Mathematics in Independent Component Analysis

1.3. Dependent component analysis 17
(a) analyzed image
0.8
0.6
0.4
0.2
0
1
1d−autocov
2d−autocov
0 50 100 150 200 250 300
|tau|
(b) autocorrelation (1d/2d)
Figure 1.6: Example of one- and two-dimensional autocovariance coefficient (b) of the gray-scale
128×128 Lena image (a) after normalization to variance 1. Clearly using local structure in both
directions (2d-autocov) guarantees that for small τ higher powers of the autocorrelations are
present than by rearranging the data into a vector (1d-autocov), thereby loosing information
about the second dimension.
1.3.2 Spatiotemporal BSS
Real-world data sets such as recordings from functional magnetic resonance imaging often possess
both spatial and temporal structure. In Theis et al. (2007b), see chapter 9, we propose an
algorithm including such spatiotemporal information into the analysis, and reduce the problem
to the joint approximate diagonalization of a set of autocorrelation matrices.
Spatiotemporal BSS in contrast to the more common spatial or temporal BSS tries to achieve
both spatial and temporal separation by optimizing a joint energy function. First proposed
by Stone et al. (2002), it is a promising method, which has potential applications in areas
where data contains an inherent spatiotemporal structure, such as data from biomedicine or
geophysics including oceanography and climate dynamics. Stone’s algorithm is based on the
Infomax ICA algorithm by Bell and Sejnowski (1995), which due to its online nature involves
some rather intricate choices of parameters, specifically in the spatiotemporal version, where
online updates are being performed both in space and time. Commonly, the spatiotemporal
data sets are recorded in advance, so we can easily replace spatiotemporal online learning by