signal at a particular time and scale. Following [@Park], the wavelet coherence matrix $\boldsymbol{\rho}_{j}(z)$ is given by $$\boldsymbol{\rho}_{j}(z)=\mathbf{D}_{j}(z)\mathbf{S}_{j}(z)\mathbf{D}_{j}(z), \label{wavecoh}$$ where $\mathbf{S}_{j}(z)$ is the LWS matrix from and $\mathbf{D}_{j}(z)$ is a diagonal matrix with entries given by $S_{j}^{(p,p)}(z)^{(-1/2)}$. It is these spectral and coherence quantities which we use to enable us to accurately classify multichannel signals with time-varying dependence and second order structure.

#### Estimation of MvLSW spectral and coherence components.

In practice, the coherence and LWS matrix are unknown for an observed multivariate series and need to be estimated. The LWS matrix of a multivariate signal can be estimated by first calculating the empirical wavelet coefficient vector $\mathbf{d}_{j,k}=\sum_{t} \mathbf{X}_{t}\psi_{j,k-t}$ at locations $k$ and scales $j$. The raw wavelet periodogram matrix is then defined as $\mathbf{I}_{j,k}=\mathbf{d}_{j,k}\mathbf{d}_{j,k}^{\top} $.

[@Park] establish that the raw wavelet periodogram is a biased and inconsistent estimator of the true LWS matrix, $ \mathbf{S}_{j}(z)$. However, they show that (asymptotically) this bias is described by the well known inner product matrix of discrete autocorrelation wavelets, $\mathbf{A}$. The elements of $\mathbf{A}$ are given by $ A_{j\ell}= \sum_{\tau} \Psi_{j}(\tau)\Psi_{\ell}(\tau) $ where $ \Psi_{j}(\tau)=\sum_{k} \psi_{jk}(0) \psi_{jk}(\tau) $ (see [@Eckley] or [@Nason00] for further information). The bias inherent within the raw wavelet periodogram 