Signal Processing - November 2017 - 155
information can be quite valuable in -determining where to
focus efforts for improve system p- erformance.
Root MSE in Beamwidths (dB)
5
Scatter matrix estimation under model misspecification
Another widely encountered inference problem is the estimation
of the correlation structure, i.e., the scatter or covariance matrix,
of a data set. The estimation of the covariance/scatter matrix is a
central component of a wide variety of SP applications [30]:
adaptive detection and DOA estimation in array processing,
principal component analysis, signal separation, interference
cancellation, and the portfolio optimization in finance, just to
name a few. Even if the data may come from disparate applications, they usually share a non-Gaussian, heavy-tailed statistical
nature, as discussed in [49]. Estimating the covariance matrix of
a set of non-Gaussian data, however, is not a trivial task. In fact,
non-Gaussian distribution characterization typically requires
additional parameters that must be jointly estimated along with
the scatter matrix. Think, for example, of the (complex) t-distribution that has been widely adopted as a suitable and flexible
model able to characterize the non-Gaussian, heavy-tailed data
behavior [26], [30], [40]. A complex, zero-mean, random vector
x m ! C N is said to be t-distributed if its pdf can be expressed as
p X (x m | R, m, h)
-^ N +mh
C ^ N + mh m m m
c m c + x mH R -1 x m m
_ N1
, tr ^ R h = N,
h
h
^
h
r R
C m
(29)
where C ($) indicates the gamma function while m and h are the
so-called shape and scale parameters, and R is the scatter matrix.
This multidimensional pdf is obtained by assuming that vector x m
follows the compound-Gaussian model with Gaussian speckle
and inverse-Gamma distributed texture [40]. For proper identifiability, a constraint on R, e.g., tr ^ R h = N, needs to be imposed.
The complex t-distribution has tails heavier than the Gaussian for
every m ! (0, 3), and it becomes the complex Gaussian distribution for m " 3. As can be clearly seen from (29), to perform
some inference on a t -distributed data set, we must jointly estimate the shape and scale parameters along with the scatter matrix.
Unfortunately, as pointed out in [26], a joint ML estimator of these
three quantities presents convergence and even existence issues.
Moreover, as discussed in the section "A Covariance Inequality in
the Presence of Misspecified Models," the t-distribution may be
only an approximation of the true heavy-tailed data model. To
overcome these problems, the SP practitioner has fundamentally
two choices: 1) to apply some robust covariance matrix estimator
(see [30] and [49] for further details) or 2) to assume a simpler, but
generally misspecified, model for characterizing the data, gaining
the possibility to derive a closed-form estimator at the cost of a
loss in the estimation performance [10], [12]. If option 2) is adopted, the most reasonable choice for the simplified data model is the
complex Gaussian distribution:
2
f X ^ x m | i h _ f X ^ x m | R, v h
=
1
^rv h R
2 N
exp e -
x mH R -1 x m
v
2
o, tr ^Rh = N.
(30)
MML
MCRB
CRB
0
-5
-10
-15
-20
0
5
10
15 20 25
SNR (dB)
30
35
40
FIGURE 1. The MSE of the MML estimator, the MCRB, for the DOA
estimation problem. Simulation parameters are set as M = 18 element
ULA, the array position errors of v e = 0.01m of standard deviation,
2
3
i t = 90c, i j = 87c, and v j = 10 (see [37]).
In fact, the joint (constrained) MML estimator of the scatter matrix and of the data power can be derived as
t CMML =
/
M
/
M
N
/
x mH x m m = 1
x m x mH ,
m =1
2
vt CMML =
M
1 / xH /
t -1 x .
NM m = 1 m CMML m
(31)
Two comments are in order:
t CMML converges to the true scatter
1) It can be shown that /
a.s.
t
matrix, i.e., / CMML M"
/; thus, it can be successfully
"3
applied to estimate it [10], [12].
2) It is computationally inexpensive and easy to implement,
t CMML feasible in real-time applicawhich makes the use of /
tions, e.g., in adaptive radar detection.
Along with knowledge of the MML estimator convergence point, the performance loss that has resulted from model
-m ismatch should also be assessed. To this purpose, since the
Gaussian model is nested in heavy-tailed t-distributed model (see
the section "An Interesting Case: A Lower Bound on the MSE
via the MCRB"), we can evaluate the MCRB for the problem at
hand and compare it with the CRB. As an example, in -Figure 2,
we compare the curves relative to the constrained CRB (CCRB)
for the estimation of the scatter matrix under matched conditions
(i.e., when the true t-distribution is assumed), the constrained
MCRB (CMCRB) [11] (i.e., when the misspecified Gaussian
model is assumed), and the MSE of the constrained MML estimator of (31) (details of the calculations can be found in [12]). The
distance between the CCRB and the CMCRB curves provides
a measure of the performance loss due to model mismatch. As
expected, the loss increases when the shape parameter m reaches
zero, i.e., when the data have an extremely heavy-tailed behavior.
However, when m " 3 , i.e., when the t-distribution tends to the
Gaussian one, the CCRB and the CMCRB tend to coincide. We
note that the constrained MML estimator of the scatter matrix
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
155
Table of Contents for the Digital Edition of Signal Processing - November 2017
Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
Signal Processing - November 2017 - 1
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Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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