Abstract
An algorithm is described to efficiently compute the cumulative distribution and probability density functions of the diffusion process (Ratcliff, 1978) with trial-to-trial variability in mean drift rate, starting point, and residual reaction time. Some, but not all, of the integrals appearing in the model’s equations have closed-form solutions, and thus we can avoid computationally expensive numerical approximations. Depending on the number of quadrature nodes used for the remaining numerical integrations, the final algorithm is at least 10 times faster than a classical algorithm using only numerical integration, and the accuracy is slightly higher. Next, we discuss some special cases with an alternative distribution for the residual reaction time or with fewer than three parameters exhibiting trialto-trial variability.
Article PDF
Similar content being viewed by others
References
Abramowitz, M., &Stegun, I. (1974).Handbook of mathematical functions. New York: Dover.
Acton, F. S. (1970).Numerical methods that work. New York: Harper & Row.
Burden, R. L., &Faires, J. D. (1997).Numerical analysis (6th ed.). Pacific Grove, CA: Brooks/Cole.
Cox, D. R., &Miller, H. D. (1970).The theory of stochastic processes. London: Methuen.
Derenzo, S. E. (1977). Approximations for hand calculators using small integer coefficients.Mathematics of Computation,31, 214–225.
Golub, G. H., &Welsch, J. H. (1969). Calculation of Gauss quadrature rules.Mathematics of Computation,23,221–230.
Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.
Myung, I. J. (2003). Tutorial on maximum likelihood estimation.Journal of Mathematical Psychology,47, 90–100.
Naylor, J. C., &Smith, A. F. M. (1982). Applications of a method for the efficient computation of posterior distributions.Applied Statistics,31,214–225.
Nelder, J. A., &Mead, R. (1965). A simple method for function minimization.Computer Journal,7, 308–313.
Press, W. H., Flannery, B. P., Teukolsky, S. A., &Vetterling, W. T. (1989).Numerical recipes: The art of scientific computing. Cambridge: Cambridge University Press.
Prudnikov, A. P., Brychkov, Y. A., &Marichev, O. I. (1986).Integrals and series. Vol. 1: Elementary functions. New York: Gordon & Breach.
Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85,59–108.
Ratcliff, R., &Rouder, J. (1998). Modeling response times for two-choice decisions.Psychological Science,9,347–356.
Ratcliff, R., &Tuerlinckx, F. (2002). Estimating parameters of the diffusion model: Approaches to dealing with contaminant reaction times and parameter variability.Psychonomic Bulletin & Review,9,438–481.
Ratcliff, R., Van Zandt, T., &McKoon, G. (1999). Connectionist and diffusion models of reaction time.Psychological Review,106,261–300.
Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times.Behavior Research Methods, Instruments, & Computers,33,457–469.
Van Zandt, T. (2000). How to fit a response time distribution.Psychonomic Bulletin & Review,7,424–465.
Wagenmakers, E.-J., Grasman, R., & Molenaar, P. C. M. (2004).On the relation between the mean and the variance of a diffusion model response time distribution. Manuscript submitted for publication.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Science Foundation Grant SES00-84368 and by the Fund for Scientific Research—Flanders.
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Tuerlinckx, F. The efficient computation of the cumulative distribution and probability density functions in the diffusion model. Behavior Research Methods, Instruments, & Computers 36, 702–716 (2004). https://doi.org/10.3758/BF03206552
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03206552