Distribution functions for the three-parameter mixture model (mixture3p)

Density, distribution, and random generation functions for the three-parameter mixture model with the location of mu, precision of memory representations kappa, probability of a target response p_mem, and probability of a non-target response p_nt.

Usage

dmixture3p(
  x,
  mu = c(0, 2, -1.5),
  kappa = 5,
  p_mem = 0.6,
  p_nt = 0.2,
  tau = 0,
  vp_nodes = 81L,
  log = FALSE
)

pmixture3p(q, mu = c(0, 2, -1.5), kappa = 5, p_mem = 0.6, p_nt = 0.2)

qmixture3p(p, mu = c(0, 2, -1.5), kappa = 5, p_mem = 0.6, p_nt = 0.2)

rmixture3p(n, mu = c(0, 2, -1.5), kappa = 5, p_mem = 0.6, p_nt = 0.2, tau = 0)

Arguments

x: Vector of observed responses
mu: Vector of locations. First value represents the location of the target item and any additional values indicate the location of non-target items.
kappa: Vector of precision values. With tau > 0 this is the concentration of the average trial, whose Fisher information J(kappa) is the mean of J.
p_mem: Vector of probabilities of a target response. For a mixture3p() fit these are thetat on its native scale (version "simple", see native_parameters()) or min(1, K / set_size) * ptarget (versions "slot" and "slot_averaging").
p_nt: Vector of probabilities of a non-target response, summed over the non-targets, which share it equally. For a mixture3p() fit these are thetant on its native scale (version "simple") or min(1, K / set_size) * (1 - ptarget) (capacity versions). Unlike ptarget, p_mem and p_nt are not conditional on a response from memory.
tau: Vector of scales for trial-to-trial variability in precision. The Fisher information J of a memory representation is drawn from gamma(mean = J(kappa), scale = tau). tau = 0, the default, is constant precision. See mixture3p().
vp_nodes: Number of quadrature nodes used when tau > 0; must be an odd number of at least 41. See mixture2p(). The random generation function needs no nodes: it draws J from its gamma distribution and then the response, which is exact.
log: Logical; if TRUE, values are returned on the log scale. Pass it by name: in bmm 1.4 it was the sixth argument, which is now tau. Until bmm 1.6.0 a logical value in that position is still read as log, with a warning.
q: Vector of quantiles
p: Vector of probability
n: Number of observations to generate data for

Value

dmixture3p gives the density of the three-parameter mixture model, pmixture3p gives the cumulative distribution function of the three-parameter mixture model, qmixture3p gives the quantile function of the three-parameter mixture model, and rmixture3p gives the random generation function for the three-parameter mixture model.

References

Bays, P. M., Catalao, R. F. G., & Husain, M. (2009). The precision of visual working memory is set by allocation of a shared resource. Journal of Vision, 9(10), Article 7.

van den Berg, R., Shin, H., Chou, W.-C., George, R., & Ma, W. J. (2012). Variability in encoding precision accounts for visual short-term memory limitations. PNAS, 109(22), 8780-8785.

Examples

# generate random samples from the mixture3p model and overlay the density
r <- rmixture3p(10000, mu = c(0, 2, -1.5), kappa = 4, p_mem = 0.6, p_nt = 0.2)
x <- seq(-pi, pi, length.out = 10000)
d <- dmixture3p(x, mu = c(0, 2, -1.5), kappa = 4, p_mem = 0.6, p_nt = 0.2)
hist(r, breaks = 60, freq = FALSE)
lines(x, d, type = "l", col = "red")