Polynomial Ensembles Research Articles

GICAR Algebras and Dynamics on Determinantal Point Processes: Discrete Orthogonal Polynomial Ensemble Case

GICAR Algebras and Dynamics on Determinantal Point Processes: Discrete Orthogonal Polynomial Ensemble Case

Analysis and classification of arrhythmia types using improved firefly optimization algorithm and autoencoder model

In the present scenario, Electrocardiogram (ECG) is an effective non-invasive clinical tool, which reveals the functionality and rhythm of the heart. The non-stationary nature of ECG signal, noise existence, and heartbeat abnormality makes it difficult for clinicians to diagnose arrhythmia. The most of the existing models concentrate only on classification accuracy. In this manuscript, an automated model is introduced that concentrates on arrhythmia type classification using ECG signals, and also focuses on computational complexity and time. After collecting the signals from the MIT-BIH database, the signal transformation and decomposition are performed by Multiscale Local Polynomial Transform (MLPT) and Ensemble Empirical Mode Decomposition (EEMD). The decomposed ECG signals are given to the feature extraction phase for extracting features. The feature extraction phase includes six techniques: standard deviation, zero crossing rate, mean curve length, Hjorth parameters, mean Teager energy, and log energy entropy. Next, the feature dimensionality reduction and arrhythmia classification are performed utilizing the improved Firefly Optimization Algorithm and autoencoder. The selection of optimal feature vectors by the improved Firefly Optimization Algorithm reduces the computational complexity to linear and consumes computational time of 18.23 seconds. The improved Firefly Optimization Algorithm and autoencoder model achieved 98.96% of accuracy in the arrhythmia type classification, which is higher than the comparative models.

Q-Pearson pair and moments in q-deformed ensembles

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little q-Laguerre weight, a particular \(_3 \phi _2\) basic hypergeometric polynomial is used to express density moments. The second approach is to study the q-Laplace transform of the un-normalised measure. Using integrability properties associated with the q-Pearson equation for the q-classical weights, a fourth-order q-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

Hard edge statistics of products of Pólya ensembles and shifted GUE’s

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.

Universal microscopic spectrum of the unquenched QCD Dirac operator at finite temperature

In the ε-regime of chiral perturbation theory the spectral correlations of the Euclidean QCD Dirac operator close to the origin can be computed using random matrix theory. To incorporate the effect of temperature, a random matrix ensemble has been proposed, where a constant, deterministic matrix is added to the Dirac operator. Its eigenvalue correlation functions can be written as the determinant of a kernel that depends on temperature. Due to recent progress in this specific class of random matrix ensembles, featuring a deterministic, additive shift, we can determine the limiting kernel and correlation functions in this class, which is the class of polynomial ensembles. We prove the equivalence between this new determinantal representation of the microscopic eigenvalue correlation functions and existing results in terms of determinants of different sizes, for an arbitrary number of quark flavours, with and without temperature, and extend them to non-zero topology. These results all agree and are thus universal when measured in units of the temperature dependent chiral condensate, as long as we stay below the chiral phase transition.

Polynomial analogy‐based software development effort estimation using combined particle swarm optimization and simulated annealing

SummarySoftware development effort estimation is an effective factor in the success or failure of software projects. There are several methods to estimate the effort of software projects, the most common of which is analogy‐based estimation (ABE). In this article, a polynomial version of ABE (named PABE) is presented, in which, the project effort is calculated based on a polynomial ensemble of different ABE models. To optimize the controllable parameters of the PABE model, a combined global–local search metaheuristic algorithm based on particle swarm optimization and simulated annealing is utilized in two steps. At the first step, for each similarity and adaptation function, the optimized ABE model is determined by exploiting the optimal value of feature weights, the number of similar projects, and other parameters of the ABE model. Then, at the second step, the amount of effort attained by the optimized models is used for estimating the final effort by the proposed polynomial equation. The proposed PABE method has been successfully executed on five well‐known software effort estimation datasets: Maxwell, Albrecht, Cocomo81, Desharnais, and Kemerer. Obtained results show the superiority of the proposed PABE model in terms of accuracy and efficiency compared to other techniques.

The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer

The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that depends on an external field V. In a recent article, the large n behavior of the associated correlation kernel at the hard edge was found for , where only few restrictions are imposed on V. In the current article we generalize the techniques and results of this article to obtain analogous results for , where r is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with r weights, which can then be related to an (r + 1) × (r + 1) Riemann–Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others.Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker–Campbell–Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero.We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

Polynomial Ensembles and Pólya Frequency Functions

We study several kinds of polynomial ensembles of derivative type which we propose to call P\'olya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of P\'olya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate P\'olya ensembles to P\'olya frequency functions. For illustration we give a number of explicit examples for our results.

Determinantal Point Processes and Fermion Quasifree States

Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes. We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations. We prove that the formalism applies to discrete N-point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Alexei Borodin and the author in [Commun. Math. Phys. 353 (2017), 853-903; arXiv:1608.01564]. As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

Universality for conditional measures of the Bessel point process

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval [Formula: see text] is almost surely an orthogonal polynomial ensemble. In this paper, we show that if [Formula: see text] tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.

Conditional Measures of Determinantal Point Processes

For a class of one-dimensional determinantal point processes including those induced by orthogonal projections with integrable kernels satisfying a growth condition, it is proved that their conditional measures, with respect to the configuration in the complement of a compact interval, are orthogonal polynomial ensembles with explicitly found weights. Examples include the sine-process and the process with the Bessel kernel. The argument uses the quasi-invariance, established in [1], of our point processes under the group of piecewise isometries of the real line.

Moments of discrete orthogonal polynomial ensembles

We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments of the randomised ensembles are precisely related to the moments of the corresponding equilibrium measures. We also briefly outline how these results can be interpreted as Cauchy-type identities for certain Schur measures.

On some nonlocal elliptic systems with multiple parameters

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Products of Complex Rectangular and Hermitian Random Matrices

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

$q$ -Racah Ensemble and Discrete Painlevé Equation

Abstract The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.