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Is it possible to use stationary wavelet decomposition as a tool to extract wavelet features for a time series? I can see how it works for image cases, but for a time series prediction problem say $x_i$, $i = 1,2,3,\cdots,n$ and you want to predict $x_{n+1}$, is it possible to use stationary wavelet decomposition to extract features and if yes why it makes sense?

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A signal is just a 1D image. So if you can make it work for images, why not for signals? Pun aside, discrete wavelets combine multiscale smoother and differential operators, so they are been used as trend and singularity detectors for a while. Wavelet coefficients can be turned into feature vectors, directly or by condensating them across subband. For instance, wavelet subband coefficients are often believed to follow generalized Laplace-Gauss model distributions: $$A_{\alpha,\beta}\exp{\left(-\frac{|x-m|}{\beta}\right)^\alpha}\,.$$

The following coefficient histogram is for instance obtained from a stationary wavelet decomposition on a seismic signal (source: A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal)

generalized-gaussian distribution of wavelet coefficients

And you can form a vector of subband features with estimated $\alpha_j $ and $\beta_j $ at each scale $j$ for classification. Using stationary wavelets adds some shift invariance (see Difference between “equivariant to translation” and “invariant to translation”). Then, wavelet coefficients can be used as time-frequency anchors as in the Shazam music matching technique (An Industrial-Strength Audio Search Algorithm), or the deep-learning mimicking technique from the scattering wavelet transform developed by S. Mallat and coauthors.

Why does this make sense with (stationary) wavelets?

  • Wavelets may act as multiscale change detectors, with a potential derivative behavior of wavelet functions at different scales, if you think about Haar (a discrete derivative) or the Laplacian of a Gaussian function with the Mexican hat wavelet,
  • They however can capture trends through the scaling function, since mother wavelets have vanishing moments,
  • They somehow can separate structured data from some noises, with orthogonality or whitening properties,
  • With stationary wavelets, the related features become integer shift invariant.
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