Dependent Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 149

Application of the spatiotemporal mixing model from equation (5.4)
together with the transformation properties equation (5.2) of the source
conditions yields

C i ( t s )= s s † C i ( x ) s s † and C i ( s s )= t s † C i ( x ) t s † (5.5)

because ∗ m n and hence ∗ s ∗ s † = I . By assumption the matrices

C i ( ∗ s ) are as diagonal as possible. In order to separate the data, we had
to find diagonalizers for both C i ( x )and C i ( x )suchthattheysatisfy
the spatiotemporal model equation (5.4). As the matrices derived from
X had to be diagonalized in terms of both columns and rows, we denoted
this by double-sided approximate joint diagonalization .
This process can be reduced to joint diagonalization [253, 254]. In or-
der to get robust estimates of the source conditions, dimension reduction
was essential. For this we considered the singular value decomposition
x , and formulated the algorithm in terms of the pseudo-orthogonal com-
ponents of X . Of course, instead of using autocovariance matrices, other
source conditions C i ( . ) from table 5.1 can be employed in order to adapt
to the separation problem at hand.
We present an application of the spatiotemporal BSS algorithm to
fMRI data using multidimensional autocovariances in chapter 8.

≥

5.3 Independent Subspace Analysis

Another extension of the simple source separation model lies in extract-
ing groups of sources that are independent of each other, but not within
the group. Thus, multidimensional independent component analysis, or
independent subspace analysis (ISA) , is the task of transforming a multi-
variate observed sensor signal such that groups of the transformed signal
components are mutually independent—however, dependencies within
the groups are still allowed. This allows for weakening the sometimes
too strict assumption of independence in ICA, and has potential appli-
cations in fields such as ECG, fMRI analysis, and convolutive ICA.
Recently we were able to calculate the indeterminacies of group ICA
for known and unknown group structures, which finally enabled us to
guarantee successful application of group ICA to BSS problems. Here,
we will review the identifiability result as well as the resulting algorithm
for separating signals into groups of dependent signals. As before, the