Final_1.pdf

The effective resistance of the circuit or the equivalent variance of the com-
bination is given as

(4.11)

Having determined the value of k, we can now compute the corrected
state and the variance of the corrected state. The process is now repeated
again for the next measurement until we reach the last of the planned n
measurements. The final outcome of taking a weighted average after mak-
ing all the measurements will work out to be the same value calculated using
the Kalman procedure.

Filtering the Random Walk

Let us now discuss the application of the Kalman filter to the random walk.
From Chapter 2, on time series, we know that a random walk series is a sim-
ple sum of white noise realizations up to the current time. In other words,
the next point in the random walk series is evaluated by adding to the cur-
rent point a random drawing from a Gaussian distribution. Also note that
this has relevance to stock prices, as the logarithm of stock prices is typically
modeled as a random walk.
Now suppose we are assigned the task of watching the random walk.
The outcome of the watching exercise is to come up with the random walk
series. In stock price terms, the watching exercise translates to coming up
with a time series of stock prices. To do that, we observe the prices at regu-
lar time intervals and record them. The resulting sequence of values consti-
tutes a random walk, and our mission is accomplished. Note that the last
traded price at each instance is known without error, and it is therefore pos-
sible to observe the series without error. And if there is no error in the ob-
servation, then the Kalman filter model does not apply. So why do we even
attempt such an exercise?
We address this matter in the following discussion. Broadly speaking,
the price at any given time instance may be construed as the price at which
the supply meets demand. Let us call this the equilibrium price. Let us now
frame the aim of the watching exercise as generating the sequence of equi-
librium prices over consecutive time intervals. In this context, the periodic
measurement approach amounts to using the observed price at a specific
time as the equilibrium price for the time interval. It is now easy to make
the case that the prices at the end of regular chunks of time are indeed ap-

Rx

RR

ii RR

= ()=

+

=

+

varˆ++ 11 |

22

22

εη

εη

εη

εη

σσ

σσ

60 BACKGROUND MATERIAL