Multiscale modelling of replicated nonstationary time series

Abstract: Within the neurosciences, to observe variability across time in the dynamics of an underlying brain process is neither new nor unexpected. Wavelets are essential in analyzing brain signals because, even within a single trial, brain signals exhibit nonstationary behaviour. However, neurological signals generated within an experiment may also potentially exhibit evolution across trials (replicates). As neurologists consider localised spectra of brain signals to be most informative, here we develop a novel wavelet-based tool capable to formally represent process nonstationarities across both time and replicate dimensions. Specifically, we propose the Replicate Locally Stationary Wavelet (RLSW) process, that captures the potential nonstationary behaviour within and across trials. Estimation using wavelets gives a natural desired time- and replicate-localisation of the process dynamics. We develop the associated spectral estimation framework and establish its asymptotic properties. By means of thorough simulation studies, we demonstrate the theoretical estimator properties hold in practice. A real data investigation into the evolutionary dynamics of the hippocampus and nucleus accumbens during an associative learning experiment, demonstrate the applicability of our proposed methodology, as well as the new insights it provides.