# WienerPDF: First-passage time probability density function of the... In WienR: Derivatives of the First-Passage Time Density and Cumulative Distribution Function, and Random Sampling from the (Truncated) First-Passage Time Distribution

 WienerPDF R Documentation

## First-passage time probability density function of the diffusion model

### Description

Calculate the first-passage time probability density function of the diffusion model.

### Usage

```WienerPDF(
t,
response,
a,
v,
w,
t0 = 0,
sv = 0,
sw = 0,
st0 = 0,
precision = NULL,
K = NULL,
n.evals = 6000
)
```

### Arguments

 `t` First-passage time. Numeric vector. `response` Response boundary. Character vector with `"upper"` and `"lower"` as possible values. Alternatively a numeric vector with `1`=lower and `2`=upper. `a` Upper barrier. Numeric vector. `v` Drift rate. Numeric vector. `w` Relative starting point. Numeric vector. `t0` Non-decision time. Numeric vector `sv` Inter-trial variability of drift rate. Numeric vector. Standard deviation of a normal distribution `N(v, sv)`. `sw` Inter-trial variability of relative starting point. Numeric vector. Range of uniform distribution `U(w-0.5*sw, w+0.5*sw)`. `st0` Inter-trial variability of non-decision time. Numeric vector. Range of uniform distribution `U(t0, t0+st0)`. `precision` Optional numeric value. Precision of the PDF. Numeric value. Default is `NULL`, which takes default value 1e-12. `K` Optional. Number of iterations to calculate the infinite sums. Numeric value (integer). Default is `NULL`. `precision = NULL` and `K = NULL`: Default `precision = 1e-12` used to calculate internal K. `precision != NULL` and `K = NULL`: `precision` is used to calculate internal K, `precision = NULL` and `K != NULL`: `K` is used as internal K, `precision != NULL` and `K != NULL`: if internal K calculated through `precision` is smaller than `K`, `K` is used. We recommend using either default (`precision = K = NULL`) or only `precision`. `n.threads` Optional numerical or logical value. Number of threads to use. If not provided (or 1 or `FALSE`) parallelization is not used. If set to `TRUE` then all available threads are used. `n.evals` Optional. Number of maximal function evaluations in the numeric integral if sv, sw, and/or st0 are not zero. Default is `6000` and `0` implies no limit and the numeric integration goes on until the specified `precision` is guaranteed.

### Value

A list of the class `Diffusion_pdf` containing

• `pdf`: the PDF,

• `logpdf`: the log-transformed PDF,

• `call`: the function call,

• `err`: the absolute error. Only provided if sv, sw, or st0 is non-zero. If numerical integration is used, the precision cannot always be guaranteed.

Raphael Hartmann

### References

Blurton, S. P., Kesselmeier, M., & Gondan, M. (2017). The first-passage time distribution for the diffusion model with variable drift. Journal of Mathematical Psychology, 76, 7–12. doi: 10.1016/j.jmp.2016.11.003

Foster, K., & Singmann, H. (2021). Another Approximation of the First-Passage Time Densities for the Ratcliff Diffusion Decision Model. arXiv preprint arXiv:2104.01902.

Gondan, M., Blurton, S. P., & Kesselmeier, M. (2014). Even faster and even more accurate first-passage time densities and distributions for the Wiener diffusion model. Journal of Mathematical Psychology, 60, 20–22. doi: 10.1016/j.jmp.2014.05.002

Hartmann, R., & Klauer, K. C. (2021). Partial derivatives for the first-passage time distribution in Wiener diffusion models. Journal of Mathematical Psychology, 103, 102550. doi: 10.1016/j.jmp.2021.102550

Navarro, D. J., & Fuss, I. G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53(4), 222–230. doi: 10.1016/j.jmp.2009.02.003

Wabersich, D., & Vandekerckhove, J. (2014). The RWiener Package: an R Package Providing Distribution Functions for the Wiener Diffusion Model. The R Journal, 6(1), 49. doi: 10.32614/rj-2014-005

### Examples

```WienerPDF(t = 1.2, response = "upper", a = 1.1, v = 13, w = .6, precision = NULL, K = NULL)
```

WienR documentation built on April 23, 2022, 9:05 a.m.