Mathematics in Independent Component Analysis

Chapter 20. Signal Processing 86(3):603-623, 2006 273
instant only k source components are allowed to vary from a previously fixed
constant (which can be different for each source).
In [3] we showed that under slight conditions k-sparseness already guarantees
identifiability of the model, even in the case of less observations than sources.
In the setting of s-EMG however we are in the comfortable situation of having
more observations than sources, so as in the ICA case we preprocess our data
using PCA projection — this dimension reduction algorithm can be applied
even to our case of non-decorrelated sources as (given low or no noise) the first
three principal components will span the source signal subspace, see comment
in section 3.2. The above uniqueness result is based on the fact that due to the
assumed sparseness the data clusters into a fixed number of hyperplanes. This
fact can also be used in an algorithm to actually reconstruct A by identifying
the set of hyperplanes. From the hyperplanes, A can be recovered by simply
taking intersections.
However, multiple hyperplane identification is non trivial, and the involved
cost function
σ(A) = 1
N
N�
t=1
n
min
i=1
|a⊤ i X(t)|
, (5)
�X(t)�
where ai denote the columns of A, is highly non-convex. In order to improve
the robustness of the proposed, stochastic identifier, we developed an identification
algorithm using a generalized Hough transform [37]. Alternatively a
generalization of k-means clustering can be used, which iteratively clusters the
data into groups belonging to the different hyperplanes, and then identifies a
hyperplane within the cluster by regression [38].
In this paper, we assume sparseness of the sources S in the sense that at
least one coefficient of S at a certain time instant has to be zero. In the
case of s-EMG, the maximum natural firing rate of a motor unit is about 30
pulses/second, lasting each pulse less than 15 ms [39]. Therefore, a motor unit
is active less than 450 ms per second; that is, at least 55% of the time each
source signal is zero. In addition, the firings of different motor units are not
synchronized (only their respective firing rate shows the tendency to change
together, the so-called common drive [40]). For these reasons, the probability
of all n sources firing at a given time instant is bounded by 0.45 n , which quickly
approaches zero for increasing n. Even in the case of only n = 3 sources, at
most 9% of the samples are fully active. Hence the source conditions for SCA
should be rather well fulfilled, an we can find isolated MUAPs inside an s-EMG
signal using SCA with high probability.
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