scholarly journals On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

2021 ◽  
Author(s):  
Alicja Dembczak-Kołodziejczyk ◽  
Anna Lytova
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Random Variable ◽  
Covariance Matrices ◽  
Convergence In Probability ◽  
Stat Phys ◽  
Sample Covariance ◽  
Arbitrary Space

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].

On the limit of the spectral distribution of large-dimensional random quaternion covariance matrices

2017 ◽  
Vol 06 (02) ◽  
pp. 1750004 ◽  
Author(s):  
Yanqing Yin ◽  
Jiang Hu
Keyword(s):  
Machine Learning ◽  
Image Processing ◽  
Adaptive Filtering ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Moment Condition ◽  
Quaternion Matrices

The use of quaternions and quaternion matrices in practice, such as in machine learning, adaptive filtering, vector sensing and image processing, has recently been rapidly gaining in popularity. In this paper, by applying random matrix theory, we investigate the spectral distribution of large-dimensional quaternion covariance matrices when the quaternion samples are drawn from a population that satisfies a mild moment condition. We also apply the result to several common models.

scholarly journals Spectrum Sensing in Cognitive Radio by Use of Volume-Based Method

2018 ◽  
Vol 7 (2.17) ◽  
pp. 34
Author(s):  
C S. Preetham ◽  
Ch Mahesh ◽  
Ch Saranga Haripriya ◽  
Ramaraju Anirudh ◽  
M S. Sireesha
Keyword(s):  
Covariance Matrix ◽  
Spectrum Sensing ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Sample Covariance Matrix ◽  
Closed Form Expression ◽  
Sample Covariance ◽  
Signal Sample ◽  
Deficiency Cases

Spectrum sensing is the mission of finding the licensed user signal situation, i.e. to determine the existence and deficiency of primary (licensed) user signal, the recent publications random matrix theory algorithms performs better-quality in spectrum sensing. The RMT fundamental nature is to make use of the distributed extremal eigenvalues of the arrived signal sample covariance matrix (SMC), specifically, Tracy-Widom (TW) distribution which is useful to certain extent in spectrum sensing but demanding for numerical evaluations because there is absence of closed-form expression in it. The sample covariance matrix determinant is designed for two novel volume-based detectors or signal existence and deficiency cases are differentiated by using volume. Under the Gaussian noise postulation one of the detectors theoretical decision thresholds is perfectly calculated by using Random matrix theory. The volume-based detectors efficiency is shown in simulation results. 

scholarly journals Unifying model for random matrix theory in arbitrary space dimensions

2018 ◽  
Vol 97 (3) ◽  
Author(s):  
Giovanni M. Cicuta ◽  
Johannes Krausser ◽  
Rico Milkus ◽  
Alessio Zaccone
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Arbitrary Space

scholarly journals Bijective evaluation of the connection coefficients of the double coset algebra

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alejandro H. Morales ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Symmetric Functions ◽  
Matrix Theory ◽  
Double Coset ◽  
Hyperoctahedral Group ◽  
Connection Coefficients ◽  
Generating Series ◽  
International Audience

International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests. Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

scholarly journals Efficient computation of limit spectra of sample covariance matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550019 ◽  
Author(s):  
Edgar Dobriban
Keyword(s):  
Covariance Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Discrete Distributions ◽  
High Dimensional ◽  
Efficient Computation ◽  
Usual Model ◽  
Sample Covariance ◽  
Distribution Of Eigenvalues

Models from random matrix theory (RMT) are increasingly used to gain insights into the behavior of statistical methods under high-dimensional asymptotics. However, the applicability of the framework is limited by numerical problems. Consider the usual model of multivariate statistics where the data is a sample from a multivariate distribution with a given covariance matrix. Under high-dimensional asymptotics, there is a deterministic map from the distribution of eigenvalues of the population covariance matrix (the population spectral distribution or PSD), to the of empirical spectral distribution (ESD). The current methods for computing this map are inefficient, and this limits the applicability of the theory. We propose a new method to compute numerically the ESD from an arbitrary input PSD. Our method, called SPECTRODE, finds the support and the density of the ESD to high precision; we prove this for finite discrete distributions. In computational experiments SPECTRODE outperforms existing methods by orders of magnitude in speed and accuracy. We apply it to compute expectations and contour integrals of the ESD, which are often central in applications. We also illustrate that SPECTRODE is directly useful in statistical problems, such as estimation and hypothesis testing for covariance matrices. Our proposal, implemented in open source software, may broaden the use of RMT in high-dimensional data analysis.

scholarly journals Variability in mRNA translation: a random matrix theory approach

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael Margaliot ◽  
Wasim Huleihel ◽  
Tamir Tuller
Keyword(s):  
Production Rate ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Protein Production ◽  
Matrix Theory ◽  
Mrna Translation ◽  
Random Variable ◽  
Theory Approach ◽  
Cellular Processes ◽  
State Production

AbstractThe rate of mRNA translation depends on the initiation, elongation, and termination rates of ribosomes along the mRNA. These rates depend on many “local” factors like the abundance of free ribosomes and tRNA molecules in the vicinity of the mRNA molecule. All these factors are stochastic and their experimental measurements are also noisy. An important question is how protein production in the cell is affected by this considerable variability. We develop a new theoretical framework for addressing this question by modeling the rates as identically and independently distributed random variables and using tools from random matrix theory to analyze the steady-state production rate. The analysis reveals a principle of universality: the average protein production rate depends only on the of the set of possible values that the random variable may attain. This explains how total protein production can be stabilized despite the overwhelming stochasticticity underlying cellular processes.

No eigenvalues outside the support of the limiting spectral distribution of quaternion sample covariance matrices

2019 ◽  
Vol 09 (04) ◽  
pp. 2150003
Author(s):  
Huiqin Li
Keyword(s):  
Random Matrix ◽  
Spectral Properties ◽  
Matrix Theory ◽  
Closed Interval ◽  
Covariance Matrices ◽  
Fourth Moment ◽  
Square Root ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Matrix Formula

In this paper, we consider the spectral properties of quaternion sample covariance matrices. Let [Formula: see text], where [Formula: see text] is the square root of a [Formula: see text] quaternion Hermitian non-negative definite matrix [Formula: see text] and [Formula: see text] is a [Formula: see text] matrix consisting of i.i.d. standard quaternion entries. Under the framework of random matrix theory, i.e. [Formula: see text] as [Formula: see text], we prove that if the fourth moment of the entries is finite, then there will almost surely be no eigenvalues that appear in any closed interval outside the support of the limiting distribution as [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document