Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 107

5.5.4 Summary of the Saketini Algorithm

The Saketini algorithm is summarized as follows. It estimates the source time course
at a pointing location, based on the data model in Eq. ( 5.134 ). It uses the VBEM algo-
rithm in which the E-step updates the posterior precision ʓ and the posterior mean ¯ z k

¯ A using Eqs. ( 5.152 )

by using Eqs. ( 5.148 ) and ( 5.149 ). The M-step updates ʨ and

and ( 5.153 ). The hyperparameter ʦ is updated using Eq. ( 5.165 ). The hyperparame-

ter is updated using Eq. ( 5.166 ).

The free energy can be computed using the same equation as in Eq. ( 5.131 ) with the
sole modification of adding log

ʱ is updated using ( 5.79 ). The noise precision ʛ

| ʦ | . That is, the free energy is expressed as

K K

K
2

log | ʛ || ʦ |
| ʓ |

1
2

1
2

M
2

log | ʱ | y k ʛ z k ʓ ¯ F = − y k +

1 ¯

z k +

| ʨ | .

(5.167)

k = 1 k =

This free energy forms the lower bound of the marginal likelihood, and it is used for
monitoring the progress of the VBEM iteration.

5.6 Numerical Examples

Computer simulation was performed to present typical results from the methods
mentioned in this chapter. An alignment of the 275-sensor array from the Omega™
(VMS Medtech, Coquitlam, Canada) neuromagnetometer was used. The coordinate
system and source-sensor configuration used in the computer simulation are depicted
in Fig. 5.1 . A vertical plane was assumed at the middle of the whole-head sensor
array ( x

=

0 cm), and three sources were assumed to exist on this plane. The time
courses shown in the upper three panels of Fig. 5.2 were assigned to the three sources.

Fig. 5.1 The coordinate
system and source-sensor
configuration used in the
computer simulation. A
vertical plane was assumed at
the middle of the whole-head
sensor array ( x

z

second source

first source

y

10

0cm), and
three sources were assumed
to exist on this plane at
x = 0 cm. The large circle
shows the source space and
the small filled circles show
the locations of the three
sources

=

third source

8

6

sensor array

-2 0 2

y (cm)