** Next:** Maximum Likelihood estimation
** Up:** Probabilistic ICA model
** Previous:** Probabilistic ICA model

##

Uniqueness

[Rao, 1969] extends the standard factor analysis model such that the
common and specific variables are independent non-degenerate random variables
and examines the implication for the minimum rank of the mixing matrix
in
equation 2. Earlier work [Rao, 1966] characterised the
multivariate normal distribution through the non-uniqueness of its
linear structure, a result which within the ICA literature has been
restated as the limitation that only
one Gaussian source process, at most, may contribute to the observations for the
ICA model to be estimable [Comon, 1994,Hyvärinen et al., 2001].
Here, a vector variable
is said to have a linear
structure if it can be decomposed as

where
is a vector of non-degenerate statistically independent
one dimensional random variables and
is a matrix of full column rank. The linear
structure is said to be *essentially unique* if all the linear
decompositions are equivalent in the sense that if the vector variable
allows for two structural representations

then every column of
is a multiple of some column of
and vice versa, i.e. the two matrices are identical modulo scaling and
permutation. This again has been noted as a standard restriction within the ICA
framework [Comon, 1994].
The main result in [Rao, 1969] is a decomposition theorem that states that if
is a -variate random variable with a linear structure
where
all the elements of
are non-Gaussian variables, then there does not exist
a non-equivalent linear structure involving the same number or a smaller
number of structural variables than that of
.

Furthermore, if
is a -vector random variable with a linear
structure
then
can be decomposed

where
and
are independent,
is
non-Gaussian and has an essentially unique linear structure and
is -variate normal with a non-unique linear structure.
The proofs involve the characteristic functions of the vector random
variables
and
and as such these results are applicable
only if the number of observations (i.e. voxels) is sufficiently
large to accurately reflect the distribution of these quantities.

The results show, however, that conditioned on knowing the number of
source signals contained in the data and under the assumption that the
data are generated according to equation 2, i.e. a
linear mixture of independent non-Gaussian source signals confounded
by Gaussian noise, there is no non-equivalent decomposition into this
number of independent non-Gaussian random variables and an associated
mixing matrix; the decomposition into independent components is
unique, provided we do not attempt to extract more than source
signals from the data.

** Next:** Maximum Likelihood estimation
** Up:** Probabilistic ICA model
** Previous:** Probabilistic ICA model
Christian F. Beckmann
2003-08-05