Wavelet periodicity detection algorithms

This paper deals with the analysis of time series with respect to certain known periodicities. In particular, we shall present a fast method aimed at detecting periodic behavior inherent in noisy data. The method is composed of three steps: 1. Non–noisy data are analyzed through spectral and wavelet methods to extract specific periodic patterns of interest. 2. Using these patterns, we construct an optimal piecewise constant wavelet designed to detect the underlying periodicities. 3. We introduce a fast discretized version of the continuous wavelet transform, as well as waveletgram averaging techniques, to detect occurrence and period of these periodicities . The algorithm is formulated to provide real time implementation. Our procedure is generally applicable to detect locally periodic components in signals s which can be modelled as s(t) = A(t)F (h(t)) + N(t) for t in I, (1) where F is a periodic signal, A is a non–negative slowly varying function, and h is strictly increasing with h′ slowly varying. N denotes background activity. For example, the method can be applied in the context of epileptic seizure detection. In this case, we try to detect seizure periodics in EEG and ECoG data. In the case of ECoG data, N is essentially 1/f noise. In the case of EEG data and for t in I, N includes noise due to cranial geometry and densities. In both cases N also includes standard low frequency rhythms. Periodicity detection has other applications including ocean wave prediction, cockpit motion sickness prediction, and minefield detection.