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Adaptiye Hierarchical Bayesian Kalman Filtering

this can be shown to be Gaussian of the form

T^CXP

1

(2^)^łfnc_<(a,)

V <=0

where c, =(P," +o l) > or the prior covariance for the estimate at time step z plus the measurement error. The notation (aj) just indicates that this calculation comes from one of the k discrete Kalman filters. While this formula looks formidable, it is really quite simple to calculate. We only need keep a running sum of the sąuared residuals from each Kalman filter divided by the C term and keep a running product of the C terms from each filter. The C terms are already calculated as part of the usual Kalman filter algorithm.

An example of an implementation of the DBAF is given in Figures 8 through 11. To generate this example, Kalman filters were constructed for five values of gain. The gain values and the error parameters for each of the filters are given in Table 1. An arbitrary measurement error variance of 0.25 was chosen. A fiat prior probability distribution (pj{a;,Y₀) = 0.2) was used for the initial prior probabilities for each of the chosen gain values. This implies that we a priori assume that any of the five possible Kalman models are eąually likely.

Table 1. Parameters for the Kalman Filters

Gain	System Error	Measurement Error
0.1	0.0028	0.25
0.3	0.032	0.25
0.5	0.125	0.25
0.7	0.408	0.25
0.9	2.025	0.25