Estimating multiresolution dependency graphs within the stationary wavelet framework

Abstract

Very recently, the locally stationary wavelet framework has provided a means to describe the dependencies of co-varying time-series over a range of multiple scale levels. However, describing the many interactions between data-streams at different scale levels with only finite data poses some serious statistical estimation challenges. We illustrate that existing approaches suffer from large variance and are sometimes difficult to interpret. We here propose a sparsity-aware estimator which furnishes a set of multiresolution, dynamic graphs that describe how the dependency structure of the variables evolves through time and over multiple levels of scale. We show that the regulariser mitigates the variance and that, since the inference is performed using convex optimisation, it converges quickly to a global optima and scales well with respect to samples and nodes. Basic properties of the new method are established on simulated data. The method is applied to inferring dependency structure in multivariate EEG data-sets during epileptic seizures where it reveals evidence of band-limited dependency structure.

Temporally Smoothed Wavelet Coherence for Multivariate Point-Processes and Neuron-Firing

“…Further work is needed here to assess the behaviour of the estimator when correlation structure is present. For instance under self-exciting Hawkes, or Cox processes [Hawkes, 1971, Cox, 1955, or even locally stationary processes [Park et al, 2014, Gibberd and Nelson, 2016, Roueff et al, 2016.…”

Regularised estimation of 2D-locally stationary wavelet processes

“…Finite Sample Performance -Whilst asymptotically, one may show that kernel smoothers akin to (1) can recover the LWS spectrum in a consistent manner [10,14], the finite sample performance of such estimators is often lacking. For example, in the multivariate case we demonstrated [5] such an estimator often produced negative estimates for the variance, a quantity that through the model construction is required to be positive.…”