Wishart Distribution Research Articles

English Teaching Ability Evaluation Algorithm Based on Random Matrix Model and Fuzzy K-Means Clustering

This paper proposes a nonelliptic extended English teaching ability evaluation algorithm based on an adaptive random matrix model. The algorithm models a nonelliptical expansion target as multiple elliptical sub-objectives, and the expansion state of each sub-target is described by an inverse Wishart distribution. The new method is combined to improve the robustness problem caused by the initialization after the algorithm expansion. This research uses the smart teacher education platform to conduct research and analysis on the evaluation of teaching practice ability of intern normal students. From the perspective of data evaluation, we explore the influence of normal students’ curriculum training on the teaching practice ability of normal students. By analyzing the correlation between normal students’ course grades and practice grades, this paper explores the influence of normal students’ prepractice training on normal students’ teaching practice ability. This paper explores the influence of the training of normal students’ course learning on the teaching practice ability of normal students. The results show that the learning level of normal students’ professional courses has a significant impact on the development of normal students’ practice performance and teaching practice ability; the impact of normal students’ pedagogical course level on normal students’ practice performance and teaching practice ability is relatively low.

Establishment of a Fuzzy Comprehensive Evaluation Random Matrix Model for the Assessment of Foreign Language Translation Level in Universities

In this paper, the random matrix method of a fuzzy comprehensive evaluation is used to conduct in-depth research and analysis on the assessment of foreign language translation levels in universities, and the corresponding model is designed for the automatic assessment of translation levels in university education. The Hessian matrix of the second-order optimization method is used to analyze the geometric properties of the loss plane of the deep neural network, and it is proved that the Hessian matrix can be constructed as a sample covariance matrix, which is the classical Wishart matrix in the random matrix theory. Using the study on the asymptotic distribution properties of the Wishart matrix in random matrix theory, the Hessian matrix is analyzed and the limiting spectral distribution, the eigenvalue extreme value distribution, and the standard conditional number distribution of the matrix are given. And the scores are basically distributed above 240, but some indicators have relatively low scores, all below 120 points, so the indicators distributed above 240 points are selected as reserved indicators. The analysis considers that the evaluation of foreign language translation in learning in colleges and universities is itself very fuzzy and constructs an evaluation model based on the fuzzy comprehensive judgment method, which provides a research strategy for the future evaluation of foreign language translation in learning in colleges and universities. Finally, the evaluation system of foreign language translation in learning in colleges and universities is designed and implemented. It is feasible to apply the fuzzy comprehensive judgment to the comprehensive evaluation of college students, the combined judgment of multiple methods has higher reliability and stability, and its research conclusions have certain reference value in the specific education management practice. The value, objectivity, feasibility, and wide adaptability of the fuzzy comprehensive evaluation method are analyzed, and some references are expected to be provided for translation quality assessment research and translation practice.

Vibrational density of states of jammed packing at high dimensions: Mean-field theory.

Several mean-field theories predict that the Hessian matrix of amorphous solids converges the Wishart matrix in the limit of the large spatial dimensions d→∞. Motivated by these results, we calculate here the density of states of random packing of harmonic spheres by mapping the Hessian of the original system to the Wishart matrix. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions d=3, 5, and 9. For small pressure p≪1 (near jamming), we find a good agreement even in d=3, and obtain better agreements in larger d, suggesting that the approximation becomes exact in the limit d→∞.

Variational Bayesian Estimator for Mobile Robot Localization With Unknown Noise Covariance

This article studies mobile robot localization with unknown noise covariance. The AprilTag is used as landmarks and observed using the onboard camera. The system model is created based on the mobile robot motion and AprilTag measurements. The unknown measurement noise covariance is considered as a random matrix satisfying an inverse Wishart distribution. A variational Bayesian estimator is proposed to estimate the robot pose, AprilTag locations, and measurement noise covariance, where the Rao–Blackwellized estimator is developed for robot pose and AprilTag location estimation, and variational Bayesian approximation is adopted for measurement noise covariance estimation. Simulations and experiments are conducted to validate the effectiveness of the proposed method.

Life-Cycle Assessment of Long-Span Bridge’s Wind Resistant Performance Considering Multisource Time-Variant Effects and Uncertainties

This paper examines the life-cycle wind resistant performance of a constructed long-span suspension bridge in the coastal region of China, aiming to quantify the multisource time-variant effects and uncertainties and offering a reference for designs of long-span bridges in the future. Randomness from modal frequencies, damping ratios, and identification uncertainty of flutter derivatives (FDs) was considered; then, their effects on probability of flutter failure and probability of exceeding the predefined buffeting response root-mean square (RMS) are discussed. Firstly, results of full-track tropical cyclone (TC) simulation under various climate warming scenarios are reviewed; then, the time-variant probability density function (PDF) of annual extreme wind speed is discussed. Secondly, 6-year modal frequencies and damping ratios of a long-span suspension bridge with a center-slotted section were extracted by fast Bayesian FFT method with structural health monitoring (SHM) data, which were utilized to explore the deterioration rules of structural properties. Thirdly, FDs were modeled from a probabilistic perspective based on complex Wishart distribution, which were identified in the turbulent flow and the frequency domain by Bayesian inference. The posterior distributions of FDs, namely identification uncertainty, were quantified by Markov chain Monte Carlo (MCMC) sampling. This paper finds that for flutter resistant performance, the time-variant effects (i.e., modal frequencies and PDFs of extreme wind speed) will make the flutter failure probability seven times larger than the initial value; for the probability of exceeding the predefined buffeting response RMS, however, the time-variant effects will make a negligible difference.

Adaptive invariant Kalman filtering for lie groups attitude estimation with biased and heavy-tailed process noise

Attitude determination is fundamental for spacecraft missions in aerospace engineering. Kalman filter (KF) is the optimal estimator in least square sense and, using the symmetry properties of matrix Lie groups system, the invariant Kalman filter (IKF) has been developed to boost the filtering performance for attitude estimation. However, due to presence of frequent and severer maneuvers, the Lie groups attitude dynamics is usually corrupted by significant biases and heavy-tailed outliers, which usually leads to decreased precision of IKF. For attitude estimation problem troubled by biased and heavy-tailed process noise, this work proposes a new invariant Kalman filter (VBAIKF) by constructing the hierarchical Gaussian system model: the probability density function of prior estimate state is first described using the student’s t distribution, while the unknown scale covariance matrix and degrees of freedom (dof) of the employed student’s t distribution are defined as the inverse Wishart distribution (IWD) and Gamma distribution. In VBAIKF, the Lie groups rotation matrix of spacecraft, the biased mean, the parameters of dof and scale covariance matrix are online estimated together by variational Bayesian fixed-point iterations. The simulation results with Lie groups attitude estimation system further verify the superior filtering adaptability and precision of proposed approach VBAIKF than other methods for attitude determination with biased mean and heavy-tailed process noise.

Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices

We consider random walks on the cone of \(m \times m\) positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann Probab 45:4101–4111, 2017), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution \(W_m(a,I_m)\), with index parameter a and matrix parameter \(I_m\), the identity matrix, we derive explicit and computable bounds for each term appearing in the Hoffmann-Jørgensen inequalities.

Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices

In [I. Dumitriu and E. Paquette, Spectra of overlapping Wishart matrices and the gaussian free field, Random Matrices: Theory Appl. 07(2) (2018) 1850003], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of [A. Borodin, CLT for spectra of submatrices of Wigner random matrices, Moscow Math. J. 14(1) (2014) 29–38] for stochastic Wigner matrices.

The partial transpose and asymptotic free independence for Wishart random matrices, II

Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.

On the singular gamma, Wishart, and beta matrix‐variate density functions

AbstractWhen a real positive definite matrix follows a Wishart or, more generally, a matrix‐variate gamma distribution with shape parameter and positive definite scale parameter matrix , one can represent as for some matrix of dimension . When , has a singular distribution whose properties can be studied via the density function of . It will be shown that when follows a matrix‐variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type‐1 and type‐2 beta‐distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.

Non-Gaussian likelihood of weak lensing power spectra

The power spectrum of weak lensing fluctuations has a non-Gaussian distribution due to its quadratic nature. On small scales the Central Limit Theorem acts to Gaussianize this distribution but non-Gaussianity in the signal due to gravitational collapse is increasing and the functional form of the likelihood is unclear. Analyses have traditionally assumed a Gaussian likelihood with non-linearity incorporated into the covariance matrix; here we provide the theory underpinning this assumption. We calculate, for the first time, the leading-order correction to the distribution of angular power spectra from non-Gaussianity in the underlying signal and study the transition to Gaussianity. Our expressions are valid for an arbitrary number of correlated maps and correct the Wishart distribution in the presence of weak (but otherwise arbitrary) non-Gaussianity in the signal. Surprisingly, the resulting distribution is not equivalent to an Edgeworth expansion. The leading-order effect is to broaden the covariance matrix by the usual trispectrum term, with residual skewness sourced by the trispectrum and the square of the bispectrum. Using lognormal lensing maps we demonstrate that our likelihood is uniquely able to model both large and mildly non-linear scales. We provide easy-to-compute statistics to quantify the size of the non-Gaussian corrections. We show that the full non-Gaussian likelihood can be accurately modelled as a Gaussian on small, non-linear scales. On large angular scales non-linearity in the lensing signal imparts a negligible correction to the likelihood, which takes the Wishart form in the full-sky case. Our formalism is equally applicable to any kind of projected field.

HIV Infection Progression Monitoring System Based on CD4 Counts and Wishart Distribution.

The human immunodeficiency virus (HIV) is a viral infection that destroys the human immune system, resulting in the acquired immunodeficiency syndrome (AIDS). The management and care of patients on antiretroviral therapy (ART) consumes a large portion of the health budget of many countries. ART improves the lives of the HIV patients. However, benefiting from the treatment remains to be low due to the nonadherence, adverse events, and treatment failure associated with the transmitted drug resistance mutations (TDRMs). Extra care is therefore required in prescribing switch of ART regimens for HIV-naive patients. We propose a disease monitoring system, which depends on how the HIV-naive patients respond to the ART regimen. We model cluster of differentiation 4 (CD4) counts data measured at every 3 months in a period of 48 weeks on a cohort of 87 HIV-naive patients on ART, from Zambia. We demonstrate how to apply the Bayesian Wishart distribution to model CD4 counts, leading to an informative HIV progression monitoring system. We found a steady increase in the average of the CD4 counts (from 219 to 315) for HIV-naive patients on the ART regimen. The average was still below the expected 500 CD4 counts for a normal person. The derived precision matrix shows an increase in probability of potency of the ART regimen, which ranges from 0.1261 to 0.8678. An early detection is crucial as it allows for timely switch of regimen from first to second line or to the third line. The proposed HIV disease progression monitoring system for HIV-naive patients on ART regimen that is based on CD4 counts could enable physicians make informed decisions on the management and care of the patients.

Poisson multi‐Bernoulli mixture filters with coloured measurement noise

To solve multitarget tracking (MTT) problems with coloured measurement noise, this study proposes a Poisson multi-Bernoulli mixture filter with coloured measurement noise (PMBM-CMN) and a robust PMBM-CMN filter. By using the measurement differencing method and state augmentation approach, the proposed PMBM-CMN filter transforms a state estimation problem with coloured measurement noise into a problem with white measurement noise. However, covariances of the true process and measurement noise in the proposed PMBM-CMN filter are time-varying and unknown, which may degrade the filtering performance. Therefore, a robust PMBM-CMN filter is proposed for estimating the augmented state, including the kinematic state, the predicted state covariance, and the white measurement noise covariance. For linear Gaussian systems, the augmented state is modelled as a Gaussian inverse Wishart inverse Wishart (GIWIW) distribution. The variational Bayesian method is also employed to guarantee the conjugacy of the GIWIW density. Simulation results demonstrate the ability of the PMBM-CMN filter to solve MTT problems with coloured measurement noise and show that the robust PMBM-CMN filter based on the GIWIW model (GIWIW-PMBM-CMN) has the best overall performance in comparison with existing state-of-the-art filters.

Efficient jamming attack against MIMO transceiver

This work addresses the issue of point-to-point intelligent jamming against MIMO architecture-based systems. The receiver uses an optimal Maximum Likelihood (ML) detector to extract the data stream from the received signal. Analyzing the receiver performance in the presence of a jamming signal, optimal attack strategies based on the pairwise error probability parameter as an objective function are designed, enabling the achievement of maximum degradation of the service quality. To perform the worst-case jamming of a legitimate transceiver, we assume that the jammer can eavesdrop on the Channel State Information (CSI). Depending on the jamming power budget constraint and the potential of jamming systems to estimate the CSI, different scenarios are envisaged, including imperfect CSI estimation cases. By minimizing the trace of Positive Semi-Definite (PSD) matrix products, the optimal jamming approaches that ensure maximum degradation of the legitimate link are designed. The disruption efficiency of the proposed approaches is comparable to the employment of the brute-force attack, but with a higher power level that is proportional to the number of jamming antennas in the case of partial knowledge of CSI, and inversely to the statistical expectation of the square root of the Wishart matrix's eigenvalues in the case of complete knowledge of CSI. To justify the relevance of the suggested policy, extensive simulations of the effect of these interventions are presented.

Role of CSN2, CSN3, and BLG genes and the polygenic background in the cattle milk protein profile

To devise better selection strategies in dairy cattle breeding programs, a deeper knowledge of the role of the major genes encoding for milk protein fractions is required. The aim of the present study was to assess the effect of the CSN2, CSN3, and BLG genotypes on individual protein fractions (αS1-CN, αS2-CN, β-CN, κ-CN, β-LG, α-LA) expressed qualitatively as percentages of total nitrogen content (% N), quantitatively as contents in milk (g/L), and as daily production levels (g/d). Individual milk samples were collected from 1,264 Brown Swiss cows reared in 85 commercial herds in Trento Province (northeast Italy). A total of 989 cows were successfully genotyped using the Illumina Bovine SNP50 v.2 BeadChip (Illumina Inc.), and a genomic relationship matrix was constructed using the 37,519 SNP markers obtained. Milk protein fractions were quantified and the β-CN, κ-CN, and β-LG genetic variants were identified by reversed-phase HPLC (RP-HPLC). All protein fractions were analyzed through a Bayesian multitrait animal model implemented via Gibbs sampling. The effects of days in milk, parity order, and the CSN2, CSN3, and BLG genotypes were assigned flat priors in this model, whereas the effects of herd and animal additive genetic were assigned Gaussian prior distributions, and inverse Wishart distributions were assumed for the respective co-variance matrices. Marginal posterior distributions of the parameters of interest were compared before and after the inclusion of the effects of the 3 major genes in the model. The results showed that a high portion of the genetic variance was controlled by the major genes. This was particularly apparent in the qualitative protein profile, which was found to have a higher heritability than the protein fraction contents in milk and their daily yields. When the genes were included individually in the model, CSN2 was the major gene controlling all the casein fractions except for κ-CN, which was controlled directly by the CSN3 gene. The BLG gene had the most influence on the 2 whey proteins. The genetic correlations showed the major genes had only a small effect on the relationships between the protein fractions, but through comparison of the correlation coefficients of the proteins expressed in different ways they revealed potential mechanisms of regulation and competitive synthesis in the mammary gland. The estimates for the effects of the CSN2 and CSN3 genes on protein profiles showed overexpression of protein synthesis in the presence of the B allele in the genotype. Conversely, the β-LG B variant was associated with a lower concentration of β-LG compared with the β-LG A variant, independently of how the protein fractions were expressed, and it was followed by downregulation (or upregulation in the case of the β-LG B) of all other protein fractions. These results should be borne in mind when seeking to design more efficient selection programs aimed at improving milk quality for the efficiency of dairy industry and the effect of dairy products on human health.

On the measure of anchored Gaussian simplices, with applications to multivariate medians

We consider anchored Gaussian ℓ-simplices in the d-dimensional Euclidean space, that is, simplices with one fixed vertex y∈Rd and the remaining vertices X1,…,Xℓ randomly sampled from the d-variate standard normal distribution. We determine the distribution of the measure of such simplices for any d, any ℓ, and any anchor point y, which is of interest, e.g., when studying the asymptotic behaviour of U-statistics based on such simplex measures. We provide two proofs of the results. The first one is short but is not self-contained as it crucially relies on a technical result for non-central Wishart distributions. The second one is a simple and self-contained proof, that also provides some geometric insight on the results. Quite nicely, variations on this second argument reveal intriguing distributional identities on products of central and non-central chi-square distributions with Beta-distributed non-centrality parameters. We independently establish these distributional identities by making use of Mellin transforms. Beyond the aforementioned use to study the asymptotic behaviour of some U-statistics, our results do find natural applications in the context of robust location estimation, as we illustrate by considering a class of simplex-based multivariate medians that contains the celebrated spatial median and Oja median as special cases. Throughout, our results are confirmed by numerical experiments.

Random Matrix Analysis of Protein Families

Proteins are vital for almost all biochemical and cellular processes. Although there is an enormous growth in the protein sequence data, the statistical characterization, structure, and function of many of these sequences are still unknown. The statistical and spectral analysis of the Pearson correlation matrices between positions based on physiochemical properties of amino acids of seven protein families is performed and compared with the random Wishart matrix model results. A detailed analysis shows that the protein families significantly diverge from the Marcenko-Pastur distribution with many eigenvalues (outliers) outside the Wishart lower and upper bound. It is shown that level spacing distribution of protein families is similar to the Gaussian orthogonal ensemble. Further, the number variance varies as log of the system size indicating the presence of long range correlations within the protein families.

Bayesian adaptive direction detector in sample‐starved environment

Here, the problem of direction detection in disturbance with unknown covariance matrix is considered. The case that the number of the training data is too small to form an effective estimate for the unknown covariance matrix is focused upon. To solve the problem, the Bayesian framework is resorted to. Precisely, the unknown covariance matrix is assumed to be ruled by an inverse Wishart distribution. Then the problem is solved by the detector design criterion of generalized likelihood ratio test. Numerical examples indicate that the proposed detector can effectively detect the target, and provide higher probability of detection than the existing detectors even when sufficient training data are available.

Consensus variational Bayesian moving horizon estimation for distributed sensor networks with unknown noise covariances

This paper deals with state estimation over distributed sensor networks with unknown process noise covariance matrix (PNCM) and measurement noise covariance matrices (MNCMs). The unknown PNCM and MNCMs are modeled with constrained inverse Wishart distributions and estimated together with the state trajectory within a moving horizon window. Each local node computes its local estimates by combining variational Bayes (VB) inference with Moving Horizon Estimation (MHE). Then, in order to diffuse the information through the network, at each time instant three consensus tasks are performed in parallel: the first one is a consensus on the parameters of the local approximated PNCM posterior; the second one is a consensus on the prior information on the state trajectory over the moving window; the third one is a consensus on the measurements over the moving window. The resulting consensus-based distributed VBMHE (D-VBMHE) algorithm generalizes existing consensus filters to the problem of jointly estimating the state trajectory over a window as well as to unknown PNCM and MNCMs. Simulation and experimental results on target tracking case-studies demonstrate the outperformance of the proposed D-VBMHE compared to state-of-the-art conventional and adaptive consensus filters.

Randomized Quaternion Singular Value Decomposition for Low-Rank Matrix Approximation

This paper presents a randomized quaternion singular value decomposition (QSVD) algorithm for low-rank matrix approximation problems, which are widely used in color face recognition, video compression, and signal processing problems. With quaternion normal distribution-based random sampling, the randomized QSVD algorithm projects high-dimensional data to a low-dimensional subspace and then identifies an approximate range subspace of the quaternion matrix. The key statistical properties of quaternion Wishart distribution are proposed and used to perform the approximation error analysis of the algorithm. Theoretical results show that the randomized QSVD algorithm can trace dominant singular value decomposition triplets of a quaternion matrix with acceptable accuracy. Numerical experiments also indicate the rationality of proposed theories. Applied to color face recognition problems, the randomized QSVD algorithm obtains higher recognition accuracies and behaves more efficient than the known Lanczos-based partial QSVD and a quaternion version of the fast frequent directions algorithm.