This paper extends the intrinsic wavelet methods for curves of Hermitian positive definite matrices of Chau and von Sachs (2017) to surfaces of Hermitian positive definite matrices, with in mind the application to nonparametric estimation of the time-varying spectral matrix of a locally stationary time series. First, intrinsic average-interpolating wavelet transforms acting directly on surfaces of Hermitian positive definite matrices are constructed 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.to allow for more flexible modeling of spectral charactaristics that vary with time. For instance, in neuroscientific experiments involving electroencephalogram (EEG) or local field potential (LFP) time series recorded during a brain seizure, our aim is to analyze the Fourier spectra locally in time to analyze the evolving spectral behavior during the experiment. There is not a unique way to relax the assumption of stationarity to define a nonstationary time series process with a timedependent Fourier spectrum. In this paper, we focus on nonparametric spectral estimation for a class of locally stationary time series as first defined in Dahlhaus (1997). The following definition generalizes the Cramér representation of a stationary time series (see e.g., (Brockwell and Davis, 2006, Section 11.8)) and is similar in definition to Guo et al. (2003), Guo and Dai (2006) and Li and Krafty (2018) among others. This is a modified version of the locally time series model in Dahlhaus (1997) or Dahlhaus (2012), where the original sequences of functions A 0 t,T (ω) in Dahlhaus (1997) and Dahlhaus (2012) are replaced by a 2-dimensional surface A(ω, t/T ), typically assumed to be smooth across frequency and time. Definition 1.1. (Locally stationary vector-valued time series) Let { Y t , t = 1, . . . , T } be a zero-mean vector-valued time series observed at time points t = 1, . . . , T . The time series Y t is said to be locally stationary if it admits the following representation with probability 1,Here, { Z(ω), −π ≤ ω ≤ π} is a vector-valued zero-mean orthogonal increment process defined as in (Brockwell and Davis, 2006, Section 11.8), with for −π ≤ λ ≤ π and 0 ≤ u ≤ 1, E λ −π A(ω, u) d Z(ω) λ −π A(ω, u) d Z(ω) * = λ −π f (ω, u) dω, such that the cumulants of d Z exist and are bounded for all orders. Moreover, f (ω, u) is the timevarying or...
This report presents the implementation of a web tool for the extension of exposure-and toxicokinetictoxicodynamic analysis that has been developed under a specific EFSA request to Open Analytics under the framework agreement (OC/EFSA/AMU/2015/02). An open source software has been developed in R as a WEB-based tool including different components for the modelling of toxicokinetic (TK) and toxicodynamic (TD) processes within a structured workflow. The workflow provides the steps to perform such TK-TD modelling for single chemicals in the human health, animal health and ecological risk assessment. The web-based tool results in the implementation of four modules: (1) Chemical specific modules, (2) Physiological and life cycle trait modules, (3) Toxicokinetic module, and (4) Toxicodynamic module.
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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