Random matrix application to correlations amongst the volatility of assets

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

Random matrix theory is applied to study the correlation between different financial correlation coefficient matrices in the financial field. Correlation coefficient matrix is a key factor for constructing a network. In this paper we relate the random matrix theory to the network construction to study the financial networks model in terms of the random matrix. We select the stock data of Shanghai stock market, and divide them into four stages. We discuss the statistical properties of eigenvalues in financial correlation coefficient matrix and random matrix based on the random matrix theory, and improve the existing denoising method to construct the correlation coefficient matrix and to make it more suitable for building financial networks. After that we can build the financial network model. Then we analyze and compare the original financial network, the denoising financial network and the noise financial network in terms of the random matrix theory and the key node of networks. It is found that the primary important information is still in the original network, and the noise information corresponds to the information which the nodes of small degree in the original network include. Finally we analyze the topological structure of the financial networks, such as the minimum spanning tree, the motif and community structure. We also find that the topological properties of the improved financial networks are more remarkable and the topological structure is more compact.

We explicate the need and pathways for systematic stock market characterisation and development (SSMCD) and the role of Random Matrix Theory (RMT) in SSMCD research. This is the first time SSMCD, itself a nascent area of empirical finance introduced by the first author, is linked to RMT. Our focus is on the Nigerian Stock Market, particularly using RMT techniques to correlate respective asset prices in the NSM. The resulting insights are combined with those from related works on stochastic-times series analyses of stylised facts and six main market issues typically explored in empirical finance (efficiency, bubbles, anomalies, volatility, valuation and predictability), to illustrate SSMCD pathways in the NSM. Specifically, the RMT analyses focus on the cross-correlation matrix C of the stock index returns in the Nigerian Stock Market (NSM) from the period 2009 to 2013. Within this purview, we test the eigenvalues of the selected assets from the NSM and use their respective eigenvectors and inverse participation ratios to determine the stocks that drive the market dynamics. A method of obtaining a realistic implied correlation matrix for a hypothetical portfolio of some given assets selected from those considered in the empirical correlation matrix of the assets is considered. The positive implied correlation matrix shows that the corresponding assets in the NSM move in the same direction, meaning that portfolio diversification is not an optimal investment strategy. Hence, investing on derivative assets like call and put options is recommended. Further SSMCD implications of the analyses are foreshadowed. Also, we develop the links among SSMCD, macroeconomic modelling of national and global economic trends amid cycles of booms and crises. Highlights of an econome derived from analogies with sequencing and editing of genomes in molecular biology are provided.

We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.

Correlations between energy levels can help distinguish whether a many-body system is of integrable or chaotic nature. The study of short-range and long-range spectral correlations generally involves quantities which are very different, unless one uses the $k$-th nearest neighbor ($k$NN) level spacing distribution. For nearest-neighbor (NN) spectral spacings, the distribution in random matrices is well captured by the Wigner surmise. This well-known approximation, derived exactly for a 2$\times$2 matrix, is simple and satisfactorily describes the NN spacings of larger matrices. There have been attempts in the literature to generalize Wigner's surmise to further away neighbors. However, as we show, the current proposal in the literature fails to accurately capture numerical data.&amp;#xD;Using the known variance of the distributions from random matrix theory, we propose a corrected surmise for the $k$NN spectral distributions. This surmise better characterizes spectral correlations while retaining the simplicity of Wigner's surmise. We test the predictions against numerical results and show that the corrected surmise is systematically more accurate at capturing data from random matrices. Using the XXZ spin chain with random on-site disorder, we illustrate how these results can be used as a refined probe of many-body quantum chaos for both short- and long-range spectral correlations.

A lock step walk is a one-dimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to infinity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek, and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials. © 2000 John Wiley & Sons, Inc.

Cross correlations in an emerging market financial data

Econophysics: financial time series from a statistical physics point of view

With the research of Random Matrix Theory (RMT), many RMT -based covariance detection algorithms have been proposed and applied to spectrum sensing. As a blind detection algorithm, the detection performance in practice is often susceptible to environmental factors due to the approximation of threshold calculation. In view of this problem, some scholars have proposed to use machine learning to improve the covariance detection, so as to obtain high environmental adaptability. In this paper, we propose a covariance spectrum sensing algorithm with judgment statistics in maximum-minimum eigenvalue algorithm (MME) and difference between the maximum eigenvalue and the minimum eigenvalue algorithm (DMM) as hybrid features, which takes the both the quotient and difference of the maximum minimum eigenvalue of the covariance matrix as the feature vector for classification, and then trained and classified by SVM. Simulation results show that the proposed algorithm improves significantly compared to the traditional covariance detection. Moreover, compared with the existing covariance in combination with SVM detection algorithm, the two features selected in this paper have better differentiation and better environmental adaptability under low SNR.

Forecasting stock volatility using after-hour information: Evidence from the Australian Stock Exchange

Analysis of Financial Correlation Matrix Using Random Matrix Theory

BackgroundThe advance in high-throughput genomic technologies including microarrays has demonstrated the potential of generating a tremendous amount of gene expression data for the entire genome. Deciphering transcriptional networks that convey information on intracluster correlations and intercluster connections of genes is a crucial analysis task in the post-sequence era. Most of the existing analysis methods for genome-wide gene expression profiles consist of several steps that often require human involvement based on experiential knowledge that is generally difficult to acquire and formalize. Moreover, large-scale datasets typically incur prohibitively expensive computation overhead and thus result in a long experiment-analysis research cycle.ResultsWe propose a parallel computation-based random matrix theory approach to analyze the cross correlations of gene expression data in an entirely automatic and objective manner to eliminate the ambiguities and subjectivity inherent to human decisions. We apply the proposed approach to the publicly available human liver cancer data and yeast cycle data, and generate transcriptional networks that illustrate interacting functional modules. The experimental results conform accurately to those published in previous literatures.ConclusionsThe correlations calculated from experimental measurements typically contain both “genuine” and “random” components. In the proposed approach, we remove the “random” component by testing the statistics of the eigenvalues of the correlation matrix against a “null hypothesis” — a truly random correlation matrix obtained from mutually uncorrelated expression data series. Our investigation into the components of deviating eigenvectors after varimax orthogonal rotation reveals distinct functional modules. The utilization of high performance computing resources including ScaLAPACK package, supercomputer and Linux PC cluster in our implementations and experiments significantly reduces the amount of computation time that is otherwise needed on a single workstation. More importantly, the large distributed shared memory and parallel computing power allow us to process genomic datasets of enormous sizes.

Course 4 Random matrices, the ulam problem, directed polymers & growth models, and sequence matching

The sinoatrial node (SAN) is the pacemaker of the heart. Recently calcium signals, believed to be crucially important in rhythm generation, have been imaged in intact SAN and shown to be heterogeneous in various regions of the SAN with a lot of analysis relying on visual inspection rather than mathematical tools. Here we apply methods of random matrix theory (RMT) developed for financial data and various biological data sets including β-cell collectives and electroencephalograms (EEG) to analyse correlations in SAN calcium signals using eigenvalues and eigenvectors of the correlation matrix. We use principal component analysis to locate signalling modules corresponding to localization properties the eigenvectors corresponding to high eigenvalues. We find that the top eigenvector captures the global behaviour of the SAN i.e. action potential (AP) induced calcium transient. In some cases, the eigenvector corresponding to the second highest eigenvalue yields a pacemaker region whose calcium signals predict the AP. Furthermore, using new analytic methods, we study the relationship between covariance coefficients and distance, and find that even inside the central zone, there are non-trivial long range correlations, indicating intercellular interactions in most cases. Lastly, we perform an analysis of nearest-neighbour eigenvalue distances and find that it coincides with universal Wigner surmise under all available experimental conditions, while the number variance, which captures eigenvalue correlations, is sensitive to experimental conditions. Thus RMT application to SAN allows to remove noise and the global effects of the AP-induced calcium transient and thereby isolate the local and meaningful correlations in calcium signalling.

This paper is divided into three disconnected parts. (i) An identity is proved which establishes an intimate connection between the statistical behavior of eigenvalues of random unitary matrices over the real field and over the quaternion field. It is proved that the joint distribution function of all the eigenvalues of a random unitary self-dual quaternion matrix of order N is identical with the joint distribution function of a set of N alternate eigenvalues of a random unitary symmetric matrix of order 2N. A corollary of this result is the following: the distribution of spacings between next-nearest-neighbor eigenvalues in a real symmetric matrix of large order is identical with the distribution of nearest-neighbor spacings in a self-dual Hermitian quaternion matrix of large order. (ii) A conjecture is made which gives an exact analytic formula for the partition function of a finite gas of N point charges free to move on an infinite straight line under the influence of an external harmonic potential. This conjecture is at the same time a statement about the statistical properties of the eigenvalues of Hermitian matrices whose elements are Gaussian random variables. (iii) A list is made of several other problems which remain unsolved in the statistical theory of eigenvalues of random matrices.