Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 96

locations in a given direction. The PCA finds a set of orthogonal axes, a base, such
that each consecutive axis spans the directions with consecutively decreasing vari-
ance. The projections of the points onto these axes constitute the components. Each
component can be visualized as a time series. On the other hand the original time
series can be recovered as a linear combination of these components. Components
corresponding to the smallest variance can be neglected and in this way a reduction
of data dimensionality can be achieved.

3.6.1.2 Computation
PCA can be performed by means of the singular value decomposition (SVD) algo-
rithm. An epoch of k channels data of length m can be represented as a m

× k matrix

x . It is always possible to find three matrices P , A ,and M such that:

PAM T

x = (3.50)

where P is the m × k matrix containing k normalized principal component waveforms

( P T P k

matrix mapping components to original data such that M ij is the contribution of
j th

= 1), A is k × k , diagonal matrix of components amplitudes, M is a k ×

component to i th

1. The SVD algorithm is implemented in
MATLAB as a function svd ;tofind the decomposition of x into P , A ,and M execute:
[P,A,M] = svd(x);
When data concerning multiple conditions or subjects are to be analyzed by PCA
the matrix x is composed by concatenating the matrixes of individual conditions or
subjects along the time dimension. Thus for example for N c conditions we have to
decompose matrix x which has dimension m

channel; M T M =

k .The n th row of resulting matrix
P represents N c concatenated time courses obtained for each condition for the n th
component. The corresponding topographical map is contained in the n th column.

· N c ×

3.6.1.3 Possible applications

PCA can be used as a tool for decomposition of multichannel data into a linear
combination of components characterized by different spatial, temporal, and ampli-
tude distribution. These components can be used as a preprocessing step in source lo-
calization techniques. However, there is no direct correspondence between the prin-
cipal components and individual sources. A single source may produce a signal that
decomposes into a number of components, while on the other hand multiple sources
can contribute to a single component. In general, the individual components should
not be ascribed directly to any physiologically meaningful phenomenon [Lamothe
and Stroink, 1991].
A similar method, which leads to the reduction of dimensionality of the data is fac-
tor analysis (FA). FA is related to PCA but not identical. PCA performs a variance-
maximizing rotation (varimax) of the variable space, thus it takes into account all
variability in the data. In contrast, factor analysis estimates how much of the vari-
ability is due to common factors. The number of common factors is usually less than
the dimension of the original variable space [Bryant and Yarnold, 1994]. FA can de-
scribe a large set of recorded wave shapes from multiple sensors in terms of a small