Kalman Filtering 67
two independent normal random variables may itself be treated as a draw-
ing from a normal distribution with an adjustment to the variance of the dis-
tribution. Therefore, the transition from time 1 to time 3 is also effected by
adding the value at time = 1 to a drawing from a normal distribution. This
therefore fits the definition of a random walk sequence. We only need to
apply this idea in an iterative fashion to see that the random walk sequence
sampled at any frequency results in a random walk.
Armed with this information, we can conclude that the Kalman smooth-
ing approach may be applied to the random walk sequence sampled at mul-
tiple frequencies to achieve varying degrees of coarseness. In our example,
we calculate the state values by sampling the random walk at half the fre-
quency of the available data. Suppose we need to calculate the value of the
state at time t= 22. We use the observations y 22 ,y 20 ,y 18 ,.... Similarly, to es-
timate the state value at time t = 23, we use the observation values
y 23 ,y 21 ,y 19 ,.... The resulting smoothing on the logarithm of the S&P prices
is shown in Figure 4.4.
SUMMARY
The Kalman filter is an optimal state estimation process applied to a dy-
namic system that involves random perturbations.
Inherent in any discussion on the Kalman filter are the notions of state
and observation.
Kalman filtering may be summarized as a three-step process comprising
prediction, observation, and correction, or reconciliation of the predic-
tion with the observation.
The simplest case of the Kalman filter reduces to finding the average of
nnumbers.
The recursive least squares method is also a special case of the Kalman
filter that may be applied to filtering random walks.
When the state and observation variances are the same, that is, the sig-
nal-to-noise ratio is unity, then the estimation of the Kalman states for
a random walk boils down to a weighted average of the observations,
with the weights formed by ratios of Fibonacci numbers.
The degree of smoothness to be achieved in a random walk can be con-
trolled by varying the sampling rate of the random walk sequence.FURTHER READING MATERIAL
Kalman Filter
Maybeck, Peter S. Stochastic Model, Estimation and Control. (San Diego, California:
Academic Press, 1979).