Deterministic sampling from uniform distributions with Sierpiński space-filling curves

Hime Aguiar E Oliveira

doi:10.1007/s00180-021-01128-w

Abstract

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves, a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarily injective, the investigation focused upon the structure of the distributions obtained when their domains are swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1] and the Sierpiński space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback–Leibler and other divergence measures are used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random. Despite this, and to be coherent with the literature, the expression “random number” will be used along the text to mean “pseudo-random number”.

Highlights

In the last decades the proliferation of low cost and faster digital computers has expanded the development and applicability of techniques using stochastic simulation
Simulations of probabilistic models need random variables with specific probability distributions and, in practice, many algorithms for generation of these non-uniform random variables are based on certain transformations of uniform random numbers
While there is some logical basis for that, this kind of approach is not very well accepted in the field of simulation. This is so because numbers generated by using a particular physical process are not truly random numbers, which exist only conceptually

Summary

Introduction

In the last decades the proliferation of low cost and faster digital computers has expanded the development and applicability of techniques using stochastic simulation. In [13] it is stated that all deterministic methods for producing randomness fail in some application, and only experience and imagination of developers and users can lead to a better understanding of this type of event Despite all those true statements, considerable progress has been made since the 1940s, when computer-generated random numbers were successfully used in the implementation of the so-called Monte Carlo methods. For the sake of efficiency, typical algorithms have been based on recurrence relations, which can be viewed as discrete dynamical systems ”equipped” with finite memory These two characteristics of discreteness and finiteness show up whenever considering any type of method for digital computer generation of random objects - present day computers are unable to exactly represent irrational numbers, leaving available only the rational ones, up to a limited numerical precision. Their most important and amazing property is the ability to completely fill compact regions of higher dimensional spaces, in the sense of (typically) being surjective and continuous at the same time

Sierpinski space-filling curves

Proposed method

Experiment description and numerical results