Adaptive Monte Research Articles

Monte Carlo algorithms for computing $$\alpha $$ α -permanents

We consider the computation of the $$\alpha $$?-permanent of a non-negative $$n \times n$$n×n matrix. This appears in a wide variety of real applications in statistics, physics and computer-science. It is well-known that the exact computation is a #P complete problem. This has resulted in a large collection of simulation-based methods, to produce randomized solutions whose complexity is only polynomial in $$n$$n. This paper will review and develop algorithms for both the computation of the permanent $$\alpha =1$$?=1 and $$\alpha >0$$?>0 permanent. In the context of binary $$n \times n$$n×n matrices a variety of Markov chain Monte Carlo (MCMC) computational algorithms have been introduced in the literature whose cost, in order to achieve a given level of accuracy, is $$\mathcal {O}(n^7\log ^4(n))$$O(n7log4(n)); see Bezakova (Faster Markov chain Monte Carlo algorithms for the permanent and binary contingency tables. University of Chicago, Chicago, 2008), Jerrum et al. (J Assoc Comput Mach 51:671---697, 2004). These algorithms use a particular collection of probability distributions, the `ideal' of which, (in some sense) are not known and need to be approximated. In this paper we propose an adaptive sequential Monte Carlo (SMC) algorithm that can both estimate the permanent and the ideal sequence of probabilities on the fly, with little user input. We provide theoretical results associated to the SMC estimate of the permanent, establishing its convergence. We also analyze the relative variance of the estimate, associated to an `ideal' algorithm (related to our algorithm) and not the one we develop, in particular, computating explicit bounds on the relative variance which depend upon $$n$$n. As this analysis is for an ideal algorithm, it gives a lower-bound on the computational cost, in order to achieve an arbitrarily small relative variance; we find that this cost is $$\mathcal {O}(n^4\log ^4(n))$$O(n4log4(n)). For the $$\alpha $$?-permanent, perhaps the gold standard algorithm is the importance sampling algorithm of Kou and McCullagh (Biometrika 96:635---644, 2009); in this paper we develop and compare new algorithms to this method; apriori one expects, due to the weight degeneracy problem, that the method of Kou and McCullagh (Biometrika 96:635---644, 2009) might perform very badly in comparison to the more advanced SMC methods we consider. We also present a statistical application of the $$\alpha $$?-permanent for statistical estimation of boson point process and MCMC methods to fit the associated model to data.

Characterization of Parameters for a Spatially Heterogenous Aquifer from Pumping Test Data

AbstractThis study considers the estimation of aquifer parameters for a spatially heterogenous aquifer from pumping test data. An approach is proposed that is based on modeling the unknown parameters as smooth Markov random fields. The associated inverse problem is formulated using the Bayesian framework and the posterior probability distribution of parameters is explored using Markov chain Monte Carlo. The method is evaluated by a numerical simulation in which measurements are taken in four observation wells. Even such a minimalist example presents significant computational challenges. Therefore, to obtain a computationally feasible solution, a model reduction is carried out and the estimation problem is reduced from over 1,000 parameters to 40 parameters. The approximate posterior distribution is then sampled using an adaptive Markov chain Monte Carlo sampler in order to quantify parameter uncertainty. This paper compares the parameter with predictive uncertainty and discusses the consequences of the mo...

Stochastic approximation Monte Carlo importance sampling for approximating exact conditional probabilities

Importance sampling and Markov chain Monte Carlo methods have been used in exact inference for contingency tables for a long time, however, their performances are not always very satisfactory. In this paper, we propose a stochastic approximation Monte Carlo importance sampling (SAMCIS) method for tackling this problem. SAMCIS is a combination of adaptive Markov chain Monte Carlo and importance sampling, which employs the stochastic approximation Monte Carlo algorithm (Liang et al., J. Am. Stat. Assoc., 102(477):305–320, 2007) to draw samples from an enlarged reference set with a known Markov basis. Compared to the existing importance sampling and Markov chain Monte Carlo methods, SAMCIS has a few advantages, such as fast convergence, ergodicity, and the ability to achieve a desired proportion of valid tables. The numerical results indicate that SAMCIS can outperform the existing importance sampling and Markov chain Monte Carlo methods: It can produce much more accurate estimates in much shorter CPU time than the existing methods, especially for the tables with high degrees of freedom.

Likelihood Inference in Non-Linear Term Structure Models: The Importance of the Lower Bound

This paper shows how to use adaptive particle filtering and Markov chain Monte Carlo methods to estimate quadratic term structure models (QTSMs) by likelihood inference. The procedure is applied to a quadratic model for the United States during the recent financial crisis. We find that this model provides a better statistical description of the data than a Gaussian affine term structure model. In addition, QTSMs account perfectly for the lower bound whereas Gaussian affine models frequently imply forecast distributions with negative interest rates. Such predictions appear during the recent financial crisis but also prior to the crisis.

Probabilistic Forecast and Uncertainty Assessment of Hydrologic Design Values Using Bayesian Theories

ABSTRACT Researches on hydrologic extreme events have great significance in reducing and avoiding the severe losses and impacts caused by natural disasters. When forecasting hydrologic design values of the hydrologic extreme events of interest by the conventional hydrologic frequency analysis (HFA) model, the results cannot take uncertainties and risks into account. In this article, in order to overcome conventional HFA model's disadvantages and to improve hydrologic design values’ forecast results, an improved HFA model named AM-MCMC-HFA is proposed by employing the AM-MCMC algorithm (adaptive Metropolis-Markov chain Monte Carlo) to HFA process. Differing with conventional HFA model, which is seeking single optimal forecast result, the AM-MCMC-HFA model can not only get the optimal but also the probabilistic forecast results of hydrologic design values. By applying to two obviously different hydrologic series, the performances of the model proposed have been verified. Analysis results show that four factors have great influence on hydrologic design values’ reliability, and also indicate that AM-MCMC-HFA has the ability of assessing the uncertainties of parameters and hydrologic design values. Therefore, by using the AM-MCMC-HFA model, hydrologic designs tasks can be operated more reasonably, and more rational decisions can be made by governmental decision-makers and public in practice.

An Adaptive $\delta f$ Monte Carlo Method

A new adaptive δf Monte Carlo method is presented with an application to radio frequency heating and transport in fusion plasmas. The method is suitable when an initial zeroth-order approximation of the distribution function is known. The difference between our method and earlier δf methods is that we model the source term, obtained from the δf ansatz, by adding particles. The rate of particles is defined by the inhomogeneous term in the Fokker-Planck equation. We develop an adaptive scheme for modifying the unperturbed part G(x) such that the number of particles used in the simulation for a fixed weight is minimized. This implicitly reduces the variance and improves computational efficiency. The method is tested on a one-dimensional Fokker-Planck model for RF-heating and compared against the analytical stationary solution.

Bayesian Causal Effects in Quantiles: Accounting for Heteroscedasticity

Testing for Granger non-causality over varying quantile levels could be used to measure and infer dynamic linkages, enabling the identification of quantiles for which causality is relevant, or not. However, dynamic quantiles in financial application settings are clearly affected by heteroscedasticity, as well as the exogenous and endogenous variables under consideration. GARCH-type dynamics are added to the standard quantile regression model, so as to more robustly examine quantile causal relations between dynamic variables. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between quantile regression and the skewed-Laplace distribution, is designed for estimation and inference of the quantile causal relations, simultaneously estimating and accounting for heteroscedasticity. Dynamic quantile linkages for the international stock markets in Taiwan and Hong Kong are considered over a range of quantile levels. Specifically, the hypothesis that these stock returns are Granger-caused by the US market and/or the Japanese market is examined. The US market is found to significantly and positively Granger-cause both markets at all quantile levels, while the Japanese market effect was also significant at most quantile levels, but with weaker effects.

Efficient Compensation of the Nonlinearity of Solid-State Power Amplifiers Using Adaptive Sequential Monte Carlo Methods

In this paper, the sequential Monte Carlo (SMC) framework is studied as a tool to compensate the nonlinear distortions caused by solid-state power amplifiers (SSPA) in M-QAM schemes. The performance of the SMC approach is shown for low- and high-order constellation schemes for different values of the input backoff (IBO). The results show that, in low-IBO regimes, the SMC method provides a significant improvement compared to conventional methods, such as the predistorter, especially for high-order constellations while the use of the predistorter is preferred in only a very limited number of cases. Moreover, the SMC framework is shown to have more robust behavior to the constellation scaling. The application of the SMC framework to multicarrier systems is also addressed and the behavior of the system in terms of the out-of-band emissions as a function of the output backoff (OBO) is investigated. Finally, an adaptive sequential Monte Carlo receiver is proposed that adapts itself efficiently to the variations in the amplifier parameters. This adaptive scheme does not require a training sequence and does not suffer from convergence problems.

Protein folding with the adaptive tempering Monte Carlo method

Characterization of the folding transition in a model protein was achieved with the recent multicanonical tempering method implemented with Monte Carlo the adaptive tempering Monte Carlo (ATMC) (X. Dong and E. Blaisten-Barojas. Adaptive tempering Monte Carlo method. J. Comput. Theor. Nanosci., 3, p. 118 2006). The folding transition temperature was successfully determined and a spread of states was observed around the interface between native and folded regions. Energy states collected from all tempering events in a series of parallel runs were used in the calculation of the free energy, internal energy, order parameter and radius of gyration as a function of temperature through the weighted histogram method. Not only the calculated thermodynamic properties are in good agreement with results from Langevin dynamics simulations (D. K. Klimov and D. Thirumalai. Native topology determines forced-induced pathways in global proteins. Proc. Natl. Acad. Sci. USA. 97, p. 7254 2000), but also this multicanonical approach is noticeably more efficient because of the adaptive manner in which the system visits states near a transition in the interface between two phases. Additionally, the ATMC is advantageous for protein simulation over regular single canonical ensemble methods because it accelerates the hopping between local energy minima on the potential energy surface.

Comparing stochastic volatility models through Monte Carlo simulations

Stochastic volatility models are important tools for studying the behavior of many financial markets. For this reason a number of versions have been introduced and studied in the recent literature. The goal is to review and compare some of these alternatives by using Bayesian procedures. The quantity used to assess the goodness-of-fit is the Bayes factor, whereas the ability to forecast the volatility has been tested through the computation of the one-step-ahead value-at-risk (VaR). Model estimation has been carried out through adaptive Markov chain Monte Carlo (MCMC) procedures. The marginal likelihood, necessary to compute the Bayes factor, has been computed through reduced runs of the same MCMC algorithm and through an auxiliary particle filter. The empirical analysis is based on the study of three international financial indexes.

Tight-binding Calcium Clusters from Adaptive Tempering Monte Carlo Simulation

The most stable structure of calcium clusters with 14 to 32 atoms is optimized by the Adaptive Tempering Monte Carlo method. The binding energy of the clusters is obtained within the tight-binding approach parameterized in a previous work. The optimization process is started at about 800 K and the tempering brings the structure to the global minimum ending the process at 1 K. It is found that six cluster sizes, 15, 16, 18, 21, 23 and 25 have a global minimum structure not reported in the literature. In this size range, Ca15, Ca21 and Ca23 are the preferred geometries that can be identified as magic numbers. The tight-binding one-electron levels in the valence band display a large energy gap of 0.5 eV between the last occupied and first unoccupied levels for the magic number clusters. This band gap is 5 to 10 times smaller for other cluster sizes.

Numerical Integration inS-PLUSorR: A Survey

This paper reviews current quadrature methods for approximate calculation of integrals within S-Plus or R. Starting with the general framework, Gaussian quadrature will be discussed first, followed by adaptive rules and Monte Carlo methods. Finally, a comparison of the methods presented is given. The aim of this survey paper is to help readers, not expert in computing, to apply numerical integration methods and to realize that numerical analysis is an art, not a science.

Adaptive sampling approach to the negative-sign problem in the auxiliary-field quantum Monte Carlo method

We propose a sampling method to calculate the ground state of interacting quantum systems. This approximate method, which we call the adaptive sampling quantum Monte Carlo (ASQMC) method, utilizes information from the short projection time density matrix derived from the Monte Carlo steps. The ASQMC method circumvents the negative-sign problem by introducing a conditional probability distribution function and guiding walkers to avoid a set of canceling fields. With the ASQMC method, the negative-sign ratio is greatly reduced. The ratio becomes zero in the limit $\ensuremath{\Delta}\ensuremath{\tau}$ goes to zero without prior knowledge of the exact ground state wave function. Comparisons with numerical results obtained by using other methods are made, and we observe that the ASQMC method gives accurate results with less than $0.5%$ error for the energy over wide regions of physical parameter values.

Adaptive Markov Chain Monte Carlo through Regeneration

Adaptive Markov Chain Monte Carlo through Regeneration