the technical analyst additional information. The question before us now is,
therefore, how can we fine-tune the coarseness of the approximations using
the Kalman filter?
To achieve varying levels of coarseness using the Kalman filter, we make
use of an important property that relates to the sampling of a random walk
sequence; that is, the random walk sequence sampled at any frequency results
in a random walk sequence. To see why that is, consider a random walk se-
quence with observations at times 1, 2, 3, and so on. By definition, the ob-
servation at time 1 plus a value drawn from a normal distribution gives the
observation at time 2, and so on. Now let us sample the random walk at half
of the original observation frequency. This results in a new sequence with
the values for the times 1, 3, 5, and so on. Note that the value at time 3 is
given by the value at time 1 plus a drawing from a normal distribution to get
it to time 2, and then again by adding another drawing from a normal dis-
tribution to get it to time 3. Thus, the transition from time 1 to time 3 is ef-
fected by summing two random drawings from independent normal
distributions and, in turn, adding it to the value at time 1. But the sum of
66 BACKGROUND MATERIAL
FIGURE 4.4 Kalman Smoothing of Random Walk.04.40
4.45
4.50
4.55
4.60
10 20 30 40 50 60 70
DaysLog of S&P daily close
Smoothing with two-period sampling