Adaptive approximate Bayesian computation

Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappe et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm.

Inferring population history with DIY ABC: a user-friendly approach to approximate Bayesian computation

Genetic data obtained on population samples convey information about their evolutionary history. Inference methods can extract part of this information but they require sophisticated statistical techniques that have been made available to the biologist community (through computer programs) only for simple and standard situations typically involving a small number of samples. We propose here a computer program (DIY ABC) for inference based on approximate Bayesian computation (ABC), in which scenarios can be customized by the user to fit many complex situations involving any number of populations and samples. Such scenarios involve any combination of population divergences, admixtures and population size changes. DIY ABC can be used to compare competing scenarios, estimate parameters for one or more scenarios and compute bias and precision measures for a given scenario and known values of parameters (the current version applies to unlinked microsatellite data). This article describes key methods used in the program and provides its main features. The analysis of one simulated and one real dataset, both with complex evolutionary scenarios, illustrates the main possibilities of DIY ABC.

An approximate Bayesian computation approach to overcome biases that arise when using amplified fragment length polymorphism markers to study population structure

There is great interest in using amplified fragment length polymorphism (AFLP) markers because they are inexpensive and easy to produce. It is, therefore, possible to generate a large number of markers that have a wide coverage of species genotnes. Several statistical methods have been proposed to study the genetic structure using AFLP's but they assume Hardy-Weinberg equilibrium and do not estimate the inbreeding coefficient, F-IS. A Bayesian method has been proposed by Holsinger and colleagues that relaxes these simplifying assumptions but we have identified two sources of bias that can influence estimates based on these markers: (i) the use of a uniform prior on ancestral allele frequencies and (ii) the ascertainment bias of AFLP markers. We present a new Bayesian method that avoids these biases by using an implementation based on the approximate Bayesian computation (ABC) algorithm. This new method estimates population-specific F-IS and F-ST values and offers users the possibility of taking into account the criteria for selecting the markers that are used in the analyses. The software is available at our web site (http://www-leca.uif-grenoble.fi-/logiciels.htm). Finally, we provide advice on how to avoid the effects of ascertainment bias.

ONeSAMP: a program to estimate effective population size using approximate Bayesian computation

The estimation of effective population size from one sample of genotypes has been problematic because most estimators have been proven imprecise or biased. We developed a web-based program, ONeSAMP that uses approximate Bayesian computation to estimate effective population size from a sample of microsatellite genotypes. ONeSAMP requires an input file of sampled individuals' microsatellite genotypes along with information about several sampling and biological parameters. ONeSAMP provides an estimate of effective population size, along with 95% credible limits. We illustrate the use of ONeSAMP with an example data set from a re-introduced population of ibex Capra ibex.

Comparative evaluation of a new effective population size estimator based on approximate Bayesian computation

We describe and evaluate a new estimator of the effective population size (N-e), a critical parameter in evolutionary and conservation biology. This new "SummStat" N-e. estimator is based upon the use of summary statistics in an approximate Bayesian computation framework to infer N-e. Simulations of a Wright-Fisher population with known N-e show that the SummStat estimator is useful across a realistic range of individuals and loci sampled, generations between samples, and N-e values. We also address the paucity of information about the relative performance of N-e estimators by comparing the SUMMStat estimator to two recently developed likelihood-based estimators and a traditional moment-based estimator. The SummStat estimator is the least biased of the four estimators compared. In 32 of 36 parameter combinations investigated rising initial allele frequencies drawn from a Dirichlet distribution, it has the lowest bias. The relative mean square error (RMSE) of the SummStat estimator was generally intermediate to the others. All of the estimators had RMSE > 1 when small samples (n = 20, five loci) were collected a generation apart. In contrast, when samples were separated by three or more generations and Ne less than or equal to 50, the SummStat and likelihood-based estimators all had greatly reduced RMSE. Under the conditions simulated, SummStat confidence intervals were more conservative than the likelihood-based estimators and more likely to include true N-e. The greatest strength of the SummStat estimator is its flexible structure. This flexibility allows it to incorporate any, potentially informative summary statistic from Population genetic data.

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

Parallel pipelined array architectures for real-time histogram computation in consumer devices

The real-time parallel computation of histograms using an array of pipelined cells is proposed and prototyped in this paper with application to consumer imaging products. The array operates in two modes: histogram computation and histogram reading. The proposed parallel computation method does not use any memory blocks. The resulting histogram bins can be stored into an external memory block in a pipelined fashion for subsequent reading or streaming of the results. The array of cells can be tuned to accommodate the required data path width in a VLSI image processing engine as present in many imaging consumer devices. Synthesis of the architectures presented in this paper in FPGA are shown to compute the real-time histogram of images streamed at over 36 megapixels at 30 frames/s by processing in parallel 1, 2 or 4 pixels per clock cycle.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.