Digital Signal Processing Reference
In-Depth Information
Figure 2.4 w(x)ofMLT(n) estimators.
By equation (2.25),
C
w
can be interpreted as a weighted covariance matrix. Hence,
a robust weight function
w
should descend to zero. This means that small weights are
given to those observations
z
i
that are highly outlying in terms of measure
z
i
C
1
w
z
i
.It
downweights highly deviating observations and consequently makes their influence in
the error criterion bounded. Note that SCM
C
is an
M
-estimator that gives unit weight
(
w
;
1) to all observations. Figure 2.4 plots the weight function (2.24) of MLT(
n
) esti-
mators for selected values of
n
. Note that weight function (2.24) tends to weight func-
tion
w
;
1 of the SCM as expected (since
T
k
,
n
tends to
F
k
distribution when
n !1
).
Thus, MLT(
n
)
C
for large values of
n
.
Some examples of
M
-estimators are given next; See [43-45, 48] for more detailed
descriptions of these estimators.
B
EXAMPLE 2.6
Huber's M
-
estimator
, labeled HUB(
q
), is defined via weight
for
x c
2
1
=b
,
w
(
x
)
¼
c
2
for
x
.
c
2
=
(
xb
),
where
c
is a tuning constant defined so that
q ¼ F
x
2
k
(2
c
2
) for a chosen
q
(0
,
q
1) and the scaling factor
b ¼ F
x
2(
kþ
1)
(2
c
2
)
þc
2
(1
q
)
=k
. The choice
q ¼
1 yields
w
;
1, that is, HUB(1) correspond to the SCM. In general, low
values of
q
increase robustness but decrease efficiency at the nominal circular
CN model. Figure 2.5. depicts weight function of HUB(
q
) estimators for selected
values of
q
.
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