18.4 Robustified Regression: the rIC filter
- {filtX, KG, PreviousPs, clipInd} =
rICfil(y, mu, Sig, H, F, Q, R, cliptyp, As, bs)
- calculates the rIC filter for a (multivariate) time series
- {A, b, ctrl} =
calibrIC(T, Sig, H, F, Q, R, typ, A0, b0, e, N, eps, itmax, expl, fact0, aus)
- calibrates the rIC filter to a given efficiency loss in the ideal model.
|
The idea to think of the filter problem as a ``regression'' problem stems from Boncelet and Dickinson (1984) and Cipra and Romera (1991),
where we write ``regression'' because the parameter in this model is
stochastic, whereas one component of the observation is deterministic,
and thus this regression is
not covered by the robustness theory for common regression.
The robustification however then uses techniques of optimal influence
curves for regression to be found in chapter VII in Rieder (1994);
instead of taking M-estimators to get a robust
regression estimator with a prescribed influence curve, we use a one-step
procedure corresponding to a
Hampel-Krasker influence curve.
18.4.1 Filtering Problem as a Regression Problem
The model suggested by Boncelet and Dickinson (1984) and Cipra and Romera (1991)
uses the innovational representation (18.2) and reads
 |
(18.9) |
where
denotes the classical Kalman filter, which in this procedure
has to be calculated aside, too.
As already indicated, assuming normality, the classical Kalman filter is the
ML-Estimator for this model, with scores function
with
 |
(18.10) |
and Fisher information
![$ {\cal I}=\mathop{\rm {{}E{}}}\nolimits [ \Lambda\Lambda']=\Sigma_{t\vert t}^{-1}.$](xploreapplichtmlimg1966.gif) |
(18.11) |
18.4.2 Robust Regression Estimates
To get a robust estimator for model (18.9) we use an
influence curve (IC) of Hampel-Krasker form:
 |
(18.12) |
where
is a Lagrange multiplyer guaranteeing that the coorelation condition
is fullfilled; due to symmetry of
,
of form (18.12) is centered automatically;
furthermore
bounds the influence of
on
.
The reader not familiar to the notion of influence curve may recur to section 18.5
and look up some of the references given there.
Instead of replacing the ML-equation by an M-Equation to be solved for
,
we use a one-step with same asymptotic properties:
We note the following properties:
- Setting
,
reproduces the classical Kalman filter.
- There is quite a numerical problem determining
, which will be roughly explained in Section 18.5.
- As in the rLS the time expensive calibration step--here to find
--can be done beforehand.
- We calibrate the IC at the ideal model, so whenever we have a situation, where
the sequence of
stabilizes,
we may, after a sufficient stabilization, stop calibration and
use the last calibrated
for all subsequent times
.
For details as to this stabilization we again refer to Anderson and Moore (1979) and Moore and Anderson (1980).
and again we already note that
the rIC has preserved the crucial features from of 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 ``almost linear'' !
- all information from the past useful for the future is captured in the
values of
and
.
- and: the correction step is now bounded in influence, as
enters bounded.
18.4.3 Variants: Separate Clipping
As already seen in (18.10),
is decomposed into a sum of two
independent variables
and
. They may be interpreted as
estimating
and
, thus they represent in some sense the sensitive point to
AO and IO respectively.
Instead of simultaneous clipping of both summands,
just clipping the ``IO-part'', i.e.
,
or ``AO-part'', i.e.
, separately will therefore lead to a
robustified version specialized to IO's or AO's.
For the AO-specialization we get
 |
(18.15) |
As to the IO-variant we have to admit that the robustification against IO's that is possible
with rIC-sep-IO is limited. Here you should better take into account more information on the process
history up to that point. Encouraging results however have been obtained in a situation with
an unfavorable signal-to-noise ratio--in one dimension the quotient of
.
18.4.4 Criterion for the Choice of
As in the rLS case, we propose the assurance criterium for
the choice of
: We adjust the procedure to a given relative efficiency loss in the ideal model.
This loss is quantified in this case as the relative degradation of the ``asymptotic''
variance of our estimator which is in our situation just
, which in
the ideal model gives again the MSE.
Of course the lower the clipping
, the higher the relative efficiency loss, so
that we may solve
 |
(18.16) |
in
for a given efficiency loss
, which is monotonous in
.
18.4.5 Examples
For better comparability the examples for the rIC will use the same setups as
those for rLS. So we just write down the modifications necessary to get from the
rLS- to the rIC-example.
18.4.5.1 Example 7
As the first example is one-dimensional,
calibrIC
uses a
simultaneous Newton procedure to determine
, so neither
a number of grid points nor a MC-sample size is needed
and parameter N is without meaning, as well as
fact and expl. Nevertheless you are to transmit
them to
calibrIC
and, beside the rLS setting we
write
fact=1.2
expl=2
A0=0
b0=-1
typ= 0 ; rIC-sim
Next we calibrate the influence curve
to
.
ergIC=calibrIC(T,Sig,H,F,Q,R,typ,A0,b0,e,N,eps,itmax,
expl,fact,aus)
A=ergIC.A
b=ergIC.b
Calling
rICfil
is then very much as calling
rlsfil--just with some extra parameters:
res= rICfil(y,mu,Sig,H,F,Q,R,typ,A,b)
frx = res.filtX
The graphical output is then done just as in Example 4.
18.4.5.2 Example 8
The second example goes through analogously with the following modifications with respect to Example 5:
N=300
eps=0.01
itmax=15
aus=4
fact=1.2
expl=2
A0=0
b0=-1
typ= 0 ; rIC-sim
Note that as we are in
dimensions, integration along the directions is
1-dimensional and is done by a Romberg procedure; so the N might
even be a bit too large. The next modifications are straightforward:
ergIC=calibrIC(T,Sig,H,F,Qid,R,typ,A0,b0,e,N,eps,itmax,
expl,fact,aus)
A=ergIC.A
b=ergIC.b
res = kfilter2(y,mu,Sig, H,F,Qid,R)
fx = res.filtX
res= rICfil(y,mu,Sig,H,F,Qid,R,typ,A,b)
frx = res.filtX
The graphical result is displayed in Figure 18.7.
18.4.5.3 Example 9
Again, as in the third rLS-example, it is shown in the next
example that we really loose
some efficiency in the ideal model, using the rIC filter instead
of the Kalman filter; the following modifications are to be done
with respect to Example 6:
e=0.05
N=300
eps=0.01
itmax=15
aus=4
fact=1.2
expl=2
A0=0
b0=-1
typ= 1 ; rIC-sep-AO
ergIC=calibrIC(T,Sig,H,F,Q,R,typ,A0,b0,e,N,eps,itmax,
expl,fact,aus)
A=ergIC.A
b=ergIC.b
res = kfilter2(y,mu,Sig, H,F,Q,R)
fx = res.filtX
res= rICfil(y,mu,Sig,H,F,Q,R,typ,A,b)
frx = res.filtX
fry=(H*frx')'
All this produces the graphics in Figure 18.8.
18.4.6 Possible Extensions
As sort of an outlook, we only want to mention here the possibility of using different
norms to assess the quality of our procedures. The most important norms besides
the euclidean are in our context those derived from the Fisher
information of the ideal model (information-standardization) and the
asymptotic Covariance of
itself (self-standardization).
Generally speaking these two have some nice properties compared
to the euclidean norm; so among others, optimal influence curves in
this norm stay invariant under smooth transformation in the
parameter space, c.f. Rieder (1994).
In the context of normal scores, they even lead to a drastic simplification
of the calibration problem even in higher dimensions, c.f. Ruckdeschel (1999).
Nevertheless the use of this norm has to be justified by the application,
and in the XploRe quantlib kalman, they have not yet been included.