Random orthogonal matrix simulation

Random orthogonal matrix (ROM) simulation is a very fast procedure for generating multivariate random samples that always have exactly the same mean, covariance and Mardia multivariate skewness and kurtosis. This paper investigates how the properties of parametric, data-specific and deterministic ROM simulations are influenced by the choice of orthogonal matrix. Specifically, we consider how cyclic and general permutation matrices alter their time-series properties, and how three classes of rotation matrices upper Hessenberg, Cayley, and exponential -influence both the unconditional moments of the marginal distributions and the behaviour of skewness when samples are concatenated. We also perform an experiment which demonstrates that parametric ROM simulation can be hundreds of times faster than equivalent Monte Carlo simulation. Crown

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Nonconvex Optimization and Its Applications, 2000

Value at Risk VaR analysis plays very important role in modern Financial Risk Management. The are two very popular approaches to portfolio VaR estimation: approximate analytical approach and Monte Carlo simulation. Both of them face some technical di culties steaming from statistical estimation of covariance matrix decribing the distribution of the risk factors. In this paper we develop a new robust method of generating scenarios in a space of risk factors consistent with a given matrix of correlations containing possible small negative eigenvalues, and nd an estimate for a change in VaR. Namely, we prove that the modi ed VaR of a portfolio VaR 0 satis es the inequality j g VaR 2 ,VaR 2 j K ; where is the maximum of the absolute values of negative eigenvalues of the approximate covariance matrix and K is an explicitly expressed constant, closely related to the market value of the portfolio.

Chemometrics and Intelligent Laboratory Systems, 2017

The simulation of multivariate data is often necessary for assessing the performance of multivariate analysis techniques. The random generation of multivariate data when the covariance matrix is completely or partly specified is solved by different methods, from the Cholesky decomposition to some recent alternatives. However, many times the covariance matrix has to be generated also at random, so that the data simulation spans different situations from highly correlated to uncorrelated data. This is the case when assessing a new multivariate analysis technique in Montercarlo experiments. In this paper, we introduce a new algorithm for the generation of random data from covariance matrices of random structure, where the user only decides the data dimension and the level of correlation. We will illustrate the application of this algorithm in several relevant problems in multivariate analysis, namely the selection of the number of Principal Components in Principal Component Analysis, the evaluation of the performance of sparse Partial Least Squares and the calibration of Multivariate Statistical Process Control systems. The algorithm is available as part of the MEDA Toolbox v1.1 1 1

2006

Model evaluation in covariance structure analysis is critical before the results can be trusted. Due to finite sample sizes and unknown distributions of practical data, existing conclusion regarding a particular statistic may not be applicable in practice. The bootstrap procedure automatically takes care of the unknown distribution and, for a given sample size, also provides more accurate results than those based on standard asymptotics. But it needs a matrix to play the role of the population covariance matrix. The closer the matrix is to the true population covariance matrix, the more valid the bootstrap inference is. The current paper proposes a class of covariance matrices by combining theory and data. Thus, a proper matrix from this class is closer to the true population covariance matrix than those constructed by any existing methods. Each of the covariance matrices is easy to generate and also satisfies several desired properties. Examples verify the properties of the matrices and illustrate the details for creating a matrix with a given amount of misspecification.

European Journal of Operational Research, 2007

In portfolio selection, there is often the need for procedures to generate "realistic" covariance matrices for security returns, for example to test and benchmark optimization algorithms. For application in portfolio optimization, such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider "realistic" for security returns. Deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings, a procedure is developed for the random generation of covariance matrices (a) whose off-diagonal (covariance) entries possess a pre-specified expected value and standard deviation and (b) whose main diagonal (variance) entries possess a likely different pre-specified expected value and standard deviation. The paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper.

The evolution of statistical inference in the last years has been induced also by the development of new computational tools which have led both to the solution of classical statistical problems and to the implementation of new methods of analysis. In this framework, a relevant computational tool is given by simulation which leads to deal with methodological problems not solvable analytically. Simulation is frequently used for the generation of variates from known distributions with Monte Carlo methods for the study of the behavior of statistics for which the sampling distribution is unknown. Moreover, Monte Carlo algorithms have been developed which can be also used to ap-proximate the stochastic integrals, as in bayesian statistical analysis and in optimization problems. The application of such algorithms needs the application of advanced com-putational tools and, particularly, computer algebra systems are surely one of the most suitable tools for the simulation of complex stochas...

Arxiv preprint physics/0507111, 2005

2020

The spectral law of random matrices developed by Wigner and Wishart converges to a deterministic law when the dimension of the matrix tends to infinity. In this paper, the objective is to apply this theory in the first degree on the returns of companies from the Paris stock exchange. This allows the theory to be approved through the use of real cases. Also, this study attempts to compare the empirical distribution of the eigenvalues resulting from the minimization of the covariance matrix (Markowitz's theory) with the law of random matrices.

This paper explores the properties of random orthogonal matrix (ROM) simulation when the random matrix is drawn from the class of rotational matrices. We describe the characteristics of ROM simulated samples that are generated using random Hessenberg, Cayley and exponential matrices and compare the computational efficiency of parametric ROM simulations with standard Monte Carlo techniques.