Random Walk Model

We derive computational models to minimize ambiguity in verbal theories. Computational models make more precise predictions and facilitate communication between scientists of the underlying model construct.

Simulation of nreps random walks in an uninformative decision-bound

#the number of random walks
nreps <- 10000
#the number of times evidence is sampled for each random walk
nsamples <- 2000


drift <- 0.0
#noise in the evidence, standard deviation of the distribution from which we willsample the evidence
sdrw <- .3
criterion <- 3


latencies <-rep(0,nreps)
responses <-rep(0,nreps)
evidence <- matrix(0, nreps, nsamples+1)
for (i in c(1:nreps)) {
  evidence[i,] <- 
    cumsum(c(0,rnorm(nsamples, drift, sdrw)))
  p <- which(abs(evidence[i,]) > criterion)[1]
  responses[i] <- sign(evidence[i, p])
  latencies[i] <- p
}


tbpn <- min(nreps, 5)
plot(1:max(latencies[1:tbpn]) + 10, type = 'n', las = 1,
           ylim = c(-criterion - .5, criterion + .5),
           ylab = 'Evidence', xlab = 'Decision time')
for (i in c(1:tbpn)){
  lines(evidence[i, 1:(latencies[i]-1)])
}
abline(h = c(criterion, -criterion), lty = 'dashed')

Histograms of decision latencies

par(mfrow = c(2,1))
toprt <- latencies[responses > 0]
topprop <- length(toprt)/nreps
hist(toprt, col = 'gray',
     xlab = 'Decision time', xlim = c(0, max(latencies)),
     main = paste('Top responses (', as.numeric(topprop),
                  ') m = ', as.character(signif(mean(toprt), 4)),
                  sep = ''), las = 1)
botrt <- latencies[responses < 0]
botprop <- length(botrt)/nreps
hist(botrt, col = 'gray',
     xlab = 'Decision time', xlim =c(0, max(latencies)),
     main = paste('Bottom responses
          (', as.numeric(botprop),
          ') m =', as.character(signif(mean(botrt), 4))))

Trial by trial variation of Initiating point and Drift rate

Note drift rate is changed to .03

nreps <- 10000
nsamples <- 2000

drift <- 0.03
sdrw <- .3
criterion <- 3
t2tsd <- c(0.0, 0.025)


latencies <-rep(0,nreps)
responses <-rep(0,nreps)
evidence <- matrix(0, nreps, nsamples+1)
for (i in c(1:nreps)) {
  sp <- rnorm(1, 0, t2tsd[1]) #various starting point
  dr <- rnorm(1, drift, t2tsd[2]) #various drift rate
  evidence[i,] <- 
    cumsum(c(sp,rnorm(nsamples, dr, sdrw)))
  p <- which(abs(evidence[i,]) > criterion)[1]
  responses[i] <- sign(evidence[i, p])
  latencies[i] <- p
}

Histograms of decision latencies for trial by trial variation

par(mfrow = c(2,1))
toprt <- latencies[responses > 0]
topprop <- length(toprt)/nreps
hist(toprt, col = 'gray',
     xlab = 'Decision time', xlim = c(0, max(latencies)),
     main = paste('Top responses (', as.numeric(topprop),
                  ') m = ', as.character(signif(mean(toprt), 4)),
                  sep = ''), las = 1)
botrt <- latencies[responses < 0]
botprop <- length(botrt)/nreps
hist(botrt, col = 'gray',
     xlab = 'Decision time', xlim =c(0, max(latencies)),
     main = paste('Bottom responses
          (', as.numeric(botprop),
          ') m =', as.character(signif(mean(botrt), 4))))