Eigenvalue Probability Density Function Research Articles

Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge

Sensitivity of an eigenvalue λ i to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as κi=⟨Li|Li⟩⟨Ri|Ri⟩ , where ⟨Li| and |Ri⟩ are left and right eigenvectors to the eigenvalue λ i . In random matrix theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer’s function.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

A novel federated learning based lightweight sustainable IoT approach to identify abnormal traffic

PurposeThis strategy significantly reduces the computational overhead and storage overhead required when using the kernel density estimation method to calculate the abnormal evaluation value of the test sample.Design/methodology/approachTo effectively deal with the security threats of botnets to the home and personal Internet of Things (IoT), especially for the objective problem of insufficient resources for anomaly detection in the home environment, a novel kernel density estimation-based federated learning-based lightweight Internet of Things anomaly traffic detection based on nuclear density estimation (KDE-LIATD) method. First, the KDE-LIATD method uses Gaussian kernel density estimation method to estimate every normal sample in the training set. The eigenvalue probability density function of the dimensional feature and the corresponding probability density; then, a feature selection algorithm based on kernel density estimation, obtained features that make outstanding contributions to anomaly detection, thereby reducing the feature dimension while improving the accuracy of anomaly detection; finally, the anomaly evaluation value of the test sample is calculated by the cubic spine interpolation method and anomaly detection is performed.FindingsThe simulation experiment results show that the proposed KDE-LIATD method is relatively strong in the detection of abnormal traffic for heterogeneous IoT devices.Originality/valueWith its robustness and compatibility, it can effectively detect abnormal traffic of household and personal IoT botnets.

Capacity Analysis of Power Beacon-Assisted Energy Harvesting MIMO System Over $\kappa -\mu$ Shadowed Fading Channels

In this paper, novel ergodic capacity (EC) performance evaluation results of a power beacon (PB)-assisted multi-input multi-output (MIMO) wireless powered communication network are presented. In the considered system, the energy harvesting node harvests energy from the radio-frequency signals sent by the dedicated PB and uses this energy to communicate with the destination node. To accurately model the combined effect of multi-path fading and shadowing, it is assumed that the energy transfer link is subject to $\kappa$ - $\mu$ shadowed fading. Performance evaluation results are presented for two cases, depending upon the availability of channel state information (CSI) at the PB, namely, $no$ CSI and $full$ CSI. In the former case, equal power allocation is assumed, whereas, in the later case, energy beamforming is employed to increase energy transfer efficiency. For the performance evaluation of EC under $full$ CSI, a closed-form approximation for the probability density function of the maximum eigenvalue of a $\kappa$ - $\mu$ shadowed distributed random matrix is derived. For both $no$ CSI and $full$ CSI cases, lower and upper bounds on the achievable EC are derived in closed-form. Moreover, in order to obtain further insights on the impact of key parameters on the system performance, asymptotic EC expressions which become very tight at low- and high-signal-to-noise ratio regimes, are obtained. Using the proposed EC lower bound as well as these asymptotic results, simple closed-form expressions for the optimal time split that maximize the achievable EC are derived. Numerically evaluated results accompanied with Monte-Carlo simulations are further presented to corroborate the theoretical analysis.

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little q-Jacobi polynomial. From their large N form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.

Classical skew orthogonal polynomials in a two-component log-gas with charges +1 and +2

There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue probability density function (PDF) for β=1 and β=4 invariant random matrix ensembles. Its solvability relies on the construction of particular skew orthogonal polynomials, with the skew inner product a linear combination of the β=1 and β=4 inner products, each involving weight functions. For suitably related classical weight functions, we seek to express the skew orthogonal polynomials as linear combinations of the underlying orthogonal polynomials. It is found that in each case (Gaussian, Laguerre, Jacobi and generalised Cauchy) the coefficients can be expressed in terms of hypergeometric polynomials with argument relating to the fugacity. In the Jacobi case, for example, the coefficients are Wilson polynomials.

Outage Probabilities of Cache and SIC Enabled Downlink MIMO NOMA Cellular Networks With Randomly Distributed Users

For non-orthogonal multiple access (NOMA) based cellular communication networks, cache enabled interference cancelation not only reduces the complexity of successive interference cancelation (SIC), but also improves the received signal-to-interference-and-noise ratios. In this paper, a cache and SIC enabled downlink multiple-input-multiple-output NOMA cellular network with randomly distributed users is studied. To theoretically derive its outage performance, the Wishart matrices are introduced to express the achievable rates of users. From the joint probability density function of the unordered eigenvalues of Wishart matrices, we obtain the moment generating function of the achievable rates. The exact outage probabilities are obtained by employing the inverse Laplace transform. Numerical results demonstrate that our theoretically derived outage probabilities match the simulation results.

A Low-Complexity AEPDF-assisted Precoding Scheme for Massive MIMO Systems with Transmit Antenna Correlation

Multi-user massive multiple-input multiple-output (MU M-MIMO) system employs array of hundred (or thousand) antennas and precoding/equalization capability at the base station (BS) has been adopted in recent 5G new radio (5G NR) standards. By taking advantage of the multiplexing, diversity and antenna gain inherited from this versatile system configuration, unprecedented enhancement in capacity, spectral efficiency and signal integrity can be realized. While many MU M-MIMO system related issues such as pilot contamination, channel estimation and precoding/detection algorithms have been extensively examined by research communities, the investigation of a low-complexity, effective precoding scheme aims to address performance degradation due to spatial correlation (i.e., packing vast number of antennas in a limited physical space) remained scarce. In this paper, an iterative precoding scheme based on Chebyshev acceleration is proposed for one-sided Kronecker model (i.e., Rayleigh flat fading channel with transmit (Tx) antenna correlation), often encountered in uniform linear array (ULA) configuration. From the asymptotic eigenvalue probability density function (AEPDF) constructed for the non-commutative Kronecker channel, near-optimal recurrence coefficients, initial and final estimates of the proposed Chebyshev iterative method can be obtained free from expensive eigenvalue computation. Simulation shows that the proposed algorithm achieves outstanding BER in all correlation scenarios (i.e., from low to high) while exhibiting much lower complexity compared to other state-of-the-art precoding schemes.

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.

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete and q q skew orthogonal polynomials, respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases in which the weights for the corresponding discrete unitary ensembles are classical. Crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones, and have a tridiagonal action on the corresponding (discrete or q q ) orthogonal polynomials.

A Novel Beamforming and Combining Scheme for Two-Way AF Satellite Systems

We propose a novel beamforming and combining scheme for a two-way amplify-and-forward (AF) satellite system. Exchange of the data in between two multiple-antenna-based Earth stations (ESs) via a single-antenna-based satellite is performed. The transmitting ES uses beamforming for transmission of the data, and the receiving ES utilizes the combining vector for detecting the transmitted signal. By using maximum eigenvalue criterion, the beamforming and combining vectors are calculated. The satellite channels (uplink and downlink) are assumed to be shadowed Rician (SR) fading. First, we assume independent and identically distributed (i.i.d.) SR fading channels and derive the probability density function (pdf) of maximum eigenvalue to obtain the expression of moment-generating function (mgf). The closed-form expression for the symbol error rate (SER) of the scheme for $M$ - ary phase-shift keying (MPSK) is obtained by using an mgf-based approach. The diversity order of the studied scheme is also derived by exploiting the asymptotic property of the Meijer G-function. Next, we consider correlated SR fading channels and derive expressions of SER and diversity order. It is shown by simulations that the proposed scheme outperforms the existing beamforming and combining based scheme in two-way AF satellite systems. It is concluded on the basis of the diversity order analysis that the diversity of the proposed scheme linearly depends upon the minimum of the number of antennas of two ESs. Furthermore, it is also shown by simulation and analysis that correlation does not affect the diversity order of the considered scheme.

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Muttalib–Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form Q (Formula presented). We study the Laguerre and Jacobi versions of this model — so named by the form of the one-body interaction terms — and show that for θ ∈ ℤ+ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general θ > 0, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlation functions contained in a paper by Adler, van Moerbeke and Wang to analyse the global density (which we also analyse by studying characteristic polynomials), and the hard edge scaled correlation functions. For the global density functional equations for the corresponding resolvents are obtained; solving this gives the moments in terms of Fuss–Catalan numbers (Laguerre case — a known result) and particular binomial coefficients (Jacobi case). For θ ∈ ℤ+ the Laguerre and Jacobi cases are closely related to the squared singular values for products of θ standard Gaussian random matrices, and truncations of unitary matrices, respectively. At the hard edge the double contour integral formulas provide a double contour integral form of the scaled correlation kernel obtained by Borodin in terms of Wright’s Bessel function.

The difference between two random mixed quantum states: exact and asymptotic spectral analysis

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson’s theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.

Stability Analysis of Grid-Interfacing Inverter Control in Distribution Systems With Multiple Photovoltaic-Based Distributed Generators

A stability-enhancement inverter controller is proposed and its impact on the small-signal stability of distribution systems with multiple photovoltaic-based distributed generators (PV-DGs) is investigated. In specific, the random variation of the power output from multiple PV-DGs is considered. Being different from a single grid-connected PV system, the spatial correlation between PV-DGs and the system stability sensitivity with respect to different PV-DGs among the feeder are also taken into account. The small-signal stability analysis of the distribution system is performed using a probabilistic approach. As the PV-DG power output is randomly variable, the distribution system is not operating in a fixed mode. Consequently, the critical eigenvalues of the system move randomly. Through the Gram–Charlier expansion method, the probability density function of the critical eigenvalues of the distribution system is approximated and the probabilities of the system stability and instability are calculated. Taking the probability of the system instability as the performance metric, the effects of different system parameter settings of the grid-interfacing inverter controller on the distribution system are analyzed. The verification of the analytical results is carried out through Monte-Carlo time-domain simulations of the distribution system. The results suggest several methods to reduce the probability of system instability, including installation of the proposed inverter controller, choosing a large substation capacitor and/or choosing large capacity load and substation transformers.

Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices [Formula: see text] where [Formula: see text] is a random [Formula: see text] ([Formula: see text]) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large [Formula: see text], large [Formula: see text] with [Formula: see text] — i.e. for quasi-square large matrices [Formula: see text] — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlevé III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first [Formula: see text] corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Péché. We also study the soft edge limit, when [Formula: see text], for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

Random matrix theory of quantum transport in chaotic cavities with nonideal leads

We determine the joint probability density function (JPDF) of reflection eigenvalues in three Dyson's ensembles of normal-conducting chaotic cavities coupled to the outside world through both ballistic and tunnel point contacts. Expressing the JPDF in terms of hypergeometric functions of matrix arguments (labeled by the Dyson index $\beta$), we further show that reflection eigenvalues form a determinantal ensemble at $\beta=2$ and a new type of a Pfaffian ensemble at $\beta=4$. As an application, we derive a simple analytic expression for the concurrence distribution describing production of orbitally entangled electrons in chaotic cavities with tunnel point contacts when time reversal symmetry is preserved.

Eigenvalue statistics for product complex Wishart matrices

The eigenvalue statistics for complex Wishart matrices , where is equal to the product of r complex Gaussian matrices, and the inverse of s complex Gaussian matrices, are considered. In the case r = s the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function . For finite N the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case s = 0. Properties of the kernel and corresponding correlation functions are discussed.

Probabilistic tuning of Power System Stabilizers considering the wind farm generation uncertainty

The increasing use of wind farms and the uncertainties in their generation levels have led to new challenges in system dynamic behaviors and Small Signal Stability (SSS). This paper proposes a probabilistic method to investigate the impact of the wind farm generation uncertainty on power system SSS, based on the Monte Carlo simulation. The proposed method is stochastic in contrast to conventional deterministic methods, which only analyze the system dynamic behaviors in one operating point. It determines several operating points, based on wind farm generation levels and calculates the Probability Density Function (PDF) of critical eigenvalues. Then, Genetic Algorithm (GA) is used to optimize the Power System Stabilizers (PSS) parameters. A case study is carried out on the IEEE 16-machine system to demonstrate the effectiveness and validity of the proposed algorithms. The results of probabilistic stability analysis are compared with those of deterministic analysis. It is shown that variation of the wind farm generation levels can cause the instability, even though the system is stable according to the deterministic analysis. Also, the time domain simulation results show how such tunings can effectively improve the SSS performance under the wind farm generation uncertainties.