nhppp is a package for simulating events from one dimensional nonhomogeneous Poisson point processes (NHPPPs). Its functions are based on three algorithms that provably sample from a target NHPPP: the time-transformation of a homogeneous Poisson process (of intensity one) via the inverse of the integrated intensity function; the generation of a Poisson number of order statistics from a fixed density function; and the thinning of a majorizing NHPPP via an acceptance-rejection scheme. It was developed to provide fast and memory efficient functions for discrete event and statistical simulations. For a description of the algorithms and a numerical comparison with other R packages, see Trikalinos and Sereda (2024), accessible at https://arxiv.org/abs/2402.00358.

You can install the release version of nhppp from CRAN with:

You can install the development version of nhppp from GitHub with:

These examples use the generic function `draw()`

, which is a wrapper for the packages specific functions. `draw()`

is a non-vectorized function, but `nhppp`

includes vectorized functions that are fast and have small memory footprint.

Consider the time varying intensity function \(\lambda(t) = e^{(0.2t)} (1 + \sin t)\), which is a sinusoidal intensity function with an exponential amplitude. To draw samples over the interval \((0, 6\pi]\) execute

```
l <- function(t) (1 + sin(t)) * exp(0.2 * t)
nhppp::draw(lambda = l, lambda_maj = l(6 * pi), range_t = c(0, 6 * pi)) |>
head(n = 20)
#> [1] 1.197587 1.238620 1.497499 1.713629 1.761914 2.256739 2.537528 3.622938
#> [9] 5.822574 6.064265 6.645696 6.651551 6.684603 6.875765 6.891348 7.130680
#> [17] 7.446557 7.453139 7.545474 7.557381
```

where `lambda_maj`

is a majorizer constant.

When available, the integrated intensity function \(\Lambda(t) = \int_0^t \lambda(s) \ ds\) and its inverse \(\Lambda^{-1}(z)\) result in faster simulation times. For this example, \(\Lambda(t) = \frac{e^{0.2t}(0.2 \sin t - \cos t)+1}{1.04} + \frac{e^{0.2t} - 1}{0.2}\); \(\Lambda^{-1}(z)\) is constructed numerically upfront (or can be calculated numerically by the function, at a computational cost).

```
L <- function(t) {
exp(0.2 * t) * (0.2 * sin(t) - cos(t)) / 1.04 +
exp(0.2 * t) / 0.2 - 4.038462
}
Li <- stats::approxfun(x = L(seq(0, 6 * pi, 10^-3)), y = seq(0, 6 * pi, 10^-3), rule = 2)
nhppp::draw(Lambda = L, Lambda_inv = Li, range_t = c(0, 6 * pi)) |>
head(n = 20)
#> [1] 0.01152846 0.23558627 0.32924742 0.49921843 0.63509297 1.36677413
#> [7] 2.38941548 3.19511655 3.28049866 4.62140995 5.96916564 6.37504015
#> [13] 6.68283108 6.76577784 7.12919141 7.29249262 7.38665270 7.92953383
#> [19] 7.94791744 7.96591106
```

- All functions whose name start with
`ppp`

or`ztppp`

sample from constant or piecewise constant intensity functions, as described below:

Functions whose names start with

`ppp_[sequential|orderstats]`

sample event times in an interval with constant intensity functions with the sequential and order statistics algorithms.Function

`ztppp()`

samples one or more event times in an interval with constant intensity, i.e., from a zero-truncated Poisson process.Functions

`ppp_n()`

and`ppp_next_n()`

sample`n`

events in an interval and the next`n`

event times after a time`t0`

.

- All functions whose name starts with
`draw`

or`vdraw`

sample from NHPPPs.

Functions with names starting with

`draw_zt`

sample at least one event in the interval, i.e., from a zero-truncated NHPPP.Functions with names starting with

`[draw|draw_zt]_intensity[_majorizer]`

expect an intensity argument. The third part (`[_majorizer]`

) denotes what, if any, majorizer function is used.Functions with names starting with

`[draw|draw_zt]_cumulative_intensity[_algorithm]`

expect a cumulative (integrated) intensity argument. The third part (`[_algorithm]`

) denotes the algorithm used, if more than one algorithms are pertinent.Functions with names starting with

`vdraw`

are vectorized.