143 1 1 10.00 UCL=11.5704
0.00 Avg=-0.0500 Residual
-10.00 1 1 1 11 LCL=-11.6704 50 100 150 200 250 300 350 403 454 504 554 604 654 704 754 804 854 905 Obs. No.
30.00 * Obs. # 144 Obs. # 379 * * * 20.00 * * * * * * * * * * * * ** * UCL=14.2772 10.00
Moving Range Moving Avg=4.3707 0.00 LCL=0.0000 50 100 150 200 250 300 350 403 454 504 554 604 654 704 754 804 854 905 Obs. No.
Figure 15. I-chart and MR-chart of the residuals from model (4.11).
2 At the end of Section 4.1.1 it was shown that the residuals, et, are IID N(0, σα ).
Thus, if the control limits are placed at ±3σˆ a in the I-Chart, it would be expected that approximately 99.73% of the residuals will fall within the control limits, and that
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approximately 0.27% of the residuals will fall outside the control limits. We expect 2 or
3 points to fall outside the limits of the I-chart (900 × 0.0027 = 2.43). A total of eight outlying residuals exceed this expectation, but not alarmingly so. Also, the points appear to randomly fall outside of both control limits, barely exceeding the control limits, and six of the eight points fall outside the CCC control limits. This seems to indicate that these residuals are not necessarily due to a special cause. However, the corresponding values in the measurement process series must be investigated to ensure that special causes are not present.
The number of moving ranges expected to fall outside the control limits in the
MR-chart cannot be determined without knowing the exact distribution of the moving ranges. Using JMP 6.03, a Gamma(1.1, 3.8) distribution was fitted to the moving ranges, as shown on Figure 16. Based on this distribution the upper control limit, 14.3, corresponds to the 96.8 percentile. Hence, we expect 28 or 29 moving range values to fall outside the upper control limit (0.032×900). Therefore, 19 points falling outside the
UCL of the MR chart is not unusual. Based on the fitted gamma distribution, the 99.7 percentile corresponds to a quantile of 23.6. Then, the UCL of the MR-chart would need to be calculated as in (2.6) but with D4 = 5.4. There are no moving range values that exceed this quantile.
Unfortunately we can never be certain that moving ranges exceeding the control limits are necessarily false. Our recommendation is to investigate moving ranges that exceed the control limits that correspond to residuals outside control limits of the I-chart and/or correspond to observations outside the CCC control limits. One must keep in
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mind that the I-chart control limits are slightly tighter than the control limits of the CCC chart (see Section 2.2.4).
99.73 percentile at 23.6
Gamma(1.1, 3.8)
UCL = 14.3
0 10 20
Figure 16. Distribution gamma(1, 4) fitted to the observed moving range values.
4.2 Phase II Implementation– Process Monitoring
In Phase II of the implementation of a monitoring scheme the stability of the process is verified using the ARIMA model selected in Phase I to generate the lead-one forecast for a new observation. The new observation, its forecast, and the corresponding upper and lower control limits are plotted on the CCC constructed in Phase I. Also, the new residual and its moving range are calculated, and plotted on the I-control chart and the corresponding moving range on the MR-control chart constructed in Phase I.
ˆ In this phase the lead-one forecast, X t (1) , is computed recursively using the fitted ARIMA model that can be coded on a spreadsheet or some other program that can be linked to or coded directly onto an existing on-line SPC software package. This recursive equation is obtained by replacing t by t + 1 in (4.1) and writing it explicitly as a
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function of previous observations and the current and previous random shocks. For example, for an ARIMA(p, 1, q) this expression is
XXXtttp+−11=+φ()1 −φ−φ( 211) ++φ−φ ( p−1) Xt−p −φpXt+1−p
+−θ−θ−−θaaattt+−−11 21 qtq a .
The conditional expectation of Xk given the Phase I observations,
EX X,, X… = E⎡ 1 +φ X − φ −φ X + + φ −φXX −φ ()ttt+−11 ( ()(1t 211 )t−( pptppt−−+1) 1−p ⎣ …⎤ +−θ−θ−−θaaattt+−−−11 21 qtqtt aXX⎦ ,,,1) yields
ˆ XXXtttp()11= ( +φ1211 ) −( φ −φ) − + +( φ −φp−− 1) Xtp −φpXt +−1p (4.18) −θ121aatt − θ−− − − θ qtq a .
Where Ea(ttt+−11 X,, X …)= 0 and random shocks for t ≥ k are estimated using the residuals, et. Expression (4.18) is the general recursive equation used to compute the lead-one forecasts in Phase II.
The recursive equation to compute the lead-one forecasts for the measurement process series is given by equation (4.15). Note that only observations at time t, t – 1, t –
2, t – 4, t – 10, and t – 11, and the estimates of φ1, φ2, φ4, φ10, and φ11, are needed to calculate the lead-one forecast. This implies that by the 12th observation in Phase II, forecasts no longer include observations from Phase I.
The procedure to monitor the process during Phase II is:
ˆ 1. Generate the lead-1 forecast, X t (1) , using equation (4.15);
2. Obtain new observation Xt+1;
3. Compute the forecast control limits using (2.1) and σˆ a from Phase I;
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ˆ 4. Plot X t ()1 , Xt+1, and the control limits from Step 3 on the CCC chart;
ˆ 5. Calculate the residual et+1 as eXXttt++11=−(1) ;
6. Plot et+1 on the I-chart generated during Phase I;
7. Compute the moving range MRt+1 as MReett++11= − t;
8. Plot MR t+1 on the MR-chart generated during Phase I.
The estimates of the coefficients in expression (4.15) are not updated with the new observations. As we have noted, in the traditional SPC chart construction and monitoring, the parameter estimates, including the mean and standard deviation, are treated as if they were the actual population parameters. Therefore, the control limits and the model parameter estimates calculated in Phase I are not updated during Phase II, despite the fact that new observations from the stable process are available to improve the precision of the coefficient estimates. The forecast variance can be updated using the smoothed variance estimate as described in Section 2.3.3.
A Bayesian approach could be used to update the parameters as well as the model
(West and Harrison; 1989). Also, a Bayesian approach may be use to update the control limits of the CCC and SCC charts (see for example Woodward and Naylor, 1992; and
Menzefricke, 2002).
Figure 17 shows the CCC chart of the last 10 observations from Phase I
(separated by the vertical line) and the 83 observations from the Phases II. The lead-one forecasts follow the general trend of the observed Phase II data and no observations are outside the control limits, but these observations are less variable. In practice, this reduction in variance must be investigated. The identification of the cause of the reduction in variance will be useful for improving the performance of the process.
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50 UCL CCC
40
30 Input Series
20
Observations LCL O Lead-one forecasts 10
895 905 915 925 935 945 955 965 975 985 Observation No.
Figure 17. CCC Chart of Phase II Data
The I-chart and MR-chart, shown in Figure 18, also display the last 10 observations from Phase I and the 83 observations collected as the process was being monitored (Phase II). None of the residuals or moving ranges associated with the Phase
II observations fall outside the control limits. This is an indication that the measurement process has remained in control, except perhaps for observations 948 to 982 that exhibit a reduction in the variation of the process. If no special cause(s) for the reduction in the process variation is found, then we can say that the 983 observations in the original data set were sampled from a stable process.
4.3 Detection of Out-of-Control Conditions
In this section we generate a single sample of data using the models in Section
4.1.1 and the control limits established in Section 4.1.3 and 4.1.4 and introduce step shifts in the process and the random shocks of different magnitudes. Our objective here is to
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illustrate the effect of these shifts on the CCC and SCC charts and which of them are detected.
15.00
10.00 UCL=11.5704
5.00
0.00 µ0=-0.0500
Residual -5.00
-10.00 LCL=-11.6704 -15.00 905 915 925 935 945 955 965 975 985
Obs. No. SCC Charts
15.00 UCL=14.2772
10.00
5.00 µ0=4.3707 Moving Range Moving 0.00 LCL=0.0000
905 915 925 935 945 955 965 975 985 Obs. No.
Figure 18. I-chart and MR-chart of Phase II data.
4.3.1 Procedure to Generate Data with Out-of-Control Conditions
We will generate out-of-control conditions in the measurement process in the form of step shifts in the mean of the process, the mean of the random shocks, and variance of the random shocks. Because of the direct dependency of the process variance on the variance of the random variable at, we will only consider changes in the variance of the random shocks. Table 9, shows the disturbances that will be introduced into the process. The performance of the CCC and SCC charts will be assessed by simply determining if these control charts detect the out-of-control condition or not. In Chapter
6 we conduct a simulation study to assess the performance of the CCC and SCC control charts for an ARMA (1, 1) model.
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The generation of the measurement process observations, the inclusion of the out-of- control conditions in Table 13, and the construction of the CCC and SCC charts proceeds as folllows:
1. Subset Phase I data (observations 1 to 900);
2 2 2. Obtain estimates of the parameters in equations (4.11) and estimates of σa , σx , and
μ0 (mean of MP series from Phase I). Save estimates;
Table 13. Out-of-Control Conditions
Step shift in the … Magnitude (δ)
Mean of the output kσx, k= 1.5, 2, 3
Mean of the random variates kσx, k= 1.5, 2, 3
Variance of the random variable k= 0.5, 2, 3
3. Compute lead-1 forecasts using (4.15), residuals using (4.16), and moving ranges
using (4.17);
4. Compute Phase I control limits for the CCC and SCC charts using (2.1), (2.4), and
(2.12). Save control limits;
5. Set initial values: Xa0 ==0, t 0 ;
6. Generate 1000 values of Xt as follows:
For a step change in the mean of the process mean: δ = kσx;
2 2 • Generate aNt~0,(σa). Use estimate of σa from Step 2;
• Generate Xt using (4.11) but written as
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XX=μ +φ( −μ) +φ( X −μ) +φ( X −μ ) tt01−− 10 2 t 2 0 4 t − 4 0 (4.19) +φ10()()XXattt−− 10 −μ 0 + φ 11 11 −μ 0 + ,
and parameter estimates from Step 2;
• Introduce disturbance as follows
o If t < 951, then set Zt = Xt;
o If t > 950, then set Zt = δ + Xt;
ˆ • Compute forecasts Zt−1 (1) using Step 2 parameter estimates and by replacing Xt
with Xt in (4.15), that is
ZZZZˆ ()1 =μ +φ( −μ) +φ( −μ) +φ( −μ ) tttt01−− 10 2 2 0 4 − 4 0 +φ10()()ZZtt−− 10 −μ 0 + φ 11 11 −μ 0 ;
• Compute CCC upper and lower control limits using (2.1) and the estimate of
2 σa from Step 2;
ˆ • Compute residuals eZZttt=−−1 (1) ;
• Compute moving range if residuals MReett++11= − t;
For a step change in the mean or in the variance of the random shocks;
• Introduce the disturbance starting at observation 950. For a step change in the
2 mean generate the random shocks as aNt~,(δ σa) , where δ = kσx. For a step
2 change in the variance generate the random shocks as aNta~0,(δσ ), where
δ = k;
• Generate Xt using (4.19) and parameter estimates from Step 2;
ˆ • Compute forecasts X t−1 (1) using (4.15) but written as
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XXXXˆ ()1 =μ +φ( −μ) +φ( −μ) +φ( −μ ) tttt01−− 10 2 2 0 4 − 4 0 +φ10()()XXtt−− 10 −μ 0 + φ 11 11 −μ 0 ,
and parameter estimates from Step 2;
2 • Compute CCC upper and lower control limits using (2.1) estimate of σa from
Step 2;
• Compute residuals using (4.16);
• Compute moving range of the residuals using (4.17);
7. Subset last 100 simulated observations (these are the new Phase II observations);
8. Construct the CCC and SCC charts;
9. Determine if the CCC and/or the SCC have detected the out-of-control condition.
The SAS code used to generate observations, forecasts, and residuals is exhibited in SAS Code 2 and Code 3 in Appendix C.
The first 900 observations of the 1000 simulated observations are discarded to
eliminate the effect of starting values of the simulation ( xa0 ==0, t 0 ). This is necessary because the starting values affect the level of the simulated series. However, as more and more observations are generated the effect of the starting values disappears
(Box, Jenkins, and Reinsel, 1994; p. 108).
4.3.2 Effect of a Step Shift in the Mean of the Process
In this section we consider the effect of a special cause, for example an accidental adjustment of the measurement instrument, that shifts the measurement process mean from μ0 to μ1 between the time period t = τ – 1 and t = τ, that is
⎧μ0 t =τ1, 2,… ,− 1 μ=t ⎨ ⎩μ=ττ+1 t ,1,… .
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The effect of this shift on the selected (4.11) can be modeled by making the mean time dependent, that is
X =μ −φμ −φ μ −φ μ −φ μ −φ μ tt1 t−− 1 2 t 2 4 t − 4 10 t − 10 11 t − 11 (4.20) +φ1122X tt−− +φXX +φ 4410101111 t − +φ X t − +φ X t − +at.
Figures 19, 21, and 22 show the CCC and SCC charts for a step change in the
2 2 2 2 process mean of 3σˆ X , 2σˆ X , and 1.5σˆ X , where σˆ X is an estimate of the variance of the measurement process series. The vertical lines on these graphs indicate the time at which the mean of the process shifted.
In every case residual 17 is outside the upper control limit of the I-chart, and moving range 18, which is associated with residuals 17 and 18, is also outside the upper control limit of the MR-chart. This is because the same seed is used to generate the random shocks at. It should be noted that 1 false alarm in 100 observations is not unusual, because if the residuals were exactly normally distributed one would expect between 0 and 1 observations in 100 to be outside the upper control limit.
In practice, it is impossible to know if observations 17 and 18 are due to an out- of-control condition. Therefore, we could obtain a new observation to verify the out of control condition or, if verification runs are expensive then these observations would be investigated.
The CCC chart and the I-chart in Figure 19 show the large jump in the observations immediately after the shift in the mean and slowly decreasing to a level slightly higher than the original level. Also, in the CCC chart we see that the lead-one forecasts slowly increase and the discrepancy between the observations and the forecasts practically disappears 11 observations later. The same pattern is observed in the I-chart.
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60 CCC
50
40
30 Simulated Data Simulated
20 O MP Observations Lead-One Forecasts Control limits 10
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 20.00
15.00 1 1 11 10.00 UCL=11.5704
5.00 0.00 µ0=-0.0500 Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000
TIME SCC 25.00 * 20.00
15.00 UCL=14.2772 10.00
of Residual 5.00 Moving Range Moving µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 19. CCC and SCC charts with a 3σX step shift in mean of the process.
The MR-chart of the residuals in Figure 19 does not detect the change in the process. If the moving range of the residuals before and after the shift in the mean is large enough it will exceed the upper control limit. However, after this initial jump the moving ranges are small because the residuals slowly decrease towards zero and the MR- chart will not have any moving ranges that exceed the upper control limit.
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The expression that describes the behavior of the residuals for a shift in the mean of the measurement process output can be determined as in Section 3.1.1. The selected model to represent the measurement process observations is given by (4.20), where
⎧μ0 t =τ1, 2,… ,− 1 EX[]t = ⎨ ⎩μ=ττ+1 t ,1,… .
The lead-one forecast of Xt, given by equation (4.15), can be written as
XXXXˆ ()1 =μ +φ( −μ) +φ( −μ) +φ( −μ ) tttt−−−1011022044−0 +φ10()()XXtt−− 10 −μ 0 + φ 11 11 −μ 0 .
The expected values of the lead-one forecast at t = τ–1 are given by
EX⎡⎤ˆ 1 =μ +φEX −μ +φEX −μ +φEX −μ ⎣⎦τ−1012023045() ()[ τ− ] ( [ τ− ] ) ()[ τ− ] 0
+φ10 ()EX[]τ−11 −μ 0 + φ 11()EXτ− 12 −μ 0
=μ0.
Proceeding in the same way for t ≥ τ and letting δ = μ1 – μ0 we obtain
EX⎡⎤ˆ 1 =μ ⎣⎦τ () 0 EX⎡⎤ˆ 1 =μ +φδ ⎣⎦τ+101() EX⎡⎤ˆ 1 =μ +φδ+φ δ ⎣⎦τ+2012() EX⎡⎤ˆ 1 =μ +φδ+φ δ ⎣⎦τ+3012() EX⎡⎤ˆ 1 =μ +φδ+φ δ+φ δ ⎣⎦τ+40124() EX⎡⎤ˆ 1 =μ +φδ+φ δ+φ δ+φ δ ⎣⎦τ+10 () 0 1 2 4 10 EX⎡⎤ˆ 1 = μ +φδ+φ δ+φ δ+φ δ++φ δ ⎣⎦τ+11 () 0 1 2 4 10 11 Then, the expected values of the residuals,
Ee=− EX E⎡ Xˆ 1 ⎤ , [ ttt] [ ] ⎣ −1 ()⎦
are
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t ≤ τ – 1 E[et] = 0
t = τ E[et] = δ
t = τ + 1 E[et] = (1 – φ1) δ
t = τ + 2 E[et] = (1 – φ1 – φ2) δ
t = τ + 3 E[et] = (1 – φ1 – φ2) δ
t = τ + 4 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 5 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 6 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 7 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 8 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 9 E[et] = (1 – φ1 – φ2 – φ4) δ
t = τ + 10 E[et] = (1 – φ1 – φ2 – φ4 – φ10) δ
t ≥ τ + 11 E[et] = (1 – φ1 – φ2 – φ4 – φ10 – φ11) δ.
2 These residuals are independent and have varianceσa .
Figure 20 shows the CCC chart with only the expected values of the behavior of the measurement process observations, the lead-one forecasts, and the lower and upper control limits before and after a step shift in the mean, where δ = 3σx. This plot was constructed using the estimated values of the parameters and assuming no error of
ˆ ˆ ˆ ˆ ˆ estimation of the model parameters; i.e. φ11=φ , φ22=φ , φ44=φ , φ10=φ 10 , and φ=φx11x 11 .
Figure 20 shows that the largest increment in the residuals occurs in the first time period after the mean shifts. The expected values of the residuals decrease in the subsequent 11 time intervals to the new level, given by
()1− φ−φ−φ−φ−φ124101110( μ−μ) .
This new level, based on the conditional least squared estimates of the model coefficients is, 0.11×(μ1 – μ0). The I-chart of the residuals has greatest likelihood of detecting the shift in the mean of the process output as soon as it occurs. The likelihood of detection decreases over time. Also, the largest discrepancy between the observed values and lead- one forecasted values occurs at the first interval after the process mean changes. Then
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the lead-one forecasts increase during the next 11 time intervals to the new level
μ+φ+φ+φ+φ+φ0() 1 2 4 10 11( μ−μ 1 0 ), which is based on the conditional least squares estimates of these coefficients. The patterns of the observed values and lead-one forecasts in the CCC chart and of the residuals in the I-chart in Figure 19 resemble the patterns of the expected values in Figure 20.
50 E[Xt]
40
EX⎡ ˆ 1 ⎤ 30 ⎣ t ( )⎦
20 Simulated Data
10 E[et]
0
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
Figure 20. Expected behavior of the residuals after a shift in the mean of the measurement process.
The CCC and SCC charts in Figure 21 are for the case when a shift in the process is equal to 2σW. The pattern of observed and forecasted values in the CCC chart is similar to the patterns of these values in Figure 19, but the discrepancy between them is smaller and barely discernable. Furthermore, there are no points outside the control limits of the CCC chart that can help us detect the out-of-control condition. The I-chart and the MR-chart do not show any points outside the control limits. The use of SCR #2
(nine points is a row on above or below the mean; see Table 1 in Chapter 1) or SCR#3
(six points is a row steadily increasing or decreasing) may be appropriate given the expected behavior of the residuals; i.e. the largest residual occurs after the shift in the
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mean and it is followed by diminishing residuals until they reach some new constant level. SCR # 3 does not detect an out-of-control condition, but SCR #2 does. The use of
SCR’s will likely result in detecting true as well as out-of-control conditions, as is the case with observations 909, 910, and 911 in the I-Chart of Figure 21.
60 CCC
50
40
30 Simulated Data Simulated
20 O MP Observations Lead-One Forecasts
10 Control limits
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
15.00 1 10.00 UCL=11.5704 2 5.00 2 2 2 2 22 0.00 µ0=-0.0500
Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000 TIME SCC 25.00
20.00
15.00 UCL=14.2772 10.00
of Residual 5.00
Moving Range µ0=4.3707 0.00 LCL=0.0000 910 920 930 940 950 960 970 980 990 1000 TIME
Figure 21. CCC and SCC Charts with a 2σX Step Shift in Mean of the Process.
Therefore, SCR’s should be activated when we know that a particular behavior of the points plotted on the control chart is likely to indicate an out-of-control condition.
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Figure 22 shows the CCC and SCC charts for the step shift in the process mean of
1.5σW. These charts do not detect the shift in the mean. The discrepancy between the observed and forecasted values in the CCC chart after the change in the mean is difficult to distinguish. Similarly, the I-chart does not display any discernable evidence of a step change in the process mean, even when SCR #2 and #3 are activated.
60 CCC
50
40
30 Simulated Data Simulated
20
O MP Observations Lead-One Forecasts Control limits 10
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
15.00 1 10.00 UCL=11.5704
5.00 22 2 0.00 µ0=-0.0500
Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000 TIME 25.00 SCC
20.00
15.00 UCL=14.2772 10.00
of Residual 5.00 Moving Range µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 22. CCC and SCC charts with a1.5σX step shift in mean of the process.
It appears that only step shifts in the process mean of 3σW or larger would be detectable by the CCC and SCC control chart scheme if we rely only on points that fall
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outside the control limits. However, if SCR #2 and #3 are activated it is likely that that shifts of 2σW may be detected. A simulation study would be necessary to quantify the actual detection capability of these charts. This is done in Chapter 6.
4.3.3 Effect of a Step Shift in the Mean of the Random Shocks
In this case we consider the cases where a special cause shifts the mean of the
random shocks from 0 to a new level, δ, between the time period t = τ – 1 and t = τ, that
is
⎧01,t =τ− τ−2, … Ea[]t = ⎨ ⎩δ=ττ+t , 1, … .
This type of shift may result when a part of the measurement instrument suffers a small irreversible change, for example a stress crack in a tension spring.
Figure 23, 24, and 25 show the CCC and SCC charts for a step change in the
mean of the random shocks of 3σˆ a , 2σˆ a , and 1.5σˆ a , respectively. The step shift in the mean of the random shocks starts at observation 951.
The pattern of the CCC chart in Figure 23 shows that the observations monotonically increase immediately after the shift in the mean of the random shocks; i.e. at observation 951. The lead-one forecasts exhibit the same behavior as the observations, except that they start to increase one time interval after the shift and stay consistently below the observations.
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110
100 CCC
90
80
70
60
50
Simulated Data Simulated 40
30 O MP Observations 20 Lead-One Forecasts Control limits 10
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 25.00 1 20.00 1 1 1 1 1 15.00 1 1 1 1 1 1 1 1 1 1 1 111 11 1 111 2 2 22 2 2 2 2 2222UCL=11.5704 10.00 2 2 2 2 2 2 2 2 5.00 22 2 2 2 0.00 Residual µ0=-0.0500 -5.00 -10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000 SCC TIME 25.00 * 20.00
15.00 UCL=14.2772 10.00
of Residual 5.00 Moving Range µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 23. CCC and SCC Charts with a 3σa step shift in the mean of the random shocks.
The behavior of the observations and lead-one forecasts can be explained using the same approach as in Section 3.1.4. Then, the effect of the change in the mean of the at can be modeled by adding the constant disturbance, δ, which is equal to the magnitude of the shift in the mean. The observations, starting at t = τ, are obtained by recursively substituting a representation of previous observations into the expression for the current observations, yielding
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XXXXXXaττ=φ0 +φ 1−τ 1 +φ 2−τ 2 +φ 4−τ 4 +φ 10−τ 10 +φ 11−τ 11 +() +δ 0 =+δX τ 0 XXτ+11=++φδ τ+ ()1 1 02 XXτ+22= τ+ +()1 +φ 112 +φ +φ δ 023 XXτ+33= τ+ +()12 +φ 111212 +φ +φ +φ + φφ δ 0 234 2 2 XXτ+44= τ+ +()12 +φ 11112212124 +φ +φ +φ +φ +φ + φφ +3 φ φ +φ δ
0 The term X τ+k , k = 0, 1, 2…, represent the undisturbed value of the observation. The expected values of the observations are
0 EX[ τ ] =μx +δ 0 EX[]τ+11=μx +()1 +φ δ 02 EX[]τ+21=μx +()1 +φ +φ1 +φ2 δ 023 EX[]τ+31= μx +()12 +φ +φ1 +φ1 +φ2 + φφ12 δ
EX =μ+012 +φ+φ+φ+φ+φ+φ+φφ+φφ+φδ234 2 32 [ τ+4] x ()111122 12124
⎡⎤00 where EX⎣⎦τ+k=μx is the level of the stable process. It follows that, after the shift in the mean of the random shocks, the observations correspond to the original level of the series plus the magnitude of the shift, δ, of the current observation and the weighted magnitude of the shift from previous observations. This is why the observations show a monotonically increasing pattern in the CCC chart.
The corresponding lead-one forecasts are also obtained by recursive substitution of previous observation and forecasted values starting at t = τ. Hence
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ˆ XXXXXXτ−1()1 =φ 0 +φ 1τ− 1 +φ 2 τ− 2 +φ 4τ− 4 +φ 10τ− 10 +φ 11τ− 11 ˆ 0 = X τ−1. ˆˆ0 XXττ()1 =+φδ+11 ˆˆ02 XXτ+12112()1 =+φ+φ+φδτ+ () ˆˆ023 XXτ+2311121()12=τ+ +φ+φ+φ+φ+φφδ()2 ˆˆ0 2342 2 XXτ+341111221()13=τ+ +φ+φ+φ+φ+φ+φ+φφ+φφ+φδ()21224
ˆ 0 The terms X τ+k , k = 0, 1, 2…, represent the lead-one forecast based on observations and random shocks of the stable process. The lead-one forecasts are constants since they are based on observations that have already occurred. Hence, the behavior of the forecasts is similar to that of the observations, except that only the weighted magnitude of the shift in the mean of previous observations accumulates onto the forecast of the stable process.
This explains the pattern of the lead-one forecasts observed in the CCC chart.
The residuals result from subtracting the lead-one forecast from the corresponding observed value and are given by
0 eeτ+kk=+δ= τ+ , k 0,1,2,…,
00ˆ 0 where eXXτ+kk=− τ+ τ+k. The expected values of the residuals are
Ee[ τ+k ] =δ, k = 0,1,2,… .
This is why the I-chart in Figure 23 exhibits a sustained shift in the mean of the residuals immediately after the shift in the mean of the random shocks. Also, adjacent residuals
after the shift are not that different from each other, since Ee[ τ+k ] =δ. Therefore, the moving ranges do not exhibit large swings and the MR-chart is unlikely to detect a shift in the mean of the random shocks.
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The CCC chart in Figure 23 detects the change in the mean by the second observation after the shift. The shift in the mean of the random shocks is confirmed by the monotonic increment of the observations and their forecasts. Furthermore, the average difference between the observations and their forecast should correspond to the change in the mean of the residuals in the I-chart.
The I-chart detects the change in the mean of the random shocks immediately after the shift occurs. Because the shift in the mean of the random shocks will cause a shift in the mean of the residuals, SCR #2 (nine points in a row, either above or below the center line) is a useful rule to detect the out-of-control condition. The I-chart in Figure
23 shows that, although SCR #2 detects the change in the mean, we have to wait for several observations before it signals. As expected the MR-chart does not detect the out- of-control condition.
Figure 24 shows the CCC and SCC chart when the mean of the random shocks suffer a step shift of 2σa starting at observation 951. The CCC chart detects the out-of- control condition four observations after the shift occurs. This is confirmed by the continuous increment of the observations and their forecasts. The I-chart detects the out- of-control condition by the fourth observations, and after few observations SCR #2 also detects the out-of-control condition. The MR-chart does not detect the out-of-control condition.
Figure 25 displays the CCC and SCC charts for the case where the mean of the random shocks suffer step shift of 1.5σa, starting at observation 951. Although there are no points outside the control limits of the CCC chart, there is a clear upward trend in the
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observations and forecasts that indicate a possible change in the mean of the random shocks.
90 CCC 80
70
60
50
40 Simulated Data Simulated 30 O MP Observations 20 Lead-One Forecasts Control limits 10
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 20.00 1 15.00 1 1 1 1 2 2 10.00 2 2 222 UCL=11.5704 22 2 22 2 22 22 2 2 2 5.00 22 2 2 2 2 2 2 2 0.00 µ0=-0.0500 Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000
TIME SCC 25.00 * 20.00
15.00 UCL=14.2772 10.00
of Residual 5.00 Moving Range µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 24. CCC and SCC charts with a 2σa step shift in the mean of the random shocks.
If we relied only on points falling outside the control limits, the I-chart will take, in this particular case, 35 observations before the out-of-control condition is detected.
SCR #2, on the other hand, detects the shift in the mean of the residuals seven observations after the shift in the men of the random shocks. As before, the moving range does not detect the shift in the mean of the random shocks.
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80 CCC 70
60
50
40
Simulated Data Simulated 30
MP Observations 20 O Lead-One Forecasts Control limits 10
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 20.00 15.00 1 1 UCL=11.5704 10.00 2 2 2 5.00 22 2 2 2 22 2 2 2 2 0.00 µ0=-0.0500 Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000
TIME SCC 25.00 * 20.00
15.00 UCL=14.2772
10.00
of Residual 5.00 Moving Range µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 25. CCC and SCC charts with a 1.5σa step shift in the mean of the random shocks.
It is important to note that the identification of the cause of a change in the mean of the random shocks might be difficult because these causes correspond to subtle changes in the process. Here we are assuming that there is a physical relationship between the random shocks and an actual change in the process. For example, detecting incremental changes in measured voltage due to accumulated contamination on a probing needle may be very difficult.
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4.3.4 Effect of a Step Shift in the Variance of the Random Shocks
In this section we determine the effect of a shift in the variance of the random shocks on the behavior of the observations, the lead-one forecasts, and the residuals. As in the previous cases, we assume that variance of the random shocks shifts from its in-
2 22 control variance,σa , to the out-of-control variance σd=δσa between the time intervals
τ – 1 and τ.
Figures 26 through 28 show the CCC and SCC charts for an out-of-control condition due to a step change in the variance of the random shocks. The shift in the variance of the random shocks starts at observation 951.
Figure 26 shows the CCC and SCC charts for the case when a special cause generates a shift in the variance of the random shocks of magnitude δ = 3σa. The CCC chart shows increasingly larger oscillations of the observed and the lead-one forecast values, whereas the residuals exhibit a clear change in variation after observation 951.
The CCC chart detects the change by the third observation after the shift. Also, the I- chart detects the shift in the variance by the third observation, but the MR chart detects the shift immediately after its occurrence due to the large oscillations between residuals.
The patterns in these plots can be explain by using model (4.11) to describe the observations after the shift in the variance of the random shocks. Thus,
X ττ=φ011224410101111 +φXXX−τ +φ−τ +φ−τ +φ X−τ +φ Xa−τ + +dτ 0 =+Xdττ 0 XXdτ+1111=++φ τ+ τ+ d τ 02 XXdτ+2221112=++φ+φ+φ τ+ τ+ d τ+ () dτ 023 XXdτ+33312121112= τ+ + τ+ +φ d τ+ +()( φ +φ dτ+ + φ +2 φφ ) dτ
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X =Xd0234 + +φ d + φ +φ d + φ +23 φφ d + φ + φ2 φ +φ2 +φ d τ+44413122112111224 τ+ τ+ τ+ ( ) τ+ ( ) τ+ ( ) τ
0 where X τ+k , k = 1, 2, … , corresponds to the in-control part of the model, dτ, dτ+1,… represent the disturbance due to a special cause and are normally distributed with mean
222 zero and variance σ=δσ−σdaa, and the random variables at and dt are independent.
60 CCC 50
40
30
20
10 Simulated Data Simulated 0 O MP Observations -10 Lead-One Forecasts Control limits -20
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 40.00 30.00 1 20.00 1 1 1 1 10.00 5 5 UCL=11.5704 5 5 0.00 5 µ0=-0.0500 -10.00 5 5 Residual 1 1 1 1 LCL=-11.6704 -20.00 1 1 -30.00 1 1 -40.00 910 920 930 940 950 960 970 980 990 1000
TIME SCC
40.00 * * 30.00 * * * * * * * * 20.00 * * ** * * * * * ** * UCL=14.2772
of Residual 10.00 Moving Range µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 26. CCC and SCC charts with a 3σa step shift in the variance of the random shocks.
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The variances of the observations are
22 Var[ X τ ] =σxd +σ 222 Var[] X τ+11=σxd +()1 +φ σ 2 2222⎡⎤ Var[] X τ+21=σ+xd1 +φ+φ+φ12 σ ⎣⎦⎢⎥()
22 222⎡⎤ 3 2 Var[] X τ+31=σ+xd12 +φ+φ+φ121 +φ+φφ12 σ ⎣⎦⎢⎥()() 22 2 222⎡⎤ 3 422 2 Var[] X τ+4=σ+x 123 +φ+φ+φ1 12 +φ+φφ 1 121 φ+φφ+φ+φ 1224 σd ⎣⎦⎢⎥()()( )
⎡⎤02 where the variance of the stable process is Var⎣⎦ X τ+k= σx. Therefore, the observations in the CCC charts will exhibit an increasing variation immediately after the special cause increases the variance of the random shocks. The sequence of weights that multiply the shift in the variance forms a convergent series because the process is stationary.
Therefore, the variance of the process eventually reaches a constant level.
The expressions for the lead-one forecast are
ˆˆ0 XXτ−1 ()1 = τ ˆˆ0 XXdττ()1 =+φ+τ11 ˆˆ02 XXdτ+121112()1 =+φ+φ+φτ+ τ+ () dτ
ˆˆ023 XXdτ+231212111()12=+φ+φ+φ+φ+φφτ+ τ+ ()( dτ+ 2 ) dτ ˆˆ02 3 422 X τ+341312211()12=Xdτ+ +φ τ+ +φ+φ()( dτ+ +φ+φφ2111 )( dτ+ +φ+φφ+φ+φ3224)dτ
The variances of these forecasts are
Var⎡⎤ Xˆ 10= ⎣⎦τ−1 ( ) Var⎡⎤ Xˆ 1 =φσ22 ⎣⎦τ () 1 d 2 Var⎡⎤ Xˆ ()1 =φ+φ+φ⎡⎤22 σ 2 ⎣⎦τ+1112⎣⎦⎢⎥()d 22 Var⎡⎤ Xˆ ()12= ⎡φ+φ+φ22 +φ+φφ 3 ⎤ σ2 ⎣⎦τ+211211⎣⎢ ()()2⎦⎥ d
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22 Var⎡⎤ Xˆ ()12=⎡⎤ φ22 + φ +φ + φ 3 + φφ + φ42 +3 φ φ +φ 2 +φ σ2 ⎣⎦τ+311211⎣⎦⎢⎥( ) ( 211) ( 224) d
ˆ 0 where X τ+k , k = 0, 1, 2, …, are constants. Then, the variance of the first lead-one forecast is zero, however the variance of subsequent forecasts increases until it reaches some stable value. The variation of the lead-one forecasts in the CCC chart exhibits this behavior.
Finally, the residuals are obtained by subtracting the expression of the lead-one forecasts from the expressions of the observations. The residuals are
0 eedτ+kkk=+ τ+ τ+ , k = 0,1,2,….
00ˆ 0 where eXXτ+kk=− τ+ τ+k. Then the variances of the residuals are
22 2 Var[ eτ+kada] =σ +σ =δσ, k = 0,1,2,….
2 2 Therefore, the variation of the residuals in the I-chart will change from σa to δσa immediately after the shift. If δ > 1 the variation of the residuals will increase, and there will be larger differences between residuals causing large moving ranges. However, if
δ < –1 the variations of the residuals will decrease, resulting in smaller differences between residuals, and consequently, moving ranges will be small.
Given the large oscillation in the residuals due to the change in the variance of the random shocks, SCR #5 (2 out of 3 points in a row, 2 standard deviations above, or below, the center line) will be a useful rule to detect a change in the variance of the random shocks or of the process. The I-chart in Figure 26 shows that SCR #5 does detect the shift in the variance a few observations after it occurred.
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It is important to note that out-of control conditions detected by the MR chart should be treated cautiously because of the interdependency of the moving ranges. To say definitively that the process is out of control, we would like to see either or both of the residuals involved in calculating the moving range that fell outside the upper control limit of the MR-chart to also be outside the control limits of the I-chart. For example moving range 53, the absolute difference between residuals 53 and 52, is outside the upper control limits of the MR-chart. Since residual 53 is outside the lower control limit of the I-chart we will say that that the MR-chart has detected an out-of-control condition.
In the case of moving range 51, the absolute difference of residuals 50 and 51 is outside the upper control limit of the MR-chart. However, we may not be certain that this is an out-of-control situation since neither of these residuals lies outside the I-chart control limits, nor do the corresponding observations lie outside the CCC chart control limits.
Indeed, it is difficult to determine if this moving range is an indication of an out-of- control condition because none of these residuals are outside the control limits of the I- chart, nor are the corresponding observations outside the control limits of the CCC chart.
In situations like this, it would be prudent to err on the side of caution and declare the process to be out of control and investigate further.
Figure 27 shows the CCC and SCC for the case when the variation of the random shocks undergoes a step increment of 2σa. The out of control condition is detected by the
CCC chart and the I-chart 12 observations after the shift in the variance occurs
(observation 962 is outside the lower control limit of both charts). Also, SCR #5 in the I- chart detects the out-of-control condition 13 observations after the shift. The MR-chart signals out-of-control 3 observations after the shift in the variance of the random shocks,
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but since none of the residuals used to compute this moving range are outside the I-chart control limits, we may doubt this is a real signal. Nevertheless, as indicated above, it may be advisable to declare the process out of control and investigate the origin of the observations that correspond to the residuals in question.
60 CCC 50
40
30
20
Simulated Data 10 O MP Observations 0 Lead-One Forecasts Control limits -10
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
20.00 1 1 10.00 UCL=11.5704
0.00 5 µ0=-0.0500
Residual -10.00 5 5 1 1 LCL=-11.6704 -20.00 1 1
910 920 930 940 950 960 970 980 990 1000
TIME SCC 30.00 * 25.00 * * 20.00 * * * * * * 15.00 UCL=14.2772 10.00
of Residual 5.00
Moving Range µ0=4.3707 0.00 LCL=0.0000 -5.00 910 920 930 940 950 960 970 980 990 1000 TIME
Figure 27. CCC and SCC charts with a 2σa step shift in the variance of the random shocks.
Figure 28 depicts the CCC and SCC charts for the special case when the variance of the random shocks decreases to 0.5σa. None of the control charts in Figure 28 show a
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point outside the control limits. This is expected because the observations, forecasts, and residuals exhibit observations with smaller variations that are unlikely to surpass the control limits based on a process that is more variable. It is clear from the patterns in all charts that the variance has decreased. After 4 or 5 observations we would begin to suspect that the variance of the process or of the random shocks has decreased.
60 CCC O MP Observations Lead-One Forecasts 50 Control limits
40
30 Simulated Data Simulated
20
10
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
15.00 1 10.00 UCL=11.5704
5.00 777 0.00 77 777 7 µ0=-0.0500 Residual -5.00
-10.00 LCL=-11.6704 -15.00 910 920 930 940 950 960 970 980 990 1000
TIME SCC 25.00 * 20.00
15.00 UCL=14.2772 10.00
of Residual 5.00 Moving Range Moving µ0=4.3707 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 TIME
Figure 28. CCC and SCC charts with a 0.5σa step shift in the variance of the random shocks.
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SCR #7 (15 points in a row within a ± 1 standard deviation around the center line) will be a useful rule for detecting this type of out-of-control condition. However, this rule will require several observations before it signals a reduction in the variation.
Reduction in variation is an important situation that a control chart should be able to detect, since we are constantly seeking the reduction of process variation to generate a product with key characteristics that are on target with minimal variation.
Effective detection of shifts in the variance of the process or the random shocks requires that we should rely not only on observations falling outside the control limits, but also on patterns in the observations, the forecasts, and the residuals. Therefore, we recommend use of SCR #5 to improve the detection capability of the I-chart and investigate any moving range that falls outside the upper control limit of the MR-chart.
4.4 Summary and Discussion
We illustrate in great detail Phase I and Phase II of the implementation of the
CCC chart and the SCC charts to monitor the measurement process introduced in Chapter
1. We show that these charts can be effective in detecting step changes in the mean of the process output or the mean of the random shocks that drive the process.
We found that several models can be used to represent the measurement process series. Each of these models behaves differently when special causes affect the mean or variance of either the process or the random shocks. In situations like this we recommend selection of the model that ensures the most efficient and effective detection of out-of-control conditions. Ideally, our knowledge of the fundamental mechanism that drives the process output should guide the identification of a model that appropriately describes its behavior. Although we infer that the process is not stationary, we find that
145
both stationary and non-stationary models adequately fit the data. In practice, detailed knowledge of the process should inform model choice. Such information was not available for the process generating the data in this example.
Another key finding is that it is important to identify the behavior of the observations, lead-one forecasts, and residuals after changes in the mean or variance of the process and the random shocks. This behavior can help us distinguish patterns associated with real out-of-control conditions and select SCR’s that can improve the detection capability of the I-chart.
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CHAPTER FIVE
Illustration of the Monitoring of a Process with Autocorrelated Output and with an Observable Autocorrelated Measurement Error
The monitoring of a process with an autocorrelated output and no observed
measurement error was presented in Chapter 2. In this chapter we will illustrate the
monitoring of a process where the autocorrelated overall process output, Yt, and the autocorrelated measurement process output, Xt, are observed. Also, in these processes it
is impractical to reduce the measurement error to a negligible level because of regulatory constraints, technological limitations, or the expense of requisite modifications. The
pharmaceutical process introduced in Chapter 1 is an example of these types of processes.
In the pharmaceutical process a measurement of the absorbance of the buffered
solution of the product, Yt, is preceded by measurements of the same characteristic on a
reference material, Xt. The measurement of the reference material is used to verify the
performance of the measurement process. We use a transfer function to relate the errors
of measurement on the product material to those on the reference material (see Section
1.3), so that
YvBXPtt= ( ) + t. (5.1)
where Yt is the overall process series, Xt is the measurement process series, Pt is the base
process series, v(B) is the transfer function defined as
ω(B) Bb vB()= , (5.2) δ()B
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2 r 2 s and δ(B) = 1 – δ1B – δ2B – … – δrB , ω(B) = ω0 – ω1B – ω2B – … – δsB , and b corresponds to the delay, in time lags, of the response of Yt to the effect of Xt .
Our objective is to monitor the behavior of the base process. Therefore, we will
assume that the measurement process is in control and that it can be described by the
model established in Chapter 4. This model is
2 4 10 11 (1−φ12B −φBB −φ 4 −φ 1011 B −φ BX)( tX −μ) =at. (5.3)
A total of 983 observations were obtained during a period when the base process
was assumed to be in control. These observations will be split into two data sets. The first set will contain observation 1 through 900 and will be used in Phase I to identify the transfer function and the base process model, estimate its parameters, and establish the control limits of the CCC and the SCC charts. The second data set, with observations
901 through 983, will be used as if they were observations obtained after the completion of Phase I to illustrate the monitoring of the base process; i.e. Phase II.
In Section 5.1 we will illustrate Phase I and in Section 5.2 Phase II of the
implementation of the monitoring scheme. In Section 5.3 we illustrate the effect of step
shifts in the process mean and in the random shocks mean and variance. We summary
and discuss our findings in Section 5.4.
5.1 Phase I of the Implementation of the CCC and SCC Charts
In Chapter 4 we provided a detailed description of how models are identified, the
procedure to obtain estimates of the model parameters, and how to obtain lead-one forecasts. In this section we will not repeat those details. Instead, we use the implementation method in Section 2.4.1 to provide a more detailed procedure. Using the
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notation from Section 2.4, the procedure to construct the CCC and SCCC charts is as follows (Wei, 2006; p. 331):
1. Prewhiten the measurement process series using the model that represents the
measurement process. Using (5.3) we have,
2 4 10 11 aBBBBBtt=−φ−φ−φ−φ−φ()1 12 4 1011X,
where XXttX=−μ();
2. Apply the prewhitening transformation to the overall process series as
24 1011 βtt =()1 −φ12B −φBB −φ 4 −φ 1011 B −φ BY;
3. Calculate the sample cross-correlation function (CCF), ρˆ aβ (k ) , between the series at
and βt as (Box, Jenkins, Reinsel, 1994; p. 411);
⎧ 1 nkI − aa−β−β= k0,1,2,… ⎪ ∑t=1 ()ttk()+ ⎪nI γ=ˆ aβ ()k ⎨ 1 n ⎪ Ik+ β−β()aak − =0, − 1, − 2,… , ⎪ ∑t=1 ()ttk− ⎩nI
where nI is the number of pairs of observations (Xt, Yt) collected during Phase I and
k is the lag (number of time periods) between pairs;
4. Calculate the estimate of vk as
σˆ β vkˆka=γˆ β (), σˆ a
where σˆ β and σˆ a are the estimates of the standard deviations of βt and at,
respectively;
1/2 5. Compare these estimates with their standard error, (nI – k) , to determine which
weights are nonzero;
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6. Identify the order b, r, and s of the polynomials in the transfer function (5.2) by
matching the nonzero vˆk ’s with theoretical patterns. See Wei (2006, p. 325), or Box,
Jenkins, Reinsel (1994, p. 389);
7. Obtain nonlinear least-squared estimates of the coefficients in the polynomials δ(B)
and ω(B) for different values of b, by setting the unobserved values of at equal to
their expected value of zero and minimizing
n Sbδω,,|,φθ = I a2 , ( bp bp ) ∑tt= t 0
where δ = (δ1, …, δr)', ω = (ω1, …, ωs)', φbp = (φ1, …, φp)', θbp = (θ1, …, θq)', and
tprbps0 =++++max{ 1, + 1} (Wei, 2006, p. 333);
8. Calculate the estimated noises series as
ωˆ (B) PY=− BXb ; ttδˆ ()B t
9. Identify the base process model using the ACF and the PACF of the series Pt;
10. Estimate the coefficients φbp and θbp of the model that represents the base process;
11. Use the residuals to determine if the identified transfer function and the base process
model adequately represent the overall and the base processes. Repeat Steps 6
through 11 until appropriate models are identified;
12. Compute the control limits of the CCC charts as
ˆ UCL= Ytr(13) +σˆ , ˆ CL= Yt ()1, (5.4) ˆ LCL= Ytr()13,−σˆ
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1/2 22 2 where σ=ωσ+σˆˆˆrau(ˆ 0 ) if b = 0, or σˆˆr=σu if b ≠ 0. The control limits (3.28) are
the limits of the 99.87% confidence interval of the lead-one forecast. Note that σˆ r
could be directly estimated from the standard deviation of the residuals;
13. Compute the control limits of the of the I-chart of the residuals as
MR UCL=+ r 3,r 1.128 CL= r , (5.5) MR LCL=− r 3;r 1.128
14. Compute the control limits of the moving ranges of the residuals, MRr= tt+1 − r
using the equations
UCL= 3.268 MRr ,
CL= MRr , (5.5) LCL = 0;
15. Construct the CCC and SCC charts;
16. Investigate any point outside the control limits. Remove or retain such points using
the following criteria:
o If there is a special cause that made the observations associated with the point
outside the control limits, then remove it from the data set and repeat the entire
process;
o If a special cause cannot be found, then the observations associated with the point
out of control are kept in the data and models, parameter estimates, and control
limits are not revised;
17. Monitor the base process using the established transfer function, base process model
and control limits of the CCC and SCC charts.
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We used PROC ARIMA in SAS V.6 to identify and estimate the transfer function and the base process model (see SAS Code 5 is in Appendix C). In the next subsections we first identify the preliminary transfer function and estimate its parameters, we then identify the base process model and estimate its parameters and adjust the transfer function, if necessary. Once the final model is established we compute lead-one forecasts, residuals, the control limits, and construct the control charts. In the last subsection we illustrate the monitoring of the processes (Phase II).
Figure 29 displays the time ordered plot of the 899 measurements of the reference a product materials, that is the measurement and overall process series, respectively.
These series have similar trends. The reference material measurements have a mean of
24.96, whereas the mean of product material observations is –5.82. Also, the product measurements exhibit larger variation, and perhaps some of the larger and smaller values may indicate that the process is affected by special causes. This will be verified later after we construct the CCC and SCC charts.
5.1.1 Identification of the Transfer Function and Estimation of its Parameters
Following the procedure outlined above we used model (5.3) to prewhiten the measurement process series (i.e. reference material measurements) and the overall process series (i.e. product measurements) to identify the preliminary transfer function.
An important step in the identification of the transfer function is to determine if the overall process series is stationary. In Chapter 4 we determined that the measurement process, Xt, is stationary and since we are assuming a causal transfer function between Xt and Yt, then we would expect the overall process, Yt, to also be stationary.
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Measurement Process Series Process Measurement Overall Process Series Overall Process Series ○ * Observation No. Observation Figure 29. Time plot of the measurement process and overall observations. 0 100 200 300 400 500 600 700 800 900 0 40 30 20 10
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The ACF and PACF of the overall process series are shown in Figure 30. The
ACF decays very slowly indicating that one of the roots of the autoregressive polynomial
φ(B) of the ARIMA model that describes the overall process is near 1, implying that overall process is stationary. However, the PACF shows that the overall process could be described by an AR model with parameters φ1, φ2, φ3, φ4, φ5, φ6, φ11, and φ12 or a subset of them. The ACF and PACF in Figure 30 resemble closely the ACF and PACF of the Xt series (see Figures 7 and 8 in Chapter 4).
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Figure 30. ACF and PACF of the overall process series.
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As we noted in Chapter 4, Box, Jenkins, and Reinsel (1994, p. 185), suggest that
ACF plots like the one in Figure 30 may indicate that the process is not stationary. If there is physical evidence that the process is indeed nonstationary, then we will need to
d determine the appropriate difference transformation such that Ztt=−()1 BY and
d WBt=−()1 Xt are stationary. In our particular case we have no physical evidence that the process is nonstationary and, as mentioned above, the PACF plot suggests that an AR model may be appropriate to represent the overall process.
Figure 31 displays the CCF of the prewhitened series Xt and Yt. The cross- correlations at lags –2, 0, 5, and 21 exceed the approximate 95% confidence interval,
given by 2 nkI − , where nI is the number of observations and k is the lag (Wei, 2006; p. 330). The cross-correlations at lags –2, 5, and 21 are small, less than half the cross- correlation at lag 0. If we assume that the only nonzero cross-correlation is at lag zero,
i.e., ρˆ aβ ()0 , then estimated transfer function in equation (5.2) reduces to vBˆˆ()= v0 so
that (5.1) becomes YXtt=ω0 +Pt.
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Figure 31. CCF of the prewhitened Xt and Yt series.
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As can be seen in Table 14, we reject the null hypothesis that the cross- correlations of the residuals ut with Xt, ρuX (k +1), ρuX (k +2), …, ρuX (k +5) , k = 0, 5, 11,
…, 41, are zero using the portmanteau lack of fit test. Therefore, the tentative transfer function ω0 is not appropriate to account for the cross-correlation between prewhitened Yt and Xt series.
Table 14. Portmanteau Test of the Cross-Correlation of the Residuals,
For the Transfer function vBˆ( ) = ωˆ 0
Cross-Correlation Check of Residuals ut with Input Xt To Lag Chi-Square DF Pr > ChiSq Cross-Correlations
5 35.24 6 <0.0001 –0.182 –0.007 0.006 –0.005 0.030 0.070 11 40.08 12 <0.0001 –0.023 0.045 0.013 0.031 0.034 0.025 17 46.15 18 0.0003 0.033 0.011 0.043 0.025 0.055 0.002 23 57.59 24 0.0001 0.040 0.033 0.031 0.079 0.032 0.043 29 64.59 30 0.0002 0.008 –0.022 0.035 0.063 –0.026 0.036 35 74.93 36 0.0002 –0.013 0.051 0.087 0.025 0.014 –0.015 41 81.52 42 0.0002 0.029 0.012 0.038 0.012 0.046 0.051 47 87.47 48 0.0004 –0.030 0.000 0.067 –0.020 0.018 0.021
A different form of the transfer function is found by noting that the pattern of the cross-correlations for k > 0 resembles an exponential decay starting at k = 0; where we had assumed that the cross-correlations at k = –2, 5 and 21 are not different from zero.
This pattern suggests that the order of the polynomial δ(B) is r = 1 and that the appropriate transfer function might be
ω vB()= 0 . (5.7) 1− δ1B
The model that represents the overall observation is
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ω0 YXt= t+ Pt. (5.8) 1−δ1B
It is important to note that a model for the base process series Pt has not yet been specified. Therefore, (5.7) is a tentative transfer function model that will be reevaluated after the model for the base process is identified and estimated.
Table 15 shows the results of the lack of fit portmanteau test of the transfer function (5.7). The p-values of the null hypotheses that the cross-correlations between the residuals ut from (5.8) and Xt are equal to zero are relatively large except for the first two groups (i.e. lags 1 to 5 and 6 to 11). At a significance level of 0.03, however, none of the hypotheses is rejected, indicating the transfer function in (5.7) may be adequate.
Other transfer functions with higher order polynomials δ(B) and ω(B) were fitted, but model (5.8) is the simplest model that yielded approximately zero cross-correlations between ut and Xt. We will proceed with the tentative transfer function (5.7) to identify a model for the base process series Pt.
Table 15. Portmanteau Test of the Cross-Correlation of the Residuals for the Transfer function (3.31)
Cross-Correlation Check of Residuals ut with Xt To Lag Chi-Square DF Pr > ChiSq Cross-Correlations
5 12.17 5 0.0326 0.094 0.028 –0.023 –0.050 0.001 0.031 11 20.73 11 0.0363 –0.076 –0.004 –0.036 –0.039 –0.001 0.031 17 22.92 17 0.1518 0.005 –0.012 0.019 0.004 0.021 –0.038 23 26.77 23 0.2658 0.007 –0.023 –0.011 0.059 0.010 0.004 29 34.84 29 0.2099 –0.016 –0.046 0.025 0.049 –0.055 0.023 35 39.81 35 0.2644 –0.035 0.029 0.051 –0.001 –0.005 –0.029 41 42.23 41 0.4176 0.004 –0.022 0.006 –0.003 0.032 0.033 47 50.48 47 0.3376 –0.063 –0.043 0.036 –0.045 0.004 0.002
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5.1.2 Identification of the Base Process Model and Estimation of its Parameters
Figure 32 shows the ACF and PACF of the residuals after fitting model (5.8).
The slowly decreasing pattern of the autocorrelations in ACF plot shows that root of the
AR part of the model that represents the base process series may be close to one. Also, the autocorrelation coefficient at lag one is the largest, perhaps suggesting MA(1)
2 3 behavior. The PACF indicates that an AR model with componentsφbp1B , φbp2 B , φbp3B ,
4 5 6 10 11 φbp4 B , φbp5B , φbp6 B , φbp10 B , and φbp11B , or a subset of these, may appropriately describe the behavior of the base process series.
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Figure 32. ACF and PACF of the residuals after model (5.8) is fitted.
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Several AR models with the components listed above were fitted, but the autocorrelations of the residuals, ut, continued to be nonzero. However, when we decided
to include the MA component, (1 – θbp1B ), in the model that describes Pt, we found the
3 5 MA coefficient, θbp1, to be non-zero. Furthermore, the AR components φbp3B , φbp5B ,
6 10 11 φbp6 B , φbp10 B , and φbp11B were either not different from zero or their addition into the model did not eliminate the autocorrelations of the residuals, ut. The model
24 ()1−φbp12B −φ bpBBP −φ bp 4( t −μ bp ) =(1 −θbp1 B)u t , (5.9) seems to appropriately represent the base process. Thus, we proceed to fit the model
ω (1−θbp1B) YX=+μ+0 u. (5.10) tt1−δ B bp 24t 1 ()1−φbp12BB −φ bp −φ bp 4 B
Table 16 shows the conditional least squared estimates of the parameters in model
(5.10) and the corresponding tests of the null hypotheses that a parameter is not different from zero. The null hypothesis is rejected at a 0.05 significance level for all parameters except for the parameter δ1, indicating that this parameter may be removed from (5.8).
Table 16. Conditional Least Squares Estimates of the parameters of Model (5.10)
Parameter Estimate Std Error t Value Approx Pr > |t| μ –14.27943 2.51006 –5.69 <0.0001
θbp 0.89461 0.02402 37.24 <0.0001
φbp1 1.06332 0.04048 26.27 <0.0001
φbp2 –0.12994 0.04079 –3.19 0.0015
φbp4 0.05757 0.02743 2.10 0.0361
ω0 0.23161 0.04500 5.15 <0.0001
δ1 0.30342 0.15958 1.90 0.0576
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Table 17 shows the results of the portmanteau lack of fit test of model (5.10). The null hypotheses that the autocorrelation of the residuals ut, ρu (k +1), ρu (k +2), …, ρu (k
+6), k = 0, 6, 12, …, 42, are not rejected because the p-values are larger than a significance level of 0.05. Thus, model (5.9) appears to adequately represent the base process.
Table 17. Portmanteau Lack of Fit Test that the Autocorrelation of the Residuals of Model (5.7) are Equal to Zero
Autocorrelation Check of Residuals ut To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 3.78 2 0.1513 –0.002 0.012 –0.003 –0.023 0.048 0.035 12 14.72 8 0.0648 -0.031 –0.038 –0.023 –0.037 0.038 0.079 18 19.87 14 0.1343 –0.045 –0.025 0.000 –0.003 0.051 0.020 24 30.93 20 0.0562 –0.052 –0.010 0.030 –0.057 0.069 –0.016 30 36.62 26 0.0808 –0.057 –0.030 –0.010 –0.004 0.041 –0.015 36 40.22 32 0.1510 –0.047 0.019 0.013 –0.003 –0.023 0.024 42 49.89 38 0.0938 –0.069 –0.002 –0.017 –0.012 0.063 –0.033 48 56.68 44 0.0952 –0.051 0.034 –0.008 –0.011 0.053 0.022
Table 18 shows the lack of fit portmanteau test that the cross-correlations between the residuals of model (5.10) and the measurement process series are equal to zero since all p-values are relatively large. This indicates that the transfer function in (5.10) appropriately accounts for the cross-correlation between Yt and Xt.
We mentioned above that the tentative transfer function (5.7) may need to be reviewed after the model for the base process series is identified and its parameters are estimated. That is the case here, where parameter δ1 may be removed, thereby simplifying (5.10) to
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(1−θbp1B) YX=ω +μ + u. (5.11) ttb0 p 24t ()1−φbp12BB −φ bp −φ bp 4 B
Table 18. Portmanteau Lack of Fit Test that the Cross-Correlation of the Residuals of Model (5.10) and the Xt series are Equal to Zero
Cross-Correlation Check of Residuals ut with Xt To Lag Chi-Square DF Pr > ChiSq Cross-correlations
5 6.97 5 0.2226 –0.007 –0.003 –0.008 –0.016 0.045 0.074 11 14.05 11 0.2303 –0.049 0.050 –0.002 0.001 0.033 0.043 17 18.18 17 0.3775 0.019 0.003 0.040 0.015 0.041 –0.027 23 24.17 23 0.3944 0.029 –0.006 0.005 0.075 0.005 0.014 29 35.07 29 0.2024 –0.013 –0.049 0.033 0.050 –0.066 0.040 35 42.93 35 0.1677 –0.037 0.044 0.066 –0.003 –0.001 –0.032 41 45.93 41 0.2751 0.011 –0.021 0.017 –0.003 0.037 0.034 47 56.72 47 0.1566 –0.071 –0.027 0.054 –0.056 0.013 0.007
Now we need to determine if model (5.11) represents the relationship between Yt and Xt, as well as the behavior of Pt. Table 19 shows the conditional least squares estimates of the parameters in model (5.11). The approximate p-values indicate that the parameters are nonzero at a significance level of 0.05.
Table 20 shows the results of the portmanteau residual test of lack of fit of model
(5.11), where the p-values for the test of the hypotheses ρu(k + 1) = ρu(k + 2) = … = ρu(k
+ 6) = 0, k = 0, 6, 12, …, 42, are relatively large, therefore the residuals are not autocorrelated.
The cross-correlation portmanteau test of lack of fit for model (5.11) is shown in
Table 21. The p-values in this table indicate that the null hypotheses ρuX(k + 1) = ρuX(k +
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2) = … = ρuX(k + 5) = 0, k = 0, 5, 11, …, 41, are all large, therefore the residuals, ut , are not cross-correlated with Xt.
Table 19. Conditional Least Squares Estimates of the parameters of Model (5.11)
Parameter Estimate Std Error t Value Approx Pr > |t| μ –12.17555 2.05891 –5.91 <0.0001
θbp 0.89401 0.02372 37.69 <0.0001
φbp1 1.06354 0.04029 26.40 <0.0001
φbp2 –0.12788 0.04082 –3.13 0.0018
φbp4 0.05631 0.02747 2.05 0.0407
ω0 0.24654 0.04469 5.52 <0.0001
Table 20. Portmanteau Lack of Fit Test that the Autocorrelation of the Residuals of Model (5.11) are Equal to Zero
Autocorrelation Check of Residuals ut To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 3.52 2 0.1719 –0.003 0.014 –0.005 –0.024 0.044 0.034 12 14.86 8 0.0620 –0.034 –0.039 –0.019 –0.033 0.038 0.082 18 19.54 14 0.1454 –0.041 –0.021 0.005 –0.004 0.052 0.015 24 30.41 20 0.0634 –0.051 –0.005 0.028 –0.061 0.067 –0.012 30 35.32 26 0.1050 –0.054 –0.025 –0.012 –0.003 0.039 –0.010 36 38.56 32 0.1972 –0.045 0.017 0.007 –0.007 –0.024 0.021 42 47.09 38 0.1481 –0.066 0.001 –0.017 –0.014 0.056 –0.033 48 53.09 44 0.1638 –0.050 0.032 –0.012 –0.013 0.045 0.022
The independence of the residuals, ut, is further validated by the ACF and PACF shown in Figure 33. The autocorrelations and partial autocorrelations are within their approximate 95% confidence interval, except at lag 12. However, the autocorrelation and
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partial autocorrelation at lag 12 are very small and can be assumed not to differ from zero. This decision is supported by the results in Table 20.
Table 21. Portmanteau Lack of Fit Test that the Cross-Correlation of the Residuals of Model (5.11) and the Xt series are Equal to Zero
Cross-Correlation Check of Residuals ut with Xt To Lag Chi-Square DF Pr > ChiSq Cross-Correlations
5 9.80 6 0.1333 –0.020 0.044 0.014 –0.006 0.049 0.077 11 15.84 12 0.1986 –0.048 0.047 –0.004 –0.001 0.026 0.038 17 20.55 18 0.3027 0.023 0.003 0.042 0.016 0.043 –0.028 23 25.95 24 0.3555 0.028 –0.006 –0.000 0.071 0.005 0.013 29 36.87 30 0.1809 –0.016 –0.049 0.032 0.053 –0.063 0.041 35 44.86 36 0.1478 –0.035 0.046 0.067 –0.006 –0.002 –0.032 41 48.04 42 0.2415 0.013 –0.019 0.015 –0.005 0.037 0.037 47 58.78 48 0.1370 –0.070 –0.027 0.053 –0.058 0.012 0.008
These results indicate that the model (5.11) adequately describes the autocorrelation between Xt and Yt and the behavior of the base process.
Next we compare models (5.10) and (5.11). Table 22 shows the Akaike information criterion (AIC), and the Shwartz’s Bayesian criterion (SBC) for both models.
We see that there is very little difference between these values, indicating that the term δ1 in model (5.10) has little influence and that model (5.11) fits the data as well as model
(5.10). It follows that model (5.11) is the better model in that it is more parsimonious.
As further validation of (5.11) we consider whether collinearity of the parameters in the model may have influenced our results (SAS/ETS User’s Guide version 6, 1993, pg. 108). If two parameters are highly correlated, then only one of them is needed in the model. Table 23 shows the correlations among the parameters in model (5.11). The
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highest estimated correlation, –0.737, is between the parameters φbp1 and φbp2. This correlation is perhaps not very high; nevertheless these parameters were removed, one by one, from the model. We found that if either parameter is removed from model (5.11) the residuals, ut, become autocorrelated. Therefore, we decided to keep both parameters in the model (5.11).
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Figure 33. ACF and PACF of the residuals ut from model (3.35).
A qualitative verification that model (5.11) does represent the overall process series appropriately is to construct a time plot of the observed values and their lead-one forecasted values. To this end, the minimum mean-squared error lead-one forecasts, obtained from model (5.11) by setting ut to its conditional expected value of zero and replacing Xt with its lead-one forecast, is (Box, Jenkins, Reinsel, 1994; p.445)
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YXXXXˆˆ()11=ω () −φ −φ −φ +Φ t−−−−−101112244()t bp t bp t bp t 0 (5.12)
+φbp11YYYr t−− +φ bp 22 t +φ bp 44 t −− −θ bp 11 t ,
ˆ where Φ=μ−φ−φ−φbp01 bp()1 bp bp2 bp4, and rYYttt−−−112=−(1) is the last residual value.
Table 22. Comparison of Variance Estimates and Information Criteria for Models (5.10) and (5.11)
Statistic Model (5.10) Model (5.11)
AIC 5637.912 5644.09
SBC 5671.513 5672.898
Table 23. Correlation between Parameter Estimates of Model (5.11)
Parameter μ θbp φbp1 φbp2 φbp4 ω0 μ 1.000 0.015 0.020 –0.039 0.030 –0.575
θbp 0.015 1.000 0.563 –0.105 –0.544 –0.008
φbp1 0.020 0.563 1.000 –0.737 –0.292 –0.027
φbp2 –0.039 –0.105 –0.737 1.000 –0.418 0.066
φbp4 0.030 –0.544 –0.292 –0.418 1.000 –0.062
ω0 –0.575 –0.008 –0.027 0.066 –0.062 1.000
Figure 34 shows that, on average, the forecasted values match the observed values fairly well, and the observed series is quite variable, with several observations that could be due to special causes. Unfortunately, no information is available from the source of this data to determine if any of these observations correspond to an out-of- control situation, therefore none of these observations can be discarded. For the sake of comparison, we conducted the analysis without observations 98, 410, 422, 482, and 523
(see Figure 34).
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Observed Values Forecasted Values Forecasted Values
○ = 523 523 = n = 482 482 = = 422 422 = n n observations and their lead-one forecasts. = 410 410 = Observation No. n = 98 = n Figure 34. Time plot of the overall process 0 100 200 300 400 500 600 700 800 900 0 30 20 10
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We found that the models as those identified using the complete data set changed little without these questionable observations. The estimated values of the model
2 2 parameters and the variance σa changed slightly. The variance σu had a larger
2 decrement than σa causing the control limits of the CCC and SCC charts to become slightly narrower.
The final step in assessing model (5.11) is to determine if indeed the residuals
ˆ rYYttt=−−1 (1) (5.13)
2 are independent and identically distributed N (0, σu ) . The independence of the residuals was verified by the ACF and PACF in Figure 33 and the portmanteau test in Table 21.
Figure 35 shows the histogram and the normal plot of the residuals. The residuals follow a normal distribution, except for outliers at the upper and lower end of the distribution.
35 R 40 e s 30 i 30 d u a 25 l 20 : P e A 10 r 20 c c t e u 15 a 0 n l t - F -10 10 o r e c -20 5 a s t -30 0 -20-16-12-8-40 4 8 12162024283236 0.01 0.1 1 5 10 25 50 75 90 95 99 99.9 99.99 Residual: Actual-Forecast Normal Percentiles Figure 35. Histogram and normal probability plot of the residuals from model (5.11).
Table 24 shows four different test of the null hypothesis that the residuals are normally distributed. In every case the null hypothesis is rejected. Not surprisingly, because of the large sample size (899 residuals), small departures from normality are detected. Indeed, if residuals 98, 410, 422, 482, and 523 (see Figure 34) are removed from the data set, all four tests fail to reject the null hypothesis. As mentioned before, no
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information is available to determine if these outliers are associated with out-of-control conditions, therefore we will not exclude them and we will assume that the residuals are approximately normal.
Table 24. Tests for Normality of the Residuals from Model (3.35)
Test Statistic p Value Shapiro-Wilk W 0.9734 Pr < W <0.0001 Kolmogorov-Smirnov D 0.033163 Pr > D 0.0175 Cramer-von Mises W-Sq 0.299341 Pr > W-Sq <0.0050 Anderson-Darling A-Sq 2.125548 Pr > A-Sq <0.0050
In conclusion we can say that model (5.11) appropriately represents the observed behavior of the overall process series, and that
1. there is no delay in the effect of measuring the reference material on the
measurement of the product,
2. the absorbance of the product solution, Yt, is a multiple of the absorbance of the
reference material solution, Xt, and
3. the base process can be represented by the ARMA model (5.9).
In the next section we will use the product measurements and the computed values from equations (5.12) and (5.13) to compute the control limits and construct the
CCC and SCC charts.
5.1.3 Construction of the CCC and SCC Charts
The requirements to compute the control limits using expressions (5.4), (5.5), and
(5.6) are that the values plotted on a control chart are independent and identically
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2 distributed N(0, σε ). Above we showed that the residuals from model (5.11) are independent, and that we can assume that they are normally distributed.
Then, continuing with Step 12 of the procedure detailed at the beginning of
Section 5.1, we compute the control limits for the CCC chart by substitutingσ=ˆ r 5.55 , the standard deviation of the residuals, into equation (5.4). These limits are
ˆˆ UCL=+σ=+ Ytrt(1) 3ˆ Y ( 1) 16.652, ˆ CL= Yt ()1, (5.14) ˆˆ LCL=−σ=− Ytrt()1 3ˆ Y () 1 16.652.
In Step 13 we calculate the control limits for the I-chart by replacing the mean of the residuals, 0.098, and the average moving range, 5.905, into equation (5.5), which yields
UCL =+0.098 3( 5.905/1.128) = 15.8, CL = 0.098, (5.15) LCL =−0.098 3() 5.905/1.128 =− 15.6.
The control limits of the MR-chart, Step 14, are obtained by replacing the average moving range into equation (5.6). The control limits are
UCL = 3.268×= 5.905 19.3, CL = 5.9, (5.16) LCL = 0.
Figure 36 shows the CCC chart and the points that fall outside the control limits, which are observations 98, 379, 410, 422, 482, 523, 554, 576, and 830. Some of these observations correspond to the residuals identified in Figure 6 as outliers. The CCC chart lends credence to the statement that these outliers are most likely associated with out-of-control conditions.
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= 830 830 = n Observed Values Forecasted Values Forecasted Values
○ = 554 554 = n = 523 523 = n = 482 482 = n = 576 576 = n rall process observations. = 422 422 = n Observation No. Observation = 410 410 = n = 379 379 = n UCL Figure 36. CCC chart for the ove LCL = 98 = n 0 100 200 300 400 500 600 700 800 900 0 30 20 10
-10 -20 -30 -40 Data
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The SCC charts, the I-chart and the MR-chart, are shown in Figure 37. The same residuals of the observations that were outside the control limits of the CCC chart fall outside the control limits of the I-chart, even though the control limits of the CCC chart are slightly wider than the control limits of the I-chart.
The MR chart indicates that the variance of the process or of the random shocks is out of control at observations 89, 98, 99, 153, 379, 408, 410, 422, 423, 456, 482, 483,
520, 524, 554, 555, 577, 816, 830, 831, and 865. We see that every observation that was outside the control limits of the I-chart is associated with a moving range outside the control limits. This indicates that the out-of-control conditions detected by the CCC chart and the I-chart are likely to be associated with changes in the variance of the base process. The MR ranges outside the control limits of the MR-chart that are not outside the control limits of the I-chart may or may not be associated with out-control conditions.
In a practical situation it is recommended to also investigate these observations to ensure that they are not associated with out-control conditions.
There may be more observations associated with out-of-control conditions, but
they are not detected because the estimates ( σˆ r and MRd2 ) of the residual variance are larger than they would be if the points outside the control limits were removed. As mentioned before, we have no information that tells us whether the points outside the control limits are due to special causes. Therefore, we will not discard any observation and use the estimates of model (5.11) and the control limits of the CCC and SCC charts to illustrate the Phase II of the implementation of the monitoring scheme.
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UCL=15.7978 Avg=0.0977 LCL=-15.6023 UCL=19.2896 Avg=5.9052 LCL=0.0000 905 905 * 854 854 1 * * * 804 804 754 754 704 704 654 654 604 604 1 * 1 * * 554 554 1 * * 504 504 1 * * * 454 454 1 * * 1 Obs. No. * * Obs. No. 403 403 1 duals of the overall process. * 350 350 300 300 250 250 200 200 * 150 150 1 * * * 100 100 Figure 37. SCC charts of the resi Figure 37. SCC 50 50 00 . 0.00 0.00
40.00 30.00 20.00 10.00 50 40.00 30.00 20.00 10.00 -10.00 -20.00 Residual
Moving Range Moving
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5.2 Phase II – Implementation of the Monitoring Scheme for the Base Process
The initial stage of Phase II relies on the models, parameter estimates and the control limits of the CCC and SCC charts established in Phase I, where it is assumed that the models represent the actual behavior of the processes and that the model parameter and control limit estimates are free of error.
The Phase II implementation of the monitoring scheme should include a plan for a regular revision of the models, the estimation of the parameters, and of the control limits using the newly acquired data. In this section we will only illustrate the initial stage of
Phase II and will address the updating of models, parameter estimates and control limits in future work.
Another requirement of the monitoring scheme of the base process is that the measurement process is monitored and maintained in control. This implies that the model that represents the measurement process, the parameter estimates, and the control limits are also regularly updated. Then, any overall process observation that falls outside the control limits of the overall process CCC and SCC charts or that violates a Special
Cause Rule indicates that the base process is out of control.
Using the notation from Sections 4.1 and 5.1, the procedure to monitor the base process is:
1. Obtain observation Xt+1 and Yt+1;
ˆ 2. Generate the measurement process lead-1 forecast, X t (1) , using
ˆ X t−−1 ()1 =Φ X0 +φ Xt 1XXX 1 +φ Xt 2− 2 +φ Xt 4− 4 +φ X 10 X t− 10 +φ X 11 X t− 11 , (5.17)
where Φ=μ−φ−φ−φ−φ−φXX0 ()1 XXXXX1 2 4 10 11 ;
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ˆ ()X 3. Compute the residual eXXttt++11=−(1) and the moving range MRett++11=−et;
4. Plot the new observation, the lead-one forecast, and the control limits established in
Phase I on the measurement process CCC chart;
5. Plot the new residual on the I-chart and the new moving range on the MR-chart of
the measurement process SCC chart. Use the control limits established in Phase I;
6. Verify that the measurement process is in control. If in control proceed to Step 7. If
out of control, stop the process. Identify the special cause of the out-of-control
condition and institute appropriate measures to permanently remove it or prevent it
from reoccurring;
ˆ 7. Generate the overall process lead-1 forecast, Yt (1) , using equation (5.12) and the
parameter estimates established in Phase I;
(Y ) 8. Compute the residual using equation (5.13) and the moving range MRrtt++11=−rt;
ˆ 9. Plot Yt ()1 and Yt+1 and the control limits (5.14) on the overall process CCC chart;
(Y ) 10. Plot rt+1 on the I-Control chart and MRt+1 on the MR-chart of the overall process
SCC charts with control limits (5.15) and (5.16), respectively;
11. Verify that the base process is stable. If in control continue normal operation. If out
of control, stop the process and follow the standard procedure to determine whether
the product material should be scrapped. Identify the special cause of the out-of
control condition and institute appropriate measures to permanently remove it or
prevent it from reoccurring.
Figure 38 shows the CCC chart of the 83 Phase II measurements on product material. In Chapter 4 we found that the measurement process was in control during
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Phase II. Therefore, this CCC chart shows that the base process went out of control, initially with observation 942 being very large and the following observations showing a clear reduction in the variation of the process. The I-chart of the residuals in Figure 39 shows the same trend as the CCC chart. Also, SCR #7 (15 points in a row within the
range CL ±σ1ˆ r ) confirms the reduction in variation.
40 Actual data n = 942 O 30 Lead -1 forecast Control limits 20
10
0
-10 Overall Process Series Process Overall -20 Phase I Phase II -30
895 905 915 925 935 945 955 965 975 985 Observation No. Figure 38. Phase II CCC chart of the overall process observations.
50.00 1 40.00 n = 942 30.00 Phase II 20.00 UCL=15.8457 10.00 7777 Residual Y 0.00 777777777 µ0=0.1423 -10.00 -20.00 LCL=-15.5612 900 905 910 915 920 925 930 935 940 945 950 955 960 965 970 975 980 985 Phase I Obs. No.
* 50.00 * 40.00 Phase II n = 943 30.00 n = 942 * 20.00 UCL=19.2937 10.00
Moving Range µ0=5.9065 0.00 LCL=0.0000 900 905 910 915 920 925 930 935 940 945 950 955 960 965 970 975 980 985 Obs. No.
Figure 39. Phase II I-chart and MR-chart of residuals.
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The MR-chart detects a large jump between observations 922 and 923 followed by a reduction of variance between observations 925 and 932. Also, the MR-chart shows two large moving ranges associated with observation 943, followed by a sequence of small moving ranges confirming the reduction in the variance of the process. In an actual implementation of a SPC system, the CCC and SCC charts would had been turned on after observation 900 (the last observation of what we considered the Phase I data set), and would have detected that the process was out-of-control as early as observation 923.
The pattern from observation 923 to observation 932 is perhaps an early indication of the special cause associated with the pattern that started with observation 943.
In the next section we will introduce different disturbances (out-of-control conditions) and see their impact on the CCC and SCC charts.
5.3 Detection of Out-of-Control Conditions
In this section we introduce disturbances in the base process to see their effect on the quantities plotted on the CCC and SCC charts and to see if the disturbances are detected. These disturbances consist of step shifts in the mean of the base process, the mean of the random shocks, and the variance of the random shocks. Table 25 shows the magnitude of these step shifts.
Table 25. Types of Step Shifts and their Magnitudes
Step shift in the … Magnitude (δ)
Mean of the base process kσu, k = 1.5, 2, 3
Mean of the random shocks kσu, k = 1.5, 2, 3
Variance of the random shocks k = 0.5, 2, 3
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In Appendix B we show that
σˆˆbp=σc u , where c is a constant that depends on the magnitude and sign of the parameters in the model that describes the base process (see Section 3.2.2). Therefore, the magnitudes of the shifts in the mean of the base process are multiples of the variance of the random shocks.
New observations, lead-one forecasts, and residuals are generated using the models and parameters established in Phase I. These new values and the Phase I control limits are used to construct the CCC and SCC charts.
The procedure to generate new observations, introduce disturbances, generate lead- one forecasts, compute residuals, and construct the CCC and SCC charts is as follows:
1. Set initial values: XaYu0000====0, 0, 0, 0 ;
2 ˆ 2. Generate 1000 values of aNt~0,( σa) , Xt using (5.3) and X t−1 ()1 using (5.17) with
parameter estimates from Phase I;
3. Generate 1000 values of Yt as follows:
For a step change in the mean of the process mean: δ = kσu;
2 2 a. Generate uNt~0,(σu). Use estimate of σu from Phase I;
b. Generate Yt using (3.35) and parameter estimates from Phase I;
c. Introduce disturbance as follows:
i. If t < 951, then set Zt = Yt;
ii. If t > 950, then set Zt = δ + Yt;
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ˆ d. Compute forecasts Yt−1 (1) using (5.12), the parameter estimates from Phase I,
and the Zt values, that is
YXXXXˆˆ()11=ω () −φ −φ −φ +Φ t−−−−−101112244()t bp t bp t bp t 0
+φbp11ZZZr t−− +φ bp 2 t 2 +φ bp 4 t −− 4 −θ bp 11 t ;
ˆ e. Compute residuals rZYttt=−−1 (1) ;
f. Compute moving range if residuals MRrtt++11= − rt;
For a step change in the mean or in the variance of the random shocks;
g. Introduce the disturbance starting at observation 950. For a step change in the
2 mean generate the random shocks as ut + δ , where δ = kσu and uNtu~0,( σ ) .
For a step change in the variance generate the random shocks as udt+ t, where
2 2 dNktu~0,()σ and uNtu~0,( σ ) ;
h. Generate Yt using (5.11) and parameter estimates from Phase I;
ˆ i. Compute forecasts Yt−1 (1) using (5.12) and Phase I parameter estimates;
ˆ j. Compute residuals rYYttt=−−1 (1) ;
k. Compute moving range if residuals MRrtt++11= − rt;
4. Subset last 100 rows of generated values (these are the new Phase II observations);
5. Construct the CCC and SCC charts;
6. Determine if the CCC and/or the SCC have detected the out-of-control condition.
5.3.1 Detection of Step Shifts in the Mean of the Base Process
Figure 40 shows the CCC and SCC charts for a step-shift in the mean of the base process equal to 3σu. The mean of the base process changes at observation 951 and it is
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indicated by the vertical line on the control charts. In the CCC chart we can clearly see the change in the mean of the overall process observations and that the first lead-one forecasts barely exceeds the upper control limits as well as observations 954 and 955.
However, we also see that the lead-one forecasts slowly increase and approach the new level of the overall process, to the point that it is difficult to know that the mean of the base process has shifted.
40 CCC 30
20
10
0
Simulated Data Simulated -10 O MP Observations -20 Lead-One Forecasts Control limits -30
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 30.00 1 20.00 1 1 1 UCL=15.7978 10.00
0.00 µ0=0.0977 Residual
-10.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000 Obs. No. SCC 25.00 * 20.00 UCL=19.2896 15.00
10.00
of Residual 5.00 µ0=5.9052 Moving Range Moving
0.00 LCL=0.0000 910 920 930 940 950 960 970 980 990 1000 n
Figure 40. CCC and SCC charts of the overall process with a 3σu shift in the mean of the base process.
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The I-chart, in Figure 40, immediately detects the out-of-control condition, but the residuals slowly decrease back to their original level, despite the fact that the mean of the base process has shifted. The MR chart, also in Figure 40, detects the disturbance immediately after it occurs, but subsequent moving ranges are well inside the control limits and there is no further indication of the out-of-control condition.
The behavior of the lead-one forecasts on the CCC Chart and the residuals in the
I-chart is explained as follows. In Section 5.1.2 we determined that the overall process can be represented as
YXtt= ω+0 Pt.
Also, in Section 5.1.2 we determined that model (5.9) represents the base process. This model can be written as
PPt=μ bp +φ bp11() t−− −μ bp +φ bp22( P t −μ bp) +φ bp44( Pu t − −μ bp) + t −θ bp1u t−1.
Then, the lead one forecast of the of the overall process can be expressed as
ˆˆ YXt−−101()11=ωt () +μbp +φ bp 11( P t − −μ bp) +φ bp 22( P t − −μ bp)( +φ bp 44 P t − −μ bp ) −θ bp 1r t−1,
and the residuals as
ˆ rYYttt=−−1 ()1
=ω+−μ−φ01ePt t bp bp()()() P t−12 −μ−φ bp bpP t−−24 −μ−φ bp bpP t4 −μ+θ bp bp1 r t−1,
ˆ where eXXttt=−−1 ()1 .
Suppose that the mean of the base process shifts between the time periods t = τ −1 and t = τ, such that
⎧ μ0 t =τ−1, τ− 2,… μ=bp ⎨ ⎩μ=μ+δ10 t =ττ+,1,… ,
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and that
⎧μ=τ−τ−0 t 1, 2,… EP[]t = ⎨ ⎩μ=ττ+1 t ,1,… .
Also, we have that Ee[ t ] = 0 for all values of t because the measurement process
in maintained in control and that Er[ t ] = 0 for t = τ−1, τ− 2,… .
Hence, the lead-one forecast at t = τ is
ˆ YPPPτ−101102204401()1 =μ +φbp (τ− −μ ) +φbp ( τ− −μ) +φbp ( τ− −μ) −θbp rτ−1, with expected value
EY⎡⎤ˆ 1 = μ . ⎣⎦τ−10()
Then, the residual at t = τ is
rePτττ=ω+00110220440 −μ−φbp ( P τ− −μ−φ) bp ( Pτ− −μ−φ) bp ( Pτ− −μ+θ) bp1 rτ−1, and its expected value is
Er[ τ ] =μ10− μ =δ.
Proceeding in the same way subsequent expected values of the lead-one forecasts and of the residuals are obtained. Table 26 summarizes these results. The expressions there indicate that the expected values of the lead-one forecasts will start to increase one time period after the mean of the base process shifted, and if θbp1< 1, they tend towards some steady-state level that may differ from the original level μ0. Similarly, if θbp1< 1, the residuals rτ+j tend towards a steady-state level as j increases, that, again, may differ from the original level. The magnitude and the sign of the parameters φbp1, φbp2, φbp4, and
θbp1 will determine the final or steady-state levels and the magnitude and sign of θbp1 will determine how fast these quantities reach their steady-state level. Figure 41 shows the
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expected behavior of the observations, the lead-one forecasts and the residuals based on the estimated values of the parameters φbp1, φbp2, φbp4, and θbp1 in Table 20. This behavior coincides with the patterns observed on the CCC and SCC chart in Figure 40.
Table 26. Expected values of the lead-one forecasts and the Residuals Of the Overall Process
EY⎡⎤ˆ Er⎡ ⎤ δ j ⎣⎦τ+ j ⎣ τ+ j ⎦
0 μ0 1
1 μ+φ011()bp −θ bp δ (1− φ+θbp11 bp ) μ+φ⎡⎤ +φ −θr δ ⎡ 11−φ +θ −φ +θ2 ⎤ 2 01211⎣⎦bp bp bp τ+ ⎣( bp112)( bp) bp bp1⎦ μ+φ +φ −θr δ ⎡ 11−φ +θ +θ23 −φ 1 +θ +θ ⎤ 3 01212()bp bp bp τ+ ⎣( bp11121)( bp bp) bp() bp bp1⎦ ⎡ 11−φ +θ +θ23 +θ ⎣( bp1)( bp 111 bp bp ) 4 μ+φ012413()bp +φ bp +φ bp −θ bp rτ+ δ −φ+θ+θ−φ+θ1 24⎤ δ bp21141() bp bp bp bp ⎦ kk−−12 ⎡ 1− φθ−φkkθ ⎣( bp112)∑∑jj==00bp bp bp1 k μ+φ+φ+φ−θr δ 012411()bp bp bp bpτ+ k − kk−4 −φ θkk + θ⎤ δ bp41∑∑jj==00bp bp 1⎦
20 Residuals Observations
10
0 Simulated Data
Lead-1 Forecasts
-10
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
Figure 41. Expected behavior of the overall process observations, lead-one forecasts, and residuals after a 3σu shift in the mean of the base process.
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The CCC and SCC charts for the cases when the magnitude of the shift in the base process mean is 2σu and 1.5σu are shown in Figures 42 and 43, respectively. The patterns on these charts resemble the behavior of the observations, the lead-one forecasts and residuals in Figure 40, except that they are less pronounced.
The CCC chart in Figure 42 detects the out-of control condition by the 4th observation after the shift, but after that the difference between the observed and forecasted value quickly decreases and no shift is evident.
30 CCC 20
10
0
-10 Simulated Data Simulated
-20 O MP Observations Lead-One Forecasts Control limits -30
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 25.00 20.00 15 15.00 5 UCL=15.7978 10.00 5 5.00 0.00 µ0=0.0977
Residual -5.00 -10.00 -15.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC
20.00 UCL=19.2896
15.00
10.00
5.00 µ0=5.9052 of Residual Moving Range 0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 42. CCC and SCC charts of the overall process with a 2σu shift in the mean of the base process.
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The same behavior is observed in the I-chart. However, the use of SCR #5 (two out of three points in a row in the region beyond 2 standard deviations from the center line) will take advantage of the fact that, when the mean of the process shifts, the residuals exhibit the largest deviations from the original level and that they decrease towards the original level. In this particular case, the disturbance is detected by the fifth observation. The MR-chart does not detect the out-of-control condition.
30 CCC
20
10
0
-10 Simulated Data
-20 O MP Observations Lead-One Forecasts Control limits -30
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 20.00 15.00 5 UCL=15.7978 5 10.00 5.00 5 0.00 µ0=0.0977 -5.00 Residual -10.00 -15.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC
20.00 UCL=19.2896
15.00
10.00
µ0=5.9052 of Residual 5.00 Moving Range
0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 43. CCC and SCC charts of the overall process with a 1.5σu shift in the mean of the base process.
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The CCC in Figure 43 does not show points outside the control limits or a clear pattern indicative of a shift in the base process mean. The only evidence of the disturbance comes from SCR #5 in the I-chart that detects the disturbance 5 time periods after the shift. Again the MR-chart does not detect the disturbance.
It is clear that the CCC and SCC charts would have difficulty identifying a shift in the mean of the process smaller than 2 standard deviations.
5.3.2 Detection of Step Shifts in the Mean of the Random Shocks of the Base Process
In this section the CCC and SCC charts in Figures 44, 45, and 46 show the effect of step shift in the mean of the random shocks equal to 3σu, 2σu, and 1.5σu, respectively.
The behavior of the observations and the lead-one forecasts in the CCC Charts, and of the residuals on the I-charts after the step shift in the mean of the random shocks can be explained (see Section 3.1.3) by recalling that
YXtt= ω+0 Pt,
where PPttb=−μp and PPtbptbptbpttbpt=φ11− +φ 22 P−− +φ 44 Puu + −θ 11−.
The step shift in the mean of the random shocks is introduced into the process by letting
⎧utt =τ−1, τ− 2,… ut = ⎨ ⎩utt +δ =τ,1, τ+ … such that,
⎧01,2t =τ− τ− ,… Eu[]t = ⎨ ⎩δ=ττ+t ,1,… .
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90
80 CCC
70
60
50
40
30
20
10 Simulated Data 0 -10 O MP Observations -20 Lead-One Forecasts
-30 Control limits
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
30.00 11 1 1 11 1 1 11 1 11 1 20.00 1 1 11 11 1 1 1 11 1 1 UCL=15.7978 10.00
Residual 0.00 µ0=0.0977
-10.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC 25.00 * 20.00 UCL=19.2896
15.00
10.00
of Residual 5.00 µ0=5.9052 Moving Range
0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 44. CCC and SCC charts of the overall process with a 3σu shift in the mean of the random shocks of the base process.
The base process values after the shift are
0 PPττ=+δ PP =++φ−θδ0 ⎡⎤1 τ+11 τ+ ⎣⎦()bp 1 bp 1
PP =02 +⎡⎤11 +φ +φ −θ +φ +φ δ τ+22 τ+ ⎣⎦()bp 111 bp bp() bp 1 bp 2 PP =02 +⎡⎤111 +φ +φ +φ3 −θ +φ +φ 2 +φ −θ +2 φ δ τ+3 τ+ 3⎣⎦bp 1111 bp bp bp() bp 11 bp bp 2() bp 1 bp 1
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0 where Pτ+ j , with j = 0, 1, 2,… represents the in-control part of the base process. Since the measurement process is maintained in control, the overall process observations are obtained directly from the expressions that describe the base process. Hence,
0 YYττ=+δ YY=++φ−θδ0 ⎡⎤1 τ+11 τ+ ⎣⎦()bp 1 bp 1 YY=02 +⎡⎤11 +φ +φ −θ +φ +φ δ τ+22 τ+ ⎣⎦()bp 11 bp bp 1() bp 1 bp 2 YY=02 +⎡⎤111 +φ +φ +φ3 −θ +φ +φ 2 +φ −θ +2 φ δ τ+3 τ+ 3⎣⎦bp 1111 bp bp bp() bp 11 bp bp 2( bp 1 bp 1) YY=02 +⎡ 11 +φ +φ +φ3 +φ4 −θ +φ +φ 2 +φ3 τ+4 τ+ 4⎣()bp 1111 bp bp bp bp 1() bp 11 bp bp1 +φ12 + φ + 3 φ2 + φ − 2 φ θ −θ + φ⎤ δ bp21121114()bp bp bp bp bp bp bp ⎦
0 0 where YXPτ+ j =ω0 τ+j + τ+ j, j = 0, 1, 2, …, represent the in-control measurement process, transfer function, base process, and overall process. These expressions show that the observed values will tend to a new level determined by magnitude and the sign of the parameters φbp1, φbp2, φbp4, and θbp1, and as time progresses, the observations will reach a new steady-state level. For the parameter estimates in Table 19, the observations will start to increase immediately after the shift in the mean of the random shocks to a steady- state level. This is in fact the behavior of the observations in the CCC charts in Figures
44, 45, and 46.
The minimum mean squared error lead-one forecast of the base process is
ˆ PPPPt−−−−11122441()1 =φbp t +φ bp t +φ bp t −θ bpr t−1,
ˆ ˆ where PPtt()11=−μ () bp. Then, the lead-one forecast of the overall process is
ˆˆ0 YXPPt−−11()1 =+φ+φ+φ−θt bp t1 bp2 t−2 bp4 P t−4 bp1r t−1.
The lead-one forecasts and residuals starting at t = τ are
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ˆˆ0 YYτ−1 ()1 = τ 0 rrττ=+δ ˆˆ0 YYττ+()1 =+φ−θδ1 ()bp bp 0 rrτ+11=+δ τ+ YYˆˆ11=02 +φ+φ−θ+φ+φδ⎡⎤ τ+1211112() τ+ ⎣⎦()bp bp bp() bp bp 0 rrτ+22=+δ τ+ YYˆˆ11=023 +φ+φ+φ−θ+φ+φ+φ−θ+φ⎡⎤ 212 δ τ+2() τ+ 3⎣⎦bp 1111 bp bp bp() bp 11 bp bp 2() bp 1 bp 1 0 rrτ+33=+δ τ+ YYˆˆ1 = 0234+⎡ φ +φ +φ +φ −θ1 +φ +φ 23 +φ τ+3 () τ+411111111⎣ bp bp bp bp bp() bp bp bp +φ12 + φ + 3 φ2 + φ − 2 φ θ − θ + φ⎤ δ bp21121114()bp bp bp bp bp bp bp ⎦
ˆ 0 00ˆ 0 where Yτ+ j , j = 0, 1, 2, … and rYYτ+τ+τ+j =−jj, j = 0, 1, 2, … are lead-one forecasts and the residuals of the in-control process, respectively. These expressions show that lead- one forecasts exhibit the same tendency as the observations, except that their change will be delayed one time interval and their magnitude will differ from the observations, on average, by an amount equal to δ. This behavior produces a distinctive pattern that can be readily observed in the CCC charts in Figures 44, 45, and 46. In Figure 46, even though the shift is relatively small, 1.5σu, and there are no points outside the control limits, we can still see the distinctive pattern of a shift in the mean of the random shocks.
The expressions of the residuals indicate that the I-chart will show a change in the level of the residuals immediately after the shift in the mean of the random shocks. This behavior is clearly observed in the I-charts in Figures 44, 45, and 46. Also, because of the sustained change in the level of the residuals, SCR #2 (nine points in a row above or below the center line) is likely to detect the shift in the mean of the residuals, as in Figure
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46 when the shift is relatively small. The disadvantage of this rule is that it requires several observations before it signals.
60 CCC 50
40
30
20
10
0 Simulated Data
-10 O MP Observations -20 Lead-One Forecasts
-30 Control limits
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 25.00 20.00 11 1 1 1 1 1 11 1 15.00 UCL=15.7978 10.00 5.00 0.00 µ0=0.0977
Residual -5.00 -10.00 -15.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC
20.00 UCL=19.2896
15.00
10.00
of Residual 5.00 µ0=5.9052 Moving Range
0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 45. CCC and SCC charts of the overall process with a 2σu shift in the mean of the random shocks of the base process.
Finally, the expression of the residuals indicate that the MR-chart will show a large value immediately after the shift, but subsequent moving ranges will very likely be within the control limits because the variance of the random shocks remains unchanged.
Then, if the shift in the mean of the random shocks is large enough, the MR-chart will detect the shift immediately when it occurs. This is the case Figure 16 when the shift is
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large enough that the first moving range after the shift is outside the control limits of the
MR-chart. In Figures 45 and 46 the shifts in the mean of the residuals are not large enough and the MR-chart does not detect the out-of-control condition.
50 CCC 40
30
20
10
0 Simulated Data -10 O MP Observations -20 Lead-One Forecasts Control limits -30
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 25.00 20.00 11 2 1 15.00 2 22 UCL=15.7978 22 2 2 10.00 2 22 2 5.00 2 22 2 0.00 µ0=0.0977
Residual -5.00 -10.00 -15.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000 Obs. No. SCC
20.00 UCL=19.2896
15.00
10.00
of Residual 5.00 µ0=5.9052 Moving Range
0.00 LCL=0.0000 910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 46. CCC and SCC charts of the overall process with a 1.5σu shift in the mean of the random shocks of the base process.
5.3.3 Detection of Step Shifts in the Variance of the Random Shocks of the Base Process
In this section we introduce step shifts in the variance of the random shocks of the base process, ut. The magnitudes of the first two shifts correspond to an increment in variation equal to 3 and 2 times the in-control standard deviation of the random shocks,
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σu. The CCC and SCC charts of these two cases are shown in Figures 47 and 48, respectively. The third step shift is a reduction in the variation of the random shocks equal to 0.5σu. Figure 49 shows the CCC and SCC charts for this situation. In every case the shift in the variation of the random shocks occurs between observations 950 and
951. The vertical line in the control charts marks the first value after the change in the variance.
30 CCC 20
10
0
-10
Simulated Data -20
MP Observations -30 O Lead-One Forecasts Control limits -40
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 40.00
30.00 1 11 1 1 20.00 1 11 1 1 1 UCL=15.7978 10.00 0.00 µ0=0.0977 -10.00 Residual 1 1 -20.00 1 1 LCL=-15.6023 1 1 1 1 1 -30.00 1 -40.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC
50.00 * * * 40.00 * * * * * * * 30.00 * * * * * * 20.00 UCL=19.2896 of Residual
Moving Range 10.00 µ0=5.9052 0.00 LCL=0.0000 910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 47. CCC and SCC charts of the overall process with a 3σu shift in the variance of the random shocks of the base process.
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30 CCC O MP Observations Lead-One Forecasts 20 Control limits
10
0
-10 Simulated Data
-20
-30
901 911 921 931 941 951 961 971 981 991 1001 Observation No.
20.00 1 11 UCL=15.7978 10.00
0.00 µ0=0.0977
Residual -10.00 1 1 1 11 LCL=-15.6023 -20.00 1
-30.00 907 914 921 928 935 942 949 956 963 970 977 984 991 998
Obs. No. SCC 35.00 30.00 * * * * * * 25.00 * * * * 20.00 UCL=19.2896 15.00 10.00 of Residual
Moving Range 5.00 µ0=5.9052 0.00 LCL=0.0000 -5.00 907 914 921 928 935 942 949 956 963 970 977 984 991 998 Obs. No.
Figure 48. CCC and SCC charts of the overall process with a 2σu shift in the variance of the random shocks of the base process.
The CCC charts in Figure 47 and 48 show a distinctive increment in the variation of the overall process observations after observation 951, but a less pronounced increment in the variation of the forecasts. In contrast, note the behavior of the CCC chart in Figure 49, where the variation of the observations and of the lead-forecasts is smaller after observation 951.
The I-charts in Figures 47 and 48 show an increment in the variation of the residuals and the large differences between subsequent residuals causing some moving
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ranges to exceed the upper control limits in the MR-charts. The I-chart and the MR-
Chart in Figure 49 show a clear reduction in the variation of the residuals.
20
10
0
-10 Simulated Data
-20 O MP Observations CCC Lead-One Forecasts -30 Control limits
901 911 921 931 941 951 961 971 981 991 1001 Observation No. 20.00 15.00 UCL=15.7978 10.00 5.00 7 77 777 0.00 7 7 7 µ0=0.0977 7 -5.00 Residual -10.00 -15.00 LCL=-15.6023 -20.00 910 920 930 940 950 960 970 980 990 1000
Obs. No. SCC 20.00 UCL=19.2896
15.00
10.00
µ0=5.9052 of Residual 5.00 Moving Range Moving
0.00 LCL=0.0000
910 920 930 940 950 960 970 980 990 1000 Obs. No.
Figure 49. CCC and SCC charts of the overall process with a 0.5σu shift in the variance of the random shocks of the base process.
The pattern of the observations, lead-one forecasts in the CCC chart, and of the residuals in the I-chart SCC charts after the step-shift in the variance of the random shocks ut can be explained as follows. First, recall that the overall process is described as
YXtt= ω+0 Pt, and the base process as
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PPtbptbptbpttbpt=φ11− +φ 2 P−− 2 +φ 4 Puu 4 + −θ 11−.
2 The variance of the random shocks shifts from its stable level, σu , to the out of control
2 level, δσu , between time intervals τ–1 and τ. This shift is modeled by adding the
222 normal random variable, dt, with mean zero and variance σdu=δσ −σu, onto the random shocks, such that
⎧φ+Pu t =τ−τ−1, 2,… ⎪ bp t−1 t Pt = ⎨ … ⎩⎪φ++=ττ+bpPudt t−1 t t ,1, , where dt is independent of ut. Then, the values of the base process after the shift in the variance are
0 PPdττ=+ τ 0 PPτ+11=+φ−θ+ τ+ ()bp 11 bp ddτ τ+ 1 02 PPτ+22=+φ−φθ+φ+φ−θ+ τ+ ()bp 1112 bp bp bp dτ ()bp 1112 bp ddτ+ τ+ 032 2 PPτ+33= τ+ +φ−φθ+φφ−φθ()bp 1111221 bp bp2 bp bp bp bp dτ +φ−φθ+φ(bp 1112 bp bp bp )dτ+1
+φ()bp1123 −θ bp ddτ+ + τ+ 0 43 2 2 PPτ+44=+φ τ+ ()bp111121212−φ bpbp θ +32 φ bpbp φ − φ bpbpbp φ θ +φ bp +φ bp 4dτ 32 2 +() φbp11112211 −φ bp θ bp +2 φ bp φ bp −φ bp θ bp ddτ+ +( φbp 11122 −φ bp θ bp +φ bp )τ+
+φ()bp1134 −θ bp ddτ+ + τ+
0 where Pτ+k , k =0, 1, 2, …, represents the in-control part of the base process.
Replacing these expressions of the base process values into the expression for the overall process yields
0 YYττ=+ d τ 0 YYτ+11=+φ−θ+ τ+ ()bp 11 bp ddτ τ+ 1 02 YYτ+22=+φ−φθ+φ+φ−θ+ τ+ ()bp 1112 bp bp bp dτ ()bp 111 bp ddτ+ τ+2
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032 YYτ+33=+φ−φθ+φφ−φθ τ+ ()bp 1111221 bp bp2 bp bp bp bp dτ 2 +φ−φθ+φ()bp11121 bp bp bp dddτ+ +φ−θ()bp 1123 bp τ+ + τ+ YY=043 +φ−φθ+φφ32 2 −φφθ +φ+φ2 d τ+44 τ+ ()bp 1111212124 bp bp bp bp bp bp bp bp bp τ 32 2 +φ−φθ+φφ−φθ()bp11112211 bp bp2 bp bp bp bp dτ+ +φ−()bp 1φθbp11 bp +φ bp 2dτ+ 2
+φ()bp1134 −θ bp ddτ+ + τ+
00 0 where YXPτ+kk=+ τ+ τ+k, k = 0, 1, 2, …, is the in-control part of the overall process.
These equations show that a shift in the variance of the random shocks of the base process will cause the overall process observations to have variances
22 2 Var() Yτ =ωσ0 au +σ δ 2 Var() Y =ωσ22 +σ 2⎡⎤ δ+ φ −θ() δ−1 τ+10au⎣⎦⎢⎥() bpbp11 2 2 Var() Y =ωσ+σδ+φ−φθ+φ22 2⎡⎤ 2 () δ−+φ−θ11() δ− τ+20a u ⎣⎦⎢⎥()bp1112 bp bp bp ()bp11 bp 2 Var() Y =ωσ22 +σ 2⎡ δ+ φ 3 −φ 2 θ +21 φ φ −φ θ() δ− τ+30a u ⎣⎢ ()bp1111221 bp bp bp bp bp bp 2 2 −φ2 −φθ +φ ()δ−11 + φ −θ() δ− ⎤ ()bp1112 bp bp bp ()bp11 bp ⎦⎥ 2 Var() Y =ωσ+σ22 2⎡ δ+φ 4 −φθ 3 +φφ322 −φφ θ +φ2 +φ() δ−1 τ+40a u ⎣⎢ ()bp1111212124 bp bp bp bp bp bp bp bp bp 22 32 2 +φ−φθ+φφ−φθ()bp1111221 bp bp21 bp bp bp bp () δ−+φ−φθ+φ()bp1112 bp bp bp () δ−1 2 +φ −θ() δ−1 ⎤ ()bp11 bp ⎦⎥
Thus, the overall process observations will have an increasing or decreasing variance that is determined by the magnitude of the shift, δ. Given the estimated values of the parameters in Table 19, the variance of the overall process observations will eventually reach some constant level.
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The expression for the lead-one forecast of the base process was found to be
ˆ PPPPt()1 =φ bp11 t− +φ bp 2 t−− 2 +φ bp 4 t 4 −θ bp 11r t−.
Hence,
ˆˆ0 YXPPt−−11()1 =+φ+φ+φ−θt bp t1 bp2 t−2 bp4 P t−4 bp1r t−1.
The expressions that represents the lead-one forecasts and the corresponding
residuals starting at t = τ are
ˆˆ0 YYτ−1 ()1 = τ 0 rrdττ=+ τ ˆˆ0 YXτ ()1 =+φ+φ+φ−θτ+11bp PPτ bp 21431τ− bp Pτ− bp rτ ˆ 0 =+φ−θYdτ+111()bp bp τ 0 rrdτ+11=+ τ+ τ+ 1 ˆˆ02 YYτ+121112111()1 =+φ−φθ+φ+φ−θτ+ ()bp bp bp bp dτ ()bp bp dτ+ 0 rrdτ+22=+ τ+ τ+ 2 ˆˆ032 YYτ+231111()12=+φ−φθ+φφτ+ ()bp bp bp bp bp221−φ bp θ bp dτ 2 +φ−φθ+φ()bp11121 bp bp bp ddτ+ +φ−θ()bp 112 bp τ+ 0 rrdτ+33=+ τ+ τ+ 3 ˆˆ043 2 2 YYτ+341111212124()13=τ+ +φ−φθ+φφ()bp bp bp bp bp −φφθ2 bp bp bp +φ+φ bp bp dτ 32 2 +φ−φθ+φφ−φθ()bp11112211 bp bp2 bp bp bp bp ddτ+τ +φ−φθ+φ(bp 1112 bp bp bp )+2
+φ()bpbp113−θ dτ+ 0 rrdτ+44=+ τ+ τ+ 4
Now, from these expressions we see that the variance of the residuals
2 Var[ rτ+ku] =δσ, k = 0,1,2,… , increases or decreases, depending on the value of δ, immediately after the shift in variance and remains constant after that. This is the behavior of the residuals found in
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the I-charts in Figures 47, 48, and 49. The decrement or increment in variation of the residuals generates the large or small moving ranges seen in the MR-charts. Also, the expressions that describe the lead-one forecasts, composed of realizations of the parameters and of the random values, dt, show that their variation will be smaller than the variation of the observed value. This is because the realizations of the dt values are multiplied by ever decreasing sums (based on the estimated values in Table 6) and do not
2 contain the term that directly adds the entire value of the shift; i.e. δσu .
In summary we see that for this particular data set, the 3σu shift in the variance of the random shocks is not detected until 4 time periods after the shift by all charts. For the
2σu shift the CCC chart detects the change 20 time periods after the shift, but the I-chart detects it by the 4th time period and the MR-chart signals at the 9th time period. This seems to indicate that a 2σu shift in the variance of the random shocks is not as easily detected. In the case of the variance reduction there are no points outside the control limits, but all charts seem to indicate a reduction in variance, especially the MR-chart because of the smaller moving ranges after observation 951. Also, SCR #7 (15 points in a row within CL ± 1σ) in the I-chart does detect the reduction in variance, but 12 observations after the shift.
The CCC and SCC charts in Section 5.3.1, 5.3.2, and 5.3.3 correspond to a single set of simulated observations. The performance of the combination of the CCC and SCC charts to detect out-of-control conditions cannot be judged by a single set of simulated data. Therefore, in Chapter 6 we will conduct a simulation study to assess the performance of these charts.
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5.4 Summary and Discussion
In this chapter we showed that the effect of shifts in the mean or variance of the base pharmaceutical process or of its random shocks is the same as for a process where the error of measurement is not observed. This happens because the measurement process in maintained in control, so that the transfer function that relates Yt and Xt remains unchanged and the CCC and the SCC charts only display what is affecting the base process. Therefore, the key insight is that the CCC and SCC chart will effectively detect out-of-control conditions as long as the measurement process is maintained in control. This raises the need to ensure that the transfer function is appropriately identified, and as it was emphasized in Chapter 4, it is necessary to take advantage of our technical knowledge to ensure its correct identification.
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CHAPTER SIX
Performance of the CCC and SCC Charts when the Mean or the Variance of the Base Process or the Random Shock Shift
In this chapter we will study the ability of the CCC/SCC charts monitoring
scheme to detect changes in the base process where a transfer function is used to relate
the product and reference material measurements. In Chapter 5 we illustrated the effect
of step shifts in the process mean, the random shocks mean and the random shocks
variance on CCC and SCC charts. We saw that shifts smaller that 2 standard deviations
may be difficult to detect. However, these illustrations corresponded to a single set of
simulated observations. The performance of the combination of the CCC and SCC charts
to detect out-of-control conditions cannot be judged by a single set of simulated data.
Therefore, we will conduct a simulation study to assess the performance of these charts to detect out-of-control conditions in the base process when the measurement process is maintained in control. Since the measurement process does not affect the performance of the CCC and SCC charts we represent the measurement process with an ARMA(1, 1) model, and the transfer function by v(B) = ω0. Also, we will use an ARMA(1, 1) model
to represent the base process and use values of φbp and θbp close to +1 and –1 to consider
the best and worst situations.
The plan of the simulation study is discussed in Section 6.1. In Section 6.2 we
show the procedure to establish the control limits for the I-chart and the MR-chart
through simulations. The performance results of the CCC and SCC charts when the
mean of the base process shifts is discussed in Section 6.3. In Section 6.4 we discuss the
199
results for shifts in the mean of the random shocks and in Section 6.5 for shifts in the variance of the random shocks. Finally, in Section 6.6 we summarize and discuss or findings.
6.1 Description of the Simulation Study
To our knowledge, Wardell, Moskowitz, and Plante (1992), hereafter WMP, are alone in considering the performance of the combination of CCC and SCC charts to detect step shifts in the mean of a process. For this reason, we follow their procedure closely in our simulation study. Nevertheless, there are several differences between our procedure and theirs:
• We include the moving range chart as part of the SCC charts.
• The type of model: We focus on a model of the form (see Section 1.3)
⎡⎤()1−θX (1−θbp ) Yau=ω⎢⎥ μ + +μ + , tX0 1−φ tbpt ⎣⎦⎢⎥()X ()1−φbp
where as WMP consider the model
(1−θX ) YwtY=μ + t. ()1−φX
• The type of disturbance: WMP only considered the step shift in the mean of the
process. In addition to this shift we also study the step shifts in the mean and variance
of the random shocks. The magnitude of these shifts is listed in Table 27.
• The calculation of the control limits: WMP calculate the control limits of the CCC
chart as
()φ−θ 2 μ ±σL 11 , Fw1−φ2
200
where μ is the process mean, the coefficient Lf is selected to match the average run
length (ARL) of different charts, φ1 and θ1 are the model parameters, and σw is the
standard deviation of the random shocks, wt. In our case we obtain the control limits
of the CCC chart as
ˆ 22 2 Ytau−10()1 ± ωσ +σ .
We set control limits for the I-chart and the MR-chart by simulation to achieve an in-
control ARL of 371. This ARL is calculated as 11( −< is the statistic plotted on the control chart and if TN~,(μ σ2 ) and the process is in control, 1−< limits of the MR-chart, calculated as UCL= D4 MR and LCL= D3 MR yield an ARL lower than 371 (Montgomery, 1996, p. 225). Table 27. Types of Step Shifts and their Magnitude Step shift Magnitude Mean of the base process kσu, k = 1.5, 2, 3 Mean of the random shocks kσu, k = 1.5, 2, 3 Variance of the random shocks k = 1.5, 2, 3 • Study levels: In our study we use fewer levels for the parameters in the simulation. For θbp we use –0.90, 0, and 0.90. For φbp, the levels are –0.95, 0, and 0.95. WMP use five levels for each of these parameters. The parameters φbp and θbp cannot be equal. As WMP note, if they are equal, the variance of the process degenerates to the 201 variance of a process with independent observations. The eight design points for our simulation study are shown in Table 28. Table 28. Simulation Design Points for the Parameters φbp, and θbp each at 3 Levels Run θbp φbp 1 –0.9 0 2 0.9 0 3 0.9 –0.95 4 0.9 –0.95 5 –0.9 0.95 6 0.9 0.95 7 0 –0.95 8 0 0.95 The performance of the CCC and SCC charts will be measured using the distribution characteristics of the run length (RL), which is the number of points plotted on the control charts before the out-of-control condition is detected. In this work we will say that an out-control-condition has been detected only when a point falls outside the control limits of the CCC chart, or of the I-chart, or of the MR-chart. Typically, the RL has a positively skewed distribution (see Section 2.2.3). Therefore, in addition to the commonly reported ARL and standard deviation of the run length (SDRL), we will also calculate the median run length (MRL), the interquartile range of the run length (IQRRL), as well as the probability that an out-of-control condition is detected before a given number of intervals after the onset of a disturbance. This last characteristic is the cumulative distribution function (CDF) suggested by Superville and Adams (1994). In our study we assume that the measurement and base processes each have an ARMA(1, 1) model. We use a stationary model because a non-stationary process can 202 always be transformed to stationarity. Furthermore, we wish to study the effect of both autoregressive and moving average components on the performance of the CCC and SCC charts. We will assume that the measurement process is in control with transfer function v(B) = ω0. In addition, we will assume that the conditional maximum likelihood estimators of the parameters of both ARMA(1, 1) models and the transfer function are consistent. Thus, in effect, because of the large sample sizes employed, we will assume that these parameters are known. Therefore, the parameters of the ARMA(1, 1) model that represents the measurement process are φX = 0.5 and θX = 0.5. Setting φX = θX results in Xt = at with probability 1. This is for convenience and does not affect the conclusions since the output of the measurement process, Xt, is assumed to be in control and does not affect the behavior of the base process. Also, the in-control transfer function parameter is ω0 = 0.3. Finally, without a loss of generality, we will assume that the mean of the measurement and base processes are equal to zero and that both white noise series at and ut are N(0, 1). Note that use of different variances would not alter the results of the simulation. For example, if were to use a variance of, say 5 instead of 1, then the simulated observations and forecasts will exhibit more variability, but the variance of the forecast errors will still be 5. Then, the control limits of the CCC and SCC charts will be wider and, if the process is in control, the expected proportion of points that fall outside the control limits is the same as if the variance had been 1. 203 6.1.1 Simulation Procedure to Obtain Run Lengths The procedure to generate new observations, introduce disturbances, generate lead- one forecasts, compute residuals, moving ranges, and determine when a chart detected the disturbance is as follows (see Section 3.2): 1. Set initial values: XaYu0000====0, 0, 0, 0 ; 2. Select a set of values of φbp, and θbp from Table 14; 3. Enter the control limits for the I-chart and the MR-chart from Table 15; 4. Generate 601 values of the following quantities: 2 2 a. aNta~0,()σ with σ=a 1, using seed 1, b. X tttt=+−0.5Xa−−11 0.5 a, ˆ c. X ttt−−−111()10.50.5=−Xa, d. If shifts are to be introduced in the random shocks, then 2 2 i. If t ≤ 101, then uNtu~0,( σ ) withσu =1, using seed 2; ii. If t > 101, then Step change in the … Mean Variance Set uktu+σ, Set udtt+ , 2 2 with uNtu~0,()σ , using seed 2; with dNktu~0,( σ ) , using seed 2; e. YXXYuuttbptbpttbpt=−φ+φ+−θ0.3()−−11 − 1; f. If a step shift in the mean of the base process is to be introduced then i. If t ≤ 101, then set Zt = Yt; ii. If t > 101, then set Zt = δ + Yt; 204 ˆˆ iii. YXXZrt−−−−−11111()10.31=−φ+φ−θ() t () bp t bp t bp t ; ˆˆ Otherwise YXXYrt−−−−−11111()10.31=−φ+φ−θ( t () bp t) bp t bp t ; ˆˆ g. YXXZrt−−−−−11111()10.31=−φ+φ−θ() t () bp t bp t bp t ; ˆ 22 21/2 h. Compute CCC chart control limits as Ytau−10()13±ω×σ+σ( ) ˆ i. Compute rZYttt=−−1 (1) ; j. Compute MRrrttt++11=−; 4. For each value of Yt outside the control limits of the CCC, for each value of rt outside the control limits of the I-chart, and for each MRt outside the control limits of the MR-chart record a 1, otherwise record a zero. Similarly, record a one if any of the charts detects a value outside the control limits, otherwise record a zero; 5. Remove the first 101 burn-in observations; 6. If no 1’s are recorded set the run length to 501, otherwise set the run length equal to the number of observations before the first recorded 1; 7. Repeat Steps 1 through 6 1000 times; 8. Save run length vectors; 9. Compute the average, median, and CDF of the run lengths for the first 5 observations after the shift. Next we will simulate the control limits for each design point in Table 28. 205 6.2 Determination of the Control Limits for the I-Chart and the MR-Chart As indicated in Section 6.1, we will calculate the control limits for the I-chart and the MR-chart to ensure that they have an in-control ARL of 371. The control limits of the SCC charts are determined for each design point in Table 28 as follows: 1. Set initial values: XaYu0000====0, 0, 0, 0 ; 2. Generate 1101 values of (see Section 3.2) 2 2 a. aNta~0,()σ with σ=a 1, using seed 1, b. X tttt=+−0.5Xa−−11 0.5 a, ˆ c. X ttt−−−111()10.50.5=−Xa, 2 2 d. uNtu~0,()σ with σ=u 1, using seed 2, e. YXXYuuttbptbpttbpt=−φ+φ+−θ0.3()−−11 − 1, ˆ f. YXXYrt−−−−1111()10.3=−φ+φ−θ() t bp t bp t bp t , ˆ g. rZYttt=−−1 ()1 , and h. MRrrttt++11=−; 5. Keep the last 1000 observations (i.e. discard the first 101 to remove the effect of the starting values); 6. Calculate the LCL and UCL of each chart using the 0.135 and the 99.865 percentiles, respectively (so that the probability that a point lies outside the control limits is 0.0027; i.e. the ARL is 371). The 0.135 percentile is located between the 2nd and 3rd ordered values ((1000+1)×0.00135 = 1.35). Hence LCL=+ T T − T 0.135 . ()232( () ()) 206 The 99.865 percentile is located between the 998th and 999th ordered observations ((1000+1)×0.99865 = 998.65). Thus UCL=+ T T − T 0.865 . ()998( () 999 () 998 ) In these expressions, T represents the residuals for the I-chart control limits, or the moving range of the residuals for the MR-chart control limits; 7. Repeat Steps 1 through 9 1000 times; 8. Calculate the average (for the I-chart), Median (for the MR-chart; the MR’s have a skewed distribution), standard error, and 95th and 5th percentiles of the UCL and LCL for the control charts. These simulations were performed using SAS IML code; see Appendix C, SAS Code 9. We use the RANNOR function in IML with Seed 1 to generate the random shocks for the measurement process and with Seed 2 to generate the random shocks for the base process. A total of 1000 samples each of 1000 observations and lead-one forecasts for the measurement and overall processes were generated. The 0.135 and 99.865 percentiles were calculated for each sample. The simulation was carried out on a portable computer running Windows XP with a 2.0GHz central processing unit (CPU). Each simulation took about 25 seconds of CPU time. Table 29 shows the results of this simulation for each pair of parameter values. This table will be used to set control limits for the I-chart and the MR-chart in the main simulation study. The repetition of the values in Table 3 is because the same seeds were used to generate the random shocks at and ut. The alternating values of the control limits for the I-chart and the MR-chart in the first six design points correspond with the alternating sign of θbp1. 207 The individual residuals are given by rYY=−ˆˆ11 =ω⎡⎤ X − X +−θ u u − r . (6.1) ttt−−−−10()⎣⎦ t t 1 () tbptt ( 11 ) Therefore, when θbp1 is negative, the discrepancy between the value of the random shock, ut, and the corresponding value of the residual, rt, is added onto the value of the residuals causing a slightly bigger dispersion and yielding wider control limits. When θbp1 is positive this component is subtracted, thus reducing the dispersion and yielding slightly narrower control limits. Table 29. Designed Experiment of the Parameters φbp, and θbp each at 3 Levels Run Parameters I-Chart UCL I-Chart LCL th th th th φbp θbp Mean Std Err 5 95 Mean Std Err 5 95 1 0 0.9 3.838 0.0126 3.257 4.526 –3.912 0.0139 –4.702 –3.282 2 0 –0.9 3.882 0.0131 3.314 4.681 –3.922 0.0132 –4.684 –3.312 3 –0.95 0.9 3.838 0.0126 3.257 4.526 –3.912 0.0139 –4.702 –3.282 4 –0.95 –0.9 3.882 0.0131 3.314 4.681 –3.922 0.0132 –4.684 –3.312 5 0.95 0.9 3.838 0.0126 3.257 4.526 –3.912 0.0139 –4.702 –3.282 6 0.95 –0.9 3.882 0.0131 3.314 4.681 –3.922 0.0132 –4.684 –3.312 7 0.95 0 3.333 0.0107 2.864 3.985 –3.357 0.0113 –3.991 –2.828 8 –0.95 0 3.333 0.0107 2.864 3.985 –3.357 0.0113 –3.991 –2.828 Run Parameters MR-Chart UCL MR-Chart LCL th th th th φbp θbp Median Std Err 5 95 Median Std Err 5 95 1 0 0.9 4.839 0.0145 4.241 5.731 0.0128 0.00005 0.00014 0.0047 2 0 –0.9 6.432 0.0206 5.610 7.685 0.0018 0.00007 0.00021 0.0066 3 –0.95 0.9 4.839 0.0145 4.241 5.731 0.0128 0.00005 0.00014 0.0047 4 –0.95 –0.9 6.432 0.0206 5.610 7.685 0.0018 0.00007 0.00021 0.0066 5 0.95 0.9 4.839 0.0145 4.241 5.731 0.0128 0.00005 0.00014 0.0047 6 0.95 –0.9 6.432 0.0206 5.610 7.685 0.0018 0.00007 0.00021 0.0066 7 0.95 0 4.954 0.0149 4.312 5.840 0.0014 0.00006 0.00017 0.0057 8 –0.95 0 4.954 0.0149 4.312 5.840 0.0014 0.00006 0.00017 0.0057 The control limits of design points 7 and 8 are narrower than the previous limits since of θbp1 = 0. In that case the residual values are given by the expression 208 rXXu= ω−⎡⎤ˆ 1 +. (6.2) tttt01⎣⎦− () Here the MA component plays no role in increasing or decreasing the dispersion of the residuals. The mean and median values of the control limits for each combination of values of φbp1 and θbp1 in Table 29 will be used in the simulations to assess the performance of the CCC and SCC charts for the corresponding design points in Table 28. In this way we will ensure an in-control run length of 371 in every case. 6.3 Performance of the CCC and SCC Charts for a Step Shift in the Mean of the Base Process SAS IML code (see Appendix C, SAS Code 10 and 11) was used to conduct these simulations. We use the RANNOR function in IML to generate white noises series values (normal variates with specified mean and variance). We use Seed 1 to generate the white series values for the measurement process and Seed 2 to generate the white noise series for the measurement process. These seeds do not change between simulations of the design points in Table 28. What changes is either the value we add to generate the step shift in the mean of the random shocks or the multiplier to generate the step shift in the variance of the random shocks. The simulations were also carried out on a portable computer running Windows XP with a 2.0GHz central processing unit (CPU). Calculation of the RL statistics for each shift type (e.g. shift in the process mean), each shift magnitude (e.g. 2σu), and for each design point took about 22 seconds of CPU time. Table 30 shows the RL summary statistics for each combination of the values of φbp and θbp and for a step shifts in the mean of the base process. In this table we provide 209 the traditional ARL and SDRL, and because the distribution of the RL is not symmetric as can be seeing in Figure 50, we also provide MRL and the IQRRL. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Cum Prob 0.3 0.2 0.1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 TRL Figure 50. Probability and cumulative probability distribution functions of the run length for all charts combined for φbp = – 0.95 and θbp = – 0.90 a step shift in the mean of the base process. The shift in the mean of the base process occurs between t = τ – 1 and t = τ. Then, RL = 0 indicates that the shift was detected immediately after it occurred, RL = 1 indicates that the shift was detected 1 time period after it occurred, and so on. In Table 30 we also provide the cumulative probability of detection for the time periods τ + k, with k = 0, 1, 2, 3, 4, and 5, immediately after the shift in the mean of the base process. For example, for the case where for φbp = – 0.95 and θbp = – 0.90 and the magnitude of the shift is 3σu, the probability that the CCC chart immediately (k = 0) detects the shift is 0.447 or 44.7% and the probability that it is detected by the 5th observation is 98.5%. Also, the cases where there is a 100% chance of detecting the shift have been highlighted. 210 Table 30. Run Length Summary Statistics for step shifts in the mean of the base process % Detections by point k after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 0.6 0.5 1 1 46.1 98.4 100 100 100 100 I 0.8 0.5 1 1 24.1 93.6 100 100 100 100 3σ u MR 82.4 101.4 41 125 10.2 17.1 21.7 24.2 27.3 29.6 Overall 0.54 0.6 1 1 47.3 98.6 100 100 100 100 CCC 1.1 0.7 1 0 16.8 75.2 98.8 100 100 100 I 1.5 0.7 1 1 7.2 53.4 83.2 93.2 99.9 100 0 0.90 2σ u MR 109.8 103.4 81 134 2.9 5.2 6.5 7.5 9.0 10.3 Overall 1.1 0.7 1 0 18.0 75.8 99.0 100 100 100 CCC 1.6 0.9 2 1 8.6 46.8 83.7 97.4 99.9 100 I 2.2 1.0 2 1 2.5 25.3 64.0 91.1 99.2 100 1.5σ u MR 113.6 101.4 88 134 2.0 3.4 4.3 5.3 6.9 7.6 Overall 1.6 0.9 2 1 9.8 48.1 84.2 97.4 99.9 100 CCC 4.9 7.8 2 7 44.7 45.6 59.9 61.3 69.5 70.6 I 25.0 34.1 10 34 22.7 22.9 32.4 32.6 38.9 39.3 3σ u MR 177.6 152.8 148.5 267.5 7.7 10.7 13.2 14.6 16.0 17.4 Overall 4.9 7.8 2 7 44.7 45.6 59.9 61.3 69.5 70.6 CCC 19.5 24.0 10 26.8 17.0 17.9 26.4 27.4 34.4 36.1 I 88.7 90.4 59 112 7.0 7.1 11.1 11.3 14.9 15.3 0 –0.90 2σ u MR 203.6 151.5 183.5 265 2.8 4.1 5.2 6.1 7.6 8.2 Overall 19.1 23.6 10 26 17.0 17.9 26.4 27.5 34.7 36.5 CCC 34.8 37.8 23 43 9.8 10.6 15.7 16.6 21.9 23.4 I 141.8 123.6 109 179.8 3.0 3.1 5.2 5.4 7.6 8.0 1.5σ u MR 213.1 147.8 196 261 1.1 1.9 2.5 3.5 4.5 4.9 Overall 34.0 37.3 23 42 9.8 10.6 15.8 16.9 22.1 23.8 CCC 0.5 0.5 1 1 46.1 100 100 100 100 100 I 0.8 0.4 1 0 24.1 100 100 100 100 100 3σ u MR 1.2 1.2 1 0 10.1 77.1 95.2 97.9 99.4 99.5 Overall 0.5 0.5 1 1 47.3 100 100 100 100 100 CCC 0.9 0.4 1 0 16.8 98.3 100 100 100 100 I 1.0 0.4 1 0 7.2 93.1 100 100 100 100 –0.95 0.90 2σ u MR 51.3 88.7 4 64 2.9 26.9 39.7 45.8 51.4 54.0 Overall 0.8 0.4 1 0 18.0 98.3 100 100 100 100 CCC 1.1 0.5 1 0 8.6 84.1 99.9 100 100 100 I 1.3 0.5 1 1 2.5 67.1 99.5 100 100 100 1.5σ u MR 91.0 102.3 52 138.5 2.0 9.7 15.3 17.9 21.4 23.6 Overall 1.1 0.5 1 0 9.8 84.4 99.9 100 100 100 CCC 1.0 1.4 1 1 44.7 77.8 87.3 94.0 96.5 98.5 I 2.8 3.2 2 3 22.7 48.2 60.2 71.4 78.5 84.2 3σ u MR 203.2 149.9 184.5 259 8.3 8.5 8.9 8.9 9.3 9.5 Overall 1.0 1.4 1 1 44.7 77.8 87.3 94.0 96.5 98.5 CCC 4.2 4.6 3 5 17.0 36.2 47.0 56.7 64.1 71.7 I 15.2 16.0 10 20 6.9 15.6 20.6 25.6 30.8 35.5 –0.95 –0.90 2σ u MR 214.8 146.2 203 254 2.9 3.5 3.5 3.9 4.9 4.9 Overall 4.2 4.5 3 5 17.0 36.4 47.2 57.0 64.6 72.2 CCC 10.1 11.1 6 12 9.8 21.0 27.0 32.5 39.6 45.2 I 38.4 38.0 28 42 2.8 6.3 8.9 11.6 14.6 17.0 1.5σ u MR 217.6 144.2 204.5 249.3 1.1 1.3 1.3 1.5 1.9 1.9 Overall 10.1 11.1 6 12 9.8 21.0 27.0 32.5 39.6 45.2 211 Table 30. Continued % Detections by point t after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 4.1 10.3 1 3 46.1 64.6 72.5 77.1 80.2 83.0 I 18.8 37.3 3 19 24.1 39.5 47.6 53.0 56.8 59.8 3σ u MR 104.3 102.7 76 133 10.3 10.8 11.3 12.0 13.0 13.7 Overall 3.5 9.0 1 3 47.3 65.9 73.9 78.5 81.5 84.4 CCC 18.4 30.6 5 22 16.8 29.5 38.0 43.1 47.9 51.4 I 86.1 101.6 49 122 7.3 14.1 16.8 19.2 21.9 24.1 0.95 0.90 2σ u MR 114.8 102.6 86 133 3.0 3.6 4.1 4.8 5.9 6.2 Overall 15.1 24.1 5 19 18.0 31.0 39.5 44.9 49.8 53.2 CCC 37.2 49.8 18 49 8.6 17.0 21.9 26.5 30.0 32.7 I 141.2 131.3 102 199 2.8 5.4 6.4 7.4 9.2 10.9 1.5σ u MR 116.6 104.6 87 138 2.0 2.5 3.7 4.7 5.7 6.0 Overall 27.6 38.6 13.5 34 9.8 18.5 24.1 29.3 33.0 35.8 CCC 26.1 66.3 1 8 44.7 56.9 63.1 66.6 69.8 71.8 I 83.3 137.3 3 129 31.9 41.9 48.6 51.3 55.5 57.0 3σ u MR 81.4 136.5 2 114 5.9 43.1 50.5 55.4 58.9 59.9 Overall 23.2 59.2 1 8 44.7 56.9 63.1 66.6 69.8 71.8 CCC 62.4 91.0 17 92 17.1 25.6 30.2 33.6 38.0 39.6 I 154.4 155.3 108 271 11.7 16.6 21.2 22.6 25.7 26.6 0.95 –0.90 2σ u MR 152.1 153.8 109 271 2.5 15.3 20.3 24.3 26.5 28.3 Overall 59.3 85.8 17 90 17.1 25.6 30.2 33.6 38.0 39.6 CCC 79.7 94.4 46 113 9.9 15.4 19.1 21.5 24.7 26.1 I 180.9 151.9 155 269 5.2 8.8 11.0 11.9 14.2 14.6 1.5σ u MR 180.9 151.6 152 265 1.1 7.1 9.6 11.5 12.9 14.3 Overall 73.3 88.0 42 106 9.9 15.4 19.1 21.5 24.7 26.1 CCC 90.8 133.5 0 160 52.7 52.8 52.9 52.9 53.3 53.7 I 100.4 147.0 0 184 53.7 53.7 53.9 53.9 54.3 54.6 3σ u MR 170.2 156.8 135 300 16.5 24.8 25.0 25.2 25.4 25.4 Overall 86.9 127.0 1 153 50.0 50.5 50.6 50.7 51.0 51.3 CCC 161.6 142.9 134 244 16.7 16.8 16.9 17.0 17.8 18.6 I 175.3 153.4 151 276 17.5 17.5 17.7 17.7 18.8 19.5 0.95 0 2σ u MR 210.5 146.9 195 253 3.8 6.6 7.0 7.2 7.2 7.2 Overall 148.9 135.6 116 218 14.8 15.1 15.3 15.5 16.2 16.9 CCC 175.3 140.8 157 235 8.9 8.9 9.4 9.5 10.6 11.4 I 195.3 151.4 178 269 9.4 9.4 9.6 9.6 10.8 11.6 1.5σ u MR 216.9 144.7 201 249 1.9 4.0 4.4 4.6 4.6 4.6 Overall 161.2 134.7 130 217 8.1 8.1 8.6 8.8 9.7 10.4 CCC 0.5 0.5 1 1 45.4 99.8 100 100 100 100 I 0.6 0.5 1 1 36.7 99.5 100 100 100 100 3σ u MR 161.6 156.4 123 290 16.3 28.3 28.5 28.7 28.7 28.7 Overall 0.5 0.5 1 1 46.0 99.8 100 100 100 100 CCC 1.1 0.7 1 0 12.9 81.0 95.9 99.4 99.8 100 I 1.3 0.8 1 0 9.6 75.5 92.8 97.9 99.1 99.8 –0.95 0 2σ u MR 208.3 147.9 194 254 3.6 7.6 8.0 8.2 8.2 8.2 Overall 1.1 0.7 1 0 12.9 81.0 95.9 99.4 99.8 100 CCC 2.3 2.0 2 2 6.6 47.3 67.7 80.7 89.5 93.5 I 2.7 2.5 2 3 4.8 39.8 59.9 63.4 73.6 79.2 1.5σ u MR 217.3 144.8 206 248 1.9 3.8 4.2 4.4 4.4 4.4 Overall 2.3 2.0 2 2 6.9 47.4 67.7 80.8 86.6 93.6 212 The first 2 cases in Table 30 correspond to a MA(1) representation of the base process. The expected value of the residuals immediately after the shift is equal to the magnitude of the shift in the process mean, but starting at k = 1 their expected value is given by (see Table 1 in Chapter 3) k 1− θbp Er[]τ+k = , k =1, 2, 3, … . 1− θbp Then, if θbp = 0.9, k Er[ τ+k ] =10×−( 1 0.9 ) δ. However, if θbp = –0.9, 10.9−−()k Er[]τ+k = δ . 1.9 Clearly, if θbp is positive the expected values of the residuals are equal to the magnitude of the shift and continue to increase as time progresses, making the shift easier to detect. However, if θbp is negative, the expected value of the residuals at the beginning is equal to the magnitude of the shift, but they decrease as time progresses and are more difficult to detect. The results in Table 30 show, that when θbp = 0.9, shifts in the mean are easily detected, even when the shift is only 1.5σu. When θbp = – 0.9, however, the probability of detecting even a shift as large as 3.0σu is more difficult, and detection is more likely immediately after the shift occurs. The detection capability of the CCC chart is similar to that of the I-chart, because the residuals are equal to the discrepancy between the lead-one forecasts and the observed values of the overall process. The difference is that the control limits of the 22 2 CCC chart are narrower (computed as ± ωσ0 au +σ ) than those of the I-chart 213 (established to obtain an ARL =371). Wardell, Moskowitz, and Plante (1992) found that the CCC and the I-charts have the same ARL when the process is a MA(1). In our case the ARL’s are not exactly the same because we did not design the two charts to have the same in-control ARL. Also, the MR-chart for a MA(1) process has a very poor detection capability because of the smooth change of the residuals from one time period to the next. The simultaneous detection capability of the CCC and SCC charts is practically equal to the detection capability of the CCC chart because the in-control ARL of the CCC chart is about 300 compared to the 371 of the I-chart. It is important to note that we chose to construct the limits of the CCC chart based on the “known” variances of the random 2 2 shocks of the measurement process, σa , and the base process, σu . We say that these variances are “known” because their maximum likelihood estimators are consistent and because they are based on a very large sample of historical data. Our approach, however, does not account for the uncertainty associated with the estimation of these variances causing the control limits of the CCC chart to be narrower than control limits based on estimates of these variances. The next four cases in Table 30 correspond to an ARMA(1, 1) representation. In this case the expected value of the residuals immediately after the shift is also equal to the magnitude of the shift, but for subsequent time periods they are k (φbp−θ bp) θ bp +1 −φ bp Er[]τ+k = , k =1, 2, 3, … . 1−θbp When φbp = – 0.95 and θbp = 0.90 the expected values of the residuals are larger than the magnitude of the shift and increase with time, facilitating the shift's detection However, when they are φbp = 0.95 and θbp = – 0.90, the expected values of the residuals decrease 214 with time, making it more difficult to detect the shift. When φbp = 0.95 and θbp = 0.90, the magnitude of the expected values of the residuals tends slowly towards the steady- state value of 0.5. When φbp = – 0.95 and θbp =– 0.90, their magnitude slowly approaches 1.03. The performance of the control charts in Table 30 reflects this behavior of the residuals. When φbp = – 0.95 and θbp = 0.90 the ARL and MRL are very small, and even the 1.5σu shift is detected by k = 3. On the other hand, when φbp = 0.95 and θbp = – 0.90, the ARL and MRL are much larger and the control charts have difficult detecting shifts below 3σu. Similarly, the probability of detecting shifts smaller than 3σu by k =5 is very low when φbp = –0.95 and θbp = – 0.90 and when φbp = 0.95 and θbp = 0.90. The MR- chart is of little value in detecting the shifts in the mean, however when φbp = 0.95 and θbp = – 0.90 the performance of the MR-chart is practically the same as the I-chart. This is because the expected value of the residuals after k = 0 have almost the same value but their sign changes from one time period to the next. The last two cases in Table 30 correspond to the AR(1) representation of the process. Here, the expected value of the residuals at k = 0 is equal to the magnitude of the shift in the mean and, for k =1, 2, 3, …, is Er[ τ+kbp] = (1−φ) δ. In this situation, when φbp = – 0.95, the expected values of the residuals at k =1 increase to a new level that is 1.95 times the magnitude of the shift from the in-control level of the residuals. However, when φbp = 0.95 the expected values decrease, one time interval after the shift, to a new level that is 0.05 times the magnitude of the shift from the in-control level of the residuals. The last case in Table 30 shows that when φbp = – 0.95, the 3σu 215 shift is quickly detected by the CCC chart and I-chart. The 2σu shift is detected less quickly, but there is a very high probability that it will be detected by k = 5. The 1.5σu shift is also detected with moderate high probability by k = 5. However, when the φbp = 0.95 there is a 50% chance that 3σu will be detected immediately after the shift; after that there is very little probability that it will be detected. Superville and Adams (1994) report similar behavior for an AR(1) with φbp = 0.9 and an I-chart with an in-control ARL = 250. These results show that in general the MR-chart does not effectively detect shifts in the mean of the process. Furthermore, the combination of the CCC-chart, the I-chart, and the MR-chart does not prove superior to the scheme where only the CCC chart and the I-chart are employed to detect changes in the mean of the base process. 6.4 Performance of the CCC and SCC Charts for a Step Shift in the Mean of the Random Shocks of the Base Process The RL summary statistics for each combination of the values of φbp and θbp and step shifts in the mean of the random shocks of the base process are shown in Table 31. The most striking feature of the results in Table 31 is that, when the process is represented by an MA(1) or an ARMA(1, 1), the RL summary statistics are determined by the sign of the AR parameter θbp, and when the process is represented by an AR(1) the RL summary statistics are the same. This remarkable feature is a consequence of two facts. First, we use fixed seeds to generate the random variates at and ut in order to replicate the simulation results and distinguish among the design points. Second, the sign of the AR parameter determines if a small quantity is added or subtracted from the expected value of the shift. In the case of the AR(1) model this effect is not present. 216 Table 31. Run Length Summary Statistics for Step Shifts in the Mean of the Random Shocks % Detections by point t after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 1.8 3.0 1 2 46.1 67.0 76.9 83.1 87.0 89.5 I 5.1 7.42 2 6 24.1 42.1 52.2 59.3 65.0 70.3 3σ u MR 102.4 102.0 74 132 10.2 10.7 11.3 11.8 12.6 13.6 Overall 1.7 2.9 1 2 47.3 68.3 78.2 84.2 87.9 90.2 CCC 7.2 9.4 4 9 16.8 31.6 41.5 49.0 55.1 60.4 I 20.4 22.5 14 25 7.2 14.6 18.9 23.8 28.3 31.7 0 0.90 2σ u MR 110.3 100.9 83 130 2.9 3.6 4.3 4.8 5.7 6.7 Overall 6.5 8.5 3 8 18.0 33.4 43.2 51.0 57.0 62.5 CCC 15.2 17.6 9 19 8.6 17.5 23.7 29.7 34.8 38.5 I 49.6 50.9 32 56 2.5 5.6 7.0 8.5 10.8 13.1 1.5σ u MR 112.3 101.5 85 131 2.0 2.7 3.4 3.9 4.8 5.8 Overall 13.0 15.3 8 17 9.8 19.3 25.6 32.0 37.5 41.6 CCC 1.1 1.6 1 1 44.7 75.7 85.6 92.4 95.7 97.8 I 3.2 3.6 2 4 22.7 43.7 55.8 65.7 73.7 79.8 3σ u MR 204.9 150.0 186 259 8.1 8.7 8.9 9.3 9.9 10.1 Overall 1.1 1.6 1 1 44.7 75.9 85.8 92.6 95.9 97.9 CCC 4.6 4.9 3 6 17.0 34.8 45.4 53.8 61.5 69.0 I 16.7 17.7 11 21 6.9 14.0 19.0 23.5 28.7 33.1 0 –0.90 2σ u MR 216.7 145.8 205 253 2.8 3.4 3.6 4.0 4.6 4.8 Overall 4.6 4.9 3 6 17.0 35.0 45.7 54.1 61.8 69.2 CCC 10.9 12.1 7 14 9.8 19.5 25.6 30.9 37.9 43.1 I 41.4 41.4 30 46 2.8 5.8 8.4 11.1 14.0 15.9 1.5σ u MR 219.8 144.4 207 248 1.1 1.7 1.9 2.3 2.9 3.1 Overall 10.8 12.0 7 14 9.8 19.7 25.9 31.2 38.2 43.4 CCC 1.8 3.0 1 2 46.1 67.0 76.9 83.1 87.0 89.5 I 5.1 7.4 2 6 24.1 42.1 52.2 59.3 65.0 70.3 3σ u MR 102.4 102.0 74 132 10.2 10.7 11.3 11.8 12.6 13.6 Overall 1.7 2.9 1 2 47.3 68.3 78.2 84.2 87.9 90.2 CCC 7.2 9.4 4 9 16.8 31.6 41.5 49.0 55.1 60.4 I 20.4 22.5 14 25 7.2 14.6 18.9 23.8 28.3 31.7 –0.95 0.90 2σ u MR 110.3 100.9 83 130 2.9 3.6 4.3 4.8 5.7 6.7 Overall 6.5 8.5 3 8 18.0 33.4 43.2 51.0 57.0 62.5 CCC 15.2 17.6 9 19 8.6 17.5 23.7 29.7 34.8 35.5 I 49.6 50.9 32 56 2.5 5.6 7.0 8.5 10.8 13.1 1.5σ u MR 112.3 101.5 85 131 2.0 2.7 3.4 3.9 4.8 5.8 Overall 13.0 15.3 8 17 9.8 19.3 25.6 32.0 37.5 41.6 CCC 1.1 1.6 1 1 44.7 75.7 85.6 92.4 95.7 97.8 I 3.2 3.6 2 4 22.7 43.7 55.8 65.7 73.7 79.8 3σ u MR 204.9 150.0 186 259 8.1 8.7 8.9 9.3 9.9 10.1 Overall 1.1 1.6 1 1 44.7 75.9 85.8 92.6 95.9 97.9 CCC 4.6 4.9 3 6 17.0 34.8 45.4 53.8 61.5 69.0 I 16.7 17.7 11 21 6.9 14.0 19.0 23.5 28.7 33.1 –0.95 –0.90 2σ u MR 216.7 145.8 205 253 2.8 3.4 3.6 4.0 4.6 4.8 Overall 4.6 4.9 3 6 17.0 35.0 45.7 54.1 61.8 69.2 CCC 10.9 12.1 7 14 9.8 19.5 25.6 30.9 37.9 43.1 I 41.4 41.4 30 46 2.8 5.8 8.4 11.1 14.0 15.9 1.5σ u MR 219.8 144.4 207 248 1.1 1.7 1.9 2.3 2.9 3.1 Overall 10.8 12.0 7 14 9.8 19.7 25.9 31.2 38.2 43.4 217 Table 31 Continued % Detections by point t after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 1.8 3.0 1 2 46.1 67.0 76.9 83.1 87.0 89.5 I 5.1 7.4 2 6 24.1 42.1 52.2 59.3 65.0 70.3 3σ u MR 102.4 102.0 74 132 10.2 10.7 11.3 11.8 12.6 13.6 Overall 1.7 2.9 1 2 47.3 68.3 78.2 84.2 87.9 90.2 CCC 7.2 9.4 4 9 16.8 31.6 41.5 49.0 55.1 60.4 I 20.4 22.5 14 25 7.2 14.6 18.9 23.8 28.3 31.7 0.95 0.90 2σ u MR 110.3 100.9 83 130 2.9 3.6 4.3 4.8 5.7 6.7 Overall 6.5 8.5 3 8 18.0 33.4 43.2 51.0 57.0 62.5 CCC 15.2 17.6 9 19 8.6 17.5 23.7 29.7 34.8 38.5 I 49.6 50.9 32 56 2.5 5.6 7.0 8.5 10.8 13.1 1.5σ u MR 112.3 101.5 85 131 2.0 2.7 3.4 3.9 4.8 5.8 Overall 13.0 15.3 8 17 9.8 19.3 25.6 32.0 37.5 41.6 CCC 1.1 1.6 1 1 44.7 75.7 85.6 92.4 95.7 97.8 I 3.2 3.6 2 4 22.7 43.7 55.8 65.7 73.7 79.8 3σ u MR 204.9 150.0 186 259 8.1 8.7 8.9 9.3 9.9 10.1 Overall 1.1 1.6 1 1 44.7 75.9 85.8 92.6 95.9 97.9 CCC 4.6 4.9 3 6 17.0 34.8 45.4 53.8 61.5 69.0 I 16.7 17.7 11 21 6.9 14.0 19.0 23.5 28.7 33.1 0.95 –0.90 2σ u MR 216.7 145.8 205 253 2.8 3.4 3.6 4.0 4.6 4.8 Overall 4.6 4.9 3 6 17.0 35.0 45.7 54.1 61.8 69.2 CCC 10.9 12.1 7 14 9.8 19.5 25.6 30.9 37.9 43.1 I 41.4 41.4 30 46 2.8 5.8 8.4 11.1 14.0 15.9 1.5σ u MR 219.8 144.4 207 248 1.1 1.7 1.9 2.3 2.9 3.1 Overall 10.8 12.0 7 14 9.8 19.7 25.9 31.2 38.2 43.4 CCC 1.2 1.6 1 2 45.4 71.5 83.6 90.8 95.0 97.1 I 1.7 2.1 1 2 36.7 62.0 75.6 84.9 90.8 94.2 3σ u MR 187.1 154.7 167 289 17.3 17.3 17.5 17.7 17.7 17.7 Overall 1.2 1.6 1 2 46.0 71.6 83.7 90.9 95.1 97.2 CCC 6.2 6.7 4 8 12.9 26.5 35.6 44.6 52.7 59.5 I 8.6 9.0 6 10 9.6 21.5 29.1 35.5 43.1 50.0 0.95 0 2σ u MR 217.5 144.6 206 249 3.8 3.8 4.2 4.4 4.4 4.4 Overall 6.2 6.7 4 8 12.9 26.5 35.6 44.7 52.8 59.6 CCC 15.5 16.2 10 19 6.6 13.1 17.8 22.2 27.5 32.1 I 24.4 24.9 17 29 4.8 8.9 12.4 14.7 18.3 21.3 1.5σ u MR 222.1 142.8 210 242 1.9 1.9 2.3 2.5 2.5 2.5 Overall 15.3 16.2 10 18 6.9 13.4 18.1 22.6 27.9 32.5 CCC 1.3 1.6 1 2 42.9 66.9 81.2 90.2 95.1 97.9 I 1.7 2.1 1 3 36.5 59.0 74.8 84.2 90.5 94.4 3σ u MR 183.1 152.8 161 276 17.2 17.4 17.7 20.0 20.0 20.2 Overall 1.3 1.6 1 2 43.2 67.2 81.5 90.4 95.2 98.0 CCC 6.2 6.7 4 8 12.9 26.5 35.6 44.6 52.7 59.5 I 8.6 9.0 6 10 9.6 20.5 28.1 34.5 42.1 49.0 –0.95 0 2σ u MR 217.5 144.6 206 249 3.8 3.8 4.2 4.4 4.4 4.4 Overall 6.2 6.7 4 8 12.9 26.5 35.6 44.7 52.8 59.6 CCC 15.5 16.2 10 19 6.6 13.1 17.8 23.2 28.5 33.1 I 24.4 24.9 17 29 4.8 8.9 13.4 15.7 19.3 22.3 1.5σ u MR 222.1 142.8 210 242 1.9 1.9 2.3 2.5 2.5 2.5 Overall 15.3 16.2 10 18 6.9 13.4 18.1 22.6 27.9 32.5 218 This is explained by noting that, immediately after the shift in the mean of the random shocks (at t = τ), the overall process observations are given by YYτ=φbp τ−10 +ω( X τ −φ bp X τ− 1) +( u t +δ) −θ bp u τ− 1, the lead-one forecasts by ˆˆ YYXXuuτ−11011()11=φbp τ− +ω( τ− () −φ bp τ−) +() t +δ −θ bp τ− 1, and the residuals by reuτ=ω011 τ +tbp +δ−θ( ur τ− − τ− ) , ˆ where eXττ−τ=−1 ()1 X. The expected value of the residuals is equal to δ, as was shown in Section 3.1.3, but the actual value of the residuals is determined by the realization of et, ut, and byθ−bp ()urτ−11 τ− . If θbp is positive then the small random discrepancy between uτ–1 and rτ–1 is subtracted from δ, and if θbp is negative it is added to δ. Therefore, we would expect slightly better performance when θbp is negative because the residuals will be slightly larger. The residual at t = τ + 1 is 2 reeuurτ+101=ω( τ+ +θbp τ) + t + 1 +δ−θ bp ( τ− 1 − τ− 1) , and at t = τ + 2 it is 23 reeeuurτ+202=ω() τ+ +θbp τ+ 1 +θ bp τ + t + 2 +δ−θ bp ( τ− 11 − τ− ) . The general expression that defines the residual for t = τ + k, k =1, 2, 3 … is k reuur=ω θjk + +δ−θ − . τ+kkjbptkbp011∑ j=0 τ+ − +() τ− τ− From this expression we see that the residuals, et, play an important role in increasing or decreasing the realizations of the individual residuals. This is important to note because, if the model selected to represent the measurement process is not appropriate, the 219 residuals will be relatively large and will cause either too many false alarms or many out- of-control conditions will go undetected. We showed in Section 3.2.1 that the expected values of the residuals is equal to δ, therefore the performance of the CCC chart, the I- Chart, and the MR-chart should be the same for a given shift in the mean or the variance of the random shocks regardless of the values of φbp and θbp. Figure 51 shows the observed, lead-one forecasts and the residual values with et = 0 and ut = 1. These graphs show that after some transition period, the residuals converge to δ, which in this case is equal to 3. In accordance with expressions (3.16) and (3.17), if φbp is positive the values of the partial sums k φk (6.3) ∑ j=0 bp increase monotonically until they converge to some limiting value, but if φbp is negative the values of (6.3) change in sign and become smaller until they reach a limiting value. These patterns can be seen in Figure 51. In the case when θbp = 0, the RL summary statistics are the same for φbp = 0.95 and for φbp = – 0.95. This is because the realizations of the residuals do not depend on φbp. Furthermore, since the expected value of the residuals is δ, the control charts perform equally well for a given shift in the mean or the variance of the random shocks. In general we see that the CCC chart and the I-chart are effective at detecting 3σu shifts, since by k = 5 the probability of detection is around 90%. These charts have moderate detection capability for 2σu shifts and poor capability for 1.5σu shifts, since by k = 5 the probability of detection is around 60% and 30%, respectively. The MR-chart has a very poor detection capability for any of the three shifts in the mean of the random shocks. 220 120 7 110 100 6 ˆ 90 Y Yt−1 (1) t 5 Yt 80 ˆ Yt−1 (1) 70 4 60 50 3 40 Simulated Data Simulated φbp = 0.95 Data Simulated 2 30 r t φbp = 0.95 20 θbp = – 0.90 rt 1 10 θbp = 0.90 0 0 50 100 150 200 250 300 350 400 450 500 550 600 50 100 150 200 250 300 350 400 450 500 550 600 Observation No. Observation No. 3.5 3 2 3.0 r Yt t 1 2.5 Yt rt 0 2.0 -1 φ = – 0.95 ˆ bp 1.5 Yt−1 (1) φbp = – 0.95 -2 θbp = 0.90 Simulated Data Simulated Data Simulated 1.0 ˆ θbp = – 0.90 -3 Yt−1 (1) 0.5 -4 0.0 -5 50 100 150 200 250 300 350 400 450 500 550 600 50 100 150 200 250 300 350 400 450 500 550 600 Observation No. Observation No. Figure 51. Behavior of the observed, lead-one forecast, and residuals values after a shift in the mean of the random shocks at observation 100 and for combinations of φbp and θbp. Figure 52 displays a sample of the CDF plots of the RL of all charts combined to illustrate how quickly a shift in the mean of the random shocks will be detected. 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 Cum Prob Cum Prob 0.3 Cum Prob 0.3 0.3 0.2 0.2 0.2 3σ shift 2σu shift 1.5σu shift 0.1 u 0.1 0.1 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 TRL TRL TRL Figure 52. CDF plots of the run length for all charts combined for different shifts in the mean of the random shocks of the base process. 221 6.5 Performance of the CCC and SCC Charts for a Step Shift in the Variance of the Random Shocks of the Base Process Table 32 shows the RL summary statistics for each combination of the values of φbp and θbp and for the step shifts of 3, 2, and 1.5 times the in-control variance of the base process. We see that the RL summary statistics for the MA(1) and the ARMA(1, 1) representations are determined by the sign of the MA parameter and, in the case of the AR(1) representation, the RL statistics are the same for both values of φbp. This behavior is expected given the discussion in the previous section. However, the variance of the residuals, as shown in Section 3.2.4, is δσu, where δ = 1.5, 2, or 3, therefore the performance of the CCC chart, the I-chart, and the MR-chart is the same in spite of the values of φbp and θbp. Figure 53 shows a sample of the RL CDF plots for all charts combined for each shift in the variance. We see that the CCC and SCC charts combined are fairly effective at detecting variance shifts of 3σu, moderately effective at detecting variance shifts of 2σu, but not effective at all for detecting variance shifts of 1.5σu. It is interesting that the MR-chart is almost as effective as the I-chart at detecting the shifts in variance. However, if we compare the performance of the combined CCC chart, I-chart, and MR-chart to the performance of the CCC chart and the I-chart together, there is little difference. In conclusion we see that the MR-chart can be omitted. We can simplify our monitoring scheme to contain only the CCC chart and the I-chart. 222 Table 32. Run Length Summary Statistics for Step Shifts in the Variance of the Random Shocks % Detections by point t after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 2.1 2.4 1 3 29.8 53.3 68.7 79.3 86.4 90.4 I 3.6 4.1 2 4 20.1 38.0 51.4 63.2 72.1 77.9 3σ u MR 3.9 3.9 3 5 12.7 30.5 46.0 57.5 68.2 74.8 Overall 1.9 2.2 1 3 30.7 55.1 71.1 81.1 88.3 92.0 CCC 6.1 6.8 4 7 13.4 26.4 36.8 45.6 55.0 61.6 I 14.3 15.6 9 16 6.9 14.3 19.6 24.6 30.9 35.6 0 0.90 2σ u MR 11.8 12.3 7 12 3.2 10.8 18.4 23.8 31.0 38.8 Overall 5.3 5.9 4 6 14.5 28.2 40.0 49.6 59.6 66.5 CCC 17.5 18.5 11 21 5.6 12.0 16.4 20.3 25.5 29.7 I 55.1 56.8 38 61 1.3 3.0 4.5 6.1 9.0 10.7 1.5σ u MR 35.0 33.9 25 40 1.0 3.8 6.5 8.8 11.1 14.4 Overall 14.2 14.8 9 16 6.3 13.6 18.7 23.5 29.5 34.4 CCC 2.2 2.6 1 3 30.2 52.2 67.4 78.9 85.0 89.2 I 3.8 4.3 2 4 20.6 38.5 50.1 59.5 69.1 74.3 3σ u MR 7.6 8.0 5 8 6.0 18.2 28.8 37.6 46.1 52.9 Overall 2.2 2.6 1 3 30.4 52.5 67.5 79.1 85.2 89.4 CCC 6.1 6.8 4 7 13.0 26.8 37.3 45.1 53.9 59.8 I 14.7 15.6 9 17 6.1 13.3 18.9 23.6 30.8 36.4 0 –0.90 2σ u MR 29.1 29.1 21 33 1.2 4.8 8.7 11.9 15.3 18.9 Overall 6.1 6.8 4 7 13.0 26.8 37.4 45.1 53.9 59.8 CCC 17.2 17.9 11 21 5.3 11.8 17.4 21.9 28.2 32.4 I 56.4 60.1 37 62 1.7 3.8 5.7 7.6 10.3 11.7 1.5σ u MR 102.1 97.0 73 114 0.2 1.3 2.6 3.6 5.1 6.2 Overall 17.0 17.7 11 20 5.4 11.9 17.7 22.2 28.4 32.6 CCC 2.1 2.4 1 3 29.8 53.3 68.7 79.3 86.4 90.4 I 3.6 4.1 2 4 20.1 38.0 51.4 63.2 72.1 77.9 3σ u MR 3.9 3.9 3 5 12.7 30.5 46.0 57.5 68.2 74.8 Overall 1.9 2.2 1 3 30.7 55.1 71.1 81.1 88.3 92.0 CCC 6.1 6.8 4 7 13.4 26.4 36.8 45.6 55.0 61.6 I 14.3 15.6 9 16 6.9 14.3 19.6 24.6 30.9 35.6 –0.95 0.90 2σ u MR 11.8 12.3 7 12 3.2 10.8 18.4 23.8 31.0 38.8 Overall 5.3 5.9 4 6 14.5 28.2 40.0 49.6 59.6 66.5 CCC 17.5 18.5 11 21 5.6 12.0 16.4 20.3 25.5 29.7 I 55.1 56.8 38 61 1.3 3.0 4.5 6.1 9.0 10.7 1.5σ u MR 35.0 33.9 25 40 1.0 3.8 6.5 8.8 11.1 14.4 Overall 14.2 14.8 9 16 6.3 13.6 18.7 23.5 29.5 34.4 CCC 2.2 2.6 1 3 30.2 52.2 67.4 78.9 85.0 89.2 I 3.8 4.3 2 4 20.6 38.5 50.1 59.5 69.1 75.3 3σ u MR 7.6 8.0 5 8 6.0 18.2 28.8 37.6 46.1 52.9 Overall 2.2 2.6 1 3 30.4 52.5 67.5 79.1 85.2 89.4 CCC 6.1 6.8 4 7 13.0 26.8 37.3 45.1 53.9 59.8 I 14.7 15.6 9 17 6.1 13.3 18.9 23.6 30.8 36.4 –0.95 –0.90 2σ u MR 29.1 29.1 21 33 1.2 4.8 8.7 11.9 15.3 18.9 Overall 6.1 6.8 4 7 13.0 26.8 37.4 45.1 53.9 59.8 CCC 17.2 17.9 11 21 5.3 11.8 17.4 21.9 28.2 32.4 I 56.4 60.1 37 62 1.7 3.8 5.7 7.6 10.3 11.7 1.5σ u MR 102.1 97.0 73 114 0.2 1.3 2.6 3.6 5.1 6.2 Overall 17.0 17.7 11 20 5.4 11.9 17.7 22.2 28.4 32.6 223 Table 32 Continued % Detections by point t after the shift φ θ Shift Chart ARL SDRL MRL IQRRL bp bp 0 1 2 3 4 5 CCC 2.1 2.4 1 3 29.8 53.3 68.7 79.3 86.4 90.4 I 3.6 4.1 2 4 20.1 38.0 51.4 63.2 72.1 77.9 3σ u MR 3.9 3.9 3 5 12.7 30.5 46.0 57.5 68.2 74.8 Overall 1.9 2.2 1 3 30.7 55.1 71.1 81.1 88.3 92.0 CCC 6.1 6.8 4 7 13.4 26.4 36.8 45.6 55.0 61.6 I 14.3 15.6 9 16 6.9 14.3 19.6 24.6 30.9 35.6 0.95 0.90 2σ u MR 11.8 12.3 7 12 3.2 10.8 18.4 23.8 31.0 38.8 Overall 5.3 5.9 4 6 14.5 28.2 40.0 49.6 59.6 66.5 CCC 17.5 18.5 11 21 5.6 12.0 16.4 20.3 25.5 29.7 I 55.1 56.8 38 61 1.3 3.0 4.5 6.1 9.0 10.7 1.5σ u MR 35.0 33.9 25 40 1.0 3.8 6.5 8.8 11.1 14.4 Overall 14.2 14.8 9 16 6.3 13.6 18.7 23.5 29.5 34.4 CCC 2.2 2.6 1 3 30.2 52.2 67.4 78.9 85.0 89.2 I 3.8 4.3 2 4 20.6 38.5 50.1 59.5 69.1 75.3 3σ u MR 7.6 8.0 5 8 6.0 18.2 28.8 37.6 46.1 52.9 Overall 2.2 2.6 1 3 30.4 52.5 67.5 79.1 85.2 89.4 CCC 6.1 6.8 4 7 13.0 26.8 37.3 45.1 53.9 59.8 I 14.7 15.6 9 17 6.1 13.3 18.9 23.6 30.8 36.4 0.95 –0.90 2σ u MR 29.1 29.1 21 33 1.2 4.8 8.7 11.9 15.3 18.9 Overall 6.1 6.8 4 7 13.0 26.8 37.4 45.1 53.9 59.8 CCC 17.2 17.9 11 21 5.3 11.8 16.4 21.9 28.2 32.4 I 56.4 60.1 37 62 1.7 3.8 5.7 7.6 10.3 11.7 1.5σ u MR 102.1 97.0 73 114 0.2 1.3 2.6 3.6 5.1 6.2 Overall 17.0 17.7 11 20 5.4 11.9 17.7 22.2 28.4 32.6 CCC 2.2 2.7 1 3 29.2 52.4 67.6 77.5 84.7 89.4 I 2.6 3.0 2 4 26.0 47.8 62.2 72.7 80.6 85.6 3σ u MR 4.1 4.1 3 5 12.0 29.7 44.7 56.4 66.7 73.4 Overall 2.1 2.5 1 3 29.6 54.0 69.7 79.5 86.4 90.5 CCC 7.2 8.1 5 8 12.3 24.1 33.5 40.8 49.4 55.4 I 9.0 9.6 6 11 9.6 19.8 27.9 34.0 42.1 47.6 0.95 0 2σ u MR 13.6 14.4 8 15 2.0 9.4 16.2 20.9 27.6 34.2 Overall 6.4 7.2 4 8 12.6 25.4 35.6 43.3 52.4 58.8 CCC 23.3 23.0 16 28 3.7 7.4 10.6 14.1 19.3 23.2 I 34.4 35.5 24 38 1.9 4.3 6.8 7.1 10.9 14.0 1.5σ u MR 46.3 47.4 32 53 0.6 3.5 5.6 7.7 10.2 12.8 Overall 19.7 19.2 14 24 4.2 8.9 12.5 16.7 22.5 26.7 CCC 2.2 2.7 1 3 29.2 52.4 67.6 77.5 84.7 89.4 I 2.6 3.0 2 4 26.0 47.8 62.2 72.7 80.6 85.6 3σ u MR 4.1 4.1 3 5 12.0 29.7 44.7 56.4 66.7 73.4 Overall 2.1 2.5 1 3 29.6 54.0 69.7 79.5 86.4 90.5 CCC 7.2 8.1 5 8 12.3 24.1 33.5 40.8 49.4 55.4 I 9.0 9.6 6 11 9.6 19.8 27.9 34.0 42.1 47.6 –0.95 0 2σ u MR 13.6 14.4 8 15 2.0 9.4 16.2 20.9 27.6 34.2 Overall 6.4 7.2 4 8 12.6 25.4 35.6 43.3 52.4 58.8 CCC 23.3 23.0 16 28 3.7 7.4 10.6 14.1 19.3 23.2 I 34.4 35.5 24 38 1.9 4.3 6.8 7.1 10.9 14.0 1.5σ u MR 46.3 47.4 32 53 0.6 3.5 5.6 7.7 10.2 12.8 Overall 19.7 19.2 14 24 4.2 8.9 12.5 16.7 22.5 26.7 224 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 Cum Prob Cum Prob 0.3 Prob Cum 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 TRL TRL TRL Figure 53. CDF plots of the run length for all charts combined for different shifts in the variance of the random shocks of the base process. 6.6 Summary and Discussion. The RL summary statistics for the performance of the I-chart for shifts in the mean of the base process are in agreement with the results of other authors (e.g. Superville and Adams, 1994). Specifically, we found that if the AR(1) parameter, φbp, is negative the CCC chart and the I-chart detect even small shifts quickly. However, if this coefficient is positive and close to one, the detection capability of the CCC chart and the I-chart is low, even for large shifts, and the best chance of detecting the shift is immediately after it occurs. For shifts in the mean of the process the MR-chart is not useful because of the smooth drifting of the residuals to their steady-state level. The performance of the CCC chart, I-chart, and the MR-chart when the there is a shift in the mean or the variance of the random shocks depends only on the magnitude of the shift and not on the model that describes the behavior of the base process or on the values of its parameters. In fact, the ARL values reported in Table 5 for the CCC chart and the I-chart are very close to the ARL values reported for an individuals control chart of independent observations for shifts in the mean of 3σ, 2σ, and 1.5σ (Montgomery, 1996, p. 210, Figure 5-17). The RL performance reported in Table 5 is correct if the measurement process is maintained in control, so that the transfer function does not 225 change and the variance of the random shocks of the measurement process have a 2 constant mean of zero and a constant varianceσa . It should be noted that the performance of the I-chart reported in Tables 5 and 6 is when only SRC #1 is applied. We noted in Chapter 5 that several SRC rules are useful to detect smaller changes in the residuals. Therefore, the performance of the I-chart could be better if we implement additional SCR’s. Also, we mentioned that perhaps the MR- chart is not needed. However, the MR-chart is a valuable visual tool that highlights potential changes in the mean or the variance that do not trigger a signal. In the author’s experience the MR-chart has been a valuable tool for discovering out-of-control conditions. 226 CHAPTER SEVEN Conclusions and Future Research The common cause (CC) chart and special cause control (SCC) chart were originally proposed by Roberts and Alwan (1988). Then, Alwan (1991) and Montgomery and Mastrangelo (1991) suggested the addition of control limits to the CC chart, becoming the common cause control (CCC) chart. The SCC chart originally contained only the I-chart of residuals. We added to the special cause chart the MR- chart, becoming the SCC charts. In this dissertation, we propose the use of CCC and SCC charts to monitor processes yielding autocorrelated output in the presence of measurement error. This measurement error may be unobserved, or observed and autocorrelated. In our study, we include an extensive survey and synthesis of the methods that have been proposed to monitor processes with autocorrelated output, finding that practically all of them address the monitoring of processes with autocorrelated output with unobserved measurement error. Most of these methods are tailored to detect the transient effect of the residuals after a shift in the mean of the process. Existing methods for the detection of shifts in the variance require monitoring functions of the residuals, such as their squares, with I-charts or other charts, such as the EWMA control chart. In light of this investigation, we believe that the method most likely to be adopted in an industrial setting is the CCC chart combined with SCC charts. The CCC/SCC charts comprise a flexible monitoring scheme capable of detecting not only changes in the 227 process mean, but also shifts in the mean and the variance of the random shocks that generate the process. The chief innovation in our work is the extension of the CCC/SCC charts to monitor an unobservable base process due to the error of measurement that contaminates the measurements of the product material. We propose the use of a transfer function to establish the relationship between the unobservable product material measurement error and the observed error of measurement on a reference material. The use of the transfer function allows us to extract the signal from the unobserved base process to identify the ARIMA model that describes it and to estimate its parameters. Then, the overall process series (product measurements) can be described by the model that includes the transfer function, the measurements of the reference material, and the model of the base process. A forecast model is obtained from the model that represents the overall process. We use the forecast model to generate lead-one forecasts and the new observations to calculate the residuals. The new observed value is plotted on the CCC chart and compared with the control limits of this chart. The new residual is compared with the control limits of the I-chart, and the moving range of the new and previous residual is compared with the control limits of the MR-chart. Then, if a value is outside the control limits of any of the charts, we say the base process is out of control. It is important to note that our methodology relies on the measurement process being in a perfect state of control. That is, the transfer function and the mean and variance of the measurement process remain constant. Another contribution of our work is the extension of the approach that other authors use to describe the transient behavior of the residuals (e.g. Vander Wiel, 1996) 228 after a shift in the process mean. These descriptions aid in modeling the behavior of the process observations, lead-one forecasts, and residuals after a shift in the mean or variance of the random shocks. The expressions derived for processes with unobserved measurement error showed that for shifts in the mean or the variance of the random shocks the residuals exhibit an immediate step shift, whereas the observations and the lead-one forecasts exhibit a transient behavior. In this case the performance of the SCC charts is the same as the I-chart/MR-chart scheme for independent data. The expressions derived for processes with observed autocorrelated measurement error showed that the observations, the lead-one forecasts, and the residuals exhibit the same behavior after step shifts in the mean and variance of the base and the random shocks of the base process as for processes where measurement error was not observed. Therefore, the performance of the CCC and SCC charts is the same when they are used to monitor a process with unobserved measurement error or with an observable and autocorrelated measurement error. Again, this conclusion is correct if the measurement process is in control and the transfer function accurately describes the relationship between the errors of measurement for the product and reference materials. Another contribution of this work is to provide a review of the methods available to monitor processes with autocorrelated output, examples of the implementation of the CCC/SCC charts, and the procedures to derive the expressions that will describe the behavior of the observations, lead-one forecasts and residuals. These expressions can be used to determine how well the different types of disturbances will be detected. 229 We conducted a limited simulation study to evaluate the performance of the CCC and SCC charts for step shifts in the mean of the base process and for step shifts in the mean and the variance of the random shocks of the base process. In this simulation, we used an AR(1), MA(1), and ARMA(1) models to describe the base process, the simple transfer function ω0. We used parameterizations for which shifts of the process mean generate residual patterns that are either easy or difficult to detect. Furthermore, we restricted use of Special Cause Rules to SCR #1. Our results, based on run length statistics, indicate that the MR-charts may not be needed, and even that perhaps we can monitor the process successfully with only the CCC charts, as suggested by Awan (1991). However, we show that the CCC chart, the I-chart and MR-chart produce distinctive patterns that can help us identify out-of-control conditions and apply specific SCR’s to the I-chart. Finally, we provide an effective and flexible methodology to monitor processes with unobserved and with observed correlated measurement error. However, there is potential for improving our new methodology requiring further investigation. Lines of additional research include the following: o Investigate what effect of variations in the transfer function on the performance of the CCC and SCC charts; o Incorporate special cause rules in the CCC chart. o Study nonlinear aggregation of the measurement error and the base process signal; o Study nonlinear relationships between the error of measurement on the product and on the reference material. 230 o Devise a different approach to better characterize the measurement error associated with the measurement of product material and reference materials. Then use this information to obtain the actual signal of the base process. o Take advantage of the transient behavior of the observations and lead-one forecasts to improve detection of shifts on the mean and variance of the random shocks of the base process. o Review the performance of the CCC and SCC charts when other SCR’s, besides SCR # 1, are used. o Devise approaches (Bayesian and non-Bayesian) to systematically update the model, the estimates of its parameters, and control limits using Phase II observations o Study the performance of the CCC chart when its control limits are established based on estimates of the variances of the random shocks of the measurement and base processes. o Implementation of production and measurement process technical knowledge and prior information in the selection of models and monitoring schemes. 231 APPENDICES 232 APPENDIX A Table A.1. Literature Summary Autocorrelation No Year Author Purpose Control charts model 1 1974 Johnson, Bagshaw Effect of autocorrelation on CUSUM AR(1), MA(1) Data CUSUM 2 1975 Johnson, Bagshaw Effect of autocorrelation on CUSUM AR(1), MA(1) Data CUSUM 3 1977 Wu Dynamic data systems- modeling (i.e. time series) ARMA(p, q) None Bethouex, Hunter, 4 1978 Sewage treatment, fit ARIMA 7-day cycle ARIMA Resid I-chart, CUSUM Pallesen 25 Vasilopoulos, 233 1978 Modify control limits AR(1), AR(2) Data X-bar, S Stamboulis ARMA(4,1), 6 1986 Notohardjono, Ermer Blast furnace- apply dynamic data systems Resid I-chart ARMA(4,0) ARMA(1,1), 7 1988 Alwan, Roberts Fit ARIMA; common & special cause charts Forecast, resid I-chart IMA(1,1) Beneke, Leemis, Mixture sine & 8 1988 Detect cycles in the mean Spectral control chart Schlegel, Foot cosine cycles AR(1), 9 1989 Maragh (PhD Thesis) Effect of autocorrelation on X-bar & EWMA Data X-bar, EWMA MA(1),IMA(1,1) Yourstone, Resid Geo MR, Geo M. 10 1989 Fit ARIMA, monitor autocorrelation of residuals ARMA(2, 1), AR(2) Montgomery avg, SACF AR(1), AR(2), 11 1990 Longnecker, Ryan Assess performance of I-chart Resid I-chart ARMA(1,1) 12 1990 Cryer, Ryan Compare MR vs S chart IMA(1, 1) Data I-chart Table A.1 Continued Autocorrelation No Year Author Purpose Control charts model Effect of autocorrelation on CUSUM, EWMA charts, Resid I-chart, Data 13 1991 Harris, Ross AR(1), IMA(1,1) fit ARIMA use residuals CUSUM & EWMA Implement CCC/SCC; Propose only CCC w/control Forecast w/control 14 1991 Alwan AR(1) limits limits, resid I-chart English, Krishnamurti, Compare I-chart w & w/o adjusted limits. Kalman Data X-chart w & w/o 15 1991 AR(1), AR(2) Sastri recursive estimation /adjusted limits 16 Montgomery, EWMA forecast instead of actual ARIMA. Use Forecast & data w/ctrl 1991 AR(1), IMA(1, 1) Mastrangelo observed & forecast w/control limits limits, Resid I-chart 234 Forecast w/limits, Resid Wardell, Moskowitz, 17 1992 Compare CCC w/ limits, SCC, EWMA, and I-chart ARMA(1,1) I-chart, Data EWMA & Plante I-chart 18 1992 Longnecker, Ryan Assess performance of I-chart AR(2), ARMA(1,1) Resid I-chart Performance of std. charts using false positives and 19 1992 Alwan AR(1), ARMA (p, q) Data I-chart, MR false negatives Padgett, Thombs, X-bar and R-chart performance with non-normal or AR(1)+subgroup 20 1992 Data X-bar and R Padgett correlated subgroups error AR(1), AR(2), 21 1993 Yaschin Transformation to eliminate autocorrelation Data CUSUM MA(1), ARMA(1,1) AR(1), ARMA(1,1), Data I-chart, EWMA, 22 1993 McGregor, Harris EWMV and EWMS control charts to monitor variance IMA(1, 1) EWMV, EWMS Compare chart performance using CDF of detection & Data I-chart, CUSUM; 23 1994 Superville, Adams AR(1) EWMA forecasts EWMA of forecasts Wardell, Moskowitz, Derive average and std. dev. of run length distribution ARMA(p, q), Data I-chart, EWMA; 24 1994a Plante for SCC ARMA(1,1) Resid I-chart Table A.1 Continued Autocorrleation No Year Author Purpose Control charts model Add likelihood ratio stat in data I-chart, EWMA, and Wardell, Moskowitz, Data I-chart, EWMA; 25 1994b resid. I-chart to detect mean shifts. Modify limits of ARMA(1,1) Plante Resid I-chart EWMA. Runger, Willemain, 26 1995 Assess performance of CUSUM chart AR(1) Resid CUSUM Prabhu Achieve independence by weighted and un-weighted Data WBM & UBM, 27 1995 Runger, Willemain AR(1) grouping of data Resid I-chart 28 Combined EWMA-Shewhart (CSE) of residuals to Resid CSE, I-chart, & 1996 Lin, Adams AR(1), IMA(1,1) 235 detect immediate and transient changes in the mean EWMA Study patterns associated with transient behavior of AR(1), ARMA(1,1), 29 1996 Hu, Roan Resid I-chart residuals ARMA (2,1) Compared performance of several charts of residuals Resid I-chart, CUSUM, 30 1996 Vander Weil IMA(1) from an EWMA forecast. EWMA, & LR Studied performance of X-bar chart of observations 31 1996 Reynolds, Arnold, Baik AR(1) Data X-bar chart with fixed and variable sampling intervals Studied CUSUM & EWMA charts of observations. AR(1)+error, Data CUSUM & 32 1997 VanBrackle, Reynolds Select chart parameters to account for autocorrelation. ARMA(1,1) EWMA Study performance of residual and observation I-chart 33 1997 Zhang AR(2) Data X, Resdi I-chart using a detection capability index. Data EWMA, Resid 34 1997 Schmid, Schöne Properties and performance of EWMA of residuals AR(1) EWMA Faltin, Mastrangelo, Review methods for monitoring processes 35 1997 N/A N/A Runger, Ryan w/autocorrelated data Table A.1 Continued Autocorrelation No Year Author Purpose Control charts model Studied properties of R and S2 charts under within or 36 1997 Amin, Schmid, Frank AR(1) R and S2 charts between group correlation Forecast/Data chart, Example of monitoring a process w/autocorrelated 37 1998 Ramírez IMA(1, 1) Resid I-chart and output CUSCORE AR(1), AR(2), Data I-chart & 38 1998 Zhang EWMA for stationary processes (EWMAST) AR(3), ARMA(1,1) EWMAST, Resid I-chart 39 Use model-based statistic (λLS) to detect level Resid I-chart , Level- 236 1998 Atienza, Tang, Ang AR(1) shifts in autocorrelated data Shift chart Study CUSUM performance. Introduced Timmer, Pignatiello, 40 1998 AR(1)CUSUM to account for autocorrelation of an AR(1) Data CUSUM Longnecker AR(1) process Model-free Run-Sum (RS) and Run-Length (RL) Data RL, RS; Resid I- 41 1998 Willemain, Runger AR(1), ARMA(1, 1) charts chart , CUSUM Resid I-chart , I-chart Study effect of parameter error of estimation on 42 1998 Adams, Tseng AR(1), IMA(1,1) w/SCR #5, CUSUM, control charts EWMA ARIMA(p=0 to 4, Resid GLRT, I-chart , 43 1999 Apley, Shi Generalized likelihood ratio test (GLRT) chart d =1 or 0, q=0 to 2) CUSUM Data EWMA; Resid I- 44 1999a Lou, Reynolds Study performance of EWMA control chart AR(1), ARMA(1,1) chart , EWMA Table A.1 Continued Autocorrelation No Year Author Purpose Control charts model EWMA log (resid2), I- Study control charts to monitor variance using log(e2), AR(1), ARMA(1,1), 45 1999b Lou, Reynolds chart of resid2, Resid e2, and moving range (MR) IMA(1, 1) MR, EWMA Literature review; Further research of control charts to 46 1999 Woodall, Montgomery N/A N/A monitor the variance Data ARMA, 47 2000 Jiang, Tsui, Woodall ARMA control chart AR(1), ARMA(1, 1) EWMAST, WBM, RS; Resid I-chart 237 48 Study performance of Resid I-chart with estimated Resid I-chart and |Resid| 2000 Kramer, Schmid AR(1) parameter values. Use modified I-chart of data I-chart Estimation of Phase I model parameters when special 49 2000 Boyles AR(1), ARMA(1, 1) Resid I-chart causes are present IMA(1, 1), 50 2000 Yang, Makis Study steady-state and transient behavior of residuals None ARIMA(p, d, q) Data I-chart, EWMAST; 51 2000 Zhang Compare performance of control charts AR(1) Resid I-chart, CUSUM, EWMA Data I-chart, CUSUM, Study performance of CUSUM for positive 52 2001 Lou, Reynolds AR(1), ARMA(1, 1) EWMA; Resid EWMA, autocorrelation I-chart, CUSUM English, Lee, Martin, 53 2000 Study detection of disturbances AR(2) Resid X-bar, EWMA Tilmon Table A.1 Continued Autocorrelation No Year Author Purpose Control charts model AR(1), MA(1), Data T2 chart; Resid I- 54 2002 Apley, Tsung T2 chart for univariate correlated data ARMA(1,1) chart, CUSUM Data CUSUM; Resid I- 55 2002a Atienza, Tang, Ang Modified CUSUM and compare performance AR(1) chart, CUSUM Monitor model adequacy using the ACF and Box- 56 2002b Atienza, Tang, Ang MA(1) X or Q statistic Ljung-Pierce Q statistic 57 Jiang, Wu, Tsung, Nair, Residual-based PID control chart tailored to detect AR(1), MA(1), 2002 Resdi I-chart, PID Tsui steady-state and transient changes in the residuals ARMA(1, 1) 238 Data I-chart, S2, Control chart to monitor the mean and variance 58 2002 Knoth, Schmid AR(1) EWMA; Resid I-chart, simultaneously S2-chart, EWMA Discussion of effect of assignable causes in residual Data I-chart, EWMA; 59 2002 Runger AR(1), IMA(1, 1) and observation control charts Resid I-chart, EWMA Detect disturbances during process transitions Use 60 2003 Nembhard, Kao AR(1) Resid EWMA adaptive EWMA (AEWMA) Worst-case EWMA chart with wider control limits to 61 2003 Apley, Lee ARMA(1, 1) Resid EWMA, I-chart account for uncertainty of estimated parameters Reverse moving average (RMA) control chart and AR(1), MA(1), Data CSE: Resid I-chart, 62 2003 Dyer, Adams, Conerly compare it ARMA (1, 1) RMA Example of monitoring a biochemical process Data EWMA, 63 2004 Winkel, Zhang AR(1) w/autocorrelated output EWMAST Table A.1 Continued Autocorrleation No Year Author Purpose Control charts model Sensitivity study of residual-based charts; standard. vs. Data EWMA; Resid 64 2004 H. C. Lee (PhD Thesis) wider limits using worst-case variance or the expected AR(1), ARMA(1, 1) EWMA variance Max-CUSUM (MCAP) chart to monitor the mean and 65 2005 Cheng, Thaga AR(1), ARMA(1, 1) Resid MCAP, CSE the variance simultaneously Improves the procedure to construct the PID Tsung, Zhao, Xiang, 66 2006 monitoring scheme by taking into account the pattern AR(1), ARMA(1, 1) Resid I-chart Jiang of the shift in the mean 67 Jensen, J-Farmer, Discussion of effect of parameter estimation on control 239 2006 N/A N/A Champ, Woodall charts 68 2006 Gruska, Kymal General discussion of SPC in an industrial setting N/A N/A Residual-based 2nd Order Filter statistic (OSLF chart) AR(1), MA(1), Resid EWMA, OSLF, 69 2006 Chin Apley optimized to provide a desired in-control ARL and ARMA(1, 1) GLF minimize out-control ARL Residual-Based general linear filter statistic (OGLF AR(1), MA(1), Resid OGLF, PID, I- 70 2007 Apley, Chin chart) for a given in-control ARL and minimal out of ARMA(1, 1) chart control ARL Box, Paniagua- Bounded-Adjustment and I-chart monitoring chart. Data adjustment, Resid 71 2007 IMA(1, 1) Quiñones Discussion of sources of noise variation I-chart Nemhard, Valverde- Use CUSCORE to adjust and monitor a process. IMA(1, 1), 72 2007 CUSCORE Ventura Discussion of several types of disturbances ARMA(1, 1) APPENDIX B Variance of the Base Process In Chapter 5 we found that model (5.11) appropriately represents the base process. The autocovariance function of the base process model is (Box, Jemkins, and Reinsel, 1994; pg. 78-80) 2 γ=φγ+φγ+φγ−θψ−011224411bp bp bp ( bp 1,) σu 2 γ=φγ+φγ+φγ−θσ11021431bp bp bp bp u , γ=φγ+φγ+φγ2112042bp bp bp , γ=φγ+φγ+φγ=φγ+31221411231bp bp bp bp a γ, γ=φγ+φγ+φγ4132240bp bp bp , where a32=φ(bp +φ bp4)and ψ1 = φbp1 – θ bp1. The process variance is obtained by sequential substitution of these expressions. Then, substituting the expressions for γ1 and γ2 into γ3 yields 2 γ=32aa( φbp 1112031 γ+φφ bp bp γ) + γ 222 =φφγ+aaaaaaaa2120bp bp ()()() 3 +φ 21bp 11011 φγ−θσ+φ bp bp u 143 bp +φ 213bp γ 2 =γ−σbb2001() bu , where −1 a12=−φ(1 bp ) −1 a24=−φ()1 bp 3 b0212131121= () aφφbp bp + aa φbp + aa φbp 2 ba111321=θ() aa +φbp 240 −1 baaa=−φ⎡1. +φ2 ⎤ 214321⎣ bp ()bp ⎦ Substituting γ3 into the equation for γ1 results in 2 γ=11a () φbp 10431 γ+φγ−θσ bp bp u 2 =φ+()aabbabba11bp 10240 φbp γ−( 1124 φ+θσbp 11 bp) u . Now, substituting γ1 and γ3 into the expression for γ2 results in γ=22a () φbp 1120 γ+φγ bp 2 2 =φ+()a22bp aa 121 φ+bp aab 1221400 φφbp bp b γ−() aabb 121214 φφ+bp bp aa 1211 φθσbp bp u . The expression for γ4, obtained from the expressions for γ1 , γ2 , and γ3 , is 22 γ=4()aaaaabb 2212121202124021 φbp + φbp φ bp + φbp φ bp φ bp + bb φbp +φ bp 4 γ0 2 −φφφ+φφθ+φσ()aa1212 bb bp 1 bp 2 bp 4aa 1 2bp 1 bp 2 bp 1bb 12bp 112 bb u . Finally, the process variance γ0 is obtained by substituting the expression of γ1 , γ2 , γ3, and γ4 into γ0. Hence, c1 2 γ=0 σu , c2 where ca1=−θ1 bp 1() φ bp 1 −θ bp 1 − 1114b φbp φ bp −a 111 φ bp θ bp −a 121212abb φbp φ bp φ bp4 22 −aa12121 φφθ−bp bp bp aabb 1212124 φφφbp bp bp −aa 121241 φφφθbp bp bp bp −φφbb 2 12bp 1 bp 4 222 caabba2=−1 11 φbp − 10214 φbp φ bp − 22 φ bp + aa 1212 φbp φ bp 22 −φaabb1202bp 1φφbp24 bp −φφaaa 224 bp bp − 12124 φφφbp bp bp 22 −φφφ−φφ−φaabb1202bp 1 bp 2 bp 4bb 02bp 1 bp 4 bp 4. The constants c1 and c2, their ratio and the square root of that ratio are calculated using the estimates in Table 6, Chapter 5, to obtain γ=σ=ˆ 0 ˆˆbp 7.5 σu . 241 APPENDIX C SAS Code 1 /*******************************************************************/ /* Phase I - Identify, Estimate, and Forecast Measurement Process */ /* Author: Jesús Cuéllar Fuentes */ /*******************************************************************/ /*Read original data*/ options nodate nonumber ps=200 ls=80 formdlim='~'; PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: *n=Obs No., x=MP data, y=OP data*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /*Plot Phase I data*/ symbol1 c=black i=join v=circle; symbol2 c=red i=join v=circle; symbol3 c=blue i=join v=star; symbol4 c=green i=join v=star; symbol5 c=blue i=spline; symbol6 c=black i=needle; symbol7 c=black i=join v=circle; symbol8 c=blue i=join w=5; symbol9 c=red i=join; axis1 label=('Observation No.') order= (0 to 901 by 100); axis2 label=(a=90 'Output'); axis3 label=(a=90 'Residuals'); axis6 label=(a=90 'Autocorrelation'); axis7 label=(a=90 'Partial Autocorrelation'); axis8 label=('Lag') order=(0 to 24 by 1); axis9 label=(a=90 'x[t]-x[t-1]'); proc gplot data=ph1; *Plot MP data; plot x*n=2/vaxis=axis2 haxis=axis1 overlay; run; /*Identify Measurement process Model*/ proc arima data=ph1; i var=x outcov=xcov; run; quit; /*ACF and PACF of the MP data*/ data xcov; *Compute approximate 95% CI limits for ACF and PACF; set xcov; AUCIL= 2*stderr; ALCIL= - AUCIL ; PUCIL= 2/sqrt(900); PLCIL=-pUCIL; run; quit; proc gplot data=xcov; *Plot ACF; plot AUCIL*lag=5 ALCIL*lag=5 corr*lag=6 / 242 overlay vref=0 haxis=axis8 vaxis=axis6; run; proc gplot data=xcov; *Plot PACF; plot PUCIL*lag=5 PLCIL*lag=5 partcorr*lag=6 / overlay vref=0 haxis=axis8 vaxis=axis7; run; /********************* Stationary Models ***************************/ proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11); f lead=1 back=0 id=N out=xfor; run; quit; /*Plot MP observed vs. lead -1 forecasts*/ proc gplot data=xfor; plot x*n=7 forecast*n=8/vaxis=axis2 haxis=axis1 overlay; run; /*Histogran and normal plot of residuals*/ proc univariate data=xfor normal; var residual; probplot residual/normal(MU=0 SIGMA=EST) ctext=black; histogram residual/normal(color=red) cbarline= ligr cframe=ligr cfill=blue; run; /*ACF and PACF of the residuals*/ proc arima data=xfor; i var=residual outcov=rcov; run; quit; data rcov; *Compute 95% CI limits of the ACF and PACF; set rcov; AUCIL= 2*stderr; ALCIL= - AUCIL ; PUCIL= 2/sqrt(900); PLCIL=-pUCIL; run; quit; proc gplot data=rcov; *Plot ACF of residuals; plot AUCIL*lag=5 ALCIL*lag=5 corr*lag=6 / overlay vref=0 haxis=axis8 vaxis=axis6; run; proc gplot data=rcov; *Plot PACF of residuals; plot PUCIL*lag=5 PLCIL*lag=5 partcorr*lag=6 / overlay vref=0 haxis=axis8 vaxis=axis7; run; /*Generate CCC chart*/ data xfor; *Compute control limits; set xfor; UCL= forecast+3*std; LCL= forecast-3*std; run; proc gplot data=xfor;*Plot CCC chart; plot x*n=7 forecast*n=8 UCL*n=9 LCL*n=9/vaxis=axis2 haxis=axis1 overlay; run; /************************ END of PROGRAM ***************************/ SAS Code 2 /*******************************************************************/ /* Phase II - Monitoring of the Measurement Process */ /* Author: Jesús Cuéllar Fuentes */ /*******************************************************************/ /*Read in data*/ PROC IMPORT OUT=rdata 243 DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MP data, y=OP data*/ data ph1x; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /* Subset Phase II data */ data ph2x; set rdata (firstobs=901); run; /*Est imate coefficients of the MP Phase I model and residuals*/ proc arima data=ph1x; i var=x noprint; e p=(1 2 4 10 11) outest=est1 noprint; *Save model estimates; f lead=1 back=0 id=N out=xfor; *Save residuals & forecasts; run; quit; /*Change generic forecast & residual names*/ data xfor (rename=(forecast=forx l95=LCL u95=UCL residual=resx)); set xfor; run; /*Control chart of residuals*/ proc shewhart data=xfor; irchart resx*N / outlimits=inctrlm; *Save control limits; run; /*Compute and save std. dev residuals of Y*/ proc univariate data=xfor normal; var resy; output out=sdxres std=residual; run; quit; /*Compute lead 1 forecast and residuals*/ proc iml; use work.xfor; read all var {n x forx resx lcl ucl} into w; use work.sdxres;read all var {residual} into sdxr; use work.ph2x; read all var {x} into u; use work.est1; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into est1; /*Coefficient estimates for the input series model*/ phi1 =est1[1,3]; phi2 =est1[1,4]; phi4=est1[1,5]; phi10=est1[1,6]; phi11=est1[1,7]; sphi=1-phi1-phi2-phi4-phi10-phi11; /*Mean of MP series*/ mu=est1[1,2]; /*Std. dev of {at} series and of residuals*/ sa=sqrt(est1[1,1]); sde=sdxr[1,1]); /*Set number of rows*/ t=nrow(w); m=nrow(u); /*Add first Ph II observation; calculate residual & control limits*/ w[t,2]=u[1,1]; w[t,4]=w[t,2]-w[t,3]; w[t,5]=w[t,3]-3*sde; w[t,6]=w[t,3]+3*sde; newobs=j(1,6,1); *Row vector of 1's to store new values; k=1; *1st row of ph2x already in matrix w; do j=1 to m-1; w=w//n ewobs; *Append row to store new values; k=k+1; *counter of new observations; t=t+1; *counter of lead 1 forecasts; w[t,1]=w[t-1,1]+1; *Update index; /*Compute forecasts Xt(1) */ 244 w[t,3]=mu*Sphi +phi1*w[t-1,2]+phi2*w[t-2,2]+phi4*w[t-4,2] +phi10*w[t-10,2]+phi11*w[t-11,2]; /*Compute residuals at t+1 */ w[t,2]=u[k,1]; *Incorporate new X values; w[t,4]=w[t,2]-w[t,3]; *Compute residual; w[t,5]=w[t,3]-3*sa; *Compute LCl of CCC chart; w[t,6]=w[t,3]+3*sa; *Compute UCl of CCC chart; end; /*Create output matrix*/ create ph2r var{n x forx resx LCL UCL}; append from w; close ph2r; run; quit; /*Subset 10 old and all new observations*/ data ph2r; set ph2r (firstobs=891); run; symbol1 c=black i=join v=circle; symbol2 c=blue i=join w=5; symbol3 c=red i =join; axis1 label=(a=90 'Measurment Process Series'); axis2 label=('Observation No.') order= (895 to 988 by 10); /* Plot MS series: Observed vs. Forecasted */ proc gplot data=ph2r; plot x*n=1 forx*n=2 LCL*n=3 UCL*n=3/vaxis=axis1 haxis=axis2 overlay; run; /*Control charts of Phase II Residuals*/ proc shewhart data=ph2r limits=inctrlm; irchart resx*n / cframe = vigb cconnect = yellow cinfill = vlib; run; /************************* END of PROGRAM ******************************/ SAS Code 3 /**********************************************************************/ /* Phase II - Measurement Process with disturbances */ /* Author: Jesús Cuéllar Fuentes */ /**********************************************************************/ /*Read in data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MP data, y=OP data*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /**************** Estimate Model Coefficients and Save ****************/ proc arima data=ph1; i var=x; e p=(1 2 4 10 11) outest=est1; *Save coeff estimates; f lead=1 back=0 id=N out=mpfor; *Save residuals & forecasts; run; quit; /**************** Generate X series with a Mean Shift *****************/ Proc iml; 245 use work.est1; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into est1; /*Std. dev. of MP series = sqrt(gamma0)*/ sdx=5.584; /*Coefficient estimates for the input series model*/ phi1 =est1[1,3]; phi2 =est1[1,4]; phi4=est1[1,5]; phi10=est1[1,6]; phi11=est1[1,7]; /*Mean of MP series*/ m0=est1[1,2]; /*Std. dev. residuals from stable MP process*/ sda=sqrt(est1[1,1]); /*Number of simulated data points + 12 trailing values*/ i=1000; k=1:i; *Row vector from 1 to i; time=k`; *Transpose k to col vector =Time index; /*generate vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; z=j(i,1); z[1:i]=0; *MP series w/disturbance; f=j(i,1); f[1:i]=0; *lead-1 forecasts; e=j(i,1); e[1:i]=0; *lead-1 residuals; a=j(i,1); a[1:i]=0; *MP white noise series; UCL=j(i,1); UCL[1:i]=0; *UCL forecast; LCL=j(i,1); LCL[1:i]=0; *LCL forecast; d3=2*sdx; *Change in mean of MP series; /*Generate MP with change in the output mean*/ do t=13 to i; a[t]=sda*rannor(100855); x[t]=m0+phi1*(x[t-1]-m0)+phi2*(x[t-2]-m0)+phi4*(x[t-4]-m0) +phi10*(x[t-10]-m0)+phi11*(x[t-11]-m0)+a[t]; if t>=i-49 then z[t]=x[t]+d3; else z[t]=x[t]; f[t]=m0+phi1*(z[t-1]-m0)+phi2*(z[t-2]-m0)+phi4*(z[t-4]-m0) +phi10*(z[t-10]-m0)+phi11*(z[t-11]-m0); UCL[t]=f[t]+3*sda; LCL[t]=f[t]-3*sda; e[t]=z[t]-f[t]; end; /*Output simulated series*/ out=time||a||z||f||e||UCL||LCL; create simdat var{time a z f e UCL LCL}; *generate data set; append from out; close simdat; run;quit; /*Remove starting values from series; save last 100 simulated values*/ data simdat; set simdat (firstobs=901); run; quit; /*Plot simulated MS and white noise series*/ symbol1 c=black i=join v=circle; symbol2 c=blue i=join w=5; symbol3 c=red i=join; symbol4 c=green i=joint v=star; axis1 label=(a=90 'Simulated Data'); axis2 label=('Observation No.') order= (901 to 1001 by 10); axis3 label=(a=90 'Residuals'); axis4 label=(a=90 'Forecasts'); proc gplot data=simdat; plot z*time=1 f*time=2 a*time=3 /vaxis=axis1 haxis=axis2 overlay; run; proc gplot data=simdat; 246 plot z*time=1 f*time=2 ucl*time=3 lcl*time=3/ vaxis=axis1 haxis=axis2 overlay; run; proc gplot data=simdat; plot e*time=1/vaxis=axis3 haxis=axis2; run; /************************* END of PROGRAM *****************************/ SAS Code 4 /**********************************************************************/ /* Phase II - Measurement Process Random Shock disturbances */ /* Author: Jesús Cuéllar Fuentes */ /**********************************************************************/ /*Read in data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MP data, y=OP data*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /**************** Estimate Model Coefficients and Save ****************/ proc arima data=ph1; i var=x; e p=(1 2 4 10 11) outest=est1; *Save coeff estimates; f lead=1 back=0 id=N out=mpfor; *Save residuals & forecasts; run; quit; /**************** Generate X series with a Mean Shift *****************/ Proc iml; use work.est1; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into est1; /*Coefficient esti m a tes for the inp u t series model* / phi1 =est1 [1,3]; phi2 =est1 [1,4]; phi4=est1[1,5]; phi10=est1[1,6]; phi11=est1[1,7]; /*Mean of MP series*/ m0=est1[1,2]; /*Std. dev. residuals from stable MP process*/ sda=sqrt(est1[1,1]); i=1000; *Number of simulated data points + 12 trailing values; k=1:i; *Row vector from 1 to i; time=k`; *Transpose k to col vector =Time index; /*Generate vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; f=j(i,1); f[1:i]=0; *lead-1 forecasts; e=j(i,1); e[1:i]=0; *lead-1 residuals; a=j(i,1); a[1:i]=0; *MP white noise series; UCL=j(i,1); UCL[1:i]=0; *UCL forecast; LCL=j(i,1); LCL[1:i]=0; *LCL forecast; d1=0*sda; *Change in mean of {at} series; d2=0.5; *Change in variance of {at} series; /*Generate MP with change in mean/varaince of a(t)*/ do t=13 to i; if t>=i-49 then a[t]=d1+d2*sda*rannor(100855); else a[t]=sda*rannor(100855); 247 x[t]=m0+phi1*(x[t-1]-m0)+phi2*(x[t-2]-m0)+phi4*(x[t-4]-m0) +phi10*(x[t-10]-m0)+phi11*(x[t-11]-m0)+a[t]; f[t]=m0+phi1*(x[t-1]-m0)+phi2*(x[t-2]-m0)+phi4*(x[t-4]-m0) +phi10*(x[t-10]-m0)+phi11*(x[t-11]-m0); UCL[t]=f[t]+3*sda; LCL[t]=f[t]-3*sda; e[t]=x[t]-f[t]; end; /*Output simulated series*/ out=time||a||x||f||e||UCL||LCL; create simdat var{time a x f e UCL LCL}; *generate data set; append from out; close simdat; run;quit; /*Remove starting values from series; save last 100 simulated values*/ data simdat; set simdat (firstobs=901); run; quit; /*Plot simulated MS and white noise series*/ symbol1 c=black i=join v=circle; symbol2 c=blue i=join w=5; symbol3 c=red i=join; symbol4 c=green i=joint v=star; axis1 label=(a=90 'Simulated Data'); axis2 label=('Observation No.') order= (901 to 1001 by 10); proc gplot data=simdat; plot x*time=1 f*time=2 e*time=3/vaxis=axis1 haxis=axis2 overlay; run; proc gplot data=simdat; plot x*time=1 f*time=2 ucl*time=3 lcl*time=3/ vaxis=axis1 haxis=axis2 overlay; run; /************************* END of PROGRAM *****************************/ SAS Code 5 /*******************************************************************/ /* Phase I - Identify, Estimate, and Forecast Overall Process */ /* Author: Jesús Cuéllar Fuentes */ /*******************************************************************/ /*Read original data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MP series, y=OP series*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; run; /*Plot specifications*/ symbol1 c=black i=join v=circle; symbol2 c=blue i=join v=star; symbol3 c=blue i=spline; symbol4 c=black i=needle; symbol5 c=blue i=join w=5; symbol6 c=red i=join; 248 axis1 label=('Observation No.') order= (0 to 901 by 100); axis2 label=(a=90 'Overall Process Series'); axis3 label=(a=90 'Autocorrelation'); axis4 label=(a=90 'Partial Autocorrelation'); axis5 label=('Lag') order=(0 to 24 by 1); axis6 label=(a=90 'Cross-Correlation'); axis7 label=('Lag') order=(-14 to 24 by 2); /*Plot MP an d OP Series*/ proc gplot data=ph1; plot x*n=1 y*n=2/vaxis=axis2 haxis=axis1 overlay; run; /***************** Characteristics of OP series ********************/ proc arima data=ph1; i var=y outcov=ycov; run; proc arima data=ph1; i var=y noprint; *OP series; e p=(1 4 2 11) q=1; *OP model; run; quit; /*ACF and PACF of the OP series*/ data ycov; *Compute approximate 95% CI for ACF & PACF; set ycov; AUCIL= 2*stderr; ALCIL= - AUCIL; PUCIL= 2/sqrt(900); PLCIL=-pUCIL; run; proc gplot data=ycov; *Plot ACF; plot AUCIL*lag=3 ALCIL*lag=3 corr*lag=4 / overlay vref=0 haxis=axis5 vaxis=axis3; run; proc gplot data=ycov; *Plot PACF; plot PUCIL*lag=3 PLCIL*lag=3 partcorr*lag=4 / overlay vref=0 haxis=axis5 vaxis=axis4; run; /************ Identificaion of the Transfer Function ***************/ *Statio nary model, X-->Y; proc arima data=ph1; i var=x noprint; *MP series; e p=(1 2 4 10 11) noprint; *MP model; i var=y crosscor=x outcov=stcorr; *Output cross-correlations; run; quit; /*Plot Crosscorrelation function (CCF)*/ data stcorr; *Compute approximate 95% CI limits for ACF and PACF; set stcorr (firstobs=36); *Cross-correlations start at k=-14; UCIL= 2*stderr; LCIL= - UCIL; run; proc gplot data=stcorr; *Plot CCF; plot UCIL*lag=3 LCIL*lag=3 corr*lag=4 / overlay vref=0 haxis=axis7 vaxis=axis6; run; /*************** Identify model of Noise series ********************/ *Stationary model; proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11) noprint; i var=y crosscor=(x) noprint outcov=stcorr; e input=( /(1) x) plot; run; quit; /*Plot ACF and PACF of BP series*/ data bpcorr; *Compute approximate 95% CI limits for ACF and PACF; set stcorr (firstobs=1 obs=25); 249 AUCIL= 2*stderr; ALCIL= -AUCIL; PUCIL= 2/sqrt(900); PLCIL=-PUCIL; run; proc gplot data=bpcorr; *Plot ACF; plot AUCIL*lag=3 ALCIL*lag=3 corr*lag=4 / overlay vref=0 haxis=axis5 vaxis=axis3; run; proc gplot data=bpcorr; *Plot PACF; plot PUCIL*lag=3 PLCIL*lag=3 partcorr*lag=4 / overlay vref=0 haxis=axis5 vaxis=axis4; run; /******** Estimate parameters of ARMA model of noise series ********/ *Stationary model; proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11) noprint; f lead=1 back=0 id=N out=xfor; i var=y crosscor=(x) noprint; e p=(1 2 4) q=1 input=(x)outest=yest;*Base process model; f lead=1 back=0 id=N out=yfor;*Forecast Overall process model; run; quit; /*********************** **Residual checks *************************/ proc arima data=bpout; i var=residual outcov=rsdcorr; run; /*ACF and PACF of the OP residuals*/ data rsdcorr; *Compute approximate 95% CI limits for ACF and PACF; set rsdcorr; AUCIL= 2*stderr; ALCIL= -AUCIL; PUCIL= 2/sqrt(900); PLCIL=-PUCIL; run; proc gplot data=rsdcorr; *Plot ACF; plot AUCIL*lag=3 ALCIL*lag=3 corr*lag=4 /overlay vref=0 haxis=axis5 vaxis=axis3; run; proc gplot data=rsdcorr; *Plot PACF; plot PUCIL*lag=3 PLCIL*lag=3 partcorr*lag=4 /overlay vref=0 haxis=axis5 vaxis=axis4; run; /*Plot observed and forecasted OP series*/ proc gplot data=yfor; plot y*n=1 forecast*n=5/vaxis=axis2 haxis=axis1 overlay; run; /*Histogran and normal plot of residuals*/ proc univariate data=yfor normal; var residual; output out=stdres std=residual; probplot residual/normal(MU=0 SIGMA=EST) ctext=black; histogram residual/normal(color=red) cbarline= ligr cframe=ligr cfill=blue; run; /*Generate CCC chart*/ data stdres (rename=(residual=rstd)); set stdres; run; data ccc; *joint st. dev of residuals; if _N_=1 then set stdres; set yfor; UCL= forecast+3*rstd; LCL= forecast-3*rstd; 250 run; proc gplot data=ccc;*Plot CC; C chart plot y*n=1 forecast*n=5 UCL*n=6 LCL*n=6 /vaxis=axis2 haxis=axis1 overlay; run; /*Control charts of residuals*/ proc shewhart data=IPoutr; irchart residual*n/ tests = 1 npanelpos = 900 outlimits=yctrlim; run; /************************ END of PROGRAM ***************************/ SAS Code 6 /*******************************************************************/ /* Phase II - Monitoring of the overall Process */ //* Author: Jesús Cuéllar Fuentes */ /*******************************************************************/ /*Read original data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MS series, y=output series*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /* Subset Phase II data */ data ph2; set rdata (firstobs=901); run; quit; /*Save Phase I parameter estimates forecasts and residuals*/ proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11) outest=xest noprint; f lead=1 back=0 id=N out=xfor; i var=y crosscor=(x) noprint; e p=(1 2 4) q=1 input=(x)outest=yest; f lead=1 back=0 id=N out=yfor; run; quit; /*Change generic forecast & residual names*/ data xfor (rename=(forecast=forx residual=resx)); set xfor; run; data yfor(rename=(forecast=fory l95=LCL u95=UCL residual=resy)); set yfor; run; /*Compute and save std. dev. residuals rt*/ proc univariate data=yfor normal; var resy; output out=sdyres std=residual; run; quit; data sdyres(rename=(residual=sdyr)); set sdyres; run; /*Compute control limits for the CCC chart*/ data yfor; if _N_=1 then set sdyres; set yfor; 251 UCL= fory+3*sdyr; LCL= fory-3*sdyr; run; quit; /*Compute lead 1 forecast and residuals*/ proc iml; Use work.xfor; read all var {n x forx resx} into xfor; use work.yfor; read all var {y fory resy lcl ucl} into yfor; use work.ph2; read all var {x y} into ph2; use work.sdyres;read all var {sdyr} into sdyr; use work.xest; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into xest; use work.yest; read all var {errorvar mu ma1_1 ar1_1 ar1_2 ar1_3 i1_1} into yest; /*Measurement process model estimates*/ mux=xest[1,2]; sa=sqrt(xest[1,1]); phx1 =xest[1,3];phx2 =xest[1,4]; phx4=xest[1,5];phx10=xest[1,6]; phx11=xest[1,7]; phx0=mux*(1-phx1-phx2-phx4-phx10-phx11); /*Transfer & BP model estimates*/ muy=yest[1,2]; su=sqrt(yest[1,1]);sdr=sdyr[1,1]; thbp1=yest[1,3]; phbp1=yest[1,4]; phbp2= yest[1, 5]; phbp4=yest[1, 6];omg0=yest[1,7]; phbp0=muy*(1-phbp1-phbp2-phbp4); /*Set number of rows*/ t=nrow(xfor); m=nrow(ph2); /*Join X and Y forecast & residual series*/ /*W:c1=n c2=x c3=forx c4=resx c5=y c6=fory c7=resy c8=LCL c9=UCL*/ w=xfor||yfor; /*Add first PhII X & Y observations and calculate residuals*/ w[t,2]=ph2[1,1]; *add X+1; w[t,5]=ph2[1,2]; *add Y+1; w[t,4]=w[t,2]-w[t,3]; *calculate resX+1; w[t,7]=w[t,5]-w[t,6]; *calculate resY+1; w[t,8]=w[t,6]-3*sdr; *CCC chart LCL of Y+1; w[t,9]=w[t,6]+3*sdr; *CCC chart UCL of Y+1; newobs=j(1,9,1); *Row vector of 1's to store new values; k=1; *1st row of new data already in matrix w; do j=1 to m-1; w=w//newobs; *Append row to store new values; k=k+1; *counter of new observations; t=t+1; *counter of lead 1 forecasts; w[t,1]=w[t-1,1]+1; *Update n index; *Compute forecasts Xt(1)and Yt(1); w[t,3]=phX0 +phx1*w[t-1,2]+phx2*w[t-2,2]+phx4*w[t-4,2]+phx10*w[t-10,2] +phx11*w[t-11,2]; w[t,6]=omg0*(w[t,3]-phbp1*w[t-1,2]-phbp2*w[t-2,2]-phbp4*w[t-4,2])+phbp0 +phbp1*w[t-1,5]+phbp2*w[t-2,5]+phbp4*w[t-4,5]-thbp1*w[t-1,7]; *Compute residuals at t+1; w[t,2]=ph2[k,1]; *Incorporate new PhII X observation; w[t,5]=ph2[k,2]; *Incorporate new PhII Y observation; w[t,4]=w[t,2]-w[t,3]; *Compute new MS residual; w[t,7]=w[t,5]-w[t,6]; *Compute new OP residual; w[t,8]=w[t,6]-3*sdr; *Compute new LCL of Y; w[t,9]=w[t,6]+3*sdr; *Compute new UCL of Y; end; /*Create output matrix*/ create gndw var{n x forx resx y fory resy lcl ucl}; append from w; close gndw; run; quit; /*Subset 10 old and all new observations*/ data neww; 252 set gndw (firstobs=891); run; /*Plot CCC Chart*/ symbol1 c=black i=join v=circle; symbol2 c=blue i=join w=5; symbol3 c=red i =join; axis1 label=(a=90 'Overall Process Series'); axis2 label=('Observation No.') order= (895 to 988 by 10); proc gplot data=neww; plot y*n=1 fory*n=2 forx*n=3/vaxis=axis1 haxis=axis2 overlay; run; proc gplot data=neww; plot y*n=1 fory*n=2 LCL*n=3 UCL*n=3/vaxis=axis1 haxis=axis2 overlay; run; /************************* END of PROGRAM ******************************/ SAS Code 7 /**********************************************************************/ /* Phase II – Base Process with Disturbances */ /* Author: Jesús Cuéllar Fuentes */ /**********************************************************************/ /*Read in data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data*/ /*n=Obs No., x=MS series, y=output series*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /*Save Phase I parameter estimates, forecasts, and residuals*/ Proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11) outest=xest noprint; f lead=1 back=0 id=N out=xfor; i var=y crosscor=(x) noprint; e p=(1 2 4) q=1 input=(x)outest=yest; f lead=1 back=0 id=N out=yfor; run; quit; /*Compute and save std. dev. residuals rt*/ Proc univariate data=yfor normal; var residual; output out=sdyres std=residual; run; quit; Proc iml; use work.sdyres; read all var {residual} into sdyr; use work.xest; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into xest; use work.yest; read all var {errorvar mu ma1_1 ar1_1 ar1_2 ar1_3 i1_1} into yest; /*Measurement process model estimates*/ mux=xest[1, 2]; sa=sqrt(xest [1,1]); phx1=xest[1,3]; phx2=xest[1,4]; phx4=xest[1,5];phx10=xest[1,6]; phx11=xest[1,7]; phx0=mux*(1-phx1-phx2-phx4-phx10-phx11); *MP model constant; /*Transfer & BP model estimates*/ 253 muy=yest[1,2]; su=sqrt(yest[1,1]); sdr=sdyr[1,1]; thbp1=yest[1,3]; phbp1=yest[1,4]; phbp2=yest[1,5]; phbp4=yest[1,6]; omg0=yest[1,7]; phbp0=muy*(1-phbp1-phbp2-phbp4); *OP model constant; /*Number of simulated data points + 12 trailing values*/ i=1000; k=1:i; *Row vector from 1 to i; time=k`; *Transpose k to col vector=Time index; /*generate vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; a=j(i,1); a[1:i]=0; *MP white noise series; fx=j(i,1); fx[1:i]=0; *Lead-1 forecasts MP series; y=j(i,1); y[1:i]=0; *OP series; u=j(i,1); u[1:i]=0; *OP white noise series; z=j(i,1); z[1:i]=0; *MP series w/disturbance; fy=j(i,1); fy[1:i]=0; *Lead-1 forecasts OP series; r=j(i,1); r[1:i]=0; *lead-1 residuals; UCL=j(i,1); UCL[1:i]=0; *UCL CCC chart; LCL=j(i,1); LCL[1:i]=0; *LCL CCC chart; /*Shift in the mean of the BP series;*/ dpt=3*su; /*Generate MP with change in the output mean*/ do t=12 to i; *Generate random shocks; *a[t]=sa*rannor(100855); *u[t]=su*rannor(100855); *Generate Xt; x[t]=mux+phx1*(x[t-1]-mux)+phx2*(x[t-2]-mux)+phx4*(x[t-4]-mux) +phx10*(x[t-10]-mux)+phx11*(x[t-11]-mux)+a[t]; *Gererate Xt forecast; fx[t]=mux+phx1*(x[t-1]-mux)+phx2*(x[t-2]-mux)+phx4*(x[t-4]-mux) +phx10*(x[t-10]-mux)+phx11*(x[t-11]-mux); *Generate Yt; y[t]=omg0*(x[t]-phbp1*x[t-1]-phbp2*x[t-2]-phbp4*x[t-4])+phbp0 +phbp1*y[t-1]+phbp2*y[t-2]+phbp4*y[t-4]+u[t]-thbp1*u[t-1]; if t>=i-49 then z[t]=y[t]+dpt; else z[t]=y[t]; *Generate Yt forecast; fy[t]=omg0*(fx[t]-phbp1*x[t-1]-phbp2*x[t-2]-phbp4*x[t-4])+phbp0 +phbp1*z[t-1]+phbp2*z[t-2]+phbp4*z[t-4]-thbp1*r[t-1]; *Compute CCC chart control limits; UCL[t]=fy[t]+3*sdr; LCL[t]=fy[t]-3*sdr; *Compute residual; r[t]=z[t]-fy[t]; end; /*Output simulated series*/ out=time||a||u||x||fx||y||z||fy||r||UCL||LCL; create simdat var{time a u x fx y z fy r UCL LCL}; *generate data set; append from out; close simdat; run; quit; /*Subset last 100 simulated values*/ data simdat; set simdat (firstobs=901); run; quit; /*Plot simulated MS and white noise series*/ symbol1 c=black i=join v=circle; symbol2 c=blue i=join w=5; symbol3 c=green i=join w=5; symbol4 c=red i=join; symbol5 c=green i=joint v=star; 254 axis1 label=(a=90 'Simulated Data'); axis2 label=('Observation No.') order= (901 to 1001 by 10); /*Plot MP series*/ proc gplot data=simdat; plot x*time=1 fx*time=2 a*time=4 /vaxis=axis1 haxis=axis2 overlay; run; /*Plot OP series*/ proc gplot data=simdat; plot z*time=1 fy*time =2 r*time=5 /vaxis=axis1 haxis=axis2 overlay; run; /*Plot CCC chart*/ proc gplot data=simdat; plot z*time=1 fy*time=2 ucl*time=4 lcl*time=4 /vaxis=axis1 haxis=axis2 overlay; run; /************************* END of PROGRAM *****************************/ SAS Code 8 /**********************************************************************/ /* Phase II – Base Process Random Shocks with Disturbances */ /* Author: Jesús Cuéllar Fuentes */ /**********************************************************************/ /*Read in data*/ PROC IMPORT OUT=rdata DATAFILE= "C:\Documents and Settings\Jesús\My Documents\Baylor U\Dissertation\SAS Files\Raw_Data.xls" DBMS=EXCEL2000 REPLACE; SHEET="Data"; GETNAMES=YES; RUN; /*Subset Phase I data: n=Obs No., x=MS series, y=output series*/ data ph1; set rdata (firstobs=1 obs=900); if _N_= 438 then delete; *remove outlier; run; /*Save Phase I parameter estimates, forecasts, and residuals*/ Proc arima data=ph1; i var=x noprint; e p=(1 2 4 10 11) outest=xest noprint; f lead=1 back=0 id=N out=xfor; i var=y crosscor=(x) noprint; e p=(1 2 4) q=1 input=(x)outest=yest; f lead=1 back=0 id=N out=yfor; run; quit; /*Compute and save std. dev. residuals rt*/ Proc univariate data=yfor normal; var residual; output out=sdyres std=residual; run; quit; Proc iml; use work.sdyres; read all var {residual} into sdyr; use work.xest; read all var {errorvar mu ar1_1 ar1_2 ar1_3 ar1_4 ar1_5} into xest; use work.yest; read all var {errorvar mu ma1_1 ar1_1 ar1_2 ar1_3 i1_1} into yest; /*Measurement process model estimates*/ mux=xest[1,2]; sa=sqrt(xest[1,1]); phx1=xest[1,3]; phx2=xest[1,4]; phx4=xest[1,5];phx10=xest[1,6]; phx11=xest[1,7]; 255 phx0=mux*(1-phx1-phx2-phx4-phx10-phx11); *MP model constant; /*Transfer & BP model estimates*/ muy=yest[1,2]; su=sqrt(yest[1,1]); sdr=sdyr[1,1]; thbp1=yest[1,3]; phbp1=yest[1,4]; phbp2=yest[1,5]; phbp4=yest[1,6]; omg0=yest[1,7]; phbp0=muy*(1-phbp1-phbp2-phbp4); *OP model constant; /*Number of simulated data points + 12 trailing values*/ i=1000; k=1:i; *Row vector from 1 to i; time=k`; *Transpose k to col vector=Time index; /*generate vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; a=j(i,1); a[1 :i]=0; *MP white noise series; fx=j(i,1); fx[1:i]=0; *Lead-1 forecasts MP series; y=j(i,1); y[1:i]=0; *OP series; u=j(i,1); u[1:i]=0; *OP white noise series; d=j(i,1); d[1 :i]=0 ; *Disturbance series; fy=j(i,1); fy [1:i] =0; *Lead-1 forecasts OP series; r=j(i,1); r[1:i]=0; *lead-1 residuals; UCL=j(i,1); UCL[1:i]=0; *UCL CCC chart; LCL=j(i,1); LCL[1:i]=0; *LCL CCC chart; /*Shift in the mean of the random shocks*/ d1=0; *Set d1=k and d2=1; /*Shift in the variance of the random shocks*/ d2=3; *Set d2=k and d1=0; /*Generate OP with changes in ut shocks*/ do t=12 to i; *Generate random shocks; a[t]=sa*rannor(100855); *Measurement process shocks; if t>=i-49 then u[t]=d1*su+d2*su*rannor(100855); *Base process + disturbance; else u[t]=su*rannor(100855); *Base process w/o disturbance; *Generate Xt; x[t]=mux+phx1*(x[t-1]-mux)+phx2*(x[t-2]-mux)+phx4*(x[t-4]-mux) +phx10*(x[t-10]-mux)+phx11*(x[t-11]-mux)+a[t]; *Gererate Xt forecast; fx[t]=mux+phx1*(x[t-1]-mux)+phx2*(x[t-2]-mux)+phx4*(x[t-4]-mux) +phx10*(x[t-10]-mux)+phx11*(x[t-11]-mux); *Generate Yt; y[t]=omg0*(x[t]-phbp1*x[t-1]-phbp2*x[t-2]-phbp4*x[t-4])+phbp0 +phbp1*y[t-1]+phbp2*y[t-2]+phbp4*y[t-4]+u[t]-thbp1*u[t-1]; *Generate Yt forecast; fy[t]=omg0*(fx[t]-phbp1*x[t-1]-phbp2*x[t-2]-phbp4*x[t-4])+phbp0 +phbp1*y[t-1]+phbp2*y[t-2]+phbp4*y[t-4]-thbp1*r[t-1]; *Compute CCC chart control limits; UCL[t]=fy[t]+3*sdr; LCL[t]=fy[t]-3*sdr; *COmpute residual; r[t]=y[t]-fy[t]; end; /*Output simulated series*/ out=time||a||u||x||fx||y||fy||r||UCL||LCL; create simdat var{time a u x fx y fy r UCL LCL}; *generate data set; append from out; close simdat; run; quit; /*Subset last 100 simulated values*/ data simdat; set simdat (firstobs=901); run; quit; /*Plot simulated MS and white noise series*/ symbol1 c=black i=join v=circle; 256 symbol2 c=blue i=join w=5; symbol3 c=green i=join w=5; symbol4 c=red i=join; symbol5 c=green i=joint v=star; axis1 label=(a=90 'Simulated Data'); axis2 label=('Observation No.') order= (901 to 1001 by 10); /*Plot MP series*/ proc gplot data=simdat; plot x*time=1 fx*time=2 a*time=4 /vaxis=axis1 haxis=axis2 overlay; run; /*Plot OP series*/ proc gplot data=simdat; plot y*time=1 fy*time =2 r*time=5 /vaxis=axis1 haxis=axis2 overlay; run; /*Plot CCC chart*/ proc gplot data=simdat; plot y*time=1 fy*time=2 ucl*time=4 lcl*time=4 /vaxis=axis1 haxis=axis2 overlay; run; /************************* END of PROGRAM *****************************/ SAS Code 9 /**********************************************************************/ /* Simulation to obtain the control limits of the I- and MR-Charts */ /* Author: Jesús Cuéllar Fuentes */ /**** *** ***************************************************************/ Proc iml; /*Measurement process model estimates*/ sa=1; phx=0.5; thx=0.5; /*Trans fer & B P m odel es ti mates* / su=1; omg0=0.3; phbp=0; thbp=0; /*Number of simulated data points*/ i=1101; *No. of Samples; k=1000; *No. of ctrl limits; n=101; *No. burn-in points; m=i-n; *No. of non-burnin points; /*Storage vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; a=j(i,1); a[1:i]=0; *MP white noise series; fx=j(i,1); fx[1:i]=0; *Lead-1 forecasts MP series; y=j(i,1); y[1:i]=0; *OP series; u=j(i,1); u[1:i]=0; *OP white noise series; fy=j(i,1); fy[1:i]=0; *Lead-1 forecasts OP series; r=j(i,1); r[1:i]=0; *lead-1 residuals; mr=j(i,1); mr[1:i]=0; *Moving ranges; iUCL=j(k,1); iLCL=j(k,1); *Cntrl limits I-Chart; mUCL=j(k,1); mLCL=j(k,1 ); *Cntrl limits MR-Chart; seed1=100855; seed2=052660; /*Generate OP with changes in ut shocks*/ do f=1 to k; do t=2 to i; a[t]=sa*rannor(seed1); *MP random shocks; u[t]=su*rannor(seed2); *BP Random shocks; x[t]=phx*x[t-1]+a[t]-thx*a[t-1]; *Xt; fx[t]=phx*x[t-1]-thx*a[t-1]; *Xt forecast; y[t]=omg0*(x[t]-phbp*x[t-1])+phbp*y[t-1]+u[t]-thbp*u[t-1];*Yt; fy[t]=omg0*(fx[t]-phbp*x[t-1])+phbp*y[t-1]-thbp*r[t-1];*Yt forecast; r[t]=y[t]-fy[t]; *Residual; 257 MR[t]=abs(r[t]-r[t-1]); *Moving ranges; end; *Sort from smallest to largest; r0 = r[(n+1):i]; r1 =r0; r0[rank(r0)] =r1; mr0=mr[(n+1):i]; mr1=mr0; mr0[rank(mr0)]=mr1; *Calcluate location of quantiles; ql0=int((m+1)*0.00135); qh0=int((m+1)*.99865); ql1=ql0+1; qh1=qh0+1; *Compute 0.135 and 99.865 quantiles; iucl[f]=r0[qh0]+(r0[qh1]-r0[qh0])*0.865; ilcl[f]=r0[ql0]+(r0[ql1]-r0[ql0])*0.035; mucl[f]=mr0[qh0]+(mr0[qh1]-mr0[qh0])*0.865; mlcl[f]=mr0[ql0]+(mr0[ql1]-mr0[ql0])*0.035; end; /*Output simulated control limits*/ out=iucl||ilcl||mucl||mlcl; create simctrl var{iucl ilcl mucl mlcl}; append from out; close simctrl; run; quit; Proc univariate data=simctrl noprint; var iucl; output out=iup mean=Mean stdmean=Ser p95=p95 p5=p5; run; quit; Proc univariate data=simctrl noprint; var ilcl; output out=ilo mean=Mean stdmean=Ser p95=p95 p5=p5; run; quit ; Proc univariate data=simctrl noprint; var mucl; output out=mrup median=median stdmean=Ser p95=p95 p5=p5; run; quit; Proc univariate data=simctrl noprint; var mlcl; output out=mrlo median=median stdmean=SEr p95=p95 p5=p5; run; quit; /************************* END of PROGRAM *****************************/ SAS Code 10 /**********************************************************************/ /* Simulation to obtain Run Lengths for the CCC & SCC Charts */ /* Step Shift in the Mean of the Base Process */ /* Author: Jesús Cuéllar Fuentes */ /**** *** ***************************************************************/ Proc iml; /*Measurement process model estimates*/ sa=1; phx=0.5; thx=0.5; /*Transfer & BP model estimates*/ su=1; omg0=0.3; phbp=-0.95; thbp=0; /*Shift in the mean of the BP series*/ dpt=1.5*su; /*Control limits*/ cof=3*sqrt((omg0**2)*(sa**2)+(su**2)); iucl=3.333; ilcl=-3.357; mucl=4.954; mlcl=0.0014; /*Number of simulated data points*/ i=601; *No. generated points/sample; k=1000; *No. generated samples; m=101; *No. burn-in runs; n=i-m; *Non burn-in points/sample; /*Storage vectors of 1's then set to 0*/ 258 x=j(i,1); x[1:i]=0; *MP series; a=j(i,1); a[1:i]=0; *MP white noise series; fx=j(i,1); fx[1:i]=0; *Lead-1 forecasts MP series; y=j(i,1); y[1:i]=0; *OP series; z=j(i,1); z[1:i]=0; *OP w/disturbance; u=j(i,1); u[1:i]=0; *OP white noise series; fy=j(i,1); fy[1:i]=0; *Lead-1 forecasts OP series; r=j(i,1); r[1:i]=0; *lead-1 residuals; mr=j(i,1); mr[1:i]=0; *Moving ranges; /*Vectors to store non-burnin out-of-control cases*/ ycount=j(i,1); ycount[1:i]=0; *CCC chart; rcount=j(i,1); rcount[1:i]=0; *I-Chart; mcount=j(i,1); mcount[1:i]=0; *MR-chart; tcount=j(i,1); tcount[1:i]=0; *All Charts; /*vectors to store run lengths*/ crl=j(k,1); crl[1:k]=0; irl=j(k,1); irl[1:k]=0;; mrl=j(k,1); mrl[1:k]=0; trl=j(k,1); trl[1:k]=0;; /*seeds random numb er gener ator*/ seed1=100855; seed2=052660; /*Loop to generate k samples*/ do f=1 to k; /*Loop to generate series of i-102 values*/ do t=2 to i; *Generate series, forecasts, residuals, moving ranges; a[t]=sa*rannor(seed1); u[t]=su*rannor(seed2); x[t]=phx*x[t-1]+a[t]-thx*a[t-1]; fx[t]=phx*x[t-1]-thx*a[t-1]; y[t]=omg0*(x[t]-phbp*x[t-1])+phbp*y[t-1]+u[t]-thbp*u[t-1]; if t>m then z[t]=y[t]+dpt; *Introduce disturbance; else z[t]=y[t]; fy[t]=omg0*(fx[t]-phbp*x[t-1])+phbp*z[t-1]-thbp*r[t-1]; r[t]=z[t]-fy[t]; MR[t]=abs(r[t]-r[t-1]); *Calculate CCC chart control limits; cucl=fy[t]+cof; clcl=fy[t]-cof; *Determine if values are outside control limits; if z[t]>cucl | z[t] 259 iRL[f]=iRL0[1,1]-1; end; *Determine run length of M-chart; if msum=0 then mrl[f]=501; else do; mRL0=loc(mrcount=1); mRL[f]=mRL0[1,1]-1; end; *Determine run length of all charts; if tsum=0 then tRL[f]=501; else do; tRL0=loc(tacount=1); tRL[f]=tRL0[1,1]-1; end; end; /*Ouput run length vectors*/ out=crl||irl||mrl||trl; create simrlength var{crl irl mrl trl}; append from out; close simrlength; run; quit; /************************* END of PROGRAM *****************************/ SAS Code 11 /**********************************************************************/ /* Simulation to obtain Run Lengths for the CCC & SCC Charts */ /* Step Shift in the Mean and Variance of the BP Random Shocks */ /* Author: Jesús Cuéllar Fuentes */ /**** *** ***************************************************************/ Proc iml; /*Measurement process model estimates*/ sa=1; phx=0.5; thx=0.5; /*Transfer & BP model estimates*/ su=1; omg0=0.3; phbp=0.95; thbp=0; /*Shift in the mean of the random shocks*/ d1=0; *Set d1=k and d2=1; /*Shift in the variance of the random shocks*/ d2=1; *Set d2=k and d1=0; /*Control limits*/ cof=3*sqrt((omg0**2)*(sa**2)+(su**2)); iucl=3.333; ilcl=-3.357; mucl=4.954; mlcl=0.0014; /*Number of simulated data points*/ i=1101; *No. generated points/sample; k=1000; *No. generated samples; m=101; *No. burn-in runs; n=i-m; *Non burn-in points/sample; /*Storage vectors of 1's then set to 0*/ x=j(i,1); x[1:i]=0; *MP series; a=j(i,1); a[1:i]=0; *MP white noise series; fx=j(i,1); fx[1:i]=0; *Lead-1 forecasts MP series; y=j(i,1); y[1:i]=0; *OP series; u=j(i,1); u[1:i]=0; *OP white noise series; fy=j(i,1); fy[1:i]=0; *Lead-1 forecasts OP series; r=j(i,1); r[1:i]=0; *lead-1 residuals; mr=j(i,1); mr[1:i]=0; *Moving ranges; /*Vectors to store non-burnin out-of-control cases*/ ycount=j(i,1); ycount[1:i]=0; *CCC chart; rcount=j(i,1); rcount[1:i]=0; *I-Chart; mcount=j(i,1); mcount[1:i]=0; *MR-chart; tcount=j(i,1); tcount[1:i]=0; *All Charts; 260 /*vectors to store run lengths*/ crl=j(k,1); crl[1:k]=0; irl=j(k,1); irl[1:k]=0;; mrl=j(k,1); mrl[1:k]=0; trl=j(k,1); trl[1:k]=0;; /*seeds random number generator*/ seed1=100855; seed2=052660; /*Loop to generate k samples*/ do f=1 to k; /*Loop to generate series of i-102 values*/ do t=2 to i; *Generate series, forecasts, residuals, moving ranges; a[t]=sa*rannor(seed1); if t>m then u[t]=d1*su+d2*su*rannor(seed2);*Introduce disturbance; else u[t]=su*rannor(seed2); x[t]=phx*x[t-1]+a[t]-thx*a[t-1]; fx[t]=phx*x[t-1]-thx*a[t-1]; y[t]=omg0*(x[t]-phbp*x[t-1])+phbp*y[t-1]+u[t]-thbp*u[t-1]; fy[t]=omg0*(fx[t]-phbp*x[t-1])+phbp*y[t-1]-thbp*r[t-1]; r[t]=y[t]-fy[t]; MR[t]=abs(r[t]-r[t-1]); *Calculate CCC chart control limits; cucl=fy[t]+cof; clcl=fy[t]-cof; *Determine if values are outside control limits; if y[t]>cucl | y[t] 261 end; end; /*Ouput run length vectors*/ out=crl||irl||mrl||trl; 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