Time Series and Forecasting a Time Seres Is a Set of Observations {Y J}
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Chapter 2 Time Series and Forecasting
Chapter 2 Time series and Forecasting 2.1 Introduction Data are frequently recorded at regular time intervals, for instance, daily stock market indices, the monthly rate of inflation or annual profit figures. In this Chapter we think about how to display and model such data. We will consider how to detect trends and seasonal effects and then use these to make forecasts. As well as review the methods covered in MAS1403, we will also consider a class of time series models known as autore- gressive moving average models. Why is this topic useful? Well, making forecasts allows organisations to make better decisions and to plan more efficiently. For instance, reliable forecasts enable a retail outlet to anticipate demand, hospitals to plan staffing levels and manufacturers to keep appropriate levels of inventory. 2.2 Displaying and describing time series A time series is a collection of observations made sequentially in time. When observations are made continuously, the time series is said to be continuous; when observations are taken only at specific time points, the time series is said to be discrete. In this course we consider only discrete time series, where the observations are taken at equal intervals. The first step in the analysis of time series is usually to plot the data against time, in a time series plot. Suppose we have the following four–monthly sales figures for Turner’s Hangover Cure as described in Practical 2 (in thousands of pounds): Jan–Apr May–Aug Sep–Dec 2006 8 10 13 2007 10 11 14 2008 10 11 15 2009 11 13 16 We could enter these data into a single column (say column C1) in Minitab, and then click on Graph–Time Series Plot–Simple–OK; entering C1 in Series and then clicking OK gives the graph shown in figure 2.1. -
Demand Forecasting
BIZ2121 Production & Operations Management Demand Forecasting Sung Joo Bae, Associate Professor Yonsei University School of Business Unilever Customer Demand Planning (CDP) System Statistical information: shipment history, current order information Demand-planning system with promotional demand increase, and other detailed information (external market research, internal sales projection) Forecast information is relayed to different distribution channel and other units Connecting to POS (point-of-sales) data and comparing it to forecast data is a very valuable ways to update the system Results: reduced inventory, better customer service Forecasting Forecasts are critical inputs to business plans, annual plans, and budgets Finance, human resources, marketing, operations, and supply chain managers need forecasts to plan: ◦ output levels ◦ purchases of services and materials ◦ workforce and output schedules ◦ inventories ◦ long-term capacities Forecasting Forecasts are made on many different variables ◦ Uncertain variables: competitor strategies, regulatory changes, technological changes, processing times, supplier lead times, quality losses ◦ Different methods are used Judgment, opinions of knowledgeable people, average of experience, regression, and time-series techniques ◦ No forecast is perfect Constant updating of plans is important Forecasts are important to managing both processes and supply chains ◦ Demand forecast information can be used for coordinating the supply chain inputs, and design of the internal processes (especially -
Moving Average Filters
CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiple- pass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In M &1 this equation, x[ ] is the input signal, y[ ] is ' 1 % y[i] j x [i j ] the output signal, and M is the number of M j'0 points used in the moving average. This equation only uses points on one side of the output sample being calculated. Where x[ ] is the input signal, y[ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: x [80] % x [81] % x [82] % x [83] % x [84] y [80] ' 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: x[78] % x[79] % x[80] % x[81] % x[82] y[80] ' 5 This corresponds to changing the summation in Eq. -
Trend Analysis and Forecasting the Spread of COVID-19 Pandemic in Ethiopia Using Box– Jenkins Modeling Procedure
International Journal of General Medicine Dovepress open access to scientific and medical research Open Access Full Text Article ORIGINAL RESEARCH Trend Analysis and Forecasting the Spread of COVID-19 Pandemic in Ethiopia Using Box– Jenkins Modeling Procedure Yemane Asmelash Introduction: COVID-19, which causes severe acute respiratory syndrome, is spreading Gebretensae 1 rapidly across the world, and the severity of this pandemic is rising in Ethiopia. The main Daniel Asmelash 2 objective of the study was to analyze the trend and forecast the spread of COVID-19 and to develop an appropriate statistical forecast model. 1Department of Statistics, College of Natural and Computational Science, Methodology: Data on the daily spread between 13 March, 2020 and 31 August 2020 were Aksum University, Aksum, Ethiopia; collected for the development of the autoregressive integrated moving average (ARIMA) 2 Department of Clinical Chemistry, model. Stationarity testing, parameter testing and model diagnosis were performed. In School of Biomedical and Laboratory Sciences, College of Medicine and Health addition, candidate models were obtained using autocorrelation function (ACF) and partial Sciences, University of Gondar, Gondar, autocorrelation functions (PACF). Finally, the fitting, selection and prediction accuracy of the Ethiopia ARIMA models was evaluated using the RMSE and MAPE model selection criteria. Results: A total of 51,910 confirmed COVID-19 cases were reported from 13 March to 31 August 2020. The total recovered and death rates as of 31 August 2020 were 37.2% and 1.57%, respectively, with a high level of increase after the mid of August, 2020. In this study, ARIMA (0, 1, 5) and ARIMA (2, 1, 3) were finally confirmed as the optimal model for confirmed and recovered COVID-19 cases, respectively, based on lowest RMSE, MAPE and BIC values. -
Time Series and Forecasting
Time Series and Forecasting Time Series • A time series is a sequence of measurements over time, usually obtained at equally spaced intervals – Daily – Monthly – Quarterly – Yearly 1 Time Series Example Dow Jones Industrial Average 12000 11000 10000 9000 Closing Value Closing 8000 7000 1/3/00 5/3/00 9/3/00 1/3/01 5/3/01 9/3/01 1/3/02 5/3/02 9/3/02 1/3/03 5/3/03 9/3/03 Date Components of a Time Series • Secular Trend –Linear – Nonlinear • Cyclical Variation – Rises and Falls over periods longer than one year • Seasonal Variation – Patterns of change within a year, typically repeating themselves • Residual Variation 2 Components of a Time Series Y=T+C+S+Rtt tt t Time Series with Linear Trend Yt = a + b t + et 3 Time Series with Linear Trend AOL Subscribers 30 25 20 15 10 5 Number of Subscribers (millions) 0 2341234123412341234123 1995 1996 1997 1998 1999 2000 Quarter Time Series with Linear Trend Average Daily Visits in August to Emergency Room at Richmond Memorial Hospital 140 120 100 80 60 40 Average Daily Visits Average Daily 20 0 12345678910 Year 4 Time Series with Nonlinear Trend Imports 180 160 140 120 100 80 Imports (MM) Imports 60 40 20 0 1986 1988 1990 1992 1994 1996 1998 Year Time Series with Nonlinear Trend • Data that increase by a constant amount at each successive time period show a linear trend. • Data that increase by increasing amounts at each successive time period show a curvilinear trend. • Data that increase by an equal percentage at each successive time period can be made linear by applying a logarithmic transformation. -
Penalised Regressions Vs. Autoregressive Moving Average Models for Forecasting Inflation Regresiones Penalizadas Vs
ECONÓMICAS . Ospina-Holguín y Padilla-Ospina / Económicas CUC, vol. 41 no. 1, pp. 65 -80, Enero - Junio, 2020 CUC Penalised regressions vs. autoregressive moving average models for forecasting inflation Regresiones penalizadas vs. modelos autorregresivos de media móvil para pronosticar la inflación DOI: https://doi.org/10.17981/econcuc.41.1.2020.Econ.3 Abstract This article relates the Seasonal Autoregressive Moving Average Artículo de investigación. Models (SARMA) to linear regression. Based on this relationship, the Fecha de recepción: 07/10/2019. paper shows that penalized linear models can outperform the out-of- Fecha de aceptación: 10/11/2019. sample forecast accuracy of the best SARMA models in forecasting Fecha de publicación: 15/11/2019 inflation as a function of past values, due to penalization and cross- validation. The paper constructs a minimal functional example using edge regression to compare both competing approaches to forecasting monthly inflation in 35 selected countries of the Organization for Economic Cooperation and Development and in three groups of coun- tries. The results empirically test the hypothesis that penalized linear regression, and edge regression in particular, can outperform the best standard SARMA models calculated through a grid search when fore- casting inflation. Thus, a new and effective technique for forecasting inflation based on past values is provided for use by financial analysts and investors. The results indicate that more attention should be paid Javier Humberto Ospina-Holguín to automatic learning techniques for forecasting inflation time series, Universidad del Valle. Cali (Colombia) even as basic as penalized linear regressions, because of their superior [email protected] empirical performance. -
Package 'Gmztests'
Package ‘GMZTests’ March 18, 2021 Type Package Title Statistical Tests Description A collection of functions to perform statistical tests of the following methods: Detrended Fluctu- ation Analysis, RHODCCA coefficient,<doi:10.1103/PhysRevE.84.066118>, DMC coeffi- cient, SILVA-FILHO et al. (2021) <doi:10.1016/j.physa.2020.125285>, Delta RHODCCA coeffi- cient, Guedes et al. (2018) <doi:10.1016/j.physa.2018.02.148> and <doi:10.1016/j.dib.2018.03.080> , Delta DMCA co- efficient and Delta DMC coefficient. Version 0.1.4 Date 2021-03-19 Maintainer Everaldo Freitas Guedes <[email protected]> License GPL-3 URL https://github.com/efguedes/GMZTests BugReports https://github.com/efguedes/GMZTests NeedsCompilation no Encoding UTF-8 LazyData true Imports stats, DCCA, PerformanceAnalytics, nonlinearTseries, fitdistrplus, fgpt, tseries Suggests xts, zoo, quantmod, fracdiff RoxygenNote 7.1.1 Author Everaldo Freitas Guedes [aut, cre] (<https://orcid.org/0000-0002-2986-7367>), Aloísio Machado Silva-Filho [aut] (<https://orcid.org/0000-0001-8250-1527>), Gilney Figueira Zebende [aut] (<https://orcid.org/0000-0003-2420-9805>) Repository CRAN Date/Publication 2021-03-18 13:10:04 UTC 1 2 deltadmc.test R topics documented: deltadmc.test . .2 deltadmca.test . .3 deltarhodcca.test . .4 dfa.test . .5 dmc.test . .6 dmca.test . .7 rhodcca.test . .8 Index 9 deltadmc.test Statistical test for Delta DMC Multiple Detrended Cross-Correlation Coefficient Description This function performs the statistical test for Delta DMC cross-correlation coefficient from three univariate ARFIMA process. Usage deltadmc.test(x1, x2, y, k, m, nu, rep, method) Arguments x1 A vector containing univariate time series. -
Spatial Domain Low-Pass Filters
Low Pass Filtering Why use Low Pass filtering? • Remove random noise • Remove periodic noise • Reveal a background pattern 1 Effects on images • Remove banding effects on images • Smooth out Img-Img mis-registration • Blurring of image Types of Low Pass Filters • Moving average filter • Median filter • Adaptive filter 2 Moving Ave Filter Example • A single (very short) scan line of an image • {1,8,3,7,8} • Moving Ave using interval of 3 (must be odd) • First number (1+8+3)/3 =4 • Second number (8+3+7)/3=6 • Third number (3+7+8)/3=6 • First and last value set to 0 Two Dimensional Moving Ave 3 Moving Average of Scan Line 2D Moving Average Filter • Spatial domain filter • Places average in center • Edges are set to 0 usually to maintain size 4 Spatial Domain Filter Moving Average Filter Effects • Reduces overall variability of image • Lowers contrast • Noise components reduced • Blurs the overall appearance of image 5 Moving Average images Median Filter The median utilizes the median instead of the mean. The median is the middle positional value. 6 Median Example • Another very short scan line • Data set {2,8,4,6,27} interval of 5 • Ranked {2,4,6,8,27} • Median is 6, central value 4 -> 6 Median Filter • Usually better for filtering • - Less sensitive to errors or extremes • - Median is always a value of the set • - Preserves edges • - But requires more computation 7 Moving Ave vs. Median Filtering Adaptive Filters • Based on mean and variance • Good at Speckle suppression • Sigma filter best known • - Computes mean and std dev for window • - Values outside of +-2 std dev excluded • - If too few values, (<k) uses value to left • - Later versions use weighting 8 Adaptive Filters • Improvements to Sigma filtering - Chi-square testing - Weighting - Local order histogram statistics - Edge preserving smoothing Adaptive Filters 9 Final PowerPoint Numerical Slide Value (The End) 10. -
Public Health & Intelligence
Public Health & Intelligence PHI Trend Analysis Guidance Document Control Version Version 1.5 Date Issued March 2017 Author Róisín Farrell, David Walker / HI Team Comments to [email protected] Version Date Comment Author Version 1.0 July 2016 1st version of paper David Walker, Róisín Farrell Version 1.1 August Amending format; adding Róisín Farrell 2016 sections Version 1.2 November Adding content to sections Róisín Farrell, Lee 2016 1 and 2 Barnsdale Version 1.3 November Amending as per Róisín Farrell 2016 suggestions from SAG Version 1.4 December Structural changes Róisín Farrell, Lee 2016 Barnsdale Version 1.5 March 2017 Amending as per Róisín Farrell, Lee suggestions from SAG Barnsdale Contents 1. Introduction ........................................................................................................................................... 1 2. Understanding trend data for descriptive analysis ................................................................................. 1 3. Descriptive analysis of time trends ......................................................................................................... 2 3.1 Smoothing ................................................................................................................................................ 2 3.1.1 Simple smoothing ............................................................................................................................ 2 3.1.2 Median smoothing .......................................................................................................................... -
Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models
mathematics Article Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models Oscar Trull 1,* , Juan Carlos García-Díaz 1 and Alicia Troncoso 2 1 Department of Applied Statistics and Operational Research and Quality, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Department of Computer Science, Pablo de Olavide University, 41013 Sevilla, Spain; [email protected] * Correspondence: [email protected] Received: 8 January 2020; Accepted: 13 February 2020; Published: 18 February 2020 Abstract: The Holt–Winters models are one of the most popular forecasting algorithms. As well-known, these models are recursive and thus, an initialization value is needed to feed the model, being that a proper initialization of the Holt–Winters models is crucial for obtaining a good accuracy of the predictions. Moreover, the introduction of multiple seasonal Holt–Winters models requires a new development of methods for seed initialization and obtaining initial values. This work proposes new initialization methods based on the adaptation of the traditional methods developed for a single seasonality in order to include multiple seasonalities. Thus, new methods to initialize the level, trend, and seasonality in multiple seasonal Holt–Winters models are presented. These new methods are tested with an application for electricity demand in Spain and analyzed for their impact on the accuracy of forecasts. As a consequence of the analysis carried out, which initialization method to use for the level, trend, and seasonality in multiple seasonal Holt–Winters models with an additive and multiplicative trend is provided. Keywords: forecasting; multiple seasonal periods; Holt–Winters, initialization 1. Introduction Exponential smoothing methods are widely used worldwide in time series forecasting, especially in financial and energy forecasting [1,2]. -
1 Simple Linear Regression I – Least Squares Estimation
1 Simple Linear Regression I – Least Squares Estimation Textbook Sections: 18.1–18.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and variance σ2. We have seen that we can write x in terms of µ and a random error component ε, that is, x = µ + ε. For the time being, we are going to change our notation for our random variable from x to y. So, we now write y = µ + ε. We will now find it useful to call the random variable y a dependent or response variable. Many times, the response variable of interest may be related to the value(s) of one or more known or controllable independent or predictor variables. Consider the following situations: LR1 A college recruiter would like to be able to predict a potential incoming student’s first–year GPA (y) based on known information concerning high school GPA (x1) and college entrance examination score (x2). She feels that the student’s first–year GPA will be related to the values of these two known variables. LR2 A marketer is interested in the effect of changing shelf height (x1) and shelf width (x2)on the weekly sales (y) of her brand of laundry detergent in a grocery store. LR3 A psychologist is interested in testing whether the amount of time to become proficient in a foreign language (y) is related to the child’s age (x). In each case we have at least one variable that is known (in some cases it is controllable), and a response variable that is a random variable. -
Indicators of Technical Analysis on the Basis of Moving Averages As Prognostic Methods in the Food Industry
Kolkova, A. (2018). Indicators of Technical Analysis on the Basis of Moving Averages as Prognostic Methods in the Food Industry. Journal of Competitiveness, 10(4), 102–119. https://doi.org/10.7441/ joc.2018.04.07 INDICATORS OF TECHNICAL ANALYSIS ON THE BASIS OF MOVING AVERAGES AS PROGNOSTIC METHODS IN THE FOOD INDUSTRY ▪ Andrea Kolkova Abstract Competitiveness is an important factor in a company’s ability to achieve success, and proper forecasting can be a fundamental source of competitive advantage for an enterprise. The aim of this study is to show the possibility of using technical analysis indicators in forecasting prices in the food industry in comparison with classical methods, namely exponential smoothing. In the food industry, competitiveness is also a key element of business. Competitiveness, however, requires not only a thorough historical analysis not only of but also forecasting. Forecasting methods are very complex and are often prevented from wider application to increase competi- tiveness. The indicators of technical analysis meet the criteria of simplicity and can therefore be a good way to increase competitiveness through proper forecasting. In this manuscript, the use of simple forecasting tools is confirmed for the period of 2009-2018. The analysis was com- pleted using data on the main raw materials of the food industry, namely wheat food, wheat forage, malting barley, milk, apples and potatoes, for which monthly data from January 2009 to February 2018 was collected. The data file has been analyzed and modified, with an analysis of indicators based on rolling averages selected. The indicators were compared using exponential smoothing forecasting.