Archive for Non-Uniform Random Variate Generation

simulating hazard

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , , , , , , on May 26, 2020 by xi'an

A rather straightforward X validated question that however leads to an interesting simulation question: when given the hazard function h(·), rather than the probability density f(·), how does one simulate this distribution? Mathematically h(·) identifies the probability distribution as much as f(·),

1-F(x)=\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}=\exp\{H(x)\}

which means cdf inversion could be implemented in principle. But in practice, assuming the integral is intractable, what would an exact solution look like? Including MCMC versions exploiting one fixed point representation or the other.. Since

f(x)=h(x)\,\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}

using an unbiased estimator of the exponential term in a pseudo-marginal algorithm would work. And getting an unbiased estimator of the exponential term can be done by Glynn & Rhee debiasing. But this is rather costly… Having Devroye’s book under my nose [at my home desk] should however have driven me earlier to the obvious solution to… simply open it!!! A whole section (VI.2) is indeed dedicated to simulations when the distribution is given by the hazard rate. (Which made me realise this problem is related with PDMPs in that thinning and composition tricks are common to both.) Besides the inversion method, ie X=H⁻¹(U), Devroye suggests thinning a Poisson process when h(·) is bounded by a manageable g(·). Or a generic dynamic thinning approach that converges when h(·) is non-increasing.

non-uniform Laplace generation

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , on June 5, 2019 by xi'an

This year, the French Statistical Society (SFDS) Prix Laplace has been granted to Luc Devroye, author of the Non-Uniform Random Generation bible. among many achievements!, prize that he will receive during the 2019 meeting in Nancy, this very week.

ziggurat algorithm

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on October 30, 2018 by xi'an

A ziggurat (Akkadian: ziqqurat, D-stem of zaqāru “to build on a raised area”) is a type of massive stone structure built in ancient Mesopotamia. It has the form of a terraced compound of successively receding stories or levels. Wikipedia

In a recent arXival, Jalalvand and Charsooghi revisit the ziggurat algorithm that simulates from a univariate distribution by finding horizontal strips that pile up on top of the target as in a ziggurat or a pyramid, hence the name. Which George Marsaglia introduced in 1963. When finely tuned the method is quite efficient. Maybe because it designs an accept-reject move for each strip of the ziggurat rather than globally. For instance, versions constructed for a Normal target are more efficient [3½ times faster] than the Box-Muller algorithm. The generalisation found in the paper divides the target into strips of equal area, rather than dominating rectangular strips of equal area, which requires some work when the target density is non-standard. For targets with unbounded support or unbounded values, a function g transforming the tail into (0,1) has to be selected. A further constraint is that the inverse cdf of the transformed g(X) has to be known. And a large part of the paper examines several scenarii towards simulating from the tail region. For unbounded densities, a similarly minute analysis is undertaken, again with requests about the target like its algebraic order.

“…the result of division of a random integer by its range is a fixed-point number which unlike a floating-point number does not enjoy increased precision near 0. When such random numbers are used in the tail algorithm they cause premature termination of the tail and large gaps between produced random numbers near the termination point.”

The paper further discusses the correction of an error common to earlier ziggurat algorithms, due to the  conversion from fixed-point to floating-point numbers, as indicated in the above quote. Although this had already been addressed by George Marsaglia in the early 1990’s.

“Ziggurat algorithm has a high setup time, so it’s not suitable for applications that require variates with frequently changing shape parameters.”

When testing the algorithm against different methods (in STL and Boost), and different distributions, the gains are between two and seven times faster, except for the Exponential target where the original ziggurat algorithm performs better. Interestingly, the gains (and the computing time) increase with the degrees of freedom for the Gamma target, in relation with Devroye’s (1986) remark on the absence of uniformly bounded execution times for this distribution. Same thing for the Weibull variates, obviously. Reflecting upon the usually costly computation of cdfs and inverse cdfs on machines and software, the inverse cdf method is systematically left behind! In conclusion, a good Sunday morning read if not of direct consequences for MCMC implementation, as warned by the authors.

 

independent random sampling methods [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on May 16, 2018 by xi'an

Last week, I had the pleasant surprise to receive a copy of this book in the mail. Book that I was not aware had been written or published (meaning that I was not involved in its review!). The three authors, Luca Martino, David Luengo, and Joaquín Míguez, of Independent Random Sampling Methods are from Madrid universities and I have read (and posted on) several of their papers on (population) Monte Carlo simulation in the recent years. Including Luca’s survey of multiple try MCMC which was helpful in writing our WIREs own survey.

The book is a pedagogical coverage of most algorithms used to simulate independent samples from a given distribution, which of course recoups some of the techniques exposed with more details by [another] Luc, namely Luc Devroye’s Non-uniform random variate generation bible, often mentioned here (and studied in uttermost details by a dedicated reading group in Warwick). It includes a whole chapter on accept-reject methods, with in particular a section on Payne-Dagpunar’s band rejection I had not seen previously. And another entire chapter on ratio-of-uniforms techniques. On which the three authors had proposed generalisations [covered by the book], years before I attempted to go the same way, having completely forgotten reading their paper at the time… Or the much earlier 1991 paper by Jon Wakefield, Alan Gelfand and Adrian Smith!

The book also covers the “vertical density representation”, due to Troutt (1991), which consists in considering the distribution of the density p(.) of the random variable X as a random variable, p(X). I remember pondering about this alternative to the cdf transform and giving up on it as the outcome has a distribution depending on p, even when the density is monotonous. Even though I am not certain from reading the section that this is particularly appealing…

Given its title, the book contains very little about MCMC. Except for a last and final chapter that covers adaptive independent Metropolis-Hastings algorithms, in connection with some of the authors’ recent work. Like multiple try Metropolis. Relating to the (unidimensional) ARMS “ancestor” of adaptive MCMC methods. (As noted in a recent blog on Holden et al., 2009 , I have trouble understanding how recycling only rejected proposed values to build a better proposal distribution is enough to guarantee convergence of an adaptive algorithm, but the book does not delve much into this convergence.)

All in all and with the bias induced by me working in the very area, I find the book quite a nice entry on the topic, which can be used in a Monte Carlo course at both undergraduate and graduate levels if one want to avoid going into Markov chains. It is certainly less likely to scare students away than the comprehensive Non-uniform random variate generation and on the opposite may induce some of them to pursue a research career in this domain.

ratio-of-uniforms [#4]

Posted in Books, pictures, R, Statistics, University life with tags , , , , on December 2, 2016 by xi'an

Possibly the last post on random number generation by Kinderman and Monahan’s (1977) ratio-of-uniform method. After fiddling with the Gamma(a,1) distribution when a<1 for a while, I indeed figured out a way to produce a bounded set with this method: considering an arbitrary cdf Φ with corresponding pdf φ, the uniform distribution on the set Λ of (u,v)’s in R⁺xX such that

0≤u≤Φοƒ[φοΦ⁻¹(u)v]

induces the distribution with density proportional to ƒ on φοΦ⁻¹(U)V. This set Λ has a boundary that is parameterised as

u=Φοƒ(x),  v=1/φοƒ(x), x∈Χ

which remains bounded in u since Φ is a cdf and in v if φ has fat enough tails. At both 0 and ∞. When ƒ is the Gamma(a,1) density this can be achieved if φ behaves like log(x)² near zero and like a inverse power at infinity. Without getting into all the gory details, closed form density φ and cdf Φ can be constructed for all a’s, as shown for a=½ by the boundaries in u and v (yellow) below

bundawhich leads to a bounded associated set Λ

ratgamp5At this stage, I remain uncertain of the relevance of such derivations, if only because the set A thus derived is ill-suited for uniform draws proposed on the enclosing square box. And also because a Gamma(a,1) simulation can rather simply be derived from a Gamma(a+1,1) simulation. But, who knows?!, there may be alternative usages of this representation, such as innovative slice samplers. Which means the ratio-of-uniform method may reappear on the ‘Og one of those days…