00061 29e7865a3a8e8c782852bef6326438

60

Hembree & Zimmer

where <l> = 1, H = 1, Q = 0, and R = oj. The Kalman filter estimate and covariance recursions for this state-space model are

[3]

where

and

An example of an application of a Kalman filter based on this model is given in Figurę 3. A seąuence of 100 observations was randomly generated according to the model outlined above with a variance oj of 0.25 and aj of 1.0. This series was then passed through a Kalman filter based on this model. The resulting estimates are also plotted.

There are several things to notę about this example. From the recursions given in [3], it is seen that the covariance and the gain converge to steady State values of zero. This is exactly what the model dictates should happen. After twenty observations or so, the mean level is estimated with a fair amount of precision so additional data does not add much information. It is seen in the plot that the data has little effect on the estimate after this point.

For our next example, lets assume that we have a process that has both white noise characteristics and some autocorrelation structure. This might arise in a situation where we have an autocorrelated process that can only be observed with some uncertainty. Again we use [21, but this time the noise term, aj, will be non-zero. We will also assume that our a priori estimate of the process State is Xo ~ N(0, oj). This then gives

^xt ~ ^xt-\ + ^w,    where w_t “ N(0,aj)

y_t = x, + v,    where v, - N(0,(j^)

The Kalman filter estimate and covariance recursions for this state-space model