# nhppp

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.

## Installation

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

install.packages("nhppp")

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

# install.packages("devtools")
devtools::install_github("bladder-ca/nhppp-fast")

## Example

These examples use the generic function draw(), which is a wrapper for the packages specific functions.

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)) |>
#>  [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)) |>
#>  [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

## Function naming conventions

1. 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.

1. 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.