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AStA Advances in Statistical Analysis

, Volume 101, Issue 1, pp 67–94 | Cite as

Control charts for multivariate spatial autoregressive models

  • Robert Garthoff
  • Philipp OttoEmail author
Original Paper

Abstract

This paper deals with spatial detection of changes in model parameters of spatial autoregressive processes. The respective sequential testing problems are formulated. Moreover, we introduce characteristic quantities to monitor means or covariances of multivariate spatial autoregressive processes. Additionally, we also take into account the simultaneous surveillance of the mean vector and the covariance matrix. The aim is to apply control charts, important tools of sequential analysis, to these quantities. The considered control procedures are based on either cumulative sums or exponential smoothing. Further, we illustrate the methodology of statistical process control studying the spectrum of additive colors in a satellite photograph. Via simulation studies, the proposed control procedures are calibrated for a predefined average run length. In addition, we compare the performance of the control procedures considering the out-of-control situation. Eventually, the control charts are applied, and the signals of the different schemes are visualized. The final results are critically discussed.

Keywords

Multivariate CUSUM charts Multivariate EWMA charts Spatial autoregressive model 

Mathematics Subject Classification

62H11 62L10 62M30 91B72 

References

  1. Besag, J.: Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. 36, 192–225 (1974)MathSciNetzbMATHGoogle Scholar
  2. Bodnar, O., Schmid, W.: CUSUM control schemes for multivariate time series. In: Lenz, H.J., Wilrich, P.T. (eds.) Frontiers of Statistical Process Control 8, vol. 8, pp. 55–73. Physica-Verlag, Heidelberg (2006)Google Scholar
  3. Bodnar, O., Schmid, W.: CUSUM charts for monitoring the mean of a multivariate Gaussian process. J. Stat. Plan. Inference 141(6), 2055–2070 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brillinger, D.: Some Aspects of the Analysis of the Evoked Response Experience. In: Saleh, A., Csorgo, M., Dawson, D., Rao, J. (eds.) Proceedings of the International Symposium on Statistics and Related Topics, North-Holland Pub. Co., Amsterdam (1981)Google Scholar
  5. Cressie, N.: Statistics for Spatial Data. Wiley, USA (1993)zbMATHGoogle Scholar
  6. Crosier, R.: Multivariate generalizations of cumulative sum quality control schemes. Technometrics 30, 291–303 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Elvidge, C.D., Baugh, K.E., Kihn, E.A., Kroehl, H.W., Davis, E.R., Davis, C.W.: Relation between satellite observed visible-near infrared emissions, population, economic activity and electric power consumption. Int. J. Remote Sens. 18(6), 1373–1379 (1997)CrossRefGoogle Scholar
  8. Fingleton, B.: A generalized method of moments estimator for a spatial model with moving average errors, with application to real estate prices. Empir. Econ. 34, 35–57 (2008)CrossRefzbMATHGoogle Scholar
  9. Garthoff, R., Otto, P.: Simultaneous surveillance of means and covariances of spatial models. In: Steland, A., Rafajłowicz, E., Szajowski, K. (eds.) Stochastic Models, Statistics and Their Applications, Springer Proceedings in Mathematics & Statistics, vol 122, pp. 271–281. Springer International Publishing. (2015)Google Scholar
  10. Garthoff, R., Okhrin, I., Schmid, W.: Statistical surveillance of the mean vector and the covariance matrix of nonlinear time series. AStA Adv. Stat. Anal. 98(3), 225–255 (2014)MathSciNetCrossRefGoogle Scholar
  11. Grimshaw, S.D., Blades, N.J., Miles, M.P.: Spatial control charts for the mean. J. Qual. Technol. 45(2) (2013)Google Scholar
  12. Hotelling, H.: Multivariate quality control—illustrated by the air testing of sample bombsights. In: Eisenhart, C., Hastay, M., Wallis, M. (eds.) Techniques of Statistical Analysis, pp. 111–184. McGraw-Hill, New York (1947)Google Scholar
  13. Kelejian, H.H., Prucha, I.R.: A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autorgegressive disturbance. J. Real Estate Fin. Econ. 17, 99–121 (1998)CrossRefGoogle Scholar
  14. Kelejian, H.H., Prucha, I.R.: A generalized moments estimator for the autoregressive parameter in a spatial model. Int. Econ. Rev. 40, 509–533 (1999)MathSciNetCrossRefGoogle Scholar
  15. Kelejian, H.H., Prucha, I.R.: Estimation of simultaneous systems of spatially interrelated cross sectional equations. J. Econom. 118(1), 27–50 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kramer, H., Schmid, W.: EWMA charts for multivariate time series. Seq. Anal. 16, 131–154 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lazariv, T., Okhrin, Y., Schmid, W.: Behavior of EWMA type control charts for small smoothing parameters. Comput. Stat. Data Anal. 89, 115–125 (2015)MathSciNetCrossRefGoogle Scholar
  18. Lowry, C., Woodall, W., Champ, C., Rigdon, S.: A multivariate exponentially weighted moving average control chart. Technometrics 34, 46–53 (1992)CrossRefzbMATHGoogle Scholar
  19. Ngai, H.M., Zhang, J.: Multivariate cumulative sum control charts based on projection pursuit. Stat. Sin. 11, 747–766 (2001)MathSciNetzbMATHGoogle Scholar
  20. Ord, K.: Estimation methods for models of spatial interaction. J. Am. Stat. Assoc. 70(349), 120–126 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Otto, P., Schmid, W.: Detection of spatial change points in the mean and covariances of multivariate simultaneous autoregressive models. Biom. J. (2016). doi: 10.1002/bimj.201500148
  22. Page, E.: Continuous inspection schemes. Biometrika 41, 100–114 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Pignatiello, J., Runger, G.: Comparison of multivariate CUSUM charts. J. Qual. Technol. 22, 173–186 (1990)Google Scholar
  24. Rafajłowicz, E.: Detection of essential changes in spatio-temporal processes with applications to camera based quality control. In: Stochastic Models, Statistics and Their Applications, pp. 433–440. Springer (2015)Google Scholar
  25. Roberts, S.: Control charts based on geometric moving averages. Technometrics 1, 239–250 (1959)CrossRefGoogle Scholar
  26. Śliwa, P., Schmid, W.: Monitoring the cross-covariances of a multivariate time series. Metrika 61, 89–115 (2005a)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Śliwa, P., Schmid, W.: Surveillance of the covariance matrix of multivariate nonlinear time series. Statistics 39, 221–246 (2005b)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sutton, P.C.: A scale-adjusted measure of urban sprawl using nighttime satellite imagery. Remote Sens. Environ. 86(3), 353–369 (2003)CrossRefGoogle Scholar
  29. Sutton, P.C., Costanza, R.: Global estimates of market and non-market values derived from nighttime satellite imagery, land cover, and ecosystem service valuation. Ecol. Econ. 41(3), 509–527 (2002)CrossRefGoogle Scholar
  30. Sutton, P.C., Elvidge, C.D., Ghosh, T.: Estimation of gross domestic product at sub-national scales using nighttime satellite imagery. Int. J. Ecol. Econ. Stat. 8(S07), 5–21 (2007)MathSciNetGoogle Scholar
  31. Theodossiou, P.: Predicting shifts in the mean of a multivariate time series process: an application in predicting business failures. J. Am. Stat. Assoc. 88, 441–449 (1993)CrossRefGoogle Scholar
  32. Wall, M.M.: A close look at the spatial structure implied by the car and sar models. J. Stat. Plan. Inference 121(2), 311–324 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Whittle, P.: On stationary processes in the plane. Biometrika, pp. 434–449 (1954)Google Scholar
  34. Woodall, W.H., Ncube, M.M.: Multivariate cusum quality-control procedures. Technometrics 27(3), 285–292 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.European University ViadrinaFrankfurt (Oder)Germany

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