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Books on the topic 'Correlation Matrix'

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Published: 4 June 2021
Last updated: 28 January 2023

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1

Bogomolny, E. B. Random matrix theory and the Riemann zeros II: N-point correlations. Bristol [England]: Hewlett Packard, 1996.

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2

Weiner, Richard M. Introduction to Bose-Einstein correlations and subatomic interferometry. Chichester, England: John Wiley, 2000.

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3

Zabrodin, Anton. Financial applications of random matrix theory: a short review. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.40.

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This article reviews some applications of random matrix theory (RMT) in the context of financial markets and econometric models, with emphasis on various theoretical results (for example, the Marčenko-Pastur spectrum and its various generalizations, random singular value decomposition, free matrices, largest eigenvalue statistics) as well as some concrete applications to portfolio optimization and out-of-sample risk estimation. The discussion begins with an overview of principal component analysis (PCA) of the correlation matrix, followed by an analysis of return statistics and portfolio theory. In particular, the article considers single asset returns, multivariate distribution of returns, risk and portfolio theory, and nonequal time correlations and more general rectangular correlation matrices. It also presents several RMT results on the bulk density of states that can be obtained using the concept of matrix freeness before concluding with a description of empirical correlation matrices of stock returns.
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4

Borodin, Alexei. Random matrix representations of critical statistics. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.12.

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This article examines two random matrix ensembles that are useful for describing critical spectral statistics in systems with multifractal eigenfunction statistics: the Gaussian non-invariant ensemble and the invariant random matrix ensemble. It first provides an overview of non-invariant Gaussian random matrix theory (RMT) with multifractal eigenvectors and invariant random matrix theory (RMT) with log-square confinement before discussing self-unfolding and not self-unfolding in invariant RMT. It then considers a non-trivial unfolding and how it changes the form of the spectral correlations, along with the appearance of a ghost correlation dip in RMT and Hawking radiation. It also describes the correspondence between invariant and non-invariant ensembles and concludes by introducing a simple field theory in 1+1 dimensions which reproduces level statistics of both of the two random matrix models and the classical Wigner-Dyson spectral statistics in the framework of the unified formalism of Luttinger liquid.
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5

Bouchaud, Jean-Philippe. Random matrix theory and (big) data analysis. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0006.

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This chapter reviews methods from random matrix theory to extract information about a large signal matrix C (for example, a correlation matrix arising in big data problems), from its noisy observation matrix M. The chapter shows that the replica method can be used to obtain both the spectral density and the overlaps between noise-corrupted eigenvectors and the true ones, for both additive and multiplicative noise. This allows one to construct optimal rotationally invariant estimators of C based on the observation of M alone. This chapter also discusses the case of rectangular correlation matrices and the problem of random singular value decomposition.
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6

Cicuta, Giovanni, and Luca Molinari. Two-matrix models and biorthogonal polynomials. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.15.

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This article considers two cases of two-matrix models that are amenable to biorthogonal polynomials: Itzykson-Zuber interaction and Cauchy interaction. The features and applications of the biorthogonal polynomials relevant to either case are discussed, but first the article provides an overview of chain-matrix models. It then describes the Itzykson-Zuber Hermitian two-matrix model and the Christoffel–Darboux identities, along with the spectral curve. It also examines the so-called mixed correlation functions that are involved in the combinatorial applications of the two-matrix model before concluding with an analysis of the Cauchy two-matrix model, which in a ‘complication scale’ turns out to lie in between the one-matrix model and the Itzykson-Zuber two-matrix model.
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7

Akemann, Gernot. Random matrix theory and quantum chromodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0005.

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This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.
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8

United States. National Aeronautics and Space Administration., ed. The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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9

The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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10

United States. National Aeronautics and Space Administration., ed. The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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11

United States. National Aeronautics and Space Administration, ed. The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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12

United States. National Aeronautics and Space Administration, ed. The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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13

United States. National Aeronautics and Space Administration., ed. The correlation of low-velocity impact resistance of graphite-fiber-reinforced composites with matrix properties. [Washington, D.C: National Aeronautics and Space Administration, 1986.

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14

National Aeronautics and Space Administration (NASA) Staff. Analytical/Numerical Correlation Study of the Multiple Concentric Cylinder Model for the Thermoplastic Response of Metal Matrix Composites. Independently Published, 2018.

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15

S, Salzar Robert, Williams Todd O, and United States. National Aeronautics and Space Administration., eds. An analytical/numerical correlation study of the multiple concentric cylinder model for the thermoplastic response of metal matrix composites. [Washington, DC]: National Aeronautics and Space Administration, 1993.

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16

Morawetz, Klaus. Approximations for the Selfenergy. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0010.

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The systematic expansion of the selfenergy is presented with the help of the closure relation of chapter 7. Besides Hartree–Fock leading to meanfield kinetic equations, the random phase approximation (RPA) is shown to result into the Lennard–Balescu kinetic equation, and the ladder approximation into the Beth–Uehling–Uhlenbeck kinetic equation. The deficiencies of the ladder approximation are explored compared to the exact T-matrix by missing maximally crossed diagrams. The T-matrix provides the Bethe–Salpeter equation for the two-particle correlation functions. Vertex corrections to the RPA are presented. For a two-dimensional example, the selfenergy and effective mass are calculated. The structure factor and the pair-correlation function are introduced and calculated for various approximations.
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17

Khoruzhenko, Boris, and Hans-Jurgen Sommers. Characteristic polynomials. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.19.

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This article considers characteristic polynomials and reviews a few useful results obtained in simple Gaussian models of random Hermitian matrices in the presence of an external matrix source. It first considers the products and ratio of characteristic polynomials before discussing the duality theorems for two different characteristic polynomials of Gaussian weights with external sources. It then describes the m-point correlation functions of the eigenvalues in the Gaussian unitary ensemble and how they are deduced from their Fourier transforms U(s1, … , sm). It also analyses the relation of the correlation function of the characteristic polynomials to the standard n-point correlation function using the replica and supersymmetric methods. Finally, it shows how the topological invariants of Riemann surfaces, such as the intersection numbers of the moduli space of curves, may be derived from averaged characteristic polynomials.
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18

Dyson, Freeman. Spectral statistics of unitary ensembles. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.4.

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This article focuses on the use of the orthogonal polynomial method for computing correlation functions, cluster functions, gap probability, Janossy density, and spacing distributions for the eigenvalues of matrix ensembles with unitary-invariant probability law. It first considers the classical families of orthogonal polynomials (Hermite, Laguerre, and Jacobi) and some corresponding unitary ensembles before discussing the statistical properties of N-tuples of real numbers. It then reviews the definitions of basic statistical quantities and demonstrates how their distributions can be made explicit in terms of orthogonal polynomials. It also describes the k-point correlation function, Fredholm determinants of finite-rank kernels, and resolvent kernels.
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19

Ferrari, Patrik L., and Herbert Spohn. Random matrices and Laplacian growth. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.39.

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This article reviews the theory of random matrices with eigenvalues distributed in the complex plane and more general ‘beta ensembles’ (logarithmic gases in 2D). It first considers two ensembles of random matrices with complex eigenvalues: ensemble C of general complex matrices and ensemble N of normal matrices. In particular, it describes the Dyson gas picture for ensembles of matrices with general complex eigenvalues distributed on the plane. It then presents some general exact relations for correlation functions valid for any values of N and β before analysing the distribution and correlations of the eigenvalues in the large N limit. Using the technique of boundary value problems in two dimensions and elements of the potential theory, the article demonstrates that the finite-time blow-up (a cusp–like singularity) of the Laplacian growth with zero surface tension is a critical point of the normal and complex matrix models.
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20

Anderson, Greg W. Spectral statistics of orthogonal and symplectic ensembles. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.5.

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This article describes a direct approach for computing scalar and matrix kernels, respectively for the unitary ensembles on the one hand and the orthogonal and symplectic ensembles on the other hand, leading to correlation functions and gap probabilities. In the classical orthogonal polynomials (Hermite, Laguerre, and Jacobi), the matrix kernels for the orthogonal and symplectic ensemble are expressed in terms of the scalar kernel for the unitary case, using the relation between the classical orthogonal polynomials going with the unitary ensembles and the skew-orthogonal polynomials going with the orthogonal and symplectic ensembles. The article states the fundamental theorem relating the orthonormal and skew-orthonormal polynomials that enter into the Christoffel-Darboux kernels
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21

Muller, Sebastian, and Martin Sieber. Resonance scattering of waves in chaotic systems. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.34.

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This article discusses some applications of random matrix theory (RMT) to quantum or wave chaotic resonance scattering. It first provides an overview of selected topics on universal statistics of resonances and scattering observables, with emphasis on theoretical results obtained via non-perturbative methods starting from the mid-1990s. It then considers the statistical properties of scattering observables at a given fixed value of the scattering energy, taking into account the maximum entropy approach as well as quantum transport and the Selberg integral. It also examines the correlation properties of the S-matrix at different values of energy and concludes by describing other characteristics and applications of RMT to resonance scattering of waves in chaotic systems, including those relating to time delays, quantum maps and sub-unitary random matrices, and microwave cavities at finite absorption.
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22

Zinn-Justin, Paul, and Jean-Bernard Zuber. Multivariate statistics. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.28.

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This article considers some classical and more modern results obtained in random matrix theory (RMT) for applications in statistics. In the classic paradigm of parametric statistics, data are generated randomly according to a probability distribution indexed by parameters. From this data, which is by nature random, the properties of the deterministic (and unknown) parameters may be inferred. The ability to infer properties of the unknown Σ (the population covariance matrix) will depend on the quality of the estimator. The article first provides an overview of two spectral statistical techniques, principal components analysis (PCA) and canonical correlation analysis (CCA), before discussing the Wishart distribution and normal theory. It then describes extreme eigenvalues and Tracy–Widom laws, taking into account the results obtained in the asymptotic setting of ‘large p, large n’. It also analyses the results for the limiting spectra of sample covariance matrices..
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23

Gallagher, Richard H. Correlation Study of Methods of Matrix Structural Analysis: Report to the 14th Meeting, Structures and Materials Panel Advisory Group for Aeronautical Research and Development, NATO, Paris, France, July 6 1962. Elsevier Science & Technology Books, 2014.

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24

Morawetz, Klaus. Spectral Properties. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0008.

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The spectral properties of the nonequilibrium Green’s functions are explored. Causality and sum rules are shown to be completed by the extended quasiparticle picture. The off-shell motion is seen to become visible in satellite structures of the spectral function. Different forms of ansatz to reduce the two-time Green’s function to a one-time reduced density matrix are discussed with respect to the consistency to other approximations. We have seen from the information contained in the correlation function that the statistical weight of excitations with which the distributions are populated are given by the spectral function. This momentum-resolved density of state can be found by the retarded and advance functions.
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25

Adler, Mark. Universality. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.6.

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This article deals with the universality of eigenvalue spacings, one of the basic characteristics of random matrices. It first discusses the heuristic meaning of universality before describing the standard universality classes (sine, Airy, Bessel) and their appearance in unitary, orthogonal, and symplectic ensembles. It then examines unitary matrix ensembles in more detail and shows that universality in these ensembles comes down to the convergence of the properly scaled eigenvalue correlation kernels. It also analyses the Riemann–Hilbert method, along with certain non-standard universality classes that arise at singular points in the limiting spectrum. Finally, it considers the limiting kernels for each of the three types of singular points, namely interior singular points, singular edge points, and exterior singular points.
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26

van Moerbeke, Pierre. Determinantal point processes. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.11.

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This article presents a list of algebraic, combinatorial, and analytic mechanisms that give rise to determinantal point processes. Determinantal point processes have been used in random matrix theory (RMT) since the early 1960s. As a separate class, determinantal processes were first used to model fermions in thermal equilibrium and the term ‘fermion’ point processes were adopted. The article first provides an overview of the generalities associated with determinantal point processes before discussing loop-free Markov chains, that is, the trajectories of the Markov chain do not pass through the same point twice almost surely. It then considers the measures given by products of determinants, namely, biorthogonal ensembles. An especially important subclass of biorthogonal ensembles consists of orthogonal polynomial ensembles. The article also describes L-ensembles, a general construction of determinantal point processes via the Fock space formalism, dimer models, uniform spanning trees, Hermitian correlation kernels, and Pfaffian point processes.
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27

Dyall, Kenneth G., and Knut Faegri. Introduction to Relativistic Quantum Chemistry. Oxford University Press, 2007. http://dx.doi.org/10.1093/oso/9780195140866.001.0001.

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This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.
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28

Orantin, Nicolas. Unitary integrals and related matrix models. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.17.

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This article examines the basic properties of unitary matrix integrals using three matrix models: the ordinary unitary model, the Brézin-Gross-Witten (BGW) model and the Harish-Chandra-Itzykson-Zuber (HCIZ) model. The tricky sides of the story are given special attention, such as the de Wit-’t Hooft anomaly in unitary integrals and the problem of correlators with Itzykson-Zuber measure. The method of character expansions is also emphasized as a technical tool. The article first provides an overview of the theory of the BGW model, taking into account the de Wit-’t Hooft anomaly and the M-theory of matrix models, before discussing the theory of the HCIZ integral. In particular, it describes the basics of character calculus, character expansion of the HCIZ integral, character expansion for the BGW model and Leutwyler-Smilga integral, and pair correlator in HCIZ theory.
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29

Morozov, Alexei. Non-Hermitian ensembles. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.18.

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This article discusses the three-fold family of Ginibre random matrix ensembles (complex, real, and quaternion real) and their elliptic deformations. It also considers eigenvalue correlations that are exactly reduced to two-point kernels in the strongly and weakly non-Hermitian limits of large matrix size. Ginibre introduced the complex, real, and quaternion real random matrix ensembles as a mathematical extension of Hermitian random matrix theory. Statistics of complex eigenvalues are now used in modelling a wide range of physical phenomena. After providing an overview of the complex Ginibre ensemble, the article describes random contractions and the complex elliptic ensemble. It then examines real and quaternion-real Ginibre ensembles, along with real and quaternion-real elliptic ensembles. In particular, it analyses the kernel in the elliptic case as well as the limits of strong and weak non-Hermiticity.
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30

Horing, Norman J. Morgenstern. Equations of Motion with Particle–Particle Interactions and Approximations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0008.

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Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ‎ and the T-matrix
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31

Morawetz, Klaus. Transient Time Period. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0019.

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The formation of correlations at short- time scales is considered. A universal response function is found which allows describing the formation of collective modes in plasmas created by femto-second lasers as well as the formation of occupations in cold atomic optical lattices. Quantum quench and sudden switching of interactions are possible to describe by such Levinson-type kinetic equations on the transient time regime. On larger time scales it is shown that non-Markovian–Levnson equations double count correlations and the extended quasiparticle picture to distinguish between the reduced density matrix and quasiparticle distribution solve this shortcoming. The problem of initial correlations and how they can be incorporated into the Green’s function technique to result into modified kinetic equations is solved and a systematic expansion is suggested.
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32

Gustafson, Karl. Operator Geometry in Statistics. Edited by Frédéric Ferraty and Yves Romain. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780199568444.013.13.

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This article discusses the essentials of operator trigonometry developed by the author as it applies to statistics, with emphasis on key elements such as operator antieigenvalues, operator antieigenvectors, and operator turning angles. Operator trigonometry started out infinite dimensional, and remains infinite dimensional, even for Banach spaces. Thus, it is in principle applicable not only to infinite-dimensional statistics but also to cases involving functional data. The article first considers how operator trigonometry gives new geometrical meaning to statistical efficiency before formalizing it in a more deductive manner. It then explains the essentials of operator trigonometry and summarizes the ensuing developments. It also describes two lemmas that are implicit and essential to operator trigonometry, Antieigenvector Reconstruction Lemma and General Two-Component Lemma, and how operator trigonometry provides new geometry to statistics matrix inequalities and canonical correlations. Finally, it presents new results applying operator trigonometry to prediction theory and to association measures.
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33

Beenakker, Carlo W. J. Classical and quantum optics. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.36.

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This article focuses on applications of random matrix theory (RMT) to both classical optics and quantum optics, with emphasis on optical systems such as disordered wave guides and chaotic resonators. The discussion centres on topics that do not have an immediate analogue in electronics, either because they cannot readily be measured in the solid state or because they involve aspects (such as absorption, amplification, or bosonic statistics) that do not apply to electrons. The article first considers applications of RMT to classical optics, including optical speckle and coherent backscattering, reflection from an absorbing random medium, long-range wave function correlations in an open resonator, and direct detection of open transmission channels. It then discusses applications to quantum optics, namely: the statistics of grey-body radiation, lasing in a chaotic cavity, and the effect of absorption on the reflection eigenvalue statistics in a multimode wave guide.
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34

Bandyopadhyay, Arindam. Basic Statistics for Risk Management in Banks and Financial Institutions. Oxford University Press, 2022. http://dx.doi.org/10.1093/oso/9780192849014.001.0001.

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The book provides an engaging account of theoretical, empirical, and practical aspects of various statistical methods in measuring risks of financial institutions, especially banks. In this book, the author demonstrates how banks can apply many simple but effective statistical techniques to analyse risks they face in business and safeguard themselves from potential vulnerability. It covers three primary areas of banking risks—credit, market, and operational risk, and in a uniquely intuitive, step-by-step manner, the author provides hands-on details on the primary statistical tools that can be applied for financial risk measurement and management. The book lucidly introduces concepts of various well-known statistical methods such as correlations, regression, matrix approach, probability and distribution theorem, hypothesis testing, Value at Risk (Vary), and Monte Carlo simulation techniques and provides a hands-on estimation and interpretation of these tests in measuring risks of the financial institutions. The books strike a fine balance between concepts and mathematics to tell a rich story of thoughtful use of statistical methods. The book will be of much interest to academics, risk managers, bankers, and consultants and general readers too. It emphasizes on specific risk measurement tools and techniques with data applications, templates required for data collection and analysis, numerous excel-based illustrations as well as analysis in econometric packages. Excel-based hands-on and use of econometric packages like STATA, EVIEWS, and @RISK will help practitioners, academia, and students to connect theory with application.
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