Regularization

In beamformer analysis, the inverse covariance matrix needs to be computed. However the measured covariance matrix may be singular or badly conditioned. If the covariance matrix

C is close to a singular matrix, calculation of C-1 will be unstable. This situation occurs for example if there are too few data samples to construct the covariance matrix, or the data have been averaged across trials. In these conditions, the covariance matrix requires regularization.

104 Regularization is introduced as a method to convert the covariance matrix to a well conditioned matrix. The simplest regularization, as shown in Equation[4-9], involves adding the same number to every element in the leading diagonal of the covariance matrix. The , which is the noise covariance matrix, is controlled by the regularization parameter . The parameter adjusts the tradeoff between the full-width half-maximum (FWHM) of the point spread function for beamformer image and the magnitude of uncorrelated noise in the timecourse. When =0, no regularization is applied. With the increase of , larger weights are clustered together in the neighbouring channels. The clustered weights lead to average of similar signals across a number of channels and improve the SNR. When a large is applied, since >> C , the Equation[4-9] reduces to:

( )

( ) ( )



L r

W

L r L r

This means the beamformer-derived weights become closer to the lead-field with a larger . In this case, the spatial filter output will have a better SNR but the spatial resolution is reduced. In practice, regularization therefore adjusts the trade off between high temporal SNR and high spatial resolution. The beamformer images and time-courses of beta ERD with different regularization parameters are shown in Figure 4-15. The regularization parameter used in this thesis is 2 based on previous studies [31-32] .

105 Figure 4-15: SAM beamformer images (left) and virtue sensor timecourses of amplitude changes (right) of beta ERD with regularization parameters ( ) of 0,2,10 and 100. [31]

106

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