Projentropy: Using entropy to optimize spatial projections

Methods for hypothesis testing on zero-mean vector-valued signals often rely on a Gaussian assumption, where the second-order statistics of the observed sample are sufficient statistics of the conditional distribution. This yields fast and simple tests, but by using information-theoretic statistics one can relax the Gaussian assumption. We propose using Rényi's quadratic entropy as an alternative to the covariance and show how a linear projection can be optimized to maximize the difference between the conditional entropies. In addition, if the observed sample is actually a window of a multivariate time-series, then the temporal structure can be exploited using the generalized auto-correlation function, correntropy, of the projected sample. This both reduces the computational complexity and increases the performance. These tests can be applied for decoding the brain state from electroencephalogram (EEG) recordings. Preliminary results are demonstrated on a brain-computer interface competition dataset. On unfiltered signals, the projections optimized with the entropy-based statistic perform better than those of common spatial pattern (CSP) algorithm in terms of classification performance.