Resolving Time-Series Structure with a Controlled Wavelet Transform
Resolving Time-Series Structure with a Controlled Wavelet Transform
Wavelet tansforms are powerful techniques that can decompose time series into both time and frequency components. Their application to experimental data has been hindered by the lack of a straightforward method to handlenoise. A noise reduction technique, developed recently for use in wavelet cluster analysis in cosmology and astronomy, is adapted here for time-series data. Noise is filtered using control surrogate data sets generated from randomized aspects of the original time series. The method is a powerful extension of the wavelet transform that is readily applied to the the detection of structure in stationary and nonstationary time series.
Controlled Wavelet Transforms for EEG Spike and Seizure Localization
Wavelet transforms are used to analyze electroencephalographic (EEG) data recorded from individual subdural electrodes (1-dimension) and electrode grids (2-dimensions) indirect contact with the brain. Structure in the data is resolved and noise filtered using surrogate data techniques. Spike and seizure detection from individual electrodes are compared with event extraction using windowed Fourier tansforms. Wavelet transforms are a powerful means to identify epileptiform activity such as spikes in 1-dimension from such data, and offer a method to localize the foci of epileptic seizures in 2-dimensions.
Fast Implementation of the Continuous Wavelet Transform with Integer Scales
We describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m. In this approach, the input signal and the analyzing wavelet are both represented by polynomial splines. The algorithm uses a combination of moving sum and zero-padded filters, and its complexity per scale is O(N), where N is the signal length. The computation is exact, and the implementation is noniterative across scales. We also present examples of spline waveletes exhibiting properties that are desirable for either singularity detection (first and second derivative operators) or Babor-like time-frequency signal analysis.
Wavelet Transforms and Surrogate Data for Elecroencephalographic Spike and Seizure Localization
Wavelet transforms are used to analyze electroencephalographic data recorded from individual subdural electrodes and 2-D electrode grids in direct contact with the brain. Surrogate data techniques are used to filter noise and resolve structure in the data. Spike and seizure detection from individual electrodes with wavelet transforms are compared with windowed Fourier transforms. Wavelet transforms are a powerful means to identify epileptiform activity such as spikes from such data and also offer a method to localize the foci of epileptic seizures from electrode grids.
Fast Wavelet Transformation of EEGWavelet tansforms offer certain advantages over Fourier transform techniques for the analysis of EEG. Recent work has demonstrated the applicability of wavelets for both spike and seizure detection, but the computational demans have been excessive. We compare the quality of feature extraction of continuous wavelet transforms using standard numerical techniques, with more rapid algorithms utilizing both polynomial splines and multiresolution frameworks. We further contrast the difference between filtering with and without the use of surrogate data to model background noise, demonstrate the preservation of feature extraction with critical versus redundant sampling, and perform the analyses with wavelets of different shape. Comparison is made with windowed Fourier transforms, similarly filtered, at different data window lengths. We here report a dramatic reduction in computational time required to perform this analysis, without compromising the accuracy of feature extraction. It now appears technically feasible to filter and decompose EEG using wavelet transforms in real time with ordinary microprocessors.
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