# Tinflex -- Universal non-uniform random number generator `Tinflex` is a universal non-uniform random number generator based on the acceptence-rejection method for all distributions that have a piecewise twice differentiable density function. Required input includes the log-density function of the target distribution and its first and second derivatives. UTF-8 package ## Details ||| |:--|:--| |Package:|Tinflex| |Type:|Package| |Version:|2.4| |Date:|2023-03-21| |License:|GPL 2 or later| Package `Tinflex` serves three purposes: 1. The installed package provides a fast routine for sampling from any distribution that has a piecewise twice differentiable density function. 2. It provides C routines functions that could be used in other packages (see the installed C header files). 3. The source (including comments) presents all details of the general sampling method which are not entirely worked out in our paper cited in the see references below. Algorithm `Tinflex`is a universal random variate generator based on transformed density rejection which is a variant of the acceptance-rejection method. The generator first computes and stores hat and squeeze functions and then uses these functions to generate variates from the distribution of interest. Since the setup procedure is separated from the generation procedure, many samples can be drawn from the same distribution without rerunning the (expensive) setup. The algorithm requires the following data about the distribution (for further details see `Tinflex.setup`): * the log-density of the targent distribution; * its first derivative; * its second derivative (optionally); * a starting partition of its domain such that each subinterval contains at most one inflection point of the transformed density; * a transformation for the density (default is the logarithm transformation). The following routines are provided. - **`Tinflex.setup`**: computes hat and squeeze. The table is then stored in a generator object of class `"Tinflex"`. - **`Tinflex.sample`**: draws a random sample from a particular generator object. - **`print.Tinflex`**: prints the properties a generator object of class `"Tinflex"`. - **`plot.Tinflex`**: plots density, hat and squeeze functions for a given generator object of class `"Tinflex"`. For further details see `Tinflex.setup`. There are variants of the method. The first one uses the second derivative to determine regions whre the transformed density is convex, concave, or has a single inflection points. The second variant estimates the signs on the second derivative by means of the first derivative. Thus it is easier to use at the expense of a more complex algorithm. There are two different implementation: Routine `Tinflex.setup` is implemented mainly in and serves (together with `Tinflex:::Tinflex.sample.R`) as a reference implementation of the published algorithm. Nevertheless, the sampling routine `Tinflex.sample` runs quite fast. Routine `Tinflex.setup.C` on the other hand is implemented entirely in C. So it also allows to link to the underlying C code from other packages. ## Warning It is very important to note that the user is responsible for the correctness of the supplied arguments. Since the algorithm works (in theory) for all distributions with piecewise twice differentiable density functions, it is not possible to detect improper arguments. It is thus recommended that the user inspect the generator object visually by means of the `plot` method (see `plot.Tinflex` for details). ## Note Routine `Tinflex.sample` is implemented both as pure code (routine `Tinflex.sample.R`) for documenting the algorithm as well as C code for fast performance. ## Author(s) Josef Leydold josef.leydold@wu.ac.at , Carsten Botts and Wolfgang . ## References C. Botts, W. , and J. Leydold (2013), Transformed Density Rejection with Inflection Points, Statistics and Computing 23(2), 251--260, tools:::Rd_expr_doi("10.1007/s11222-011-9306-4") . See also Research Report Series / Department of Statistics and Mathematics Nr. 110, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, [https://epub.wu.ac.at/id/eprint/3158](https://epub.wu.ac.at/id/eprint/3158). W. , and J. Leydold (2022), A Generalized Transformed Density Rejection Algorithm, in: Advances in Modeling and Simulation, Ch. 14, tools:::Rd_expr_doi("10.1007/978-3-031-10193-9_14") , accepted for publication.. See also Research Report Series / Department of Statistics and Mathematics Nr. 135, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, [https://research.wu.ac.at/de/publications/a-generalized-transformed-density-rejection-algorithm](https://research.wu.ac.at/de/publications/a-generalized-transformed-density-rejection-algorithm). ## See Also See `Tinflex.setup` for further details. Package `Runuran` provides a set of many other automatic non-uniform sampling algorithms. ## Examples ```r ## Bimodal density ## f(x) = exp( -|x|^alpha + s*|x|^beta + eps*|x|^2 ) ## with alpha > beta >= 2 and s, eps > 0 alpha <- 4.2 beta <- 2.1 s <- 1 eps <- 0.1 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha + s*abs(x)^beta + eps*abs(x)^2 } dlpdf <- function(x) { (sign(x) * (-alpha*abs(x)^(alpha-1) + s*beta*abs(x)^(beta-1) + 2*eps*abs(x))) } d2lpdf <- function(x) { (-alpha*(alpha-1)*abs(x)^(alpha-2) + s*beta*(beta-1)*abs(x)^(beta-2) + 2*eps) } ## Parameter cT=0 (default): ## There are two inflection points on either side of 0. ib <- c(-Inf, 0, Inf) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=c(-Inf,0,Inf), rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-2.5, to=2.5) ## ... and in transformed (log) scale. plot(gen, from=-2.5, to=2.5, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=c(-Inf,0,Inf), rho=1.1) Tinflex.sample(gen, n=10) ```

Tinflex-package