efore we proceed with derivations for a linear version of the Kalman filter
we perform some general calculation we will use repeatedly. We would like to
evaluate the following
integral
where the
are matrixes and
are vectors. We
calculate
The
integral
is evaluated via completion of the
square:
for some
.
Hence, we
require
The expression
is positive and symmetrical, therefore such
exists. We
continue
We
have
We perform the change
in the integral. The Jacobian is
.
The integral is
We collect our results
together,
Observe that if
is such
that
then
with
Hence,
Therefore
is the mean of the resulting normal distribution. Consequently, we introduce
such that
and
calculate
The
is a quadratic function and we already know that
,
hence
We
have
Hence, the covariance of the result is the inverse of the above
expression:
Therefore, we may state the result
