Graph-based representation enables to outline efficiently interactions between sensors and as such has encountered a growing interest. For example in neurosciences, the graph of interactions between brain regions has shed lights on evolution of diseases. In this paper, we describe a whole procedure which estimates the graph from multivariate time series. First correlations using wavelet decomposition of the signals are estimated. Bonferroni (1935)'s procedure on multiple correlation testing is then used. We prove theoretically that the Family Wise Error Rate (FWER) is asymptotically controlled for any graph structures. We implement our approach on smallworld graph structures, with signals possibly having long-memory properties. This structure is inspired by real data examples from resting-state functional magnetic resonance imaging. The control is confirmed graphically. Numerical simulations illustrate the behavior of the bias and the power of our proposed approach.
Many applications fields deal with multivariate long-memory time series. A challenge is to estimate the long-memory properties together with the coupling between the time series. Real wavelets procedures present some limitations due to the presence of phase phenomenons. A perspective is to use analytic wavelets to recover jointly long-memory properties, modulus of long-run covariance between time series and phases. Approximate wavelets Hilbert pairs of Selesnick (2002) fullfilled some of the required properties. As an extension of Selesnick (2002)’s work, we present some results about existence and quality of these approximately analytic wavelets.