Signal Processing - November 2017 - 151

Power Estimation in Correlated Data

	D (p X fX |i) = 1 6tr (i -1 vr 2 R) - N + ln i - ln det (vr 2 R)@ .
2
(S8)
Keeping in mind that tr (R) = N, it is immediately verified
that the minimum is given by i 0 = vr 2, i.e., the pseudotrue
parameter is equal to the true power. After some basic calculus, the terms A i0 and B i0 are obtained as

	

A i0 _ E p )

2 2 ln fX (x m i)
i = i0 3
2i 2

= N2 - 13 E p " x Tm x m , =- N 4 ,
2vr
2i 0 i 0

0.1

0

0.2

0.4
0.6
0.8
One-Lag Coefficient: ρ

1

FIGURE S3. The MSE of the MML estimator, the MCRB, the estimated
MCRB, and the CRB as functions of t. Simulation parameters are set
as N = 8, M = 3N, and vr 2 = 4.

The CRB for the estimation of the statistical power
of the true model can be easily obtained as CRB (vr 2) =
2vr 4 MN . As expected, the CRB is always greater
that the MCRB on vr 2 , and they are equal if, and only if,
R = I N, i.e., when the model is correctly specified. We
can go on to investigate the properties of the MML estimator. Unlike the example in "Variance Estimation,"
the MML estimator of the statistical power is consistent
since, from (11), it converges to i 0 that is equal to the
true power vr 2:
it MML (x) = (MN ) -1

	

(S9)

M

/

m =1

a.s.

x Tm x m M"
i 0 = vr 2 .(S12)
"3

2 ln fX (x m i) 2
m i = i0 1
2i
Ni 20 + E p " (x Tm x m) 2 , - 2Ni 0 E p " x Tm x m ,

=
4i 40
tr ^R 2 h
.
=
2vr 4

E p " it MML (x) , = (MN) -1

	

M

/

m =1

E p " x Tm x m ,

= N -1 vr 2 tr (R) = vr 2 = i 0.

(S10)

Finally, from (5), we get
	

1

Moreover, the MML estimator is MS-unbiased, since

B i0 _ E p 'c
	

MML
MCRB
MCRB_est
CRB

MSE and Bounds

Another example that clarifies the theory concerns the estimation of the statistical power of a set of zero mean
Gaussian vectors. Let x = {x m} M
m = 1 be a set of M i.i.d. real
N-dimensional random vectors sampled from a multivariate
Gaussian pdf with a zero mean value and covariance
matrix given by M = vr 2 R, i.e., p X (x m) / N (0, vr 2 R) ! P,
where vr 2 is the statistical power and R is a symmetric, positive definite matrix whose trace is equal to N, i.e.,
tr (R) = N. For simplicity, we assume that [R] ij = t i -j ,
i, j = 1, f, N, where t 1 1 is the one-lag correlation coefficient (this is the typical correlation matrix of an autoregressive process of order 1). Suppose now that the user is not
aware of the data correlation structure and decides to
assume the following parametric Gaussian model:
F = {fX fX (x m i) = N (0, iI N) 6i ! R +}, where I N is the
identity matrix of dimension N. Note that, as long as t ! 0,
the true pdf p X (x m) does not belong to the assumed model
F . We will proceed exactly as in "Variance Estimation" by
checking the Assumptions 1 and 2 and then by evaluating
the MML estimator and the relative MCRB.
To evaluate the pseudotrue parameter i 0, we need
to find the minimum of the KLD between the true and
assumed model. Following [7] again, the KLD between
N (0, vr 2 R) and N (0, iI N) is given by

r4
MCRB (i 0) = MCRB (vr 2) = 2v 2 tr ^R 2 h .(S11)
MN


(S13)

Then, according to Theorem 1, its MSE is lower bounded
by the MCRB in (5). Figure S3 shows the MSE of the MML
estimator, the MCRB, the sample estimate of the MCRB,
and the CRB as a function of the one-lag coefficient t. The
MCRB is a tight bound for the MSE of the MML estimator,
and the sample MCRB accurately predicts it. Finally, we
note that the MCRB is equal to the CRB only when t = 0,
i.e., when R = I N.

IEEE SIGNAL PROCESSING MAGAZINE

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November 2017

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151
Table of Contents for the Digital Edition of Signal Processing - November 2017
