Random Simplex Matrix

Overview

This repository contains a function that utilises the Dirichlet distribution method to generate points on the (n−1)-dimensional simplex. The randomSimplexMatrix.m generates m x n matrices where each row is a random sample from the (n−1)-dimensional simplex, i.e., it produces vectors where each element is a non-negative number and the sum of all elements in each vector is 1.

About

Mathematica Link

Methodology

1. Generation of K Unit-Exponential Distributed Random Draws:

   • In each row (sample), the function generates K uniform random numbers y_i from the open interval (0,1].
   • These are transformed to unit-exponential distributed random numbers x_i = -log(y_i).

2. Normalization:

   • Compute the sum S of all x_i values.

3. Calculation of Simplex Coordinates:

   • The coordinates t_1, ..., t_K of the final point on the unit simplex are computed as t_i = x_i / S.

4. Output:

   • Returns a matrix, where each row is a vector on the (n-1)-dimensional simplex.

Some Applications

Stability Analysis

• Polynomial Stability/Stability of discrete-time control systems:

  Use simplex sampling to generate coefficients for polynomials and analyse their stability by checking if all roots lie within the unit circle.

• Lyapunov Functions:

  Construct Lyapunov functions with randomly sampled coefficients to study the stability of equilibrium points in dynamical systems.

Bifurcation Analysis

• Parameter Space Exploration:

  Investigate the behavior of dynamical systems under different parameter regimes by sampling parameters from a simplex. Identify bifurcation points where system behavior changes qualitatively.

• Nonlinear Dynamics:

  Model/simulate nonlinear systems to study chaos, where initial conditions or parameters are sampled from a simplex.

Additional Scripts

1. simplexSpace.m

   Demonstrates various plots and visualisations of simplex sampling.

2. MultivariateND.m

   Showcases an application of simplex sampling for sampling from a multivariate normal distribution.

3. VoronoiDiagram.m

   Visualise the Voronoi diagram of random points on a 2-dimensional simplex, divides regions based on proximity.

Example Usage

n = 100;  % Number of columns (dimensionality of simplex)
m = 1500;  % Number of rows (number of samples)
y = randomSimplexMatrix(n, m);
disp('Generated simplex matrix:');
disp(y);