# On multivariate control charts

## Sobre gráficos de controle multivariados

## Frisén, Marianne

http://dx.doi.org/10.1590/S0103-65132011005000010
Prod, vol.21, n2, p.235-241, 2012

## Abstract

Industrial production requires multivariate control charts to enable monitoring of several components. Recently there has been an increased interest also in other areas such as detection of bioterrorism, spatial surveillance and transaction strategies in finance. In the literature, several types of multivariate counterparts to the univariate Shewhart, EWMA and CUSUM methods have been proposed. We review general approaches to multivariate control chart. Suggestions are made on the special challenges of evaluating multivariate surveillance methods.

## Keywords

Surveillance. Monitoring. Quality control. Multivariate evaluation. Sufficiency.

## Resumo

A produção industrial requer o uso de gráficos de controle para permitir o monitoramento de vários componentes. Recentemente tem havido um aumento de interesse também em outras áreas como a detecção do bioterrorismo, vigilância espacial e estratégias de operação na área financeira. Na literatura, vários tipos de gráficos multivariados têm sido propostos contrapondo-se aos gráficos univariados de Shewhart, EWMA e CUSUM. Uma revisão geral sobre os gráficos de controle multivariados é apresentada. Sugestões são dadas em especial aos desafios em avaliar métodos multivariados em vigilância.

## Palavras-chave

Vigilância. Monitoramento. Controle de qualidade. Avaliação multivariada. Suficiência.

## References

ALT, F. B. Multivariate quality control. In: JOHNSON, N. L.; KOTZ, S. (Ed.). Encyclopedia of Statistical Science. New York: Wiley, 1985.

BASSEVILLE, M.; NIKIFOROV, I. Detection of abrupt changes - Theory and application. Englewood Cliffs: Prentice Hall, 1993.

BENJAMINI, Y.; HOCHBERG, Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society B, v. 57, p. 289-300, 1995.

BOCK, D. Aspects on the control of false alarms in statistical surveillance and the impact on the return of financial decision systems. Journal of Applied Statistics, v. 35, p. 213-227, 2008.

BODNAR, O.; SCHMID, W. Year. CUSUM control schemes for multivariate time series. In: FRONTIERS in statistical quality control. Intelligent statistical quality control. Warsaw, 2004.

CHAN, L. K.; ZHANG, J. Cumulative sum control charts for the covariance matrix. Statistica Sinica, v. 11, p. 767-790, 2001.

CROSIER, R. B. Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, v. 30, p. 291-303, 1988.

FRICKER, R. D. Directionally sensitive multivariate statistical process control procedures with application to syndromic surveillance. Advances in Disease Surveillance, v. 3, p. 1-17, 2007.

FRISÉN, M. Evaluations of methods for statistical surveillance. Statistics in Medicine, v. 11, p. 1489-1502, 1992.

FRISÉN, M. Statistical surveillance. Optimality and methods. International Statistical Review, v. 71, p. 403-434, 2003.

FRISÉN, M.; ANDERSSON, E.; SCHIÖLER, L. Evaluation of multivariate surveillance. Journal of Applied Statistics, 2010a.

FRISÉN, M.; ANDERSSON, E.; SCHIÖLER, L. Sufficient reduction in multivariate surveillance. Communications in Statistics - Theory and Methods, 2010b.

FRISÉN, M.; DE MARÉ, J. Optimal surveillance. Biometrika, v. 78, p. 271-80, 1991.

FUCHS, C.; BENJAMINI, Y. Multivariate profile charts for statistical process control. Technometrics, v. 36, p. 182-195, 1994.

GOLOSNOY, V.; SCHMID, W.; OKHRIN, I. Sequential monitoring of optimal portfolioweights. In: FRISÉN, M. (Ed.). Financial surveillance. Chichester: Wiley, 2007.

HAUCK, D. J.; RUNGER, G. C.; MONTGOMERY, D. C. Multivariate statistical process monitoring and diagnosis with grouped regression-adjusted variables. Communications in Statistics. Simulation and Computation, v. 28, p. 309-328, 1999.

HAWKINS, D. M. Multivariate quality control based on regression-adjusted variables. Technometrics, v. 33, p. 61-75, 1991.

HEALY, J. D. A note on multivariate CUSUM procedures. Technometrics, v. 29, p. 409-412, 1987.

HOCHBERG, Y.; TAMHANE, A. C. Multiple comparison procedures. New York: Wiley, 1987.

HOTELLING, H. Multivariate quality control. In: EISENHART, C.; HASTAY, M. W.; WALLIS, W. A. (Ed.). Techniques of statistical analysis. New York: McGraw-Hill, 1947.

JACKSON, J. E. Multivariate quality control. Communications in statistics. Theory and methods, v. 14, p. 2657-2688, 1985.

KNOTH, S.; SCHMID, W. Monitoring the mean and the variance of a stationary process. Statistica Neerlandica, v. 56, p. 77-100, 2002.

KOURTI, T.; MACGREGOR, J. F. Multivariate SPC methods for process and product monitoring. Journal of Quality Technology, v. 28, p. 409-428, 1996.

LIU, R. Y. Control charts for multivariate processes. Journal of the American Statistical Association, v. 90, p. 1380-1387, 1995.

LOWRY, C. A. et al. A multivariate exponentially weighted moving average control chart. Technometrics, v. 34, p. 46-53, 1992.

LOWRY, C. A.; MONTGOMERY, D. C. A review of multivariate control charts. IIE Transactions, v. 27, p. 800-810, 1995.

LU, X. S. et al. Control chart for multivariate attribute processes. International Journal of Production Research, v. 36, p. 3477-3489, 1998.

MARAVELAKIS, P. E. et al. Identifying the out of control variable in a multivariate control chart. Communications in Statistics. Theory and Methods, v. 31, p. 2391-2408, 2002.

MARSHALL, C. et al. Statistical issues in the prospective monitoring of health outcomes across multiple units. Journal of the Royal Statistical Society A, v. 167, p. 541-559, 2004.

MASON, R. L.; TRACY, N. D.; YOUNG, J. C. Decomposition of T2 for multivariate control chart interpretation. Journal of Quality Technology, v. 27, p. 99-108, 1995.

MASTRANGELO, C. M.; RUNGER, G. C.; MONTGOMERY, D. C. Statistical process monitoring with principal components. Quality and Reliability Engineering International, v. 12, p. 203-210, 1996.

MOHEBBI, C.; HAVRE, L. Multivariate control charts: A loss function approach. Sequential Analysis, v. 8, p. 253-268, 1989.

MOSTASHARI, F.; HARTMAN, J. Syndromic surveillance: a local perspective. Journal of Urban Health, v. 80, p. 11-I7, 2003.

NGAI, H. M.; ZHANG, J. Multivariate cumulative sum control charts based on projection pursuit. Statistica Sinica, v. 11, p. 747-766, 2001.

OKHRIN, Y.; SCHMID, W. Surveillance of univariate and multivariate nonlinear time series. In: FRISÉN, M. (Ed.). Financial surveillance. Chichester: Wiley, 2007.

PIGNATIELLO, J. J.; RUNGER, G. C. Comparisons of multivariate CUSUM charts. Journal of Quality Technology, v. 22, p. 173-186, 1990.

ROGERSON, P. A. Surveillance systems for monitoring the development of spatial patterns. Statistics in Medicine, v. 16, p. 2081-2093, 1997.

ROLKA, H. et al. Issues in applied statistics for public health bioterrorism surveillance using multiple data streams: research needs. Statistics in Medicine, v. 26, p. 1834-1856, 2007.

ROSOLOWSKI, M.; SCHMID, W. EWMA charts for monitoring the mean and the autocovariances of stationary Gaussian processes. Sequential Analysis, v. 22, p. 257-285, 2003.

RUNGER, G. C. et al. Optimal monitoring of multivariate data for fault detection. Journal of Quality Technology, v. 39, p. 159-172, 2007.

RUNGER, G. C. Projections and the U2 chart for multivariate statistical process control. Journal of Quality Technology, v. 28, p. 313-319, 1996.

RYAN, T. P. Statistical methods for quality improvement. New York: Wiley, 2000.

SAHNI, N. S.; AASTVEIT, A. H.; NAES, T. In-Line process and product control using spectroscopy and multivariate calibration. Journal of Quality Technology, v. 37, p. 1-20. 2005.

SHEWHART, W. A. Economic control of quality of manufactured product. London: MacMillan and Co., 1931.

SHIRYAEV, A. N. On optimum methods in quickest detection problems. Theory of Probability and its Applications, v. 8, p. 22-46, 1963.

SKINNER, K. R.; MONTGOMERY, D. C.; RUNGER, G. C. Process monitoring for multiple count data using generalized linear model-based control charts. International Journal of Production Research, v. 41, p. 1167-1180, 2003.

SONESSON, C.; FRISÉN, M. Multivariate surveillance. In: LAWSON, A.; KLEINMAN, K. (Ed.). Spatial surveillance for public health. New York: Wiley, 2005.

STEINER, S. H.; COOK, R. J.; FAREWELL, V. T. Monitoring paired binary surgical outcomes using cumulative sum charts. Statistics in Medicine, v. 18, p. 69-86, 1999.

SULLIVAN, J. H.; JONES, L. A. A self-starting control chart for multivariate individual observations. Technometrics, v. 44, p. 24-33, 2002.

TIMM, N. H. Multivariate quality control using finite intersection tests. Journal of Quality Technology, v. 28, p. 233-243, 1996.

TSUI, K. L.; WOODALL, W. H. Multivariate control charts based on loss functions. Sequential Analysis, v. 12, p. 79-92, 1993.

TSUNG, F.; LI, Y.; JIN, M. Statistical process control for multistage manufacturing and service operations: a review and some extensions International Journal of Services Operations and Informatics, v. 3, p. 191-204, 2008.

WÄRMEFJORD, K. Multivariate quality control and diagnosis of sources of variation in assembled products. Thesis (Licentiat)-Göteborg University, 2004.

WESSMAN, P. Some Principles for surveillance adopted for multivariate processes with a common change point. Communications in Statistics. Theory and Methods, v. 27, p. 1143-1161, 1998.

WOODALL, W. H. Current research on profile monitoring. Produção, v. 17, p. 420-425, 2007.

WOODALL, W. H.; AMIRIPARIAN, S. On the economic design of multivariate control charts. Communications in Statistics - Theory and Methods, v. 31, p. 1665-1673, 2002.

WOODALL, W. H.; NCUBE, M. M. Multivariate cusum quality control procedures. Technometrics, v. 27, p. 285-292, 1985.

YASHCHIN, E. Monitoring variance components. Technometrics, v. 36, p. 379-393, 1994.

YEH, A. B. et al. A multivariate exponentially weighted moving average control chart for monitoring process variability. Journal of Applied Statistics, v. 30, p. 507-536, 2003.