# Statistical Process Control

## Statistical Process Control

The objective of Statistical Process Control (SPC), is to detect changes in a process that may result from uncontrollable an predictable causes at unknown times. The most important tools for monitoring data are control charts. In principle, a control chart is a decision rule which signals that the process has changed if the control statistic lies outside the control limits. The first control charts were introduced by Shewhart (1926, 1931). Since this pioneering work many other control schemes have been introduced, such as the cumulative sum (CUSUM) chart of Page (1954) and the exponentially weighted moving average (EWMA) chart of Roberts (1959).

Up to the end of the 1980s one of the basic assumptions of nearly all papers was that the underlying observations are independent over time. Only some papers dealt with dependence problem. Goldsmith and Whitfield (1961) investigated the influence of correlation on the CUSUM chart by Monte Carlo simulations. In the same year Page (1961) mentioned: '*The effects of correlated observations have been examined in one case by Goldsmith and Whitfield; wider study is desirable.* ' The problem was addressed but one had to wait over 10 years more for the next papers. Then, Bagshaw and Johnson (1974, 1975, 1977) approximated the CUSUM statistic for correlated data by Brownian motion. Rowlands (1976) studied the performance of CUSUM schemes for AR(1) and ARMA (1,1) data (cf. Rowlands and Wetherill (1991)). Furthermore, Nikiforov (1975, 79, 83, 84) wrote a lot about CUSUM charts for ARIMA data. Finally, Vasilopoulos and Stamboulis (1978) proposed modified alart limits for the Shewhart chart in case of dependent data.

After the more theoretical oriented period of dealing with dependence of the data, in recent years it has been pointed out that quite often the independence assumption is incorrect. Wetherill (1977) reminded in his monography, that observations from modern industrial process are often autocorrelated. Then, in the end of the 1980s treating the dependence problem became very popular. Alwan (1989) analyzed 235 datasets which were frequently used as (standard) SPC examples. It turned out that in about 85% the control limits were misplaced. In nearly the half of these cases neglected correlation was responsible. Now, the problem was realized as severe.

Besides assessing these strong effects of correlation on the control charts as quality control tools in industrial statistics, one has to consider correlation as a typical pattern of data which we can find in medicine, environmental and financial statistics, and in automatic control in engineering. Here time series models are standard.

Many of our papers deal with '*Statistical process control for time series*' where recently research is done in the following directions:

- Monitoring highdimensional time series (O.Bodnar, W.Schmid)
- Application of control charts for finance (V. Golosnoy, Y. Okhrin, W. Schmid, I. Yatsyshynets, S. Ragulin)
- Characterization of control charts using stochastic ordering (Y. Okhrin, W. Schmid, M. Morais, A. Pacheco)