Random Matrix Approach Research Articles

Random matrix approach to categorical data analysis.

Correlation and similarity measures are widely used in all the areas of sciences and social sciences. Often the variables are not numbers but are instead qualitative descriptors called categorical data. We define and study similarity matrix, as a measure of similarity, for the case of categorical data. This is of interest due to a deluge of categorical data, such as movie ratings, top-10 rankings, and data from social media, in the public domain that require analysis. We show that the statistical properties of the spectra of similarity matrices, constructed from categorical data, follow random matrix predictions with the dominant eigenvalue being an exception. We demonstrate this approach by applying it to the data for Indian general elections and sea level pressures in the North Atlantic ocean.

Coordinated Multicell Beamforming for Massive MIMO: A Random Matrix Approach

We consider the problem of coordinated multicell downlink beamforming in massive multiple input multiple output (MIMO) systems consisting of $N$ cells, $N_{t}$ antennas per base station (BS) and $K$ user terminals (UTs) per cell. In particular, we formulate a multicell beamforming algorithm for massive MIMO systems that requires limited amount of information exchange between the BSs. The design objective is to minimize the aggregate transmit power across all the BSs subject to satisfying the user signal-to-interference-noise ratio (SINR) constraints. The algorithm requires the BSs to exchange parameters which can be computed solely based on the channel statistics rather than the instantaneous channel state information (CSI). We make use of tools from random matrix theory to formulate the decentralized algorithm. We also characterize a lower bound on the set of target SINR values for which the decentralized multicell beamforming algorithm is feasible. We further show that the performance of our algorithm asymptotically matches the performance of the centralized algorithm with full CSI sharing. While the original result focuses on minimizing the aggregate transmit power across all the BSs, we formulate a heuristic extension of this algorithm to incorporate a practical constraint in multicell systems, namely the individual BS transmit power constraints. Finally, we investigate the impact of imperfect CSI and pilot contamination effect on the performance of the decentralized algorithm, and propose a heuristic extension of the algorithm to accommodate these issues. Simulation results illustrate that our algorithm closely satisfies the target SINR constraints and achieves minimum power in the regime of massive MIMO systems. In addition, it also provides substantial power savings as compared with zero-forcing beamforming when the number of antennas per BS is of the same orders of magnitude as the number of UTs per cell.

Capacitance and charge relaxation resistance of chaotic cavities —Joint distribution of two linear statistics in the Laguerre ensemble of random matrices

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance and the charge relaxation resistance Rq, where C is the geometric capacitance and Cq the quantum capacitance. We study the limit of a large number of conducting channels N with a Coulomb gas method. We obtain and show that the relative fluctuations are of order 1/N both for Cq and Rq, with strong correlations . The detailed analysis of large deviations involves a second-order phase transition in the Coulomb gas. The two-dimensional phase diagram is obtained.

Alternative Random Matrix Approach in Analysis of Correlations in Financial Data

We present an alternative method based on random matrix approach that enables to distinguish the respective role of temporal autocorrelations inside given time series and cross correlations between various time series. The proposed algorithm is based on the properties of Wigner eigenspectrum of random matrices instead of commonly used Wishart eigenspectrum methodology. It is then qualitatively and quantitatively applied to financial data of stocks building WIG 30 — the main Warsaw Stock Exchange Index.

Random Matrix Approach to Correlation Matrix of Financial Data (Mexican Stock Market Case)

The main purpose of this work is to reproduce the method used for U.S. market which consists in the approach of random matrices to crossed correlation matrices built with financial data taken from a Mexican stock market database. First we built a cross correlation empirical matrix with these financial data. Eigenvalue spectrum was obtained from this matrix. We made the same spectrum analysis for a random matrix, and finally we compared both eigenvalue sets, and we tried to set up a hypothesis of how risk was related to this random matrix-correlation matrix approach. We used financial data over a period of six months and time series where made upon three hours measures for crossed correlation matrix.

Zooming into market states

We analyze the daily stock data of the Nasdaq Composite index in the 22 year period 1992–2013 and identify market states as clusters of correlation matrices with similar correlation structures. We investigate the stability of the correlation structure of each state by estimating the statistical fluctuations of true correlations due to their non-stationarity. Our study is based on a random matrix approach recently introduced to model the non-stationarity of correlations by an ensemble of random matrices. This approach reduces the complexity of the correlated market to two parameters, one of which measures the fluctuations of the correlations and can be determined directly from the empirical return distributions. This parameter characterizes the stability of the correlation structure of each market state as well as the dynamics of the correlation structure in the whole observation period. We find clear indications for correlation fluctuations within market states. The analysis reveals an intriguing relationship between average correlation and correlation fluctuations. The strongest fluctuations occur during periods of high average correlation which is the case particularly in times of crisis.

Random Matrix Approach: Toward Probabilistic Formulation of the Manipulator Jacobian

In this paper, we formulate the manipulator Jacobian matrix in a probabilistic framework based on the random matrix theory (RMT). Due to the limited available information on the system fluctuations, the parametric approaches often prove to be inadequate to appropriately characterize the uncertainty. To overcome this difficulty, we develop two RMT-based probabilistic models for the Jacobian matrix to provide systematic frameworks that facilitate the uncertainty quantification in a variety of complex robotic systems. One of the models is built upon direct implementation of the maximum entropy principle that results in a Wishart random perturbation matrix. In the other probabilistic model, the Jacobian matrix is assumed to have a matrix-variate Gaussian distribution with known mean. The covariance matrix of the Gaussian distribution is obtained at every time point by maximizing a Shannon entropy measure (subject to Jacobian norm and covariance positive semidefiniteness constraints). In contrast to random variable/vector based schemes, the benefits of the proposed approach now include: (i) incorporating the kinematic configuration and complexity in the probabilistic formulation; (ii) achieving the uncertainty model using limited available information; (iii) taking into account the working configuration of the robotic systems in characterization of the uncertainty; and (iv) realizing a faster simulation process. A case study of a 2R serial manipulator is presented to highlight the critical aspects of the process.

A random matrix approach to detect defects in a strongly scattering polycrystal: How the memory effect can help overcome multiple scattering

We report on ultrasonic imaging in a random heterogeneous medium. The goal is to detect flaws embedded deeply into a polycrystalline material. A 64-element array of piezoelectric transmitters/receivers at a central frequency of 5 MHz is used to capture the Green's matrix in a backscattering configuration. Because of multiple scattering, conventional imaging completely fails to detect the deepest flaws. We utilize a random matrix approach, taking advantage of the deterministic coherence of the backscattered wave-field which is characteristic of single scattering and related to the memory effect. This allows us to separate single and multiple scattering contributions. As a consequence, we show that flaws are detected beyond the conventional limit, as if multiple scattering had been overcome.

A random matrix approach to credit risk.

We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory. We demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the correlations are not identically zero. The existence of correlations alters the tails of the loss distribution considerably, even if their average is zero. Under the assumption of randomly fluctuating correlations, a lower bound for the estimation of the loss distribution is provided.

Random Matrix Approach to the Distribution of Genomic Distance

The cycle graph introduced by Bafna and Pevzner is an important tool for evaluating the distance between two genomes, that is, the minimal number of rearrangements needed to transform one genome into another. We interpret this distance in topological terms and relate it to the random matrix theory. Namely, the number of genomes at a given 2-break distance from a fixed one (the Hultman number) is represented by a coefficient in the genus expansion of a matrix integral over the space of complex matrices with the Gaussian measure. We study generating functions for the Hultman numbers and prove that the two-break distance distribution is asymptotically normal.

Orbital entanglement and electron localization in quantum wires

We study the signatures of disorder in the production of orbital electron entanglement in quantum wires. Disordered entanglers suffer the effects of localization of the electron wave function and random fluctuations in the entanglement production. This manifests in the statistics of the concurrence, a measure of the produced two-qubit entanglement. We calculate the concurrence distribution as a function of the disorder strength within a random-matrix approach. We also identify significant constraints on the entanglement production as a consequence of the breaking/preservation of time-reversal invariance. Additionally, our theoretical results are independently supported by simulations of disordered quantum wires based on a tight-binding model.

Random Matrix Approach for Portfolio Optimization Problem

Random Matrix Approach for Portfolio Optimization Problem

Quantifying randomness in protein–protein interaction networks of different species: A random matrix approach

We analyze protein–protein interaction networks for six different species under the framework of random matrix theory. Nearest neighbor spacing distribution of the eigenvalues of adjacency matrices of the largest connected part of these networks emulate universal Gaussian orthogonal statistics of random matrix theory. We demonstrate that spectral rigidity, which quantifies long range correlations in eigenvalues, for all protein–protein interaction networks follow random matrix prediction up to certain ranges indicating randomness in interactions. After this range, deviation from the universality evinces underlying structural features in network.

Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach

To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.

Physical model of differential Mueller matrix for depolarizing uniform media

In this article, we address the question of significance of the parameters of differential Mueller matrix formalism. We show how the concept of mean value and uncertainty of the optical properties recently introduced to depict this differential matrix can be related to the random fluctuations of these optical properties. From the layered-medium interpretation introduced by Jones [J. Opt. Soc. Am. 38, 671 (1948)] and extended to Mueller-Jones matrix by Azzam [J. Opt. Soc. Am. 68, 1756 (1978)], a generalization to depolarizing Mueller matrices is proposed. Based on the random Mueller-Jones matrix approach, the obtained parameterization perfectly fits the previous results from the literature. Necessary conditions of positivity on specific coefficients imposed in order to have physical Mueller matrix are introduced in a natural way and not inferred a posteriori. Interpretations of the underlying physical processes are also presented. An illustrative experimental example is provided from literature data.

Chaotic scattering on individual quantum graphs

For chaotic scattering on quantum graphs, the semiclassical approximation is exact. We use this fact and employ supersymmetry, the color-flavor transformation, and the saddle-point approximation to calculate the exact expression for the lowest and asymptotic expressions in the Ericson regime for all higher correlation functions of the scattering matrix. Our results agree with those available from the random-matrix approach to chaotic scattering. We conjecture that our results hold universally for quantum-chaotic scattering.

Random matrix approach to scalar fields on fuzzy spaces

We formulate theory of interacting scalar field on the fuzzy sphere as a random matrix model. We then analyze the expectation values of observables of the theory in the large N limit and we demonstrate that the eigenvalue distribution of the matrix M remains the polynomially deformed Wigner semicircle. We also compute distributions involving the matrix Laplacian of M and we show that the correlation between the eigenvalues of these two is different from the free field case.

Random matrix approach to the dynamics of stock inventory variations

It is well accepted that investors can be classified into groups owing to distinct trading strategies, which forms the basic assumption of many agent-based models for financial markets when agents are not zero-intelligent. However, empirical tests of these assumptions are still very rare due to the lack of order flow data. Here we adopt the order flow data of Chinese stocks to tackle this problem by investigating the dynamics of inventory variations for individual and institutional investors that contain rich information about the trading behavior of investors and have a crucial influence on price fluctuations. We find that the distributions of cross-correlation coefficient Cij have power-law forms in the bulk that are followed by exponential tails, and there are more positive coefficients than negative ones. In addition, it is more likely that two individuals or two institutions have a stronger inventory variation correlation than one individual and one institution. We find that the largest and the second largest eigenvalues (λ1 and λ2) of the correlation matrix cannot be explained by random matrix theory and the projections of investors' inventory variations on the first eigenvector u(λ1) are linearly correlated with stock returns, where individual investors play a dominating role. The investors are classified into three categories based on the cross-correlation coefficients CVR between inventory variations and stock returns. A strong Granger causality is unveiled from stock returns to inventory variations, which means that a large proportion of individuals hold the reversing trading strategy and a small part of individuals hold the trending strategy. Our empirical findings have scientific significance in the understanding of investors' trading behavior and in the construction of agent-based models for emerging stock markets.

Random matrix approach and combinatorial topology approach to sorting by transpositions

Random matrix approach and combinatorial topology approach to sorting by transpositions

Universality in chaotic quantum transport: The concordance between random-matrix and semiclassical theories

Electronic transport through chaotic quantum dots exhibits universal, system-independent properties, consistent with random-matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semiclassical diagrams, through a connection to factorizations of permutations. We show agreement between the semiclassical and random matrix approaches to the moments of the transmission eigenvalues. The result is valid for all moments to all orders of the expansion in inverse channel number for all three main symmetry classes (with and without time-reversal symmetry and spin-orbit interaction) and extends to nonlinear statistics. This finally explains the applicability of random-matrix theory to chaotic quantum transport in terms of the underlying dynamics as well as providing semiclassical access to the probability density of the transmission eigenvalues.