On-line multiscale filtering of random and gross errors without process models
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Data Rectification by univariate filtering is popular for processes lacking an accurate model. Linear filters are most popular for online filtering; however, they are single-scale best suited for rectifying data containing features and noise that are at the same resolution in time and frequency. Consequently, for multiscale data, linear filters are forced to trade off the extent of noise removal with the accuracy of the features retained. In contrast, nonlinear filtering methods, such as FMH and wavelet thresholding, are multiscale, but they cannot be used for online rectification. A technique is presented for online nonlinear filtering based on wavelet thresholding. OLMS rectification applies wavelet thresholding to data in a moving window of dyadic length to remove random errors. Gross errors are removed by combining wavelet thresholding with multiscale median filtering. Theoretical analysis shows that OLMS rectification using Haar wavelets subsumes mean filters of dyadic length, while rectification with smoother boundary corrected wavelets is analogous to adaptive exponential smoothing. If the rectified measurements are not needed online, the quality of rectification can be further improved by averaging the rectified signals in each window, overcoming the boundary effects encountered in TI rectification. Synthetic and industrial data show the benefits of the online multiscale and boundary corrected translation invariant rectification methods.
