can therefore be removed using the inverse of this inner product matrix. To obtain consistency, the resulting estimate must be smoothed in some way, for example using a rectangular kernel smoother [@Park]. This results in an (asymptotically) unbiased, and consistent, estimator of the LWS matrix, $\mathbf{S}_{j,k}$, given by $ \hat{\mathbf{S}}_{j,k}= (2M+1)^{-1} \sum_{m=k-M}^{k+M} \sum_{l} A_{jl}^{-1}\mathbf{I}_{lm} $, where $M$ denotes the kernel bandwidth corresponding to a smoothing window of length $2M+1$. Estimation of the wavelet coherence matrix is then straightforward, simply using a plug-in estimator, substituting $\hat{\mathbf{S}}$ into Equation .

With the key modelling notation established, we now briefly summarise an approach to dynamic classification based upon the MvLSW framework. This will be the cornerstone of the approach that we propose in Section 3.

Dynamic Classification {#subsec:DC}
----------------------

Following their work on MvLSW processes, [@ParkDC] introduced an approach to dynamically classify a Multivariate Locally Stationary Wavelet signal $\mathbf{X}_{t}$ whose class membership may change over time. The approach assumes that at any time $t$, the signal $\mathbf{X}_{t}$ can belong to one of $N_{c} \geq 2$ different classes, where $N_{c}$ is known. Let $C_{X}(t)$ denote the class membership of $X_{t}$ at time $t$ where $C_{X}(t) \in \{1,2,\hdots, N_{c}\}$. Following [@ParkDC], the signal $\mathbf{X}_{t}$ in 