Linear Profile Forecasting using Regression Analysis

Transcription

1 Proceedings of the 2012 International Conference on Industrial Engineering and Operations Management Istanbul, Turkey, July 3 6, 2012 Linear Profile Forecasting using Regression Analysis Hamideh Razavi and Sanaz Sharifi Ghazvini Faculty of Engineering Ferdowsi University of Mashhad Mashhad, Iran Abstract In this paper, the status of a single variable linear profile has been predicted for quality control, using regression analysis with maximum likelihood approach. In the proposed model, the parameters of the profile including the slope and the intercept are estimated using collected data. Afterwards, the control limits for parameters are calculated and introduced to the predicted values. Therefore it can be checked whether the profile is under statistical control or not. Although time series analytical approach is applied in this research, the profile characteristics do not necessarily depend on time and can be defined according to any other variable. Keywords Quality control, Profile, Forecasting, Maximum likelihood, Linear regression 1. Introduction There have been many research activities recently on the monitoring of profiles. In many practical situations, a quality variable (response variable) is functionally dependent on one or more explanatory variables. This relationship is named profile. This relationship is widely found in real applications such as automotive engineering and calibration processes. [1-6] Profile monitoring has been discussed in two phases. Phase I includes determination of the control limits and phase II monitors the condition of the profile. Many techniques have been used for the control of linear and multiple linear profiles. [5-12] An approach for establishing control charts involves a multivariate T 2 chart in order to monitor parameters. Another approach uses statistics based on deviations from the in-control line and exponentially weighted moving average chart (EWMA). [2] By this technique a method has been proposed for phase 1 monitoring based on F-test. [4] Some studies focused on change point model and defined control charts for monitoring the linear profile when the process parameters are all unknown. [8, 9] The effect of non-normality on monitoring linear profiles and also presence of profile autocorrelation are considered in some other studies [13, 14]. In many practical situations, the quality profile needs to be predicted for a requested interval. While profile monitoring has been widely studied for phase I and phase II of quality control, profile forecasting or phase III has hardly been discussed. In this paper, profile forecasting is discussed as phase III of profile control. The formulation of the model is presented and the solution method is described in detail. A case model is then solved using the presented methodology and SAS software. [15] Hence, the condition of the profile can be checked for any desired interval. 2. Problem Formulation Assuming the process is under statistical control, the mathematical representation of the model is defined with n observations y 1, y 2,, y n and n corresponding independent variable x 1, x 2,, x n. According to the nature of the process, the range of measurements is divided into m intervals in order to obtain the linear profile of the process in each interval. For the j th (j=1, 2,,m) interval, the relationship between response variable (Y), and explanatory variable X can be modeled as Equation (1). Y(j) = A 0j + A 1j X(j) (1) Where A 0j and A 1j are profile parameters related to each interval. It is assumed that A 0 and A 1 follows a simple linear regression model with first-order autoregressive errors as in Equations (2) and (3). However this assumption does not violate the methodology and other polynomial or even nonlinear representations can be applied. A 0j = β 0 + β 1 j + ε j (2) ε j = ρ 0 ε j 1 + a j A 1j = γ 0 + γ 1 j + ε j (3) ε j = ρ 1 ε j 1 + a j 1639

2 In above equations a j and a j are normally and independently distributed with mean equal to zero and variance σ 2 and ρ 0 and ρ 1 are the autocorrelation parameters. β 0 and β 1 are estimated according to A 0j and A 1j respectively which are obtained for each interval. It should be noted that each of the parameters A 0 and A 1 should remain under control through the process. The aim of this research is to predict these parameters in any period in future and check if those are under control. 3. Forecasting Method The main problem can be broken into two problems: 1. Obtaining the value of the parameters in each interval (A 0j,A 1j ) 2. Estimating the parameters of simple linear regression model with first-order autoregressive errors In order to solve the first problem, regression analysis is used in order to determine the slope and intercept of the profile related to each interval. Figure 1 shows three example of the regression for parameters. 1 2 m A 01 + A 11 X(1) A 02 + A 12 X(2) A 0m + A 1m X(m) Figure 1: schematic estimated parameters in any interval Therefore, the required data for forecasting the values of Y at the k th interval after the interval m, would be classified as in Equations (4). A 01, A 02,, A 0m A 11, A 12,, A 1m (4) X(m + 1), X(m + 2),, X(m + k) For solving the second problem, the maximum likelihood approach is used. [16] By using this technique, it is possible to estimate the parameters β 0, β 1, γ 0, γ 1, ρ 0, ρ 1 of Equations (2) and (3). This can be performed by a simple model in SAS (Statistical Analysis System) software package. [15] In general, using Equation (5), τ period ahead of T can be predicted. [16] Y T+τ (T) = ρ Y T+τ 1 (T) + (1 ρ )β 0 + β (X 1 T+τ ρ X T+τ 1 ) (5) Equation (5) can be used for predicting the slope and intercept (A 0,A 1 ), at the interval m + k. In fact, there are two sections of equation (5) that should be solved such as the set of Equations (6) and (7). and A 0(m+k) (m) = ρ 0 A 0(m+k 1) (m) + (1 ρ 0 )β 0 + β 1((m + k) ρ 0 (m + k 1)) A 0(m+k 1) (m) = ρ 0 A 0(m+k 2) (m) + (1 ρ 0 )β 0 + β 1((m + k 1) ρ 0 (m + k 2)) (6) A 0(m+1) (m) = ρ 0 A 0(m) (m) + (1 ρ 0 )β 0 + β ((m 1 + 1) ρ 0 (m)) A 1(m+k) (m) = ρ 1 A 1(m+k 1) (m) + (1 ρ 1 )γ 0 + γ 1 ((m + k) ρ 1 (m + k 1)) A 1(m+k 1) (m) = ρ 1 A 1(m+k 2) (m) + (1 ρ 1 )γ 0 + γ 1 ((m + k 1) ρ 1 (m + k 2)) (7) A 1(m+1) (m) = ρ 1 A 1(m) (m) + (1 ρ 1 )γ 0 + γ 1 ((m + 1) ρ 1 (m)) By solving above equations and estimating the A 0 and A 1 at the interval m + k, the general profile equation in that interval is obtained as Equation (8). Y = A 0 + A 1 X (8) 1640

3 In order to control the slope and intercept, it is necessary to apply control charts. Due to the nature of the data, it is preferred to use control charts for x and R. Equations (9) are employed in this research in which σ is estimated by the range of the samples, according to A 0m and A 1m by the historical data. [17] UCL = X + A 2 R LCL = X A 2 R (9) The tabulated constant A 2 can be found for various sample sizes in reference [17]. 4. Case study A set of historical data are employed in this study in order to verify the proposed method. (See Table 1) [15, 18] Although these data are taken from real applications (market share of a particular brand of toothpaste and the corresponding selling price), there are no limitations as to definitions of the variables. The question is the state of statistical control at the 9 th interval. Initially, data are divided into five intervals signified by p as in Table 1. For each interval linear regression method is applied and profile parameters are obtained. (See Table 2) In order to predict the slope and intercept at the 9 th period, maximum likelihood method is applied using SAS software and the results are shown in Table 3 and 4. Table 1: Historical data Time Y X P Table 2: Profile parameters P Linear Regression Intercept (A 0 ) Slope (A 1 ) Table 3: Interpolated line for A 0 Intercept ( β 0 ) Slope (β 1 ) Autocorrelation parameter (ρ 0 ) Table 4: Interpolated line for A 1 Intercept ( γ 0 ) Slope (γ 1 ) Autocorrelation parameter (ρ 1 ) The UCL and LCL according to Equation (9) are as in Equations (10) and (11). For the Intercept (A 0 ): UCL = A 0 + A 2 R = LCL = A 0 A 2 R = (10) For the slope (A 1 ): UCL = A 1 + A 2 R = LCL = A 1 A 2 R = (11) According to Table 3 and 4 and Equations (6) and (7), estimated intercept and slope in 9 th interval are equal to and respectively. It concludes that the profile will be in control in the 9 th period based on control charts x in figures 2 and 3. The R control charts have been similarly made that ensured the estate of control in both cases. 1641

4 Xbar Chart of Intercept 35 UCL=35.60 Intercept Mean _ X= Sample 4 5 LCL=11.42 Figure 2: Intercept Control chart Xbar Chart of Slope UCL= Slope Mean _ X= LCL= Sample 4 5 Figure 3: Slope control chart 5. Conclusion In this paper, a methodology was proposed for forecasting profile conditions within requested intervals. It uses the maximum likelihood approach similar to time series forecasting. However, the characteristics of linear profiles are such that do not necessarily depend on time. When the parameters of the model including the slope and intercept are estimated, the profile measures can be predicted for any intervals beyond the historical range. Therefore, the actions can be pre-planned. The model is implemented for a set of data collected in a case study. The results show that for a particular interval the profile will become out of control and precautions must be made. In future research, other regression models and estimation methods such as fuzzy regression or Cochrane- Orcutt can be used. Furthermore, other types of profiles can be studied for more complex processes or products. References 1. Lawless, J. F.; Mackay, R. J.; and Robinson, J. A. 1999, Analysis of Variation Transmission in Manufacturing Processes-Part I, Journal of Quality Technology 31, Kang, L. and Albin, S. L 2000, On-line Monitoring When the Process Yields a Linear Profile, Journal of Quality Technology 32, Stover, F. S., and Brill, R. V. 1998, Statistical Quality Control Applied to Ion Chromatography Calibrations, Journal of Chromatography A, 804,

5 4. Mahmoud, M. A., and Woodall, W. H. 2004, Phase I Monitoring of Linear Profiles with Calibration Application, Technometrics 46, Woodall, W. H., Spitzner, D. J., Montgomery, D. C., and Gupta, S. 2004, Using Control Charts to Monitor Process and Product Quality Profiles, Journal of Quality Technology 36, Noorossana, R., Amiri, A., and Soleimani, P. 2008, On the Monitoring of autocorrelated Linear Profile, Communications in Statistics, Theory and Method 37, Kim, K., Mahmoud, M. A., and Woodall, W. H. 2003, On the Monitoring of Linear Profiles, Journal of Quality technology 35, Zou, C., Zhang, Y., and Wang, Z. 2006, Control Chart Based on Change-point Model for Monitoring Linear Profiles, IIE Transactions 38, Mahmoud, M. A., Parker, P. A., Woodall, W, H., and Hawkins, D.M. 2007, A change-point Method for Linear Profile data, Quality and Reliability Engineering International 23, Mahmoud, M. A. 2008, Phase I Analysis of Multiple Linear Regression Profiles, Communications in Statistics, Simulation and Computation 37, Jensen, W. A., Birch, J. B., and Woodall, W. H. 2008, Monitoring Correlation Within Linear Profiles Using Mixed Models, Journal of Quality Technology. 12. Zou, C., Tsung, F., and Wang, Z. 2008, Monitoring Profiles based on Nonparametric Regression Methods, Technometrics 50, Noorossana, R., Vaghefi, S. A., and Amiri, A. 2004, The Effect of non-normality on Monitoring Linear Profiles, Proceeding of the 2 nd International Engineering Conference, Riyadh, Saudi Arabia. 14. Soleimani, P., Noorossana, R., and Amiri, A. 2009, Simple Linear Profiles Monitoring in the Presence of within Profile autocorrelation, Computer and Industrial Engineering 57, Der, G., and Everitte, B. S. 2008, A Handbook of Statistical Analyses using SAS, 3 rd Edition 16. Montgomery, D. C., Jennings, C. L., Kulahci, M. 2008, Introduction to Time series Analysis and forecasting, Wiley Series in Probability and Statistics, Montgomery, D. C., Introduction to statistical Quality Control, 5 th Edition. 18. Montgomery, D. C., Peck, and Vining, 2006, Introduction to Linear Regression Analysis, 4 th Edition. 1643