Mathematics in Independent Component Analysis

18 Chapter 1. Statistical machine learning of biomedical data
=
X t A t S
X
=
(a) temporal BSS
s S ⊤
(c) spatiotemporal BSS
t S
=
X ⊤ s A s S
(b) spatial BSS
Figure 1.7: Temporal, spatial and spatiotemporal BSS models. The lines in the matrices ∗ S
indicate the sample direction. Source conditions apply between adjacent such lines.
batch optimization. This has the advantage of greatly reducing the number of parameters in the
system, and leads to more stable optimization algorithms. In Theis et al. (2007b), we extended
Stone’s approach by generalizing the time-decorrelation algorithms to the spatiotemporal case,
thereby allowing us to use the inherent spatiotemporal structures of the data.
For this, we considered data sets x(r, t) depending on two indices r and t, where r ∈ R n
can be any multidimensional (spatial) index and t indexes the time axis. In order to be able to
use matrix-notation, we contracted the spatial multidimensional index r into a one-dimensional
index r by row concatenation. Then the data set x(r, t) =: xrt can be represented by a data
matrix X of dimension s m× t m, where the superscripts s (.) and t (.) denote spatial and temporal
variables, respectively.
Temporal BSS implies the matrix factorization X = t A t S, whereas spatial BSS implies the
factorization X ⊤ = s A s S or equivalently X = s S ⊤s A ⊤ . Hence X = t A t S = s S ⊤s A ⊤ . So both
source separation models can be interpreted as matrix factorization problems; in the temporal
case restrictions such as diagonal autocorrelations are determined by the second factor, 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 (1.8)
with spatial source matrix s S and temporal source matrix t S, which both have (multidimensional)
autocorrelations being as diagonal as possible. The three models are illustrated in figure
1.7.
Concerning conditions for the sources, we interpreted Ci(X) := Ci( t x(t)) as the i-th temporal
autocovariance matrix, whereas Ci(X ⊤ ) := Ci( s x(r)) denoted the corresponding spatial
autocovariance matrix. Application of the spatiotemporal mixing model from equation (1.8)