Biomedical Engineering Reference
In-Depth Information
FIgURE 3.3:
Illustrations of condition number and nonstationary properties of the input autocorrela-
tion matrix. The dotted lines in the bottom panel indicate the reference maximum eigenvalue that is
computed over all the data samples.
input autocorrelation matrix shown in Figure
3.3
. The autocorrelation matrix of the multidimen-
sional input data (here 104 neurons) is estimated based on the assumption of ergodicity.
1
In order
to monitor the temporal change, the autocorrelation matrix is estimated for a sliding time window
(4000-sample length), which slides by 1000 samples (100 second). For each estimated autocorrela-
tion matrix, the condition number and the maximum eigenvalue are computed as approximations
of the properties of the matrix. The experimental results of these quantities for three BMI data sets
(three different animals) are presented in Figure
3.3
. As we can see, the temporal variance of the
input autocorrelation matrix is quiet large. Moreover, the inverse of the autocorrelation is poorly
conditioned. Notice that in the formulation presented here,
R
must be a nonsingular matrix to obtain
the optimal solution from (
3.10
). However, if the condition number of
R
is very large, the optimal
weight matrix
w
Wiener
may be inadequately determined due to round off errors in the arithmetic. This
usually happens when the number of samples is too small or the input variables are linearly depen-
dent on each other (as may occur with neural recordings in a small volume of tissue). In such a case,
we can reduce the condition number by adding an identity matrix multiplied by some constant to
R
before inversion. This procedure is called ridge regression in statistics [
31
], and the solution obtained
1
A property of some stationary random processes in which the statistical moments obtained by statistical operators
equal the moments obtained with time operators.
