Abstract
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.
Article PDF
Similar content being viewed by others
References
Bhattacharya, R. N., &Waymire, E. C. (1990).Stochastic processes with applications. New York: Wiley.
Bouleau, N., &Lépingle, D. (1994).Numerical methods for stochastic processes. New York: Wiley.
Burden, R. L., &Faires, J. D. (1997).Numerical analysis (6th ed.). Pacific Grove, CA: Brooks/Cole.
Busemeyer, J. R., &Townsend, J. T. (1992). Fundamental derivations from decision field theory.Mathematical Social Sciences,23, 255–282.
Cox, D. R., &Miller, H. D. (1970).The theory of stochastic processes. London: Methuen.
Diederich, A. (1997). Dynamic stochastic models for decision making under time constraints.Journal of Mathematical Psychology,41, 260–274.
Fahrmeir, L. (1976). On the simulation of stochastic processes with continuous state and parameter space. In L. Dekker (Ed.),Simulation of systems (pp. 67–71). Amsterdam: North-Holland.
Feller, W. (1968).An introduction to probability theory and its applications (Vol. 1, 3rd ed.). New York: Wiley.
Hanes, D. P., &Schall, J. D. (1996). Neural control of voluntary movement initiation.Science,274, 427–430.
Heath, R. A., &Kelly, L. (1988). An application of a discriminability index for the assessment of individual differences. In R. A. Heath (Ed.),Current issues in cognitive development and mathematical psychology (pp. 180–196). Newcastle, NSW: University of Newcastle.
Karlin, S., &Taylor, H. M. (1981).A second course in stochastic processes. New York: Academic Press.
Lichters, R., Fricke, T., & Schnakenberg, J. (1995).Stochastic simulation of diffusion with absorbing and reflecting boundary conditions [On-line]. Available: http://www.physik.rwth-aachen.de/group/thphys/ tpd/Werke/absrefl.html
Luce, R. D. (1986).Response times. New York: Oxford University Press.
Mood, A. M., Graybill, F. A., &Boes, D. C. (1974).Introduction to the theory of statistics. New York: McGraw-Hill.
Press, W. H., Flannery, B. P., Teukolsky, S. A., &Vetterling, W. T. (1986).Numerical recipes: The art of scientific computing. New York: Cambridge University Press.
Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85, 59–108.
Ratcliff, R. (1980). A note on modeling accumulation of information when the rate of accumulation changes over time.Journal of Mathematical Psychology,21, 178–184.
Ratcliff, R., &Rouder, J. (1998). Modeling response times for two-choice decisions.Psychological Science,9, 347–356.
Ratcliff, R., &Rouder, J. (2000). A diffusion model account of masking in two-choice letter identification.Journal of Experimental Psychology: Human Perception & Performance,26, 127–140.
Ratcliff, R., & Tuerlinckx, F. (2001).Estimating the parameters of the diffusion model. Manuscript submitted for publication.
Ratcliff, R., Van Zandt, T., &McKoon, G. (1999). Connectionist and diffusion models of reaction time.Psychological Review,106, 261–300.
Ross, S. M. (1996).Stochastic processes (2nd ed.). New York: Wiley.
Schwarz, W. A. (1993). A diffusion model of early visual search: Theoretical analysis and experimental results.Psychological Research,55, 200–207.
Smith, P. L. (1990). A note on the distribution of response times for a random walk with Gaussian increments.Journal of Mathematical Psychology,34, 445–459.
Smith, P. L. (2000). Stochastic dynamic models of response time and accuracy: A foundational primer.Journal of Mathematical Psycholog y,44, 408–463.
Tanner, M. A. (1996).Tools for statistical inference: Methods for the exploration of posterior distributions and likelihood functions (2nd ed.). New York: Springer-Verlag.
Wald, A. (1947).Sequential analysis. New York: Wiley.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was a research assistant of the Fund for Scientific Research (Flanders). The research in this paper was also supported by BOF Grant GOA/02/2000 from the University of Leuven.
Rights and permissions
About this article
Cite this article
Tuerlinckx, F., Maris, E., Ratcliff, R. et al. A comparison of four methods for simulating the diffusion process. Behavior Research Methods, Instruments, & Computers 33, 443–456 (2001). https://doi.org/10.3758/BF03195402
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03195402