Evaluating methods for approximating stochastic differential equations

https://doi.org/10.1016/j.jmp.2006.03.004Get rights and content

Abstract

Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler's method for solving ordinary differential equations. The accuracy of Euler's method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matrix method proposed by Diederich and Busemeyer (2003) yielded significant improvements. The accuracy of all methods depended critically on the size of the approximating time step. The large (∼10 ms) step sizes often used by psychological researchers resulted in large and systematic errors in evaluating RT distributions.

Section snippets

Tracking differential equations

Consider the leaky accumulator model presented above. The RT distributions and response accuracy can be estimated by Monte Carlo integration of repeated simulated decisions. Simulating a single decision could proceed as follows:

  • 1.

    Decide on values for the parameters I1, I2, k, σ and C.

  • 2.

    Initialize time to t=0 and each accumulators’ activation to x1=x2=0.

  • 3.

    Choose a time “step size”, Δt.

  • 4.

    Set t=tt.

  • 5.

    Sample two random numbers, η1 and η2, from a standard normal distribution.

  • 6.

    For i=1,2, set: xi:=xi+(Ii-kxi)Δt+

Evaluation methods

To match the kind of parameter settings and tasks faced during real RT modeling, we used a fixed set of parameters for the model, along with factorial combinations of four different input strengths (drift rates, I) and two different response criteria (C). This simulated four experimental conditions of different difficulty, each of which is given in both speed- and accuracy-emphasis conditions (see, e.g., Ratcliff & Rouder, 1998). The particular model we used was dX(t)=(I-kX(t))dt+σdW(t). The

Implementation and computation times

The computation time for each of the methods we investigated is inversely proportional to the step size parameters, at least approximately. Using a standard desktop computer (32 bit CPU, about 2 GHz clock speed, 1 GB memory), the matrix approximation method required about 2 s to evaluate distributions associated with a single set of parameters when Δt=0.1ms, and about 0.2 s when Δt=1ms. Euler's method, using 20,000 Monte Carlo repetitions, required about 20 s when Δt=0.1ms and about 2 s when Δt=1 ms.

A note on parameter scaling

The stochastic differential equation models used in psychology are sufficiently complex that tradeoffs between various model parameters are not always well understood. As an example, the model we have investigated has two different parameter scaling properties that must both be understood to allow accurate parameter estimation. Firstly, there is a scaling that adjusts RT distributions. If the following parameters are all scaled by some factor, say w, the predicted RT distributions will all

Conclusions

Our results are important for working with sequential sampling models. For those researchers using Euler's method who are interested only in the goodness-of-fit of various models, a large and computationally fast step size (e.g., Δt=20 ms) is quite adequate. However, if the goal is to compare or interpret estimated parameter values, any step size larger than Δt=1 ms may lead to significant bias. There are two possible solutions to this problem. The simplest solution is to use a small step size

References (37)

  • Brown, S. (2002). Quantitative approaches to skill acquisition in choice RT. Unpublished doctoral dissertation,...
  • S. Brown et al.

    A ballistic model of choice response time

    Psychological Review

    (2005)
  • K. Burrage et al.

    Numerical methods for strong solutions of SDES

    Proceeding of the Royal Society London

    (2004)
  • J.R. Busemeyer et al.

    Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment

    Psychological Review

    (1993)
  • R.H.S. Carpenter et al.

    Letters to the editor (reply)

    Nature Neuroscience

    (2001)
  • E.P. Cook et al.

    Attentional modulation of behavioral performance and neuronal responses in middle temporal and ventral intraparietal areas of Macaque monkey

    The Journal of Neuroscience

    (2002)
  • T.C. Gard

    Introduction to stochastic differential equations

    (1988)
  • P.W. Glimcher

    The neurobiology of visual-saccadic decision-making

    Annual Review of Neuroscience

    (2003)
  • Cited by (35)

    • A new framework for modeling decisions about changing information: The Piecewise Linear Ballistic Accumulator model

      2016, Cognitive Psychology
      Citation Excerpt :

      This is because, to evaluate the choice and associated RT for a trial requires, for each accumulator, only a single sample from the uniform distribution, two samples from the normal distribution, and then some simple linear calculations based on these samples and threshold and delay parameters. In contrast, models with moment-to-moment variability in accumulation rates, such as the DDM, require fine-grained Euler methods using a separate normal sample for at least each 10 ms of decision time (Brown, Ratcliff, & Smith, 2006). Similarly fine-grained approximation methods for solving differential equations are required for non-linear models like the Ballistic Accumulator model (Brown & Heathcote, 2005b), and both types of approximation are required for the Leaky Competitive Accumulator model (Usher & McClelland, 2001).

    • Modeling confidence and response time in associative recognition

      2016, Journal of Memory and Language
      Citation Excerpt :

      Because there are no exact solutions for this model, simulations are used to generate predicted values from the model. To simulate the process of accumulation given by Eqs. (1) and (2), we used the simple Euler’s method with 1-ms steps (cf. Brown, Ratcliff, & Smith, 2006; Usher & McClelland, 2001). For each millisecond step, one accumulator was chosen randomly, and the evidence in it was incremented or decremented according to Eq. (1) and opposite accumulators were incremented or decremented according to Eq. (2) (e.g., if the selected accumulator was for one of the ‘intact’ responses, then the evidence in the ‘rearranged’ accumulators would be adjusted according to Eq. (2) and the other ‘intact’ accumulators would be unchanged).

    View all citing articles on Scopus
    View full text