Directed wavelet covariance for locally stationary processes

• Autor(a) principal: Lopes, Kim Samejima Mascarenhas
• Data de Publicação: 2018
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/45/45133/tde-14032018-174950/
• topic: Coerência direcionada; Covariância de ondaletas; Cross spectrum; Directed coherence; Espectro cruzado; Locally stationary processes; Processos localmente estacionários; Séries temporais; Time series; Wavelet covariance

The main goal of this study is to propose a methodology that measures directed relations between locally stationary processes. Unlike stationary processes, locally stationary processes may present sudden pattern changes and have local characteristics in specific intervals. This behavior causes instability in measures based on Fourier transforms. The relevance of this study relies on considering these processes and propose robust methodologies that are not affected by outliers, sudden pattern changes or local behavior. We start reviewing the Partial Directed Coherence (PDC) and the Wavelet Coherence. PDC measures the directed relation between components of a multivariate stationary Vector Autoregressive (VAR) model in the frequency domain, while Wavelet Coherence is based on complex wavelets decomposition. We then propose a causal wavelet decomposition of the covariance structure for bivariate locally stationary processes: the Directed Wavelet Covariance (DWC). Compared to Fourier-based quantities, wavelet-based estimators are more appropriate for non-stationary processes and processes with local patterns, outliers and rapid regime changes like in EEG experiments with the introduction of stimuli. We then propose its estimators and calculate its expectation and analyze its variance. Next we propose a decomposition for the variance of multivariate processes with more than two components: the Partial Directed Wavelet Covariance (pDWC). Considering a N-variate locally stationary process, the pDWC calculates the Directed Wavelet Covariance of X_1(t) with X_2(t) eliminating the effect of the other components X_3(t), ... ,X_N(t). We propose two approaches to this situation. First we filter the multivariate process to remove all the exogenous influences and then we calculate the directed relation between the components. In the second case, as in Partial Directed Coherence, we consider the multivariate process as a time-varying Vector Autoregressive Model (tv-VAR) and use its coefficients in the decomposition of the covariance function to isolate the effects of the other components. We also compare results of the PDC, Wavelet Coherence and Directed Wavelet Covariance with simulated data. Finally, we present an application of the proposed Directed Wavelet Covariance and Partial Directed Wavelet Covariance on EEG data. Simulation results show that the proposed measures capture the simulated relations. The pDWC with linear filter has shown more stable estimations than the proposed pDWC considering the tv-VAR. Future studies will discuss the DWC\'s and pDWC\'s asymptotic distributions and significance tests. The proposed Directed Wavelet Covariance decomposition is a different approach to deal with non-stationary processes in the context of causality. The use of wavelets is a gain and adds to the number of studies that can be addressed when Fourier transform does not apply. The pDWC is an alternative for multivariate processes and it removes linear influences from observed external components.