Evaluating methods for approximating stochastic differential equations

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.Over the past 40 years, models of response time (RT) for simple decision making have become very successful at capturing the details of observed data (Audley & Pike, 1965;Brown & Heathcote, 2005;Busemeyer & Townsend, 1993;Diederich, 1997;Heath, 1981;LaBerge, 1962;Lacouture and Marley, 1991;Laming, 1966;Link & Heath, 1975;Ratcliff, 1978; Ratcliff & Smith, 2004, Appendix;Ratcliff, Van Zandt, & McKoon, 1999;Smith, 1995;Vickers, 1970;Vickers & Lee, 2000). More recently, the same models have also become quite successful at explaining decision making at a neural level (Carpenter & Reddi, 2001;Cook & Maunsell, 2002;Glimcher 2003;Gold & Shadlen, 2001;Ratcliff, Cherian & Seagraves, 2003;Reddi & Carpenter, 2000;Roitman & Shadlen, 2002;Sato, Murthy, Thompson & Schall, 2001;Sato & Schall, 2003;Shadlen, Britten, Newsome & Movshon, 1996;Wang, 2002). The most successful models of decision making in both cognitive and neural domains are the sequential sampling models. These models are based on the idea that noisy stimulus information is accumulated progressively over time until sufficient information for one of the response alternatives has been obtained. The predicted decision time in such models is obtained mathematically by solving a first-passage-time (FPT) problem, that is, the time taken for the accumulated information to reach a criterion and trigger a response. For some models, there exist explicit analytic methods or highly accurate numerical methods for solving the FPT problem (see Ratcliff & Smith, Appendix, for a survey). For other models, this problem may be complex or intractable. In such situations, researchers must resort to Monte Carlo simulation techniques to obtain predicted RT distributions and choice probabilities. We investigate the properties of such simulation techniques in this article.One of the best-known sequential sampling models, and one that has been applied to a wide range of experimental data, is the diffusion model of Ratcliff (1978;Ratcliff & Rouder, 1998). This model assumes that evidence accumulation begins at some initial value (z) and

Efficient simulation of diffusion-based choice RT models on CPU and GPU

“…Based on this graph, we suggest using a EulerMaruyama time step dt of at least 0.001 (which is in line with the findings of Brown et al (2006)) and a fast-dm accuracy of at least 2.5. When using approximate intermediaries in an analysis (e.g., estimations based on approximate distributions), it is wise to at least check the convergence of this approximation for the final outcome (by comparing with higher accuracy approximations).…”

“…For high dimensionality, straightforward deterministic numerical integration becomes problematic, as the underlying matrices grow in size with a power equal to the dimensionality of the model. The main drawback of simulation-based methods (e.g., Brown et al, 2006) on the other hand, is that smooth first-passage time distributions, generally require a large number of simulations.…”

Unconscious information changes decision accuracy but not confidence

“…Model predictions were generated by simulation using the Euler method (34). There were six free parameters in the model (a, Ter, β 0 , β 1 , S Ter , and K), and they were estimated using differential evolution optimization (35).…”

Fast and accurate calculations for first-passage times in Wiener diffusion models

“…Our approach is somewhat different to recent proposals by Tuerlinckx (2004) and Voss and Voss (2008), since we focus on the WFPT distribution itself, not the full model with the additional random variables. It also differs from the approach taken by Wagenmakers, van der Maas, and Grasman (2007), in that we do not place any restrictions on the parameters (but see, Grasman, Wagenmakers, and van der Maas (in press), for a more general approach), and from that of Brown, Ratcliff, and Smith (2006) and Diederich and Busemeyer (2003) who focus on general simulation methods. Our choice to focus on the model at this level of generality is deliberate-in the ''narrow'' context of modeling choice and RT in simple two-alternative decision tasks, 1 Ratcliff (1978) fixes σ = 0.1, which has some practical advantages but is mathematically inconvenient for our purposes.…”