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15/02/2002, 14:30 — 15:30 — Room P3.31, Mathematics Building

Wolfgang Schmid, *Europa-Universität Viadrina , Frankfurt-Oder*

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Simultaneous and Multivariate Control Charts for Time Series Data

In this talk we present several new control charts for univariate as well as multivariate time series. All control schemes are EWMA (exponentially weighted moving average) charts.

First, simultaneous control schemes for the mean and the autocovariances of a univariate stationary process are introduced. A multivariate quality characteristic is considered. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The test statistic is obtained by smoothing this variable. Another control chart is based on a multivariate EWMA attempt which is directly applied to our quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts, too. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1,1) process with normal white noise.

EWMA control charts for multivariate time series were discussed by Kramer and Schmid (1997). Their aim was to find deviations in the mean behaviour. Here we focus on charts detecting changes between the cross-covariances of a multivariate stationary process. The starting point is again a multivariate characteristic. To introduce control charts a similar procedure is chosen as described above. In our comparison study the target process is taken as a 4-variate AR(1) process.

#### References

- Kramer, H. and Schmid, W. (1997). EWMA charts for multivariate time series. Sequential Analysis 16, 131-154.
- Rosolowski, M. and Schmid, W. (2001). EWMA charts for monitoring the mean and the autocovariances of stationary Gaussian process. Submitted for publication.
- Schmid, W. and Sliwa, P. (2001). Monitoring the cross-covariances of a multivariate time series. Technical Report.