Random numbers from the tails of probability distributions using the transformation method

The speed of many one-line transformation methods for the production of, for example, Levy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, for the class of decreasing probability densities fast rejection implementations like the Ziggurat by Marsaglia and Tsang promise a significant speed-up if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transform maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.Comment: 17 pages, 7 figures, submitted to a peer-reviewed journa

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation

Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively the derivation of an analytical expression for the spectra of covariance matrices approximated by free Levy stable random variables and validate it by Monte Carlo simulation.Comment: 10 pages, 1 figure, submitted to Eur. Phys. J.

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.Comment: 7 pages, 5 figures, 1 table. Presented at the Conference on Computing in Economics and Finance in Montreal, 14-16 June 2007; at the conference "Modelling anomalous diffusion and relaxation" in Jerusalem, 23-28 March 2008; et

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity