The estimation of event related potentials affected by random shifts and scalings

There is considerable evidence for trial to trial variability of the event related potentials (ERPs) within a given subject's recording. This variability influences the outcome of usual procedures in ERP analysis. Better results may be obtained if the sources of variability are explicitly taken into account in an appropriate model. This paper considers a probabilistic model, the random shift and scaling (RSS) model, where the response is modified by a random time shift and a random scale factor. In addition to this, an additional random scale factor which affects both the response and the background noise is taken into account. This time shift and these scale factors are handled as nuisance parameters. Maximum likelihood and least squares estimators of these parameters and the waveform of response are derived for the RSS model. It is shown that the Woody estimate of the ERP reported in earlier work can be derived by restricting the assumptions for the RSS model. Test statistics for hypotheses on means are obtained for the RSS model and a new type of discriminant function. The usefulness of the method is illustrated by means of simulation studies. Receiver operating characteristic (ROC) curves are used to demonstrate that the new type of discriminant performs better than the usual Fisher's Linear Discriminant.