Karjalainen, Pasi A. Regularization and Bayesian methods for ...

16 1. Introduction
deterministic signal in independent additive background noise of zero mean.
However, for over three decades it has been evident that the nature of evoked
potentials is more or less stochastic. In particular, the latencies and the amplitudes
of the peaks in the potentials can have stochastic variation between the
repetitions of the stimuli [22]. In addition the variations can be trend-like and
the means of the latencies or the amplitudes can change during the test. The
information about these kinds of variations in evoked potentials vanishes when
the signal is averaged. The resulting estimates for the potentials possibly do not
correspond to any physical or neuroanatomical situation and thus inference about
the neurophysical structure is difficult. A concrete example is the use of average
evoked potentials in source estimation, the estimation of the location and other
parameters of the electrical sources in brain.
Statistically the average evoked potential, the sample mean, is an example of
the use of the first order statistics, in which only the first moment of the population
parameters is estimated. The next evident improvement is to use the second
order statistics, that is, covariance analysis. Typical examples of the second order
methods are e.g. principal component analysis [90] and factor analysis [173] of the
potentials. With these methods also a measure of the variation of the potentials
can be obtained. However, also with these methods the information carried by
a single measurement is lost. Also all the information about the time variation
between the repetitions of the test is lost.
Currently the goal in the analysis of the evoked potentials is to get information
about single potentials e.g. obtain the best possible estimate for a single potential.
The notion of single trial analysis can be used in this context. The most common
way to do single trial analysis is to form an estimator (filter) with which the
unwanted contribution of the on-going background activity can be filtered out
from the evoked potential. A major difficulty in this task is often the very low
signal-to-noise ratio, typically SNR < −10 dB.
In any such filtering or estimation method the performance of the estimator
depends on the properties of the underlying signals. In the most realistic methods
some models for the measurements and the signals are assumed. The estimator
that yields the minimum mean square criterion is then derived. The performance
of the estimators then depends strongly on how realistic the assumptions are.
The most common assumptions concern the second order statistics of the evoked
potentials and the on-going background EEG. In addition, we sometimes have
prior information about the evoked potentials. The information can relate e.g. to
the shape or the location of the peaks in potential. Generally, we should take this
prior information into account to obtain best possible estimates.
In literature we can find two approaches to take the prior information or
beliefs about the parameters (signal) to be estimated into account. The first
is the Bayesian approach with which the prior information can be taken into
account in the statistical sense. This procedure usually necessitates some prior
statistical information about the parameters in the form of prior distributions.
The other approach, the use of the regularization methods, arises from the field of
ill-posed problems. In this class of the methods the prior information about the