A comparison of four methods for simulating the diffusion process

Four methods for the simulation of the Wiener process with constant drift and variance are described. These four methods are (1) approximating the diffusion process by a random walk with very small time steps; (2) drawing directly from the joint density of responses and reaction time by means of a (possibly) repeated application of a rejection algorithm; (3) using a discrete approximation to the stochastic differential equation describing the diffusion process; and (4) a probability integral transform method approximating the inverse of the cumulative distribution function of the diffusion process. The four methods for simulating response probabilities and response times are compared on two criteria: simulation speed and accuracy of the simulation. It is concluded that the rejection-based and probability integral transform method perform best on both criteria, and that the stochastic differential approximation is worst. An important drawback of the rejection method is that it is applicable only to the Wiener process, whereas the probability integral transform method is more general.

[1]  J. Wolfowitz,et al.  An Introduction to the Theory of Statistics , 1951, Nature.

[2]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[3]  J. Miller.  Numerical Analysis , 1966, Nature.

[4]  文魚 長谷川.  D.R. Cox and H.D. Miller: The Theory of Stochastic Processes, Methuen. London, 1965, 398頁, 24×16cm, 4,200円. , 1966 .

[5]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[6]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[7]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[8]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 1 , 1967 .

[9]  J. Andel.  Sequential Analysis , 2022, The SAGE Encyclopedia of Research Design.

[10]  Franklin A. Graybill,et al.  Introduction to the Theory of Statistics, 3rd ed. , 1974 .

[11]  I. I. Gikhman.  Theory of stochastic processes , 1974 .

[12]  Steven M. Melimis.  Numerical methods for stochastic processes , 1978 .

[13]  Roger Ratcliff,et al.  A Theory of Memory Retrieval. , 1978 .

[14]  Roger Ratcliff,et al.  A note on modeling accumulation of information when the rate of accumulation changes over time , 1980 .

[15]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[16]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[17]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[18]  J. E. Glynn,et al.  Numerical Recipes: The Art of Scientific Computing , 1989 .

[19]  Philip L. Smith.  A note on the distribution of response times for a random walk Gaussian increments , 1990 .

[20]  I. G. MacKenzie,et al.  Stochastic Processes with Applications , 1992 .

[21]  J. Townsend,et al.  Fundamental derivations from decision field theory , 1992 .

[22]  W Schwarz,et al.  A diffusion model of early visual search: Theoretical analysis and experimental results , 1993, Psychological research.

[23]  J. Schall,et al.  Neural Control of Voluntary Movement Initiation , 1996, Science.

[24]  Diederich,et al.  Dynamic Stochastic Models for Decision Making under Time Constraints , 1997, Journal of mathematical psychology.

[25]  Jeffrey N. Rouder,et al.  Modeling Response Times for Two-Choice Decisions , 1998 .

[26]  M. A. Tanner,et al.  Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd Edition , 1998 .

[27]  R. Ratcliff,et al.  Connectionist and diffusion models of reaction time. , 1999, Psychological review.

[28]  Philip L. Smith,et al.  Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer. , 2000, Journal of mathematical psychology.

[29]  Jeffrey N. Rouder,et al.  A diffusion model account of masking in two-choice letter identification. , 2000, Journal of experimental psychology. Human perception and performance.

[30]  R. Ratcliff,et al.  Estimating parameters of the diffusion model: Approaches to dealing with contaminant reaction times and parameter variability , 2002, Psychonomic bulletin & review.