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We provide a probabilistic framework for the analysis of the restricted isometry constants (RICs) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distribution of the extreme eigenvalues of Wishart matrices, or on its approximation based on the gamma distribution. In particular, we derive tight lower bounds on the cumulative distribution functions (CDFs) of the RICs. The presented framework provides the tightest lower bound on the maximum sparsity order, based on sufficient recovery conditions on the RICs, which allows signal reconstruction with a given target probability via different recovery algorithms.

In this paper, we consider a generalized multivariate regression problem where the responses are monotonic functions of linear transformations of predictors. We propose a semi-parametric algorithm based on the ordering of the responses which is invariant to the functional form of the transformation function. We prove that our algorithm, which maximizes the rank correlation of responses and linear transformations of predictors, is a consistent estimator of the true coefficient matrix. We also identify the rate of convergence and show that the squared estimation error decays with a rate of o(1/√n). We then propose a greedy algorithm to maximize the highly non-smooth objective function of our model and examine its performance through simulations. Finally, we compare our algorithm with traditional multivariate regression algorithms over synthetic and real data.

This paper introduces a new Estimation of Distribution Algorithm (EDA) based on the multivariate Boltzmann distribution. In this work, the design variables and the energy function of the Boltzmann distribution are continuous. Note that since the population has finite size, it can only approximate a continuous Boltzmann distribution with some error. In order to tackle this issue, the parameter estimators for the mean vector and covariance matrix of a Multivariate Normal Density that approximate the Boltzmann density, are derived by minimizing the Kullback-Leibler divergence. The algorithm introduced here uses one energy function for the mean estimator and another for the covariance matrix estimator. The first function places the probability mass around the most promising regions by assigning larger weights to individuals with higher fitness. However, the second function orients the covariance matrix along improving directions by assigning larger weights to individuals with lower fitness. Our proposal combines the conveniences of linear weights with a simple annealing schedule to regulate the exploration and exploitation of the search process. The resulting algorithm is named the Boltzmann Estimation of Multivariate Normal Algorithm (BEMNA). By applying the developed formulae the BEMNA is capable of adapting the structure of a density model to the promisory search directions. BEMNA is tested with a benchmark of 16 functions and contrasted with the AMaLGaM algorithm, a state of the art EDA. Statistical tests of the experimental data show the competitiveness of the proposed algorithm.

We propose and investigate a particle filtering method for multivariate state-space models. In the literature, the most studied state-space model is the linear Gaussian model, which includes known matrices and known noise covariance matrices. In our work, we drop the assumption of knowing these matrices, which produces a nonlinear model. In tracking the dynamic states, we propose to integrate out all the static unknowns and therefore, we sample particles only from the space of the dynamic states. In computing the particle weights, again, we only use the sampled states. The sampling distribution of the states is a multivariate Student t distribution, and the computation of the weights is based on another multivariate Student t distribution. The performance of the proposed method is examined by computer simulations.

We address the problem of multivariate data classification by the nonparametric Bayesian methodology, where the priors are modeled as Dirichlet processes. Sets of series of multivariate data are observed, where each vector in the series is modeled by a Gaussian linear regression. Each class is defined by the unknown matrices of linear coefficients of the model and the covariance matrices of the errors of the model. The number of different classes is unknown. For the unknown coefficients and covariance matrices we adopt a conjugate prior, the matrix-normal - inverse Wishart distribution. We implement the classification by Markov chain Monte Carlo sampling. The proposed approach is demonstrated by extensive computer simulations.

Based on a characterization of multivariate normal distribution and the goodness-of-fit test for uniformity on the surface of a unit sphere, we present a new test for multivariate normal distribution. The new test possesses symmetry which improves on the combination tests based on the center of mass and the moment of inertia. The asymptotic chi-squared distribution of the test statistic is obtained, the test statistic has nice properties and it can be easily computed.

Copula is a powerful tool for multivariate probability analysis. Estimation of distribution algorithms are a class of optimization algorithms based on probability distribution model. This paper introduces a new estimation of distribution algorithm with multivariate Gaussian copulas. In the algorithm, Gaussian copula parameters are firstly estimated by estimating Kendall's tau and using the relationship of Kendall's tau and correlation matrix, thus, joint distribution is estimated. Then, the Monte Carte simulation is used to generate new individuals. The relative experimental results show that the new algorithm is effective.

Firstly, from the external structure of the multivariate time series ( ) matrix, the article defines the similarity function based on the Range and the number of the different samples of the two matrixes difference; Afterward, considering the correlations between the column vectors of the matrix, the internal factors of the matrix, defines the similarity function based on the weighted square norm of the corresponding covariance matrix; and constructs the similarity function by weighted the two defined functions. Lastly, applying the similarity function of two s to the clustering the database, the algorithm is very effective.

The condition number distribution of random matrices is crucial to understand for statistical characterization of various communication systems. In this paper, we propose a general approximation methodology for the condition number distribution of finite dimensional Wishart matrices. This approximation leads to a computable expression for condition number distributions. The proposed method can be applied to both uncorrelated and correlated central Wishart matrices. This approximation is shown to be most accurate when the multivariate sample matrix is close to a square matrix and when there is correlation in the smaller of the dimensions of the sample matrix. The accuracy of the proposed approximation formula is validated by comparing with simulation results, with the achieved accuracy being remarkably good.

The complex central semi-correlated Wishart matrix is the product of a zero-mean row-wise correlated (or columnwise correlated) complex Gaussian matrix and its Hermitian transposition. In this paper, we investigate the distribution of the smallest eigenvalue of the Wishart matrix, which is generated from the square Gaussian matrix. A closed-form expression for distribution of the smallest eigenvalue is derived. Compared with previous results, our expression is very concise, since the distribution has been known in terms of matrix determinants. Numerical results that confirm the expression are also presented. And a complicated equality is proved due to the fact that our distribution is mathematically equivalent to those reported.

In this paper, a new infinite series representation for the multivariate alpha-mu joint probability density function is derived allowing for an arbitrary correlation matrix and non-identically distributed variates. The formulation is general and exact and comprises all of the other joint densities that arise from alpha-mu distribution published in the literature.

We derive the exact cumulative distribution function (c.d.f.) of the minimum eigenvalue of a correlated complex non-central Wishart matrix. This result is in the form of a simple infinite series with fast convergence, and applies for the important case where the non-centrality matrix has rank one. Simplified asymptotic expressions for the c.d.f. are given for large matrix dimensions, as well as first-order expansions around the origin. The eigenvalue distributions in this paper have various important applications to multiple-input multiple-output (MIMO) communication systems, as well other scientific areas such as econometrics and multivariate statistics.

In this paper, an exact new closed formula for the multivariate alpha-mu joint probability density function (PDF) is derived allowing for an arbitrary correlation matrix. In addition, the bit error rate (BER) and the outage probability for the selection combining (SC) technique in a correlated alpha-mu fading channel is obtained.

The Fisher information matrix determines how much information a measurement brings about the parameters that index the underlying probability distribution for the measurement. In this paper we assume that the parameters structure the mean value vector in a multivariate normal distribution. The Fisher matrix is then a Gramian constructed from the sensitivity vectors that characterize the first-order variation in the mean with respect to the parameters. The inverse of the Fisher matrix lower bounds the covariance of any unbiased estimator of the parameters. This inverse has several geometrical properties that bring insight into the problem of identifying multiple parameters. For example, it is the angle between a given sensitivity vector and the linear subspace spanned by all other sensitivity vectors that determines the variance bound for identifying a given parameter. Similarly, the covariance for identifying the linear influence of two different subsets of parameters depends on the principal angles between the linear subspaces spanned by the sensitivity vectors for the respective subsets. These geometrical interpretations may also be given filtering interpretations in a related multivariate normal estimation problem.