Racing Diffusion Model

The Racing Diffusion Model (RDM; Tillman, Van Zandt, & Logan, 2020) is a sequential sampling model in which evidence for options races independently. The RDM is similar to the Linear Ballistic Accumulator, except it assumes noise occurs during the within-trial evidence accumulation process, but the drift rate is constant across trials.

Example

In this example, we will demonstrate how to use the RDM in a generic two alternative forced choice task.

Load Packages

The first step is to load the required packages.

using SequentialSamplingModels
using Plots
using Random

Random.seed!(8741)

Random.TaskLocalRNG()

Create Model Object

In the code below, we will define parameters for the RDM and create a model object to store the parameter values.

Drift Rates

The drift rates control the speed with which information accumulates. Typically, there is one drift rate per option.

ν = [1.0,0.50]

2-element Vector{Float64}:
 1.0
 0.5

Maximum Starting Point

The starting point of each accumulator is sampled uniformly between $[0,A]$.

A = 0.80

0.8

Threshold - Maximum Starting Point

Evidence accumulates until accumulator reaches a threshold $\alpha = k +A$. The threshold is parameterized this way to faciliate parameter estimation and to ensure that $A \le \alpha$.

k = 0.50

0.5

Non-Decision Time

Non-decision time is an additive constant representing encoding and motor response time.

τ  = 0.30

0.3

RDM Constructor

Now that values have been assigned to the parameters, we will pass them to RDM to generate the model object.

dist = RDM(;ν, k, A, τ)

RDM
┌───────────┬────────────┐
│ Parameter │ Value      │
├───────────┼────────────┤
│ ν         │ [1.0, 0.5] │
│ k         │  0.50      │
│ A         │  0.80      │
│ τ         │  0.30      │
└───────────┴────────────┘

Simulate Model

Now that the model is defined, we will generate $10,000$ choices and reaction times using rand.

 choices,rts = rand(dist, 10_000)

(choice = [2, 1, 1, 2, 1, 1, 1, 1, 1, 1  …  2, 1, 1, 1, 1, 2, 1, 1, 2, 1], rt = [0.4261252869183349, 0.7972780554571337, 0.5626104946052586, 0.709356356770205, 0.47920902310738883, 1.8152686671350737, 1.0603968132096429, 0.38991099679639674, 0.6206523528546303, 1.6505907453116413  …  0.7888321659958766, 0.5921379499413524, 1.2366558717646412, 0.44494442746499724, 0.945687762519527, 0.8134378742077351, 0.7190376150001438, 0.3551308652948572, 1.103503275499708, 0.8243406918226641])

Compute PDF

The PDF for each observation can be computed as follows:

pdf.(dist, choices, rts)

10000-element Vector{Float64}:
 0.6812870492546116
 0.5418501066765354
 0.9505477125260924
 0.4434948183057127
 1.0282226389507736
 0.05632148757361826
 0.27698646477960737
 0.7441997519730676
 0.845351352480957
 0.07688104603728428
 ⋮
 0.8988136318313844
 0.18297381769392043
 0.9918778088571767
 0.36868563648003216
 0.3296695179829067
 0.6644503768117955
 0.38738935575430167
 0.1495938427500118
 0.504737288418596

Compute Log PDF

Similarly, the log PDF for each observation can be computed as follows:

logpdf.(dist, choices, rts)

10000-element Vector{Float64}:
 -0.38377155025649884
 -0.6127658717384906
 -0.05071692102144627
 -0.8130691609406383
  0.027831718438863104
 -2.8766791545973174
 -1.2837866376001894
 -0.29544579634961177
 -0.1680029362939447
 -2.5654959083085482
  ⋮
 -0.10667957208439144
 -1.69841220908741
 -0.008155355839384558
 -0.9978109317017789
 -1.109664586976242
 -0.40879508110146934
 -0.9483250044984833
 -1.8998313723769338
 -0.6837172060219201

Compute Choice Probability

The choice probability $\Pr(C=c)$ is computed by passing the model and choice index to cdf.

cdf(dist, 1)

0.6069322520678416

To compute the joint probability of choosing $c$ within $t$ seconds, i.e., $\Pr(T \leq t \wedge C=c)$, pass a third argument for $t$.

Plot Simulation

The code below overlays the PDF on reaction time histograms for each option.

histogram(dist; xlims=(0,2.5))
plot!(dist; t_range=range(.301, 2.5, length=100))

References

Tillman, G., Van Zandt, T., & Logan, G. D. (2020). Sequential sampling models without random between-trial variability: The racing diffusion model of speeded decision making. Psychonomic Bulletin & Review, 27, 911-936.