Dependent Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 148

=

t A t s

x

(a) temporal BSS

=

x

s A s s

(b) spatial BSS

x =

s s

t s

(c) spatiotemporal BSS

Figure 5.2
Temporal, spatial and spatiotemporal BSS models. The lines in the matrices ∗ S
indicate the sample direction. Source conditions apply between adjacent such lines.

els can be interpreted as matrix factorization problems; in the temporal
case, restrictions such as diagonal autocorrelations are determined by
the second factor, and in the spatial case, by the first one. In order to
achieve a spatiotemporal model, we required these conditions from both
factors at the same time. Therefore, the spatiotemporal BSS model can
be derived from the above as the factorization problem

x = s s t s (5.4)

with spatial source matrix s s and temporal source matrix t s ,which
both have (multidimensional) autocorrelations that are as diagonal as
possible. The three models are illustrated in figure 5.2.
Concerning conditions for the sources, we interpreted C i ( x ):=
C i ( t x ( t )) as the i -th temporal autocovariance matrix, whereas C i ( x ):=
C i ( s x ( r )) denoted the corresponding spatial autocovariance matrix.