The global Beta test for hidden periodicities in signals and its extensions to multivariate systems

There are many phenomena that lead to changes in the power spectrum of a given signal, and their detection has been a challenge that has received considerable attention over the years. Objective Response Detection (ORD) techniques are a set of tools that perform automated tests for such a task, allowing thus to automatically track changes in the spectrum. The performance of these detectors is affected by the signal-to-noise ratio (SNR) of the recorded signal as well as the length of the available data. The Global F Test (GFT) is a promising detector that can be used to test whether there is a statistically significant difference between the spectrum before and during an event. In fact, this detector has proved useful in the detection of event-related desynchronization/synchronization (ERD/ERS), where only amplitude, but not the phase, changes are locked to the stimulus. In order to improve the statistical power of the GFT (for the same length of recording), multiple channels recorded simultaneously can be included. This concept is called Multivariate Response Detection. The aim of the current work is to extend the GFT to the multivariate (multichannel) case.

Firstly, the single channel normalization of the GFT is presented as a new ORD detector - the global Beta test (GBT). After that, three multivariate extensions of this new test are derived. The critical values used in the detection of spectral changes are obtained by using theoretical distributions, and where this is intractable, by means of Monte Carlo simulations. The probability of detection (PD) of each technique was estimated using simulation and was used in order to compare the detectors performance. A practical example with the electroencephalogram (EEG) from 10 volunteers under intermittent photic stimulation was also provided.

The statistics under both the null and alternative hypothesis could be obtained for all detectors. Simulated results for PD demonstrate the strong potential of the proposed method and the performances in EEG data are always improved with increasing number of signals.

If more than one signal is available, then the multivariate extensions may provide significant benefit compared to the original GFT.