Techniques of EEG Recording and Preprocessing - Quantitative EEG Analysis Methods and Clinical Applications - page 40

− 1

xWu = (2.3)

Here, x is considered the linear superposition or mixture of basis functions (i.e.,
columns of W -1 ), each of which is activated by an independent component, u i .We
call the rows of W filters because they extract the independent components. In
orthogonal transforms such as PCA, the Fourier transform, and many wavelet trans-
forms, the basis functions and filters are the same (because W T = W -1 ), but in ICA
they are different.
The algorithm for learning W is commonly accomplished by formulating a cost
function and running an optimization process. There are many possible cost func-
tions and many more optimization processes. Thus, there are many somewhat dif-
ferent algorithmic approaches to solving the blind source separation problem.
Information maximization [31, 35], maximum likelihood [36, 37], FastICA [38],
and Joint Approximate Decomposition of Eigen matrices (JADE) [39] are just some
of the widely used algorithms whose cost functions and optimization processes are
recommended for further reading.

2.4.2 Applying ICA to EEG/ERP Signals
More than a decade ago, the authors first explored and reported the application of
ICA to multiple-channel EEG and averaged ERP data recorded from the scalp for
separating joint problems of source identification and source localization [24]. Fig-
ure 2.7(a) presents a schematic illustration of the ICA decomposition. For EEG or
ERP data, the rows of the input matrix x in (2.1) and (2.3) are EEG/ERP signals
recorded at different electrodes and the columns are measurements recorded at dif-
ferent time points [Figure 2.7(a), left]. ICA finds an “unmixing” matrix W that
decomposes or linearly unmixes the multichannel scalp data into a sum of tempo-
rally independent and spatially fixed components, u

Wx [Figure 2.7(a) right]. The
rows of this output data matrix, u , called the component activations, are the time
courses of relative strengths or levels of activity of the respective independent com-
ponents through the input data. The columns of the inverse of the unmixing matrix,
W -1 , give the relative projection strengths of the respective components onto each of
the scalp sensors. These may be interpolated to show the scalp map [Figure 2.7(a),
far right] associated with each component. These scalp maps provide very strong
evidence as to the components' physiological origins (for example, vertical eye
movement artifacts project principally to bilateral frontal sites), and may be sepa-
rately input into any inverse source localization algorithm to estimate the actual cor-
tical distributions of the cortical area or areas generating each source.
Note that each independent component of the recorded data is specified by both
component activation and a component map—neither alone is sufficient. Note also
that ICA does not solve the inverse (source localization) problem. Instead, ICA,
when applied to EEG data, reveals what distinct, for example, temporally independ-
ent activities compose the observed scalp recordings, separating this question from
the question of where exactly in the brain (or elsewhere) these activities arise. How-
ever, ICA facilitates answers to this second question by determining the fixed scalp
projection of each component alone.

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