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Adaptive Hierarchical Bayesian Kalman Filtering

Relation to Statistical Process Control

A key part of any statistical process control program consists of the employment of various control charting schemes that track process performance history. By employing these schemes, we are able make decisions about process ąuality based on clearly defined sets of mathematical procedures and action rules.

The most popular control charting methodologies in use today are the Shewart, CUSUM and Exponentially Weighted Moving Average (EWMA) control charts. In these methodologies, the underlying assumption is that we have removed all assignable causes of process variation and are left with a process that varies randomly about sonie constant level. For scal ar processes, this is usually represented mathematically by

x,=ju + w, where w, ~ n(o,<t²)    [1]

The control chart procedurę estimates the current State of the process and utilizes that State estimate in a set of control rules to decide if the process has changed from the model in some meaningful way.

The estimates of the current process State are obtained as weighted functions of the observed data, Hunter (1986). In particular the equation for the estimate is given by

t

i=0

For the Shewart control chart the estimate for the current State is just the most recent observation. This gives the weighting function

w = 1 for i = t w = 0 for 0śi <t

For the CUSUM, the estimate of the current State is the average of all observations. This gives a weighting function,

¹ w

w, = - Vi

Finally, for the EWMA, the estimate of the current State is given by