18.2 Classical Method: Kalman Filter
- {filtX, KG, PreviousPs} =
kfilter2
(y, mu, Sig, H, F, Q, R)
- calculates the classical Kalman filter
for a (multivariate) time series
|
For the definition and notation of the Kalman filter we refer to
Härdle, Klinke, and Müller (2000, Section 10.2).
As for our purposes,
the state is more interesting, we have modified the quantlet
kfilter
by changing the output.
18.2.1 Features of the Classical Kalman Filter
At this point, we only recall the features that are--in our point of
view--central for the Kalman filter:
- an easy, understandable structure with an initialization step, a prediction step
and a correction step,
- the correction step is an easily evaluable
function--it is linear !
- all information from the past useful for the future is captured in the
value of
.
- the correction step is linear and thus not robust, as
enters unbounded;
These features--except for the last one--are to be preserved in
our filtering procedures.
18.2.2 Optimality of the Kalman Filter
The classical Kalman filter is characterized by an optimality property that
coincides with various different notions of optimality in the case of a normal
state-space model.
- best linear filter:
-
The Kalman filter is obtained as the linear filter minimizing the mean squared error
(MSE).
This is shown with Hilbert space theory, considering the closed linear spaces
generated by the observations
and
orthogonal decompositions as follows:
where
denotes the orthogonal projection onto the closed linear space
generated by
and
is called the innovation induced by
.
This decomposition will be the basis for the quantlet
rlsfil
.
- conditional expectation (under normality assumptions):
-
A very nice property is that under normality, classical Kalman filter and conditional expectation
coincide, so that
is not only optimal among all linear filters based on
, but among all
-measurable
filters.
- posterior mode (under normality):
-
Again under normality,
coincides with the posterior mode of
,
which is the basis for robustifications done by Fahrmeir and Künstler (1999).
- ML-Estimator in a Regression Model (under normality):
-
Finally, under normality the Kalman filter is also the
maximum likelihood estimator (MLE) for a
certain regression model with random parameter which is the basis for the
rICfil
.