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Adaptiye Hierarchical Bayesian Kalman Filtering

For the covariance analysis, the expected covariances were calculated for DBAFs consisting of 2, 3, 5, and 9 filter elements. Each of the filters had equal initial prior weight for each of the respective filter elements. The measurement error was arbitrarily set at 0.25 in all cases. The truth model was varied over a rangę of gains from 0.05 through 0.95. Table 2 summarizes the covariance analysis setup.

The results of the analysis are shown in Figures 14 and 15. The expected covariances are plotted against the gains for the truth models. It is clearly seen that the 2 and 3 element filters do not do well except where the elements are located. The 5 element filter does however do quite well and the 9 element filter is practically indistinguishable from the plot of the actual covariances. On the basis of this analysis the 5 element DBAF is clearly sufficient.

Conclusions

We have derived a discrete adaptive form of the Kalman filter, assuming the additive error was known and constant and that error that contributes to autocorrelation in the data was unknown and constant. We illustrated, through a model that defined our process as a random walk buried in white noise, the DBAF's applicability for adaptive State estimation in univariate quality control settings.

Through analysis of the weighting function, it was demonstrated that the filter generally behaves as desired. That is, morę weight is given to estimates based on filter elements that correspond to the truth model and less weight is given to those that differ from the truth model. The weight given to estimates derived from filter elements that differ from the truth model was shown to be asymmetrical. Models with gain values lower than the truth model (e.g., where we have over specified the contribution of the additive error relative to the autocorrelated error) are rejected morę easily than those with gain values greater than the truth model.

Table 2. Coyariance Analysis Setup

A Priori Filter Weights for Each Gain Yalue

u filters	.1	.2	.3	.4	.5	.6	.7	.8	.9
2	0.5	0.0	Ó.Ó	Ó.Ó	Ó.Ó	Ó.Ó	0.0	0.0	0.5
3	0.3333	0.0	0.0	0.0	0.3334	0.0	0.0	0.0	0.3333
5	0.2	0.0	0.2	0.0	0.2	0.0	0.2	0.0	0.2
9	0.1111	0.1111	0.1111	0.1111	0.1112	0.1111	0.1111	0.1111	0.1111