Signal Processing - January 2016 - 28

where B ! R N # R ^ R % M % N h, and U ! R R # M = {z jk ($)} `
GP (0, C 0) consists of a covariance dictionary. Such GP modeling of
dictionary elements allows the MEG signals' long-range dependen-
cies to be captured. Each row of B will be assigned with a sparse
shrinkage prior (that penalizes a large coefficient). The proposed
hierarchical model is able to characterize nonstationarity (via the
time-varying factor loading matrix) and to share information (via
coupling) between single trials.
The goal of Bayesian modeling is to infer the latent variable
and parameters of the hierarchical factor analysis model. First, the
componentwise observation x (nl,) t can be written as
x

(l)
n, t

M

=

/

m=1

z

(l)
m, t

R

/ b nr z rm (x t) + e

(l)
i, t

As seen in Figure 2(b), spatial patterns {a (ms)} are drawn from
Gaussian distributions N (a (ms) | n (ms), (U (ms)) -1), where the Gaussian
parameters (i (ms) = {n (ms), U (ms)}) are further drawn from a random
measure G in a DP, i (ms) + G, G + DP (a, G 0) for m = 1, f, M
and s = 1, f, S, where a 2 0 is a positive concentration param-
eter (which specifies how strong this discretization is: when
a " 0, the realizations are all concentrated on a single value,
when a " 3, the realizations become continuous), and the base
measure G 0 = p (n, U) is set to be a Gaussian-Wishart distribu-
tion, which is the conjugate prior for Gaussian likelihood
p (n, U) = N (n | m 0, (b 0 U) -1) W (U | o 0, W 0),

r=1

Marginalizing the dynamic and measurement noise v (tl) and
(l)
e t induces the following time-varying mean and covariance
structure for the observed signal x (tl) + N (n (l) (x t), R (x t)), where

where W (U | o 0, W 0) denotes a Wishart distribution with the
degree of freedom o 0 and the scale matrix W 0 . The random
measure G has the following stick-breaking representation
G=

n (l) (x t) = BU (x t) } (l) (x t)

3

/ ri d

i *i

, ; ri = vi

i=1

R (x t) = BU (x t) U (x t) < B < + R 0 .
The time-varying covariance structure captures the heterosce-
dasticity of time series, and the overcomplete representation pro-
vides flexibility and computational advantages. Bayesian
inference (posterior computation and predictive likelihood) of
this hierarchical GP model is achieved by MCMC sampling meth-
(l)
(l)
ods of {} j , v 1: T, } (0), U, B, R 0} [38]. As shown in [38], such hier-
archical NB modeling provides a powerful framework to
characterize a signal noisy MEG recording that allows word cate-
gory classification.
DP MODELING
Mixture modeling has been commonly used time-series data analy-
sis. Unlike finite mixture models, NB models define a prior distribu-
tion over the set of all possible partitions, where the number of
clusters or partitions may grow as the data samples increase. This is
particularly useful for EEG/MEG applications in clustering, partition,
segmentation, and classification. Examples of static or dynamic mix-
ture models include DP mixtures, the infinite hidden Markov model,
and hierarchical DP. As an illustration, we consider an example of
multisubject EEG classification in the context of NB CSP, which
defines the Bayesian CSP model with either a DP prior or an Indian-
buffet-process prior [18], [39].
The NB CSP model extends the probabilistic CSP (see [17]) and
further introduces the DP prior, which was referred to as BCSPDP [18]. The BCSP-DP uses a DP mixture model to learn the
number of spatial patterns among multiple subjects. The spatial
patterns with the same hyperparameter are grouped in the same
cluster [see the graphical model illustration in Figure 2(b)],
thereby facilitating the information transfer between subjects with
similar spatial patterns. Essentially, BCSP-DP is a nonparametric
counterpart of VB-CSP.
For condition k ! {1, 2} and subject s ! {1, f, S}, the multi-
subject CSP model is rewritten as
X (sk) = A (s) Z (sk) + E (sk) .

(50)

.

(49)

i-1

% (1 - v i),

(51)

j=1

where v i + B (v i | 1, a) and i *i ! G 0 are independent and random
variables drawn from a beta distribution and the beta measure G 0,
respectively. The mixing proportions {r i} are given by successively
breaking a unit-length stick into an infinite number of pieces.
Assuming hierarchical priors for z (tsk) and x (tsk) [see legend of
Figure 2(b) for the priors for a, v i, A (s)]
z (tsk) + N (0, R (zsk))
x (tsk) + N (A (s) z (tsk), R (esk)),
where R z and R e denote two diagonal covariance matrices with
diagonal elements drawn from inverse-Gamma distributions.
Finally, let H define the collective set of unknown variables of
parameters and hyperparameters
H = {{A (s)}, c (ms), {Z (sk)}, {v i}, a, {R (zsk)}, {R (esk)}, {n *i , U *i }} .
The authors of [18] employed VB inference by assuming a fac-
torized variational distribution q (H)
q (H) = q ({A (s)}) q (c (ms)) q ({Z (sk)}) q ({v k})
# q (a) q ({R (zsk)}) q ({R (esk)}) q ({n *i , U *i }) .
The variational posterior q (Z (sk)) = % t q (z (tsk)), q (z (tsk)) =
(sk)
N (n (tsk), R * ) has an analytic form, with
N

(R * ) -1 = E q [(R (zsk)) -1] + / E q [(R (esk)) -1] i E q [[A (s)] 



Table of Contents for the Digital Edition of Signal Processing - January 2016

Signal Processing - January 2016 - Cover1

Signal Processing - January 2016 - Cover2

Signal Processing - January 2016 - 1

Signal Processing - January 2016 - 2

Signal Processing - January 2016 - 3

Signal Processing - January 2016 - 4

Signal Processing - January 2016 - 5

Signal Processing - January 2016 - 6

Signal Processing - January 2016 - 7

Signal Processing - January 2016 - 8

Signal Processing - January 2016 - 9

Signal Processing - January 2016 - 10

Signal Processing - January 2016 - 11

Signal Processing - January 2016 - 12

Signal Processing - January 2016 - 13

Signal Processing - January 2016 - 14

Signal Processing - January 2016 - 15

Signal Processing - January 2016 - 16

Signal Processing - January 2016 - 17

Signal Processing - January 2016 - 18

Signal Processing - January 2016 - 19

Signal Processing - January 2016 - 20

Signal Processing - January 2016 - 21

Signal Processing - January 2016 - 22

Signal Processing - January 2016 - 23

Signal Processing - January 2016 - 24

Signal Processing - January 2016 - 25

Signal Processing - January 2016 - 26

Signal Processing - January 2016 - 27

Signal Processing - January 2016 - 28

Signal Processing - January 2016 - 29

Signal Processing - January 2016 - 30

Signal Processing - January 2016 - 31

Signal Processing - January 2016 - 32

Signal Processing - January 2016 - 33

Signal Processing - January 2016 - 34

Signal Processing - January 2016 - 35

Signal Processing - January 2016 - 36

Signal Processing - January 2016 - 37

Signal Processing - January 2016 - 38

Signal Processing - January 2016 - 39

Signal Processing - January 2016 - 40

Signal Processing - January 2016 - 41

Signal Processing - January 2016 - 42

Signal Processing - January 2016 - 43

Signal Processing - January 2016 - 44

Signal Processing - January 2016 - 45

Signal Processing - January 2016 - 46

Signal Processing - January 2016 - 47

Signal Processing - January 2016 - 48

Signal Processing - January 2016 - 49

Signal Processing - January 2016 - 50

Signal Processing - January 2016 - 51

Signal Processing - January 2016 - 52

Signal Processing - January 2016 - 53

Signal Processing - January 2016 - 54

Signal Processing - January 2016 - 55

Signal Processing - January 2016 - 56

Signal Processing - January 2016 - 57

Signal Processing - January 2016 - 58

Signal Processing - January 2016 - 59

Signal Processing - January 2016 - 60

Signal Processing - January 2016 - 61

Signal Processing - January 2016 - 62

Signal Processing - January 2016 - 63

Signal Processing - January 2016 - 64

Signal Processing - January 2016 - 65

Signal Processing - January 2016 - 66

Signal Processing - January 2016 - 67

Signal Processing - January 2016 - 68

Signal Processing - January 2016 - 69

Signal Processing - January 2016 - 70

Signal Processing - January 2016 - 71

Signal Processing - January 2016 - 72

Signal Processing - January 2016 - 73

Signal Processing - January 2016 - 74

Signal Processing - January 2016 - 75

Signal Processing - January 2016 - 76

Signal Processing - January 2016 - 77

Signal Processing - January 2016 - 78

Signal Processing - January 2016 - 79

Signal Processing - January 2016 - 80

Signal Processing - January 2016 - 81

Signal Processing - January 2016 - 82

Signal Processing - January 2016 - 83

Signal Processing - January 2016 - 84

Signal Processing - January 2016 - 85

Signal Processing - January 2016 - 86

Signal Processing - January 2016 - 87

Signal Processing - January 2016 - 88

Signal Processing - January 2016 - 89

Signal Processing - January 2016 - 90

Signal Processing - January 2016 - 91

Signal Processing - January 2016 - 92

Signal Processing - January 2016 - 93

Signal Processing - January 2016 - 94

Signal Processing - January 2016 - 95

Signal Processing - January 2016 - 96

Signal Processing - January 2016 - 97

Signal Processing - January 2016 - 98

Signal Processing - January 2016 - 99

Signal Processing - January 2016 - 100

Signal Processing - January 2016 - 101

Signal Processing - January 2016 - 102

Signal Processing - January 2016 - 103

Signal Processing - January 2016 - 104

Signal Processing - January 2016 - 105

Signal Processing - January 2016 - 106

Signal Processing - January 2016 - 107

Signal Processing - January 2016 - 108

Signal Processing - January 2016 - 109

Signal Processing - January 2016 - 110

Signal Processing - January 2016 - 111

Signal Processing - January 2016 - 112

Signal Processing - January 2016 - 113

Signal Processing - January 2016 - 114

Signal Processing - January 2016 - 115

Signal Processing - January 2016 - 116

Signal Processing - January 2016 - 117

Signal Processing - January 2016 - 118

Signal Processing - January 2016 - 119

Signal Processing - January 2016 - 120

Signal Processing - January 2016 - 121

Signal Processing - January 2016 - 122

Signal Processing - January 2016 - 123

Signal Processing - January 2016 - 124

Signal Processing - January 2016 - 125

Signal Processing - January 2016 - 126

Signal Processing - January 2016 - 127

Signal Processing - January 2016 - 128

Signal Processing - January 2016 - 129

Signal Processing - January 2016 - 130

Signal Processing - January 2016 - 131

Signal Processing - January 2016 - 132

Signal Processing - January 2016 - 133

Signal Processing - January 2016 - 134

Signal Processing - January 2016 - 135

Signal Processing - January 2016 - 136

Signal Processing - January 2016 - 137

Signal Processing - January 2016 - 138

Signal Processing - January 2016 - 139

Signal Processing - January 2016 - 140

Signal Processing - January 2016 - 141

Signal Processing - January 2016 - 142

Signal Processing - January 2016 - 143

Signal Processing - January 2016 - 144

Signal Processing - January 2016 - 145

Signal Processing - January 2016 - 146

Signal Processing - January 2016 - 147

Signal Processing - January 2016 - 148

Signal Processing - January 2016 - 149

Signal Processing - January 2016 - 150

Signal Processing - January 2016 - 151

Signal Processing - January 2016 - 152

Signal Processing - January 2016 - 153

Signal Processing - January 2016 - 154

Signal Processing - January 2016 - 155

Signal Processing - January 2016 - 156

Signal Processing - January 2016 - 157

Signal Processing - January 2016 - 158

Signal Processing - January 2016 - 159

Signal Processing - January 2016 - 160

Signal Processing - January 2016 - 161

Signal Processing - January 2016 - 162

Signal Processing - January 2016 - 163

Signal Processing - January 2016 - 164

Signal Processing - January 2016 - 165

Signal Processing - January 2016 - 166

Signal Processing - January 2016 - 167

Signal Processing - January 2016 - 168

Signal Processing - January 2016 - Cover3

Signal Processing - January 2016 - Cover4


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