arXiv subjects

Joris Chau

Publications and source records attributed to Joris Chau.

4 recordsLinked to original sources

arXiv2019

Intrinsic wavelet regression for curves of Hermitian positive definite matrices

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment.

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arXiv2019

Intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

This paper develops intrinsic wavelet denoising methods for surfaces of Hermitian positive definite matrices, with in mind the application to nonparametric estimation of the time-varying spectral matrix of a multivariate locally stationary time series. First, we construct intrinsic average-interpolating wavelet transforms acting directly on surfaces of Hermitian positive definite matrices in a curved Riemannian manifold with respect to an affine-invariant metric. Second, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces of Hermitian positive definite matrices, and investigate practical nonlinear thresholding of wavelet coefficients based on their trace in the context of intrinsic signal plus noise models in the Riemannian manifold. The finite-sample performance of nonlinear tree-structured trace thresholding is assessed by means of simulated data, and the proposed intrinsic wavelet methods are used to estimate the time-varying spectral matrix of a nonstationary multivariate electroencephalography (EEG) time series recorded during an epileptic brain seizure.

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arXiv2018

Intrinsic data depth for Hermitian positive definite matrices

Nondegenerate covariance, correlation and spectral density matrices are necessarily symmetric or Hermitian and positive definite. The main contribution of this paper is the development of statistical data depths for collections of Hermitian positive definite matrices by exploiting the geometric structure of the space as a Riemannian manifold. The depth functions allow one to naturally characterize most central or outlying matrices, but also provide a practical framework for inference in the context of samples of positive definite matrices. First, the desired properties of an intrinsic data depth function acting on the space of Hermitian positive definite matrices are presented. Second, we propose two computationally fast pointwise and integrated data depth functions that satisfy each of these requirements and investigate several robustness and efficiency aspects. As an application, we construct depth-based confidence regions for the intrinsic mean of a sample of positive definite matrices, which is applied to the exploratory analysis of a collection of covariance matrices associated to a multicenter research trial.

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arXiv2016

Functional mixed effects wavelet estimation for spectra of replicated time series

Motivated by spectral analysis of replicated brain signal time series, we propose a functional mixed effects approach to model replicate-specific spectral densities as random curves varying about a deterministic population-mean spectrum. In contrast to existing work, we do not assume the replicate-specific spectral curves to be independent, i.e. there may exist explicit correlation between different replicates in the population. By projecting the replicate-specific curves onto an orthonormal wavelet basis, estimation and prediction is carried out under an equivalent linear mixed effects model in the wavelet coefficient domain. To cope with potentially very localized features of the spectral curves, we develop estimators and predictors based on a combination of generalized least squares estimation and nonlinear wavelet thresholding, including asymptotic confidence sets for the population-mean curve. We derive risk bounds for the nonlinear wavelet estimator of the population-mean curve, a result that reflects the influence of correlation between different curves in the replicate-population, and we derive consistency of the estimators of the inter- and intra-curve correlation structure in an appropriate sparseness class of functions. To illustrate the proposed functional mixed effects model and our estimation and prediction procedures, we present several simulated time series data examples and we analyze a motivating brain signal dataset recorded during an associative learning experiment.

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