Signal Processing - January 2016 - 25

stages of mental processes and differentiating among various events or
stimuli. In multichannel EEG recordings, since the number of ERP
components can be considerably smaller than the number of the sen-
sors, SBL is suited to automatically infer the sparse ERP components.
In [29], a Bayesian model was proposed to estimate the compo-
nents specific to each experimental condition for ERPs from mul-
ticondition and multitrial EEG data. More specifically, we adapted
(2) to the following linear latent variable model to accommodate
the ERP estimation:
x t(kl) =

M

/

c (mk) a m z m, t + x (mk) + e t(kl),

the noise-plus-interference activities e (tkl) are modeled as ran-
dom effects in (34), which is also in agreement with existing
experimental observations. Crucially, e (tkl) is allowed to be corre-
lated and nonisotropic among sensors, with the covariance
matrices R (ek) following noninformative inverse Wishart distri-
butions for o -1 " 0:
[R (ek)] -1 ~ W (oI, M) _

(34)

m=1

where x (tkl) ! R N and e (tkl) ! R N denote the EEG signal and the
noise-plus-interference term from the lth ^l ! {1, g, L} h trial of
condition k ! {1, g, K}, respectively; z m, t denotes the waveform
of the stereotyped mth ERP component; a m ! R N denotes the
scalp map of the mth ERP component; and c (mk) and x (mk) denote
the amplitude factor and the latency of the mth ERP component
for condition k, respectively. We further assume the noise terms
are independent and identically Gaussian distributed across time
and trials e (tkl) ~ N (0, R (ek)), where R (ek) are the spatial covariance
matrices. Note that no particular structure is assumed for R (ek)
(e.g., R (ek) can be nondiagonal).
Two key assumptions are made in (34): 1) the waveform and
spatial pattern of each ERP component are invariant across tri-
als and conditions, and 2) intercondition ERPs differ only in
their amplitudes and latencies. These assumptions are moti-
vated by the existing experimental observations that ERPs are
approximately time- and phase-locked across trials and that
ERP variability is typically far less within conditions than across
conditions. Furthermore, unlike fixed-effect ERP components,

M2
2

1

CM ` M j
2
1
(k) 1
| R e | 2 exp (- trace [o -1 [R (ek)] -1]) . (35)
2
2

| oI |

M
2

To address the inherent scaling ambiguity in (34), z m and a m are
endowed with standard Gaussian priors:
a m ~ N (0, I), z m, t ~ N (0, 1) .

(36)

To allow for automatic determination of the component number
M, a noninformative hierarchical prior akin to (32) can be
imposed on c (mk) . Here, at the first level, a half-Gaussian is
assumed to account for a fixed polarity of each ERP component
across conditions
c (mk) ~ N + (0, a -m1) _ 2
a m ~G (u, v) _

am
exp (- 1 a m [c (mk)] 2), c (mk) $ 0
2r
2

v u a u - 1 exp (- va ), u " 0, v " 0.
m
C (u) m

(37)

Moreover, the latency shifts x (mk) are integers within a preset inter-
val [t 1, t 2] . The probabilistic graphical model is shown in Figure 5.
In [29], a VB-EM algorithm termed Bayesian estimation of
ERPs (BEEP) was developed for inferring model (34) based on the
following mean-field approximation:

Algorithm 3: The BEEP algorithm.
Input: {X (kl)}
Output: q (a m), q (z m, t), q (c (mk)), q (a m), q (R (ek)), x (mk)
1: Initialization
2: repeat
3:

q (a m) = N (E [a m], C [a m]), where C [a m] = [L / E [[c (mk)] 2] E [z 2m, t] E [[R (ek)] -1] + I] -1,
k, t

(k)
E [a m] = R / C [a m] E [c (mk)] E [z m, t] E [[R (ek)] -1] rs (mk), t, rs (mk), t _ / r [ x (mk), t - / i ! m a i c i z i, t - x )li]
k, t

4:

k 2
q (z m, t) = N (E (z m, t), C (z m, t)), where C [z m, t] = [L / k E ([c m
] ) ^trace (C [a m] E [[R (ek)] -1]) + E < [a m] E [[R (ek)] -1] E [a m] h + K] -1,

E [z m, t] = [L / k [ s m, t - x lj] < E [[R (ek)] -1] E [a m] E [c (mk)]] C [z m, t]
(k)

5:

q (c (mk)) = l · N (n (mk), v (mk)), c (mk) $ 0, where v (mk) _ 8a m + L / t E ([z m, t] 2) ^trace (C [a m] E [[R (ek)] -1]) + E < [a m] E [[R (ek)] -1] E [a m] hB ,
-1

(k)

n m _ [L

/ t [rs (mk), t -

x lj

(k)

nm

] < E [[R (ek)] -1] E [a m] E [z m, t]] v (mk), l _ 1/ (1 - U (-

6:

q (a m) = G (uu , vu ), where uu _ K , vu _ 1 / k E [[c (mk)] 2] .
2
2

7:

q ([R (ek)] -1) = W ([C (k)] -1, LT + M), where

(k)

vm

)) .

C (k) _ / l / t ^ x t(kl) [x t(kl)] < h- 2 / l / t ` x t(kl) / Mj = 1 [h j, t] < j+ L / t / iM= 1 / Mj = 1, j ! i h i, t [h j, t] < + L / t / j = 1 E [a j a 



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

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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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