Wavelet Coherence Analysis with an Application of Brain Images

Wavelet analysis has become an emerging method in a wide range of applications with non-stationary data. In this work, we apply wavelets to tackle the problem of estimating dynamic association in a collection of multivariate non-stationary time series. Coherence is a common metric for linear dependence across signals. However, it assumes static dependence and does not sufficiently model many biological processes with time-evolving dependence structures. We explore continuous wavelet analysis for modeling and estimating such dynamic dependence under the replicated multivariate time series settings. Wavelet transformation provides a decomposition of signals that localizes in both time and frequency domains, hence extracts time-evolving and scale-specific features. Under this setting, we rigorously define and estimate wavelet coherency, standardized wavelet coherency, and partial wavelet coherency. While the first two coherency measure marginal linear dependence for bivariate time series, the last one measures the conditional linear dependence in multivariate settings, which identifies the direct association. The properties and consistency of the proposed estimators are established. We further develop the permutation test and bootstrap confidence intervals for the power spectrum of coherency and partial coherency. The presented methodology is demonstrated via simulations and applied to the fMRI data in a study of Alzheimer’s Disease