Requeijo, José Gomes 1. Souza, Adriano Mendonça , ,

Transcription

1 T 2 CONTROL CHART TO PROCESSES WITH CROSS-AUTOCORRELATION Requeijo, José Gomes 1 Souza, Adriano Mendonça 2 1 UNIDEMI, Faculty of Science and Technology, New University of Lisbon, Portugal, , jfgr@fct.unl.pt 2 Department of Statistics, Federal University of Santa Maria, Brasil, , amsouza@smail.ufsm.br Abstract: Today's society is characterized by consumers becoming more educated and demanding for products and services they use. Wherever, organizations are structured to respond the explicit or implicit needs of consumers, which lead to increased competitiveness of organizations. This competitiveness is reflected in the ongoing quest to provide high quality products and excellence process development. The monitoring of productive processes is essential to respond adequately to these objectives and therefore the implementation of statistical techniques suited to different situations is crucial, as is the case of statistical control charts introduced in the 192s by Walter Shewhart. The Statistical Process Control (SPC), initially applied to a single variable, proved inadequate in studying simultaneously several quality characteristics of the same product. T 2 Charts, introduced in 1985 by Alt, are the appropriate statistical techniques to the simultaneous control of the averages of several product characteristics, advantages over the univariate study, once on the other hand it prevents the production of a large number of documents and leads to more accurate analysis by considering the correlation among variables. As for the univariate study, the multivariate control is based on the assumptions of Normality and data independence of all characteristic. When the assumption of independence of at least one characteristic is violated, the construction of T 2 charts directly on the data is an inadequate solution, being necessary to isolate the autocorrelation for each variable and construct T 2 charts based on residuals/forecast errors. Another relevant question, that has not deserved much attention by the scientific community, is related with possibility of cross-autorrelation existence among variables, i.e., the autocorrelation occurring in a variable is not exclusive of the influence of that variable but its effect may be transmitted for part or all variables. This paper presents a methodology for implementation of multivariate statistical control, considering the cross-autocorrelation, both for Phase I (preliminary or retrospective) and Phase II (monitoring) statistical control. In Phase I, it seems that the process is in-control and estimates the process parameters, the mean vector and covariance matrix. In addition, the multivariate process capability using specific indicators will be study. One suggestion is the use of capability indices C pm, PV, and LI. Verified the stability and capability of the process, Phase II begins consisting of monitoring the process in real time based on estimates performed in Phase I. When there is significant autocorrelation at least in one of the variables, it is necessary to model the process in order to contemplate cross-autocorrelation effect acting in the variables. Variables are estimated by a VAR(p) model, considering only the significant parameters. An example application for Phase I of multivariate statistical process control is presented, as well as multivariate process capability. Keywords: Multivariate Statistical Process Control (MSPC); Multivariate Process Capability; Autocorrelation; Model VAR(p). 1. Introduction The multivariate statistical process control (MSPC) deserved attention of many researchers, such as Alt (1985), Doganaksoy et al. (1991), Mason and Young (1998), Hayter and Tsui (1994), Lowry and Montgomery (1995), Mason et al. (1997), Nedumaran and Pignatiello (1998), Egyptian clover et al (27), 822

2 Montgomery (28 ), Pereira and Requeijo (28) and Young-il and Adams (29), among others. Due to the further article publication by Alt (1985), its great coverage is considered by many one of the greatest contributions to the study of this issue, being an indispensable reference to be checked. In many cases the assumptions of normality and independence are not always verified, so that traditional developments seem to be inadequate. To address these possible situations, it is suggested the data transformation when distributions do not follow the Normal distribution and modeling of the process if there is autocorrelation of quality characteristics. As for the univariate study, multivariate control is based on the assumptions of Normality and independence of data of all characteristic. When the assumption of independence of at least one characteristic is violated, the construction of T 2 charts directly on the data is an inadequate solution, being necessary to isolate the autocorrelation for each variable and construct T 2 charts based on residuals/forecast errors. Another relevant question that has not deserved much attention by the scientific community is related to the possibility of cross-autorrelation existence among variables, i.e., the autocorrelation occurring in a variable is exclusive of the influence of that variable but its effect may be transmitted for part or all variables. This paper presents a methodology for implementation of multivariate statistical control considering the crossautocorrelation, both for Phase I (preliminary or retrospective) and for Phase II (monitoring) statistical control. Phase I culminates with the multivariate study of process capability. If this proves ability to produce according to technical specifications defined in the stage of design and product/process development is initiated the Phase II, which consists of monitoring the process in real time. In Phase II, the main focus is the identification of possible special causes of variation and subsequent corrective actions to be implemented in the process. About multivariate analysis of process capability, are referred the studies of Taam et al. (1993), Chen (1994), Shahriari et al. (1995), Wang et al. (2) and Kotz and Johnson (22). 2. Methodology After collecting data for analysis, the existence of autocorrelation among observations is verified and since it is working on multivariate data the cross-correlation among variables is investigated. Determined the existence of correlation between variables and the autocorrelation, it is estimated the model of autoregression vectors (VAR), so that only the significant coefficients are present in the model, thus enabling the estimation of residuals with blank noise characteristics, i.e. with zero mean, constant variance and non-auto-correlated. For the implementation of multivariate statistical control of the process mean vector, when there is significant autocorrelation of one or more characteristics of quality is suggested modeling the process through a VAR (p) in detriment of integrated autoregressive models and mobile means (ARIMA), so that to be contemplated the possibility of cross-autocorrelation between variables under study. This study will be based on developments of Box, Jenkins and Reinsel (28), Morettin (26), Souza et al. (24), Enders (1995), Maddala (1992), Alexander (25) and Bueno (28). After obtaining a new data set represented by blank noise residuals originated from the modeling VAR (p) and meeting the requirements of control charts for independence and normality is applied the multivariate control charts. Like the univariate statistical process control (SPC), the multivariate statistical process control (MSPC) is also divided into two different phases, Phase I and Phase II. In Phase I is verified the process stability and in MSPC s Phase II future data are monitored. The traditional MSPC can only be implemented if the joint 823

3 distribution of p variables is a multivariate normal distribution N p ( µ,σ) with mean vector µ = (µ 1, µ 2,..., µ p ) and covariance matrix Σ. It is also assumed that variables considered in isolation are independent and identically distributed according to normal distributions. 2. Case Study The application refers to the multivariate statistical process control represented by the analysis of three quality characteristics, the cooling temperatures related to the tile firing process of a Brazilian ceramics company. 92 hourly measurements were carried out simultaneously to these three temperatures named as "RESF1", "RESF2" and "RESF3." Two approaches are presented to compare the results. The first one (approach 1) consists of verifying the existence of autocorrelation in the process and applying the methodology of VAR modeling to determine the residues and analyzed them by the Hotelling T 2 chart. The second one (approach 2) consists of implementing the T 2 charts directly to the original data series related to the three quality characteristics, with no concern to check the existence of autocorrelation in the process. 3.1 Approach 1 Based on 92 observations was carried out the MSPC s Phase 1 following the methodology of residues estimation in order to address the cross-autocorrelation between variables by VAR estimation methodology. In this context, it was verified by the FACE and FACPE that the process has significant auto-correlation, so that its modeling should be made. It was found that the process can be adjusted by a VAR (1) model, where the coefficients showed in equation (1) were estimated, having been considered in modeling only statistically significant coefficients. X X 1t 121, 626, 849 1,t 1 ε1t X 2t = 66, 466 +, 216, 6233 X 2,t 1 + ε2t X 3t 11538,, 5255, 4621 X,t ε 3 1 3t where "X 1 " "X 2 " and "X 3 " represent, respectively, the quality characteristics "RESF1", "RESF2" and "RESF3", Φ is the vector of coefficients defining the means of the process, Φ1 is the matrix of autoregressive coefficients and εt is the vector of residues (blank noise). Φ 121, 626 = 66, 466, 11538, Φ 1, 849 =, 216, 6233, 5255, 4621, ε t = ε ε ε 1t 2t 3t Subsequently, it was verified through residues analysis by means of the estimate correlation sample function (FACE) and partial autocorrelation sample function (FACPE) that residues are not auto-correlated and follow the normal distribution. Based on the residues determined for each time t was constructed its T 2 chart presented in Figure

4 12 T 2 CHART Phase 1 9 T Observation n. T ² LCL UCL Figure 1. T 2 control chart of residues The analysis of residues T 2 chart shows that the process on three temperatures under study is stable, ie, under statistical control, since no statistical value is the upper limit of control. It is also verified that at the 2 time 53 ( T = value) the point is near the upper limit of control ( UCL T 2 = ), although still with lower value Approach 2 With the original values from the 92 observations was constructed the T 2 chart, which is presented in Figure T 2 CHART Phase T Observation n. T ² LIC LSC Figure 2: T 2 control chart of original daa The analysis of original data s T 2 chart wrongly reveals the existence of four special causes of variation, with upper limit statistical values of control (items 3, 6, 53 and 54). Comparison of Approach 1 and Approach 2 Comparing T 2 control charts of residues and original data (Figure 1 and Figure 2) is verified that the effect of ignoring the verification of auto-correlation in the process leads to a drastic increase of false alarms. This is 825

5 an undesirable situation, since the investigation of possible causes leading to eventual (false) anomalies in the process will not lead to the identification of any problem. 3. Conclusions This paper aims to develop a methodology for implementation of multivariate statistical control to processes that exhibit significant autocorrelation. Overall, the main conclusion from this research lies in the application of T 2 control charts for auto-correlated data in detriment of control charts for traditional multivariate. The multivariate SPC is more appropriate than the univariate when is necessary to simultaneously control several variables. The multivariate study allows detecting possible changes in the correlation of variables with less information for the analysis (i.e. a single graph is sufficient). However, T 2 charts applied directly to the original data, ignoring the process autocorrelation, show many signs of special causes when actually, these signs are natural in the process. After modeling the process through VAR (p) models, the application of T 2 control charts to residues is revealed to be an effective method to the statistical control of auto-correlated processes. These models have advantages over standard models of ARIMA modeling for beholding the cross autocorrelation of different variables under study. Some of these mistakes can be enumerated by not being considered autocorrelation, such as: i) Considering a stable process, when actually there are special causes of variation, which corresponds to an unstable process, ii) considering a process out of statistical control, when it is really statistically stable (increase of false alarms), iii) incorrectly estimating parameters of the process, iv) performing inaccurate and/or incorrect analyses about process capability and v) loss of resources by undue interference in the production process in order to solve problems that really do not exist (false alarms). References Alexander, C. (25). Modelos de mercados: um guia para a análise de informações Finaceiras (Tradução José Carlos de Souza Santos), São Paulo: Bolsa de Mercadorias & Futuros, 25. Alt, F. B. (1985). Multivariate Quality Control, in Encyclopaedia of the Statistical Sciences, edited by S. Kotz and N. L. Johnson, Wiley, New York, 6, Bersimis, S., Psarakis, S. & Panaretos, J. (27). Multivariate Statistical Process Control Charts: An Overview, Quality and Reliability Engineering International, Vol. 23, pp Box, G. E., Jenkins, G. M., & Reinsel, G. C. (28). Time Series Analysis - Forecasting and Control, 4ª Edição. New Jersey: John Wiley & Sons. Chen, H. (1994). A Multivariate Process Capability Index Over a Rectangular Solid Tolerance Zone, Statistica Sinica, Vol. 4, pp Bueno, R. D. L. S. (28). Econometria de Séries Temporais. Cengade Learning Edições Lda., São Paulo, Brasil. Doganaksoy, N., Faltin, F. W. & Tucker, W. T. (1991). Identification of Out of Control Quality Characteristics in a Multivariate Manufacturing Environment, Communications in Statistics - Theory and Methods, 2, Enders, W. (1995) Applied econometric time series, New York: John Wiley and Sons, Inc. 826

6 Hayter, A. J. & Tsui, K. (1994). Identification and Quantification in Multivariate Quality Control Problems. Journal of Quality Technology, 26, Kotz, S. & Johnson, N. L. (22). Process Capability Indices - A Review, , Journal of Quality Technology, 34, Lowry, C. A. & Montgomery, D. C. (1995). A Review of Multivariate Control Charts, IIE Transactions, 27, Maddala, G. S. (1992). Introduction to econometrics, 2.ed. New Jersey: Prentice Hall. Mason, R. L. & Young, J. C. (1998). Hotelling's T 2 :A Multivariate Statistics for Industrial Process Control, Proceedings of the 52nd Annual Quality Congress, pp Mason, R. L., Tracy, N. D. & Young, J. C. (1997). A Practical Approach for Interpreting Multivariate T 2 Control Chart Signals,, Journal of Quality Technology, 29, Montgomery, D. (28). Introdution to Statistical Quality Control, 6 th Edition, Jonh Wiley & Sons, New York. Morettin, P. A. (26). Econometria Financeira -ABE Associação Brasileira de Estatística, São Paulo, Brasil. Nedumaran, G. & Pignatiello, J. J. (1998). Diagnosing Signals from T 2 and χ 2 Multivariate Control Charts, Quality Engineering, 1(4), Pereira, Z. L., & Requeijo, J. G. (28). Qualidade: Planeamento e Controlo Estatístico de Processos (Quality: Planning and Statistical Process Control), Editora Prefácio, Lisboa. Shahriari, H., Hubele, F. N. & Lawrence, F. P. (1995). A Multivariate Process Capability Vector. Proceedings of the 4th Industrial Engineering Research Conference, pp Nashville. Souza, A. M.; Samohyl, R. W.; Malavé, C. O. (24). Multivariate feedback control: an application in a productive process, Computers & Industrial Engineering, v. 46, p Taam, W., Subbaiah, P. & Liddy, J. W. (1993). A Note on Multivariate Capability Indices, Journal of Applied Statistics, Vol. 2, pp Wang, F. K., Hubele, N. F., Lawrence, F. P., Miskulin, J. D. & Shahriari, H. (2). Comparison of Three Multivariate Process Capability Indices, Journal of Quality Technology, 32, Young-il, K. & Adams, B. M. (29). Multivariate SPC for Recipe Preservation of Batch Processes. Quality and Reliability Engineering International, Vol. 26, pp