decimation step is not carried out, resulting in $2^{J}$ smooth and detail coefficients at each level of the transform. This allows for a fuller description of the local characteristics of the data in the decomposition, a feature that turns out to be particularly helpful for describing time series. A more detailed treatment of the NDWT can be found in [@NSDWT], see also [@coifman95:translation; @percival95:on]. In the context of streaming data, the transform is such that only a small number of coefficients need to be recomputed at each time step, recycling previously evaluated coefficients. This computationally efficient algorithm will be used within our online dynamic classification technique described in Section \[sec:method\].

Multivariate locally stationary wavelet (MvLSW) processes {#subsec:MvLSWmodel}
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We now turn to consider the application of wavelets within non-stationary time series models. Specifically we focus on the recently proposed multivariate locally stationary wavelet (MvLSW) framework introduced by [@Park], which we later use to model the DAS data described in Section \[intro\]. This approach provides a flexible model for multivariate time series that is able to capture (second order) nonstationarity, as well as temporally inhomogeneous dependence structure between channels of a multivariate series.

Following [@Park], a $P$-variate locally stationary wavelet process 