$\{\mathbf{X}_{t,T}\}_{t=1}^{T} $ can be represented as $$\mathbf{X}_{t,T}=\displaystyle\sum_{j=1}^{\infty} \displaystyle\sum_{k} \mathbf{V}_{j}(k/T)\psi_{j,t-k}\mathbf{z}_{j,k}, \label{MvLSWeq}$$ where $T=2^{J} \geq 1$ and $\mathbf{V}_{j}(k/T)$ is the lower-triangular transfer function matrix. Each element of the transfer function matrix is assumed to be a Lipschitz continuous function with Lipschitz constants, $L_{j}$, that satisfy $\sum_{j=1}^{\infty} 2^{j}L_{j}^{(p,q)} < \infty$ for each pair of channels $(p,q)$. The vectors $\psi_{j}=(\psi_{j,0},\hdots,\psi_{j,(N_{j}-1)})$ are discrete non-decimated wavelets associated to a low- / high-pass filter pair, $\{\mathcal{H}, \mathcal{G}\}$, constructed according to [@Nason00] as $$\begin{aligned}
\psi_{1,n}&=&\sum_{k} g_{n-2k} \delta_{0,k}=g_{n}\quad \text{for}~n=0,1,\hdots,N_{1}-1, \\
\psi_{j+1,n}&=&\sum_{k} h_{n-2k} \psi_{j,k} \qquad \quad \text{for}~n=0,1,\hdots,N_{j+1}-1.\end{aligned}$$ In the equations above, $\delta_{0,k}$ is the Kronecker delta function and $ N_{j}=(2^{j}-1)(N_{h}-1)+1 $ where $N_{h}$ is the number of non-zero elements of the filter $\mathcal{H}=\{h_{k}\}_{k\in\Z}$. The random vectors $\mathbf{z}_{j,k}$ in are defined such that $ E(\mathbf{z}_{j,k})=\mathbf{0}$ and $\text{cov}\big(z_{j,k}^{(i)},z_{j',k'}^{(i')}\big)=\delta_{i,i'}\delta_{j,j'}\delta_{k,k'}$.

The local wavelet spectral (LWS) matrix and the wavelet coherence of a multivariate signal are key quantities of interest in the dynamic classification problem. Given a MvLSW signal $\mathbf{X}_{t}$ with associated transfer function matrix, $\mathbf{V}_{j}(z)$, the local wavelet spectral matrix $\mathbf{S}_{j}(z)$ is defined as $$\label{eq:spec}
\mathbf{S}_{j}(z)=\mathbf{V}_{j}(z)\mathbf{V}_{j}(z)^{\top}.$$ This quantity describes the cross-covariance between channels at each scale and (rescaled) location $z$. The coherence is a measure of the dependence between the channels of a multivariate 