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Multivariate Control Chart Soo King Lim 1.0 Multivariate Control Chart............................................................. 3 1.1 Multivariate Normal Distribution ............................................................. 5 1.1.1 Estimation of the Mean and Covariance Matrix ............................................... 6 1.2 Hotelling’s T2 Control Chart ..................................................................... 6 1.2.1 Hotelling’s T Square ............................................................................................. 8 1.2.2 T2 Average Value of k Subgroups ...................................................................... 10 Example 1 ...................................................................................................................... 13 1.2.3 T2 Value of Individual Observation .................................................................. 14 Example 2 ...................................................................................................................... 14 1.3 Two-Sample Hotelling’s T Square .......................................................... 16 Example 3 ...................................................................................................................... 17 Example 4 ...................................................................................................................... 19 1.4 Confidence Level of Two-Sample Difference Mean .............................. 20 1.5 Principal Component Analysis ................................................................ 22 1.6 Disadvantage of Using Multivariate Control Chart .............................. 22 - 1 - Soo King Lim Figure 1: An n by p matrix ..................................................................................................... 3 Figure 2: 5 by 3 measurement matrix for parameters length, width, and height of a simple for five times .......................................................................................................... 3 Figure 3: Variance-covariance matrix of results shown in Fig. 2 .......................................... 4 Figure 4: Calculation of mean, variance, and covariance of data shown in Fig. 2 ................ 5 2 2 Figure 5: (a) T mean and (b) Td dispersion of multivariate control charts .......................... 7 Figure 6: k subgroup of n by p measurement matrices ........................................................ 11 Figure 7: Measurement results of five samples each from (a) an old facility and (b) a new facility ................................................................................................................... 17 - 2 - Soo King Lim 1.0 Multivariate Control Chart Multivariate analysis is a branch of statistics concerning with the analysis of multiple measurements made on one or several samples. It is an array representing n measurement in row on each of p parameter in column. Figure 1 shows an n by p matrix. The matrix has row value from j = 1 to n and row value from i = 1 to p. X11 X12 . X1p X21 X22 . X 2 p X . . X n1 Xn2 . Xnp Figure 1: An n by p matrix Take for an example, measuring the length, width, and height of a box five times, the results in X matrix shown in Fig. 2. Column one is the length measurement, column two is the width measurement, and column three is the height measurement. 4.0 2.00 0.61 4.2 2.10 0.59 X 4.0 2.03 0.58 4.3 2.10 0.62 4.1 2.20 0.63 Figure 2: 5 by 3 measurement matrix for parameters length, width, and height of a simple for five times This set of five measurements, measuring three parameters can be described by its mean vector X and variance-covariance matrix S. Each row vector Xj is has measurement of the three parameters. The mean vector consists of the mean of each parameter. Using the example shown in Fig. 2, the mean is defined as n Xi Xij (1) j1 4.1 X 2.086 Thus, the mean vector matrix for three parameters is . 0.606 - 3 - Soo King Lim The variance-covariance matrix consists of the variances of the parameters along the main diagonal and the covariance between each pair of parameters in the other matrix positions. Covariance between p parameter is defined as 1 n Covariance Xij Xi Yij Yi (2) n 1 j1 The variance of p parameter is defined 1 n 2 Variance Xij Xi (3) n 1 j1 The variance-covariance S matrix is defined as n 1 T S X ij X i X ij X i (4) n 1 j1 Using the results shown in Fig. 2, the X ij X i matrix is equal to 0.12 0.08 0.004 0.08 0.02 0.005 X ij X i 0.12 0.08 0.006 , while the transpose of the matrix is 0.18 0.02 0.016 0.02 0.12 0.026 0.12 0.08 0.12 0.18 0.02 T X X i 0.08 0.02 0.08 0.02 0.12 . Thus, using equation (4), the ij 0.004 0.005 0.006 0.016 0.026 0.12 0.08 0.004 0.08 0.02 0.005 1 variance-covariance matrix S is S 0.12 0.08 0.006 5-1 0.18 0.02 0.016 0.02 0.12 0.026 0.12 0.08 0.12 0.18 0.02 0.08 0.02 0.08 0.02 0.12 and the results are shown in Fig. 3. 0.004 0.005 0.006 0.016 0.026 0.0170 0.0046 0.00085 S 0.0046 0.0059 0.00095 0.00085 0.00095 0.00043 Figure 3: Variance-covariance matrix of results shown in Fig. 2 - 4 - Soo King Lim The calculation of mean, variance, and covariance are shown in Fig. 4. T Parameter X Xi X Xi X Xi n ij ij ij L W H L W H L-W L-H W-H - 1 4.00 2.00 0.61 3.983 -0.086 0.004 -0.3425 0.01593 0.00034 - - 2 4.20 2.10 0.59 4.183 0.014 0.05856 -0.0669 0.016 0.00022 - 3 4.00 2.03 0.58 3.983 -0.056 -0.2230 -0.1035 0.00145 0.026 4 4.30 2.10 0.62 4.283 0.014 0.014 0.05996 0.05996 0.00019 5 4.10 2.20 0.63 4.083 0.114 0.024 0.46546 0.09799 0.00273 Mean 4.12 2.086 0.606 Covariance Variance 0.0170 0.0059 0.00043 0.00460 0.00085 0.00095 Figure 4: Calculation of mean, variance, and covariance of data shown in Fig. 2 1.1 Multivariate Normal Distribution Multivariate normal distribution is the common model for multivariate data analysis. The multivariate normal distribution model is an extension of univariate normal distribution model to fit vector observation. A p-dimensional vector of random parameters is X = X1, X2, X3, …., Xp for - < Xi < , for i = 1, 2, 3,….., p. It is said to have a multivariate normal distribution if its probability density function f(X) is in this form. f (X) f (X1,X2 ,....., X p ) (5) ρ/2 1 1/ 2 1 Σ exp X m)' 1(X m) 2π 2 where m = (m1, m2, ….., mp) is the vector of mean, is the correlation between two parameters, and is the variance-covariance matrix of the multivariate normal distribution. The shortcut notation of multivariate normal distribution is X = Np(m, ) (6) When p = 1 that is one dimensional vector, thus, X = X1, whereby it has normal distribution with mean m and variance 2. The distribution is 1 2 2 f (X) exp (X m) /(2 ) for - < X < . When p = 2, X = (X1, X2), it has 2 bivariate normal distribution with two dimensional vector means m = (m1, m2) and - 5 - Soo King Lim 2 1 12 variance-covariance matrix Σ 2 and the correlation between the two 21 2 random parameters given by 21 . 12 1.1.1 Estimation of the Mean and Covariance Matrix Let X1, X2, …, Xn be n p-dimensional vectors of observations that are sampled independently from Np(m, Σ), where p < n - 1, and Σ is the variance-covariance matrix of X. The observed mean vector X and the sample dispersion matrix are shown in equation (7). 1 n Covar iance (Xi X)(Yi Y) (7) n 1 i1 They are the unbiased estimators of m and Σ respectively. 1.2 Hotelling’s T2 Control Chart In 1947 Harold Hotelling introduced a statistic which uniquely lends itself to plotting multivariate observations. This statistic is named as Hotelling’s T2, which is scalar combining information from the dispersion and mean of several parameters. Owing to the fact that calculation is laborious and fairly complex coupled with requiring knowledge of matrix algebra, acceptance of multivariate control chart by industry low. In this modern computer age, with the help of computer to calculate complex solution, multivariate control charts began to attract attention. Indeed, the multivariate chart which displays Hotelling T2 statistic became so popular that it is at time being name as Shewhart control chart. As for the univariate case, when data are grouped, the T2 chart can be paired with a chart that displays a measure of variability within the subgroups for all the analyzed characteristics. The combined 2 2 T mean and Td dispersion control charts are the multivariate counterpart of the univariate X and S or X and R control charts. 2 2 Example of a Hotelling T mean and Td dispersion pair of control charts are shown in Fig. 5. - 6 - Soo King Lim 2 Td (a) (b) Figure 5: (a) T2 mean and (b) dispersion of multivariate control charts Each chart represents 12 consecutive subgroup measurements on the means of four parameters. The T2 chart for means indicates an out-of-control state for subgroup 2 2, 9 and 10, and 11. The Td dispersion chart indicates that subgroup 10 is also out of control. Based on the results, multivariate system is suspect. However, to find an assignable cause, one has to resort to the individual univariate control charts or some other univariate procedure that should accompany this multivariate chart. The Hotelling T2 distance is a measure that accounts for the covariance structure of a multivariate normal distribution. It was proposed by Harold Hotelling in 1947 and is called Hotelling T. It may be thought of as the multivariate counterpart of the Student’s t-statistic. The T2 distance is a constant multiplied by a quadratic form. This quadratic form is obtained by multiplying the three quantities, which are the vector of deviations between the observations and the mean m that is expressed by (X - m)T, the inverse of the variance-covariance matrix S-1, and the vector of deviation (X - m).
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