Mathematics

The Ziggurat Method for Generating Gaussian Random Numbers

Introduction

The Ziggurat method for generating random numbers is a special case of the rejection method, where the density of the proposal distribution looks like the silhouette of a Ziggurat. The method can be used to produce very fast random number generators for distributions like the normal distribution or the exponential distribution. The main advantage comes from the fact that for a high percentage of the generated numbers no slow floating point operations are necessary.

Mathematical Homepage of Jochen Voss

This page gives a short summary of my mathematical interests.

Contents

• Conditional Path Sampling of SDEs
• Diffusion Processes
• Large Deviations
• Simulations and Numerical Analysis
• Miscellaneous

Conditional Path Sampling of SDEs

With Andrew Stuart and Martin Hairer I am exploring how infinite dimensional Langevin sampling can be used to sample from conditioned distributions of solutions of stochastic differential equations. Our method is based on constructing a partial stochastic differential equation which has the conditioned target density as its stationary distribution.

The Mathematical Theory of Juggling

The purpose of this text is to explain enough of the theory of juggling, that you can understand the juggling patterns which are explained on my juggling web pages. This page does not intend to give a complete explanation of site-swaps and indeed there are many interesting facts omitted. If you want to know more, Burkhard Polster's book The Mathematics of Juggling is a good reference. The references section below also contains some links to online resources.

Fast-DM: Fast Diffusion Model Analysis

This page contains some information about the program fast-dm. Please note that this is not the official homepage.

Introduction

Fast-dm is a programm for fast parameter estimation in Ratcliff's (1978) diffusion model. With a Diffusion model data analysis it is possible to analyse data from any fast binary decision task. A diffusion-model data analysis is based on the distributions of both correct and erroneous responses. From these distributions a set of parameters is estimated that allows to draw conclusions about the underlying cognitive processes (Voss, Rothermund, & Voss, 2004).

Mathematische Bildergalerie

Diese Seite enthält einige Bilder, die ich zu unterschiedlichen Zwecken irgendwann einmal erzeugt habe. Jedermann darf diese Bilder gerne für eigene Zwecke verwenden und weiterverbreiten, aber natürlich freue ich mich, wenn ich als Quelle genannt werde. Durch Anklicken der kleinen Bilder erhält man größere Versionen.

• Ein schönes Zufallsprodukt.
• Ein Pfad einer stochastischen Differentialgleichung mit vorgegebenem Anfangs- und Endpunkt:
• Einige "klassische" Fraktale:

  Die Koch-Kurve	Das Sierpinsky-Dreieck	Viele Quadrate

• Anfangsstücke von Brownschen Pfaden. Der äußere Rand ist rot markiert. Dies sind zufällige Fraktale. Alle drei Bilder sind nach demselben Mechanismus erzeugt.
• Zustände des Ising-Modells. Dies sind zufällige Bilder, die mit mit meinem XIsing Programm erzeugt sind. Der Parameter β ist die inverse Temperatur.

  Matrix Analysis and Algorithms

  This page contains lecture notes about numerical linear algebra which I wrote together with Andrew Stuart. The notes form the basis of the MA398 numerical linear algebra module at the university of Warwick.

  Contents

  • Vector and Matrix Analysis
    • Vector Norms and Inner Products
    • Eigenvalues and Eigenvectors
    • Dual Spaces
    • Matrix Norms
    • Structured Matrices
  • Matrix Factorisations
    • Diagonalisation
    • Jordan Canonical Form
    • Singular Value Decomposition
    • QR Factorisation
    • LU Factorisation
    • Cholesky Factorisation
  • Stability and Conditioning
    • Conditioning of SLE
    • Conditioning of LSQ
    • Conditioning of EVP
    • Stability of Algorithms
  • Complexity of Algorithms
    • Computational Cost
    • Matrix-Matrix Multiplication
    • Fast Fourier Transform
    • Bidiagonal and Hessenberg Forms
  • Systems of Linear Equations
    • Gaussian Elimination
    • Gaussian Elimination with Partial Pivoting
    • The QR Factorisation
  • Iterative Methods
    • Linear Methods
    • The Jacobi Method
    • The Gauss-Seidel and SOR Methods
    • Nonlinear Methods
    • The Steepest Descent Method
    • The Conjugate Gradient Method
  • Least Squares Problems
    • LSQ via Normal Equations
    • LSQ via QR factorisation
    • LSQ via SVD
  • Eigenvalue Problems
    • The Power Method
    • Inverse Iteration
    • Rayleigh Quotient Iteration
    • Simultaneous Iteration
    • The QR Algorithm for Eigenvalues
    • Divide and Conquer for Symmetric Problems

  Download

  The script is available for download here.

  XIsing

  Introduction

  The Ising model is a simple model from statistical mechanics, which describes a system of coupled spins. The spins are modelled as cells which can be in either of two states. In the pictures below each pixel corresponds to one cell, and the states are represented by the colors black and white.

  The program implements two iterative methods to simulate states of the Ising model: the Gibbs sampler and the Metropolis algorithm.