New version of PLNoise: a package for exact numerical simulation of power-law noises

In a recent paper I have introduced a package for the exact simulation of power-law noises and other colored noises (E. Milotti, Comput. Phys. Commun. {\bf 175} (2006) 212): in particular the algorithm generates $1/f^\alpha$ noises with $0 < \alpha \leq 2$. Here I extend the algorithm to generate $1/f^\alpha$ noises with $2 < \alpha \leq 4$ (black noises). The method is exact in the sense that it produces a sampled process with a theoretically guaranteed range-limited power-law spectrum for any arbitrary sequence of sampling intervals, i.e., the sampling times may be unevenly spaced.Comment: 3 figures, submitted to Computer Physics Communication

Self-similar prior and wavelet bases for hidden incompressible turbulent motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201

Levy process simulation by stochastic step functions

We study a Monte Carlo algorithm for simulation of probability distributions based on stochastic step functions, and compare to the traditional Metropolis/Hastings method. Unlike the latter, the step function algorithm can produce an uncorrelated Markov chain. We apply this method to the simulation of Levy processes, for which simulation of uncorrelated jumps are essential. We perform numerical tests consisting of simulation from probability distributions, as well as simulation of Levy process paths. The Levy processes include a jump-diffusion with a Gaussian Levy measure, as well as jump-diffusion approximations of the infinite activity NIG and CGMY processes. To increase efficiency of the step function method, and to decrease correlations in the Metropolis/Hastings method, we introduce adaptive hybrid algorithms which employ uncorrelated draws from an adaptive discrete distribution defined on a space of subdivisions of the Levy measure space. The nonzero correlations in Metropolis/Hastings simulations result in heavy tails for the Levy process distribution at any fixed time. This problem is eliminated in the step function approach. In each case of the Gaussian, NIG and CGMY processes, we compare the distribution at t=1 with exact results and note the superiority of the step function approach.Comment: 20 pages, 18 figure

Subsampling MCMC - An introduction for the survey statistician

The rapid development of computing power and efficient Markov Chain Monte Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics, making it a highly practical inference method in applied work. However, MCMC algorithms tend to be computationally demanding, and are particularly slow for large datasets. Data subsampling has recently been suggested as a way to make MCMC methods scalable on massively large data, utilizing efficient sampling schemes and estimators from the survey sampling literature. These developments tend to be unknown by many survey statisticians who traditionally work with non-Bayesian methods, and rarely use MCMC. Our article explains the idea of data subsampling in MCMC by reviewing one strand of work, Subsampling MCMC, a so called pseudo-marginal MCMC approach to speeding up MCMC through data subsampling. The review is written for a survey statistician without previous knowledge of MCMC methods since our aim is to motivate survey sampling experts to contribute to the growing Subsampling MCMC literature.Comment: Accepted for publication in Sankhya A. Previous uploaded version contained a bug in generating the figures and reference

SKIRT: the design of a suite of input models for Monte Carlo radiative transfer simulations

The Monte Carlo method is the most popular technique to perform radiative transfer simulations in a general 3D geometry. The algorithms behind and acceleration techniques for Monte Carlo radiative transfer are discussed extensively in the literature, and many different Monte Carlo codes are publicly available. On the contrary, the design of a suite of components that can be used for the distribution of sources and sinks in radiative transfer codes has received very little attention. The availability of such models, with different degrees of complexity, has many benefits. For example, they can serve as toy models to test new physical ingredients, or as parameterised models for inverse radiative transfer fitting. For 3D Monte Carlo codes, this requires algorithms to efficiently generate random positions from 3D density distributions. We describe the design of a flexible suite of components for the Monte Carlo radiative transfer code SKIRT. The design is based on a combination of basic building blocks (which can be either analytical toy models or numerical models defined on grids or a set of particles) and the extensive use of decorators that combine and alter these building blocks to more complex structures. For a number of decorators, e.g. those that add spiral structure or clumpiness, we provide a detailed description of the algorithms that can be used to generate random positions. Advantages of this decorator-based design include code transparency, the avoidance of code duplication, and an increase in code maintainability. Moreover, since decorators can be chained without problems, very complex models can easily be constructed out of simple building blocks. Finally, based on a number of test simulations, we demonstrate that our design using customised random position generators is superior to a simpler design based on a generic black-box random position generator.Comment: 15 pages, 4 figures, accepted for publication in Astronomy and Computin