WTM: Weibull Threshold Model

In ream: Density, Distribution, and Sampling Functions for Evidence Accumulation Models

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)

Arguments

rt	vector of response times
resp	vector of responses ("upper" and "lower")
phi	parameter vector in the following order: 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. 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). Stimulus strength (\mu). Strength of the stimulus and used to set the drift rate. For changing threshold models v(x,t) = \mu. Noise scale (\sigma). Model noise scale parameter. Initial decision threshold location (b_0). Sets the location of each decision threshold at time t = 0. Log10-decision threshold scale (log_{10}(\lambda)). Sets the approximate time for threshold collapse or rise. Log10-decision threshold shape (log_{10}(\kappa)). Sets the threshold shape. \kappa > 1 produces logistic-like thresholds, \kappa < 1 produces exponential-like thresholds. Collapse parameter (c). Sets the amount of collapse. c = -1 gives collapse to zero, c = 1 gives no collapse, and c > 1 gives rise. 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). Lower bound of contamination distribution (g_l). See parameter g. 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)

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).

Author(s)

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)

ream documentation built on Oct. 7, 2024, 1:20 a.m.