Approximate Sampling Methods Research Articles

Hybrid Iterative Ensemble Smoother for History Matching of Hierarchical Models

The choice of a prior model can have a large impact on the ability to assimilate data. In standard applications of ensemble-based data assimilation, all realizations in the initial ensemble are generated from the same covariance matrix with the implicit assumption that this covariance is appropriate for the problem. In a hierarchical approach, the parameters of the covariance function, for example, the variance, the orientation of the anisotropy and the ranges in two principal directions, may all be uncertain. Thus, the hierarchical approach is much more robust against model misspecification. In this paper, three approaches to sampling from the posterior for hierarchical parameterizations are discussed: an optimization-based sampling approach (randomized maximum likelihood, RML), an iterative ensemble smoother (IES), and a novel hybrid of the previous two approaches (hybrid IES). The three approximate sampling methods are applied to a linear-Gaussian inverse problem for which it is possible to compare results with an exact “marginal-then-conditional” approach. Additionally, the IES and the hybrid IES methods are tested on a two-dimensional flow problem with uncertain anisotropy in the prior covariance. The standard IES method is shown to perform poorly in the flow examples because of the poor representation of the local sensitivity matrix by the ensemble-based method. The hybrid method, however, samples well even with a relatively small ensemble size.

Parameter sampling capabilities of sequential and simultaneous data assimilation: I. Analytical comparison

We assess the parameter sampling capabilities of some Bayesian, ensemble-based, joint state-parameter (JS) estimation methods. The forward model is assumed to be non-chaotic and have nonlinear components, and the emphasis is on results obtained for the parameters in the state-parameter vector. A variety of approximate sampling methods exist, and a number of numerical comparisons between such methods have been performed. Often, more than one of the defining characteristics vary from one method to another, so it can be difficult to point out which characteristic of the more successful method in such a comparison was decisive. In this study, we single out one defining characteristic for comparison; whether or not data are assimilated sequentially or simultaneously. The current paper is concerned with analytical investigations into this issue. We carefully select one sequential and one simultaneous JS method for the comparison. We also design a corresponding pair of pure parameter estimation methods, and we show how the JS methods and the parameter estimation methods are pairwise related. It is shown that the sequential and the simultaneous parameter estimation methods are equivalent for one particular combination of observations with different degrees of nonlinearity. Strong indications are presented for why one may expect the sequential parameter estimation method to outperform the simultaneous parameter estimation method for all other combinations of observations. Finally, the conditions for when similar relations can be expected to hold between the corresponding JS methods are discussed. A companion paper, part II (Fossum and Mannseth 2014 Inverse Problems 30 114003), is concerned with statistical analysis of results from a range of numerical experiments involving sequential and simultaneous JS estimation, where the design of the numerical investigation is motivated by our findings in the current paper.

Fast MCMC sampling for hidden markov models to determine copy number variations

BackgroundHidden Markov Models (HMM) are often used for analyzing Comparative Genomic Hybridization (CGH) data to identify chromosomal aberrations or copy number variations by segmenting observation sequences. For efficiency reasons the parameters of a HMM are often estimated with maximum likelihood and a segmentation is obtained with the Viterbi algorithm. This introduces considerable uncertainty in the segmentation, which can be avoided with Bayesian approaches integrating out parameters using Markov Chain Monte Carlo (MCMC) sampling. While the advantages of Bayesian approaches have been clearly demonstrated, the likelihood based approaches are still preferred in practice for their lower running times; datasets coming from high-density arrays and next generation sequencing amplify these problems.ResultsWe propose an approximate sampling technique, inspired by compression of discrete sequences in HMM computations and by kd-trees to leverage spatial relations between data points in typical data sets, to speed up the MCMC sampling.ConclusionsWe test our approximate sampling method on simulated and biological ArrayCGH datasets and high-density SNP arrays, and demonstrate a speed-up of 10 to 60 respectively 90 while achieving competitive results with the state-of-the art Bayesian approaches.Availability: An implementation of our method will be made available as part of the open source GHMM library from http://ghmm.org.

Critical Evaluation of the Ensemble Kalman Filter on History Matching of Geologic Facies

Summary The objective of this paper is to compare the performance of the ensemble Kalman filter (EnKF) to the performance of a gradient-based minimization method for the problem of estimation of facies boundaries in history matching. The EnKF is a Monte Carlo method for data assimilation that uses an ensemble of reservoir models to represent and update the covariance of variables. In several published studies, it outperforms traditional history-matching algorithms in adaptability and efficiency. Because of the approximate nature of the EnKF, the realizations from one ensemble tend to underestimate uncertainty, especially for problems that are highly nonlinear. In this paper, the distributions of reservoir-model realizations from 20 independent ensembles are compared with the distributions from 20 random-ized-maximum-likelihood (RML) realizations for a 2D waterflood model with one injector and four producers. RML is a gradient-based sampling method that generates one reservoir realization in each minimization of the objective function. It is an approximate sampling method, but its sampling properties are similar to the Markov-chain Monte Carlo (McMC) method on highly nonlinear problems and are relatively more efficient than McMC. Despite the nonlinear relationship between the data (such as production rates and facies observations) and the model variables, the EnKF was effective at history matching the production data. We find that the computational effort to generate 20 independent realizations was similar for the two methods, although the complexity of the code is substantially less for the EnKF.

Teacher’s Corner: Negative Information and The Three-Parameter Logistic Model

The 3-parameter logistic model is commonly used to describe the relationship among an unobserved latent trait (ability), unobserved item properties, and an observed binary outcome. We show that for certain values of the item properties and latent ability, the observed information about ability contained in the binary response is negative. This result has implications for maximization procedures, such as Newton-Raphson; approximate sampling methods, such as the Metropolis-Hastings algorithm; and Bayesian adaptive testing. All of these typically utilize the observed information. This result is contrasted with the fact that observed negative information does not occur in the limiting case with no guessing (2-parameter logistic model). The probability of negative information is expressed by a simple formula. This research extends the work of Samejima (1973) and Yen, Burket, and Sykes (1991).

Negative Information and the Three-Parameter Logistic Model

The 3-parameter logistic model is commonly used to describe the relationship among an unobserved latent trait (ability), unobserved item properties, and an observed binary outcome. We show that for certain values of the item properties and latent ability, the observed information about ability contained in the binary response is negative. This result has implications for maximization procedures, such as Newton-Raphson; approximate sampling methods, such as the Metropolis-Hastings algorithm; and Bayesian adaptive testing. All of these typically utilize the observed information. This result is contrasted with the fact that observed negative information does not occur in the limiting case with no guessing (2-parameter logistic model). The probability of negative information is expressed by a simple formula. This research extends the work of Samejima (1973) and Yen, Burket, and Sykes (1991). In many educational tests, dichotomously scored multiple-choice items are given to examinees in order to obtain estimates of their true underlying ability. A widely adopted tenet is that a test with more items must yield estimates of ability with greater precision than the corresponding test with some items deleted. We show that not to be the case for the three-parameter logistic (3PL) model when the standard errors of ability estimates are based on the total observed information function. We focus on this issue by computing the additional information about a single examinee's ability from a single binary item response (generalizations to multiple responses are straightfor

Trajectory studies of energy transfer: CH 3NC with He, Xe, H 2 and N 2

A trajectory program was used to simulate the collisions of CH 3NC with He, Xe, H 2 and N 2. Calculated energy transfer is in accord with experiment. The pattern of CH 3NC vibrational mode energization is found to be noticeably non-random. The approximate sampling methods used in thermal unimolecular trajectory studies produce a more uniform state distribution.