Signal Processing - July 2017 - 74
a uniform prior U (R m) for the position [K 1, K 2] <, over a predefined support, and a uniform prior for c j, also over a preset
range, R c. Thus, the posterior pdf is
ru ^ m, c Y h \ , ^ y m 1, m 2, c 1, f, c M h
No
2
M
i=1
m=1
% p ^m ih %
p ^c mh,
1 exp - 1 ^ y - 20 log ^ m - h h2
i, m
m
c
mG
2c 2m
2rc m2
(30)
# I ^R mh I ^R ch,
= =%
M
%
i=1 m=1
where N o is the number of observations, y i, m is the i th observation of the mth sensor, and I c (S) is an indicator function
that takes a value equal to one if c ! S, and is equal to
zero otherwise. Thus, in this problem x = [m <, c <] <, and
d x = M + 2.
Our goal is to compute the minimum mean square error
(MMSE) estimate, which corresponds to the expected value
of the posterior ru ^m, c y 1, y 2, f, y M h, where the y m s are vectors whose elements are the measurements of the mth sensor.
Because the MMSE estimate cannot be computed analytically,
we applied several AIS methods to approximate it via MC
quadrature. In particular, we worked with the standard PMC
method [19], two different DM-PMC techniques [24], AMIS
[21], and LAIS [23].
In our experiment, we had M = 6 sensors, and the locations of the sensors were at h 1 = [3, - 8] <, h 2 = [8, 10] <,
h 3 = [- 4, - 6] <, h 4 = [- 8, 1] <, h 5 = [10, 0] <, and h 6 =
[0, 10] < . In all of the cases, we employed Gaussian proposal densities, q n, j (x n n, j, C n, j) = N (x n n, j, C n, j) with
n n, 1 ~ U ^61, 4@d x h for n = 1, f, N. The target was located at
<
m = [m 1 = 2.5, m 2 = 2.5] , and the vector of standard deviations
was c = 6c 1 = 1, c 2 = 2, c 3 = 1, c 4 = 0.5, c 5 = 3, c 6 = 0.2@ .
We generated N o = 20 observations for each sensor according
Table 7. The results of standard PMC [19] (localization example).
MSE
25.12
3.96
1.35
1.08
0.72
0.61
0.70
N
5
10
50
100
500
1,000
2,000
J
2000
1,000
200
100
20
10
5
4
E
S = NJ = 10
Range
MMSE = 0.61 ___________ Maximum MSE = 25.12
to the model given by (29). The uniform prior U (R m) over the
position [m 1, m 2] < had a support R m = [- 30 # 30] 2, and the
uniform prior of the c i s was U ([0.01, 20]). Thus, the overall prior of c was U (R c) with R c = [0.01, 20] M. Then, we
obtained the measurement vectors y 1, f, y M, where y i ! R N o.
Note that, regarding the dimension of the observations, we
have d y = N o M = 120.
For t he PMC, t he DM-PMCs a nd LA IS we set
C n, j = C n = C = v 2 I with v = 1. In AMIS, we have
N = 1 and C n, j = C j = v 2j I, and we set v 1 ! {1, 2}. In
the adaptation layer of LAIS, to obtain {n n, j} nN= 1, from
the previous population {n n, j - 1} nN= 1, we employ parallel
MH chains with a Gaussian random-walk proposal pdf,
2
2
{ n (n n, j | n n, j - 1, v I) = N (n n, j | n n, j - 1, v I) with v = 1.
Moreover, we also test the application of N independent
parallel MH algorithms with the same Gaussian random-walk
proposal pdf, { n (n n, t | n n, j - 1, v 2 I), employed in the adaptation
of LAIS.
We fix the total number of evaluations of the posterior
density to E = 10 4, because this is usually the most costly
step in MC algorithms. Let us recall that J denotes the total
number of iterations and K the number of samples drawn
from each proposal at each iteration. Moreover, we denote
as S the total number of samples employed in the final IS
estimator. In LAIS, the total number of evaluations of the
target pdf is E = NJ (K + 1), whereas S = NJK (i.e., E 2 S
due to the use of the Markov adaptation process). For the
rest of the methods, we have E = S = NKJ (note that N = 1
in AMIS, while K = 1 in standard PMC and MH). Several combinations of N, J, and K are tested for the fixed
E = 10 4 evaluations.
We computed the mean square error (MSE) of the different estimators obtained with respect to the ground truth,
x = [m <, c <] <. The results, averaged over 500 independent
runs, are provided in Tables 7-12 (one table per technique)
with the best and worst MSE values highlighted in boldface.
In this particular experiment, with a unimodal posterior pdf
and a good initialization n n, 1 ~U ([1, 5] d x), the PMC techniques and the AMIS method provide the smallest MSE
values. The standard PMC method seems to perform better if one uses a larger value of N and a smaller number of
iterations J. In fact, the use of a small number of proposal
pdfs can lead to catastrophic results in this case. The DMPMC techniques substantially mitigate this problem, with
Table 8. The results of GR-DM-PMC [24] (localization example).
MSE
0.96
0.89
0.75
0.84
0.85
1.47
0.81
0.76
0.79
0.84
0.80
0.81
N
5
5
5
10
10
10
50
50
100
100
500
1,000
J
50
100
10
10
5
200
5
10
5
10
5
5
K
40
20
200
100
200
5
40
20
20
10
4
2
E
S = NTM = 10 4
MMSE = 0.75 ___________ Maximum MSE = 1.47
Range
74
IEEE SIGNAL PROCESSING MAGAZINE
|
July 2017
|
Table of Contents for the Digital Edition of Signal Processing - July 2017
Signal Processing - July 2017 - Cover1
Signal Processing - July 2017 - Cover2
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Signal Processing - July 2017 - Cover3
Signal Processing - July 2017 - Cover4
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