Probabilistic ICA model

Similar to the square noise-free case, the probabilistic ICA model is
formulated as a generative linear latent variables model. It is
characterised by assuming that the -variate vector of observations
is generated from a set of statistically independent non-Gaussian
sources via a linear instantaneous mixing process corrupted by
additive Gaussian noise
**:
**

Here,
** denotes the -dimensional column vector of individual
measurements at voxel location ,
**** denotes the -
dimensional column vector of non-Gaussian source signals contained in
the data and
**** denotes Gaussian noise
****. We assume
that , i.e. that there are fewer source processes than
observations in time. The covariance
of the noise is allowed to
be voxel dependent in order to allow for the vastly different noise
covariances observed in different tissue types [Woolrich et al., 2001].
**

The vector
** defines the mean of the observations
**** where
the index is over the set of all voxel locations and
the matrix
**** is assumed to be non-degenerate, i.e. of
rank . Solving the blind separation problem requires finding a
linear transformation matrix
**** such that
**

The PICA model is similar to the standard GLM with the difference that,
unlike the design matrix in the GLM, the mixing matrix
** is no longer
pre-specified prior to model fitting but will be estimated from the data as part
of the model fitting. The spatial source signals correspond to parameter
estimates in the GLM with the additional constraint of being statistically
independent.
**

The model of equation 2 is closely related to Factor
Analysis (FA) [Bartholomew, 1987]. There, the sources are assumed to have a
Gaussian distribution and the noise is assumed to have a diagonal covariance
matrix. In Factor Analysis, the sources are known as common factors
and
** is a vector of random variables called specific factors.
In FA the assumption of independence between the individual source processes
reduces to assuming that sources are mutually uncorrelated.
**