Multivariate Autocorrelated Processes: Data and Shift Generation - ASQ

Multivariate Autocorrelated Processes: Data and Shift Generation

The comparison of out-of-control performance of multivariate control chart methods on autoregressive processes requires a consistent method of generating a multivariate process shift. By applying the shift to the mean vector of the noise series, the covariance structure of the data may be maintained. We present a program for generating multivariate autoregressive data with a shift in the mean vector of the noise series. The program can be used to generate multivariate data from a first order vector autoregressive model with a shift in the mean vector of the noise series. The data can then be used to compare the shift detection properties of multivariate control chart methods.

Key Words: Autocorrelation, Multivariate Quality Control.

By CHRISTINA M. MASTRANGELO and DAVID R. FORREST, University of Virginia, Charlottesville, VA 22904-4747

INTRODUCTION

A KEY assumption underlying many traditional univariate process control techniques is that the observations are independent, and serially correlated data clearly violate this assumption. Autocorrelation results in a number of problems, including an increase in the false alarm rate (Alwan (1992), Montgomery and Mastrangelo (1991), and Wardell et al. (1994)). This problem also extends to multivariate systems (Kramer and Schmid (1997) and Lowry and Montgomery (1995)).

To compare monitoring multivariate methods, such as Hotelling's T2 chart (Hotelling (1947) and Montgomery (2001, pp. 512 525)), the multivariate CUSUM charts (Pignatiello and Runger (1990)), the multivariate EWMA chart (Prabhu and Runger (1997) and Lowry et al. (1992)), and the MEWMA residuals chart (Mastrangelo and Forrest (2002)) in the presence of autocorrelation, the rst requirement is to be able to generate data for a speci ed model. The program presented here can be used to generate multivariate vector autoregressive data with a shift, where the shift is correspondingly propogated to the cross-correlated variables. Data sets from this program may be used to compare shift detection properties of competing multivariate quality control methods.

Several forms of multivariate process disturbances that can occur are:

  1. Additive shifts (e.g., a data transcription error).
  2. Innovational shifts which a ect the underlying time series, while maintaining the cross-correlation structure ( e.g., a steam valve opens, increasing temperature throughout the process).
  3. Innovational shifts which a ect the underlying time series, while changing the cross-correlation structure (e.g., a leak occurs, changing the cross-correlation structure between two variables, say Temperature 1 and Temperature 2).

In the uncorrelated univariate realm, process shifts can be added either as a mean shift or as a shift in the mean of the noise series. When auto-correlation is considered, the distinction between a mean shift and a noise shift becomes important (Del Castillo (1994)). In multivariate processes, the correlation structure can be maintained or broken (Choe (1997)). This program provides a tool for inducing innovational shifts, as described in point 2 above.

Suppose the process under consideration consists of p coded process variables, denoted by the p × 1 column vector Xt at time t. Asimple model with correlations between the variables, and also between lagged observations of variables, is the vector autoregressive VAR(1) model given by


where Xt is the observed process, µX is the process mean vector, is a vector of normally distributed random variates with a mean vector of zeroes and covariance matrix , and is the autocorrelation parameter matrix of the selected VAR(1) model.

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