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Register a new userSpectral Density of Sample Covariance Matrices of Colored Noise
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We study the dependence of the spectral density of the covariance matrix ensemble on the power spectrum of the underlying multivariate signal. The white noise signal leads to the celebrated Marchenko-Pastur formula. We demonstrate results for some colored noise signals.
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The problem of information extraction from discrete stochastic time series, produced with some finite sampling frequency, using flicker-noise spectroscopy, a general framework for information extraction based on the analysis of the correlation links between signal irregularities and formulated for continuous signals, is discussed. It is shown that the mathematical notions of Dirac and Heaviside functions used in the analysis of continuous signals may be interpreted as high-frequency and low-frequency stochastic components, respectively, in the case of discrete series. The analysis of electroencephalogram measurements for a teenager with schizophrenic symptoms at two different sampling frequencies demonstrates that the power spectrum and difference moment contain different information in the case of discrete signals, which was formally proven for continuous signals. The sampling interval itself is suggested as an additional parameter that should be included in general parameterization procedures for real signals.
The heuristic identification of peaks from noisy complex spectra often leads to misunderstanding of the physical and chemical properties of matter. In this paper, we propose a framework based on Bayesian inference, which enables us to separate multipeak spectra into single peaks statistically and consists of two steps. The first step is estimating both the noise variance and the number of peaks as hyperparameters based on Bayes free energy, which generally is not analytically tractable. The second step is fitting the parameters of each peak function to the given spectrum by calculating the posterior density, which has a problem of local minima and saddles since multipeak models are nonlinear and hierarchical. Our framework enables the escape from local minima or saddles by using the exchange Monte Carlo method and calculates Bayes free energy via the multiple histogram method. We discuss a simulation demonstrating how efficient our framework is and show that estimating both the noise variance and the number of peaks prevents overfitting, overpenalizing, and misunderstanding the precision of parameter estimation.
This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $mathbf{B}_n=n^{-1}sum_{j=1}^{n}mathbf{Q}mathbf{x}_jmathbf{x}_j^{*}mathbf{Q}^{*}$ where $mathbf{Q}$ is a nonrandom matrix of dimension $ptimes k$, and ${mathbf{x}_j}$ is a sequence of independent $k$-dimensional random vector with independent entries, under the assumption that $p/nto y>0$. A key novelty here is that the dimension $kge p$ can be arbitrary, possibly infinity. This new model of sample covariance matrices $mathbf{B}_n$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $k=p$ and $mathbf{Q}=mathbf{T}_n^{1/2}$ for some positive definite Hermitian matrix $mathbf{T}_n$. Also with $k=infty$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (2004). Applications of this new CLT are proposed for testing the structure of a high-dimensional covariance matrix. The derived tests are then used to analyse a large fMRI data set regarding its temporary correlation structure.
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these results, we develop a new statistic to test the equality of two spiked population covariance matrices. Numerical studies show that the new test procedure is more powerful than some existing methods.
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