of a multivariate signal in a fast and efficient manner.

Before introducing our proposed method in Section \[sec:method\], we first outline some details of the locally stationary wavelet framework of [@Park] together with the (offline) dynamic classification method introduced in [@ParkDC] that forms the basis of our online approach. We begin with some introductory concepts of wavelets. For a more comprehensive introduction to the area, see [@NasonTB] or [@Vidakovic].

Discrete wavelet transforms {#subsec:wavelets}
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Succinctly, wavelets can be seen as oscillatory basis functions with which one can represent signals in a multiscale manner. More specifically, for a function (signal) $f\in L^2(\R)$, we can write $$f(x) =\sum_{k\in\Z} c_{0,k} \phi_{0,k}(x) + \sum_{j\leq J}\sum_{k\in\Z} d_{j,k} \psi_{j,k}(x),$$ for scales $j$ and locations $k$, and where the wavelet $\psi_{j,k}(x) = 2^{-j/2}\psi(2^{-j}x-k)$ is a basis function formed as a dilation and translation of a “mother” wavelet $\psi$; scaling functions $\phi_{j,k}$ are similarly formed as dilated and translated versions of a father wavelet $\phi$. The wavelet coefficients $d_{j,k}$ capture local oscillatory behaviour of the signal at a scale (frequency) $j$, whereas the scaling coefficients $c_{j,k}$ represent the signal’s smooth (mean) behaviour at different scales. More specifically, fine scale coefficients capture the local characteristics of a signal; coarse 