Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 82

the model order L is set according to some prior knowledge, and appropriate initial
values are given to A and

ʛ ʓ ¯ = ,...,

L ) are updated
according to Eqs. ( 5.10 ) and ( 5.11 ). In the M-step, parameters A and

. In the E-step, and u k ( k 1

ʛ are updated

using Eqs. ( 5.18 ) and ( 5.23 ). The marginal likelihood p in Eq. ( 5.30 ) can

be used for monitoring the progress of the EM iteration. If the likelihood increase
becomes very small with respect to the iteration count, the EM iteration may be
stopped.
The BFA can be applied to denoising the sensor data. Using the BFA algorithm,
the estimate of the signal component,

( y | A , ʛ )

y k , is given by

y k = E u [ Au k ]= A E u [ u k ]= A u k . ¯ (5.36)

y k may be used for further analysis such as source localization. We can compute
the sample data covariance using only the signal component Au k , which is

This

E u

T

A E u u k u k A T

K K

1
K

1
K

R

= ( Au k )( Au k ) =

k = 1 k = 1

A 1
K

A T

K

A + ʓ − 1 A T

K

1
K

u k u k ʓ − 1 A T = u k ¯ ¯ =

1 ¯

u k ¯ + A . (5.37)

k = 1 k =

This R can be used in source imaging algorithms such as adaptive beamformers.
Note that, on the right-hand side of Eq. ( 5.37 ), the second term A ʓ − 1 A T works

R has a form in which the regularization is already

as a regularization term and

incorporated.

5.3 Variational Bayes Factor Analysis (VBFA)

5.3.1 Prior Distribution for Mixing Matrix

In the BFA algorithm described in the preceding section, a user must determine
the model order L , according to prior knowledge of the measurement. However,
determination of the model order is not easy in most practical applications. We
describe here an extension of the Bayesian factor analysis based on the variational
Bayesian method [ 6 ], The method is called the variational Bayesian factor analysis
(VBFA), in which the model order determination is embedded in the algorithm. On
the basis of the factor analysis model in Eq. ( 5.2 ), the VBFA algorithm estimates the
posterior probability distributions not only for the factor activity u k but also for the
mixing matrix A .