Matching pursuit by undecimated discrete wavelet transform for non-stationary time series of arbitrary length

We describe how to formulate a matching pursuit algorithm which successively approximates a periodic non-stationary time series with orthogonal projections onto elements of a suitable dictionary. We discuss how to construct such dictionaries derived from the maximal overlap (undecimated) discrete wavelet transform (MODWT). Unlike the standard discrete wavelet transform (DWT), the MODWT is equivariant under circular shifts and may be computed for an arbitrary length time series, not necessarily a multiple of a power of 2. We point out that when using the MODWT and continuing past the level where the filters are wrapped, the norms of the dictionary elements may, depending on N, deviate from the required value of unity and require renormalization.We analyse a time series of subtidal sea levels from Crescent City, California. The matching pursuit shows in an iterative fashion how localized dictionary elements (scale and position) account for residual variation, and in particular emphasizes differences in construction for varying parts of the series.