ream (version 1.0-5)

WTM: Weibull Threshold Model

Description

SDDM with thresholds that change with time. Thresholds are Weibull functions of the form \(b_u(t) = -b_l(t) = b_0 - b_0*(1 – c)*[1 - exp(-(t/\lambda)^{\kappa})].\)

Usage

dWTM(rt, resp, phi, x_res = "default", t_res = "default")
pWTM(rt, resp, phi, x_res = "default", t_res = "default")
rWTM(n, phi, dt = 1e-05)

Value

For the density a list of PDF values, log-PDF values, and the sum of the log-PDFs, for the distribution function a list of of CDF values, log-CDF values, and the sum of the log-CDFs, and for the random sampler a list of response times (rt) and response thresholds (resp).

Arguments

rt

vector of response times

resp

vector of responses ("upper" and "lower")

phi

parameter vector in the following order:

1. Non-decision time (\(t_{nd}\)). Time for non-decision processes such as stimulus encoding and response execution. Total decision time t is the sum of the decision and non-decision times.

2. Relative start (\(w\)). Sets the start point of accumulation as a ratio of the two decision thresholds. Related to the absolute start z point via equation \(z = b_l + w*(b_u - b_l)\).

3. Stimulus strength (\(\mu\)). Strength of the stimulus and used to set the drift rate. For changing threshold models \(v(x,t) = \mu\).

4. Noise scale (\(\sigma\)). Model noise scale parameter.

5. Initial decision threshold location (\(b_0\)). Sets the location of each decision threshold at time \(t = 0\).

6. Log10-decision threshold scale (\(log_{10}(\lambda)\)). Sets the approximate time for threshold collapse or rise.

7. Log10-decision threshold shape (\(log_{10}(\kappa)\)). Sets the threshold shape. \(\kappa > 1\) produces logistic-like thresholds, \(\kappa < 1\) produces exponential-like thresholds.

8. Collapse parameter (\(c\)). Sets the amount of collapse. \(c = -1\) gives collapse to zero, \(c = 1\) gives no collapse, and \(c > 1\) gives rise.

9. Contamination (\(g\)). Sets the strength of the contamination process. Contamination process is a uniform distribution \(f_c(t)\) where \(f_c(t) = 1/(g_u-g_l)\) if \(g_l <= t <= g_u\) and \(f_c(t) = 0\) if \(t < g_l\) or \(t > g_u\). It is combined with PDF \(f_i(t)\) to give the final combined distribution \(f_{i,c}(t) = g*f_c(t) + (1-g)*f_i(t)\), which is then output by the program. If \(g = 0\), it just outputs \(f_i(t)\).

10. Lower bound of contamination distribution (\(g_l\)). See parameter \(g\).

11. Upper bound of contamination distribution (\(g_u\)). See parameter \(g\).

x_res

spatial/evidence resolution

t_res

time resolution

n

number of samples

dt

step size of time. We recommend 0.00001 (1e-5)

Author

Raphael Hartmann & Matthew Murrow

References

Hawkins, G. E., Forstmann, B. U., Wagenmakers, E.-J., Ratcliff, R., & Brown, S. D. (2015). Revisiting the Evidence for Collapsing Boundaries and Urgency Signals in Perceptual Decision-Making. The Journal of Neuroscience, 35(6), 2476-2484.

Palestro, J. J., Weichart, E., Sederberg, P. B., & Turner, B. M. (2018). Some task demands induce collapsing bounds: Evidence from a behavioral analysis. Psychonomic Bulletin & Review, 25(4), 1225-1248.

Examples

# Probability density function
dWTM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
     phi = c(0.3, 0.5, 1.0, 1.0, 1.5, 0.2, 0.5, -1.0, 0.0, 0.0, 1.0))

# Cumulative distribution function
pWTM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
     phi = c(0.3, 0.5, 1.0, 1.0, 1.5, 0.2, 0.5, -1.0, 0.0, 0.0, 1.0))

# Random sampling
rWTM(n = 100, phi = c(0.3, 0.5, 1.0, 1.0, 1.5, 0.2, 0.5, -1.0, 0.0, 0.0, 1.0),
     dt = 0.0001)