Kalman Filtering 53
(^1) Kalman, R. E. ( 1960). “A New Approach to Filtering and Prediction Problems.”
Transaction of the ASME Journal of Basic Engineering, 82(Series D), 35–45.
It was under these circumstances that the Kalman filter was proposed by
R. E. Kalman. He addressed these issues in a direct and practical manner and
presented his ideas in a ground-breaking article^1 titled “A New Approach to
Linear Filtering and Prediction Problems.” The approach caught the atten-
tion and imagination of the engineering community, and the ideas found ap-
plication in multiple domains.
One of the key contributions of Kalman’s article is the notion of system
state,” or the current state of the system. It is represented as a vector of the
current values of various system parameters. The vector itself is deduced
from a set of measurements on the system that is in turn translated into
system-state terms. This translation is typically modeled as a linear equation.
Thus, the means to make observations and the equation translating the ob-
servation into the system state fully characterize the notion of system state.
Having established the notion of system state, Kalman proceeded to
cast a dynamical system in terms of system states. A dynamical system in
the Kalman-filtering approach is modeled as a sequence of transitions from
one system state to another. These transitions are also modeled as linear
equations.
Next, he asserted that to monitor the system effectively (for purposes of
control) it makes sense to make an assessment of the state that we are cur-
rently in and the state that we expect to transition to in the next time step.
In other words, we are in a situation where we are constantly predicting the
next system state and taking measurements to verify the predictions. The
Kalman filter provides a prescription to reconcile this sequence of predic-
tions followed by measurements to arrive at a sequence of optimal estimates
for system states. This approach to thinking of systems as a sequence of state
transitions was a radical departure from the thinking at the time. It started
a revolution of sorts in the field of control theory and marks the beginning
of a new era in the field commonly referred to by many as Modern Control
Theory.
So, how does this fit into our scheme of things? For one, we use the
Kalman-filtering technique to filter the noise from the observed spread in the
case of risk arbitrage. Describing this in the introduction therefore helps pro-
vide the context for its application later in the book.
To help to illustrate the Kalman-filtering ideas and also as a matter of
interest, we apply the Kalman-filtering concepts to smooth out a random
walk. Now, many practitioners of technical analysis make use of so-called
moving averages to smooth out or filter price series. This method of using
moving averages may be thought of as an attempt to estimate the sequence
of stock prices (states) after filtering out the noise. The common peeve