Evaluating methods for approximating stochastic differential equations
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=t+Δt.
- 5.
Sample two random numbers, η1 and η2, from a standard normal distribution.
- 6.
For i=1,2, set:
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 . 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)
- et al.
High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations
Applied Numerical Mathematics
(1996) Dynamic stochastic models for decision making under time constraints
Journal of Mathematical Psychology
(1997)- et al.
Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time
Journal of Mathematical Psychology
(2003) - et al.
Neural computations that underlie decisions about sensory stimuli
Trends in Cognitive Sciences
(2001) A general nonstationary diffusion model for two choice decision making
Mathematical Social Sciences
(1992)- et al.
Search efficiency but not response interference affects visual selection in frontal eye field
Neuron
(2001) - et al.
Effects of stimulus–response compatibility on neural selection in frontal eye field
Neuron
(2003) Stochastic, dynamic models of response times and accuracy: A foundational primer
Journal of Mathematical Psychology
(2000)Probabilistic decision-making by slow reverberation in cortical circuits
Neuron
(2002)- et al.
Some stochastic models of choice
British Journal of Mathematical and Statistical Psychology
(1965)
A ballistic model of choice response time
Psychological Review
Numerical methods for strong solutions of SDES
Proceeding of the Royal Society London
Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment
Psychological Review
Letters to the editor (reply)
Nature Neuroscience
Attentional modulation of behavioral performance and neuronal responses in middle temporal and ventral intraparietal areas of Macaque monkey
The Journal of Neuroscience
Introduction to stochastic differential equations
The neurobiology of visual-saccadic decision-making
Annual Review of Neuroscience
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2016, Cognitive PsychologyCitation 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).
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2016, Journal of Memory and LanguageCitation 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).