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Wang, Zheng, Shanxiang Lyu, and Ling Liu. "Learnable Markov Chain Monte Carlo Sampling Methods for Lattice Gaussian Distribution." IEEE Access 7 (2019): 87494–503. http://dx.doi.org/10.1109/access.2019.2925530.
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Ahmadian, Yashar, Jonathan W. Pillow, and Liam Paninski. "Efficient Markov Chain Monte Carlo Methods for Decoding Neural Spike Trains." Neural Computation 23, no. 1 (January 2011): 46–96. http://dx.doi.org/10.1162/neco_a_00059.
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Stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity. We discuss Bayesian decoding methods based on an encoding generalized linear model (GLM) that accurately describes how stimuli are transformed into the spike trains of a group of neurons. The form of the GLM likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is log concave so long as the prior is. This allows the maximum a posteriori (MAP) stimulus estimate to be obtained using efficient optimization algorithms. Unfortunately, the MAP estimate can have a relatively large average error when the posterior is highly nongaussian. Here we compare several Markov chain Monte Carlo (MCMC) algorithms that allow for the calculation of general Bayesian estimators involving posterior expectations (conditional on model parameters). An efficient version of the hybrid Monte Carlo (HMC) algorithm was significantly superior to other MCMC methods for gaussian priors. When the prior distribution has sharp edges and corners, on the other hand, the “hit-and-run” algorithm performed better than other MCMC methods. Using these algorithms, we show that for this latter class of priors, the posterior mean estimate can have a considerably lower average error than MAP, whereas for gaussian priors, the two estimators have roughly equal efficiency. We also address the application of MCMC methods for extracting nonmarginal properties of the posterior distribution. For example, by using MCMC to calculate the mutual information between the stimulus and response, we verify the validity of a computationally efficient Laplace approximation to this quantity for gaussian priors in a wide range of model parameters; this makes direct model-based computation of the mutual information tractable even in the case of large observed neural populations, where methods based on binning the spike train fail. Finally, we consider the effect of uncertainty in the GLM parameters on the posterior estimators.
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Wang, Zheng. "Markov Chain Monte Carlo Methods for Lattice Gaussian Sampling: Convergence Analysis and Enhancement." IEEE Transactions on Communications 67, no. 10 (October 2019): 6711–24. http://dx.doi.org/10.1109/tcomm.2019.2926470.
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Whiley, Matt, and Simon P. Wilson. "Parallel algorithms for Markov chain Monte Carlo methods in latent spatial Gaussian models." Statistics and Computing 14, no. 3 (August 2004): 171–79. http://dx.doi.org/10.1023/b:stco.0000035299.51541.5e.
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Geweke, John, and Hisashi Tanizaki. "On markov chain monte carlo methods for nonlinear and non-gaussian state-space models." Communications in Statistics - Simulation and Computation 28, no. 4 (January 1999): 867–94. http://dx.doi.org/10.1080/03610919908813583.
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Jiao, Zhun, and Rong Zhang. "Improved Particle Filter for Integrated Navigation System." Applied Mechanics and Materials 543-547 (March 2014): 1278–81. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1278.
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As a new method for dealing with any nonlinear or non-Gaussian distributions, based on the Monte Carlo methods and Bayesian filtering, particle filters (PF) are favored by researchers and widely applied in many fields. Based on particle filtering, an improved particle filter (IPF) proposal distribution is presented. Evaluation of the weights is simplified and other improved techniques including the residual resampling step and Markov Chain Monte Carlo method are introduced for SINS/GPS integrated navigation system. The simulation results confirm that the improved particle filter outperforms the others.
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Lu, Dan, Daniel Ricciuto, Anthony Walker, Cosmin Safta, and William Munger. "Bayesian calibration of terrestrial ecosystem models: a study of advanced Markov chain Monte Carlo methods." Biogeosciences 14, no. 18 (September 27, 2017): 4295–314. http://dx.doi.org/10.5194/bg-14-4295-2017.
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Abstract. Calibration of terrestrial ecosystem models is important but challenging. Bayesian inference implemented by Markov chain Monte Carlo (MCMC) sampling provides a comprehensive framework to estimate model parameters and associated uncertainties using their posterior distributions. The effectiveness and efficiency of the method strongly depend on the MCMC algorithm used. In this work, a differential evolution adaptive Metropolis (DREAM) algorithm is used to estimate posterior distributions of 21 parameters for the data assimilation linked ecosystem carbon (DALEC) model using 14 years of daily net ecosystem exchange data collected at the Harvard Forest Environmental Measurement Site eddy-flux tower. The calibration of DREAM results in a better model fit and predictive performance compared to the popular adaptive Metropolis (AM) scheme. Moreover, DREAM indicates that two parameters controlling autumn phenology have multiple modes in their posterior distributions while AM only identifies one mode. The application suggests that DREAM is very suitable to calibrate complex terrestrial ecosystem models, where the uncertain parameter size is usually large and existence of local optima is always a concern. In addition, this effort justifies the assumptions of the error model used in Bayesian calibration according to the residual analysis. The result indicates that a heteroscedastic, correlated, Gaussian error model is appropriate for the problem, and the consequent constructed likelihood function can alleviate the underestimation of parameter uncertainty that is usually caused by using uncorrelated error models.
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Durante, Daniele. "Conjugate Bayes for probit regression via unified skew-normal distributions." Biometrika 106, no. 4 (August 25, 2019): 765–79. http://dx.doi.org/10.1093/biomet/asz034.
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Summary Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically taken to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, including Markov chain Monte Carlo routines and algorithms that approximate the posterior. Despite being implemented routinely, Markov chain Monte Carlo strategies have mixing or time-inefficiency issues in large-$p$ and small-$n$ studies, whereas approximate routines fail to capture the skewness typically observed in the posterior. In this article it is proved that the posterior distribution for the probit coefficients has a unified skew-normal kernel under Gaussian priors. This result allows efficient Bayesian inference for a wide class of applications, especially in large-$p$ and small-to-moderate-$n$ settings where state-of-the-art computational methods face notable challenges. These advances are illustrated in a genetic study, and further motivate the development of a wider class of conjugate priors for probit models, along with methods for obtaining independent and identically distributed samples from the unified skew-normal posterior.
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Peng, Li Feng, Guo Shao Su, and Wei Zhao. "Fast Analysis of Structural Reliability Using Gaussian Process Classification Based Dynamic Response Surface Method." Applied Mechanics and Materials 501-504 (January 2014): 1067–70. http://dx.doi.org/10.4028/www.scientific.net/amm.501-504.1067.
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The performance function of large-scale complicated engineering structure is always highly nonlinear and implicit, and its reliability needs to be evaluated through a time-consuming Finite Element method (FEM). A new method, Gaussian process classification (GPC) dynamic response surface based on Monte Carlo Simulation (MCS) was proposed. Small training samples were created using FEM and Markov chain. Then, the most probable point (MPP) is predicted quickly using MCS without any extra FEM analysis. Furthermore, an iterative algorithm is presented to reduce the errors of GPC by using information of MPP to improve the reconstructing precision constantly. Then, Monte Carlo method combined with GPC surface is applied to get the probability of failure. Several examples results demonstrate the efficiency and robustness of the proposed method, compared with the results of common reliability methods.
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Tilmann, F. J., H. Sadeghisorkhani, and A. Mauerberger. "Another look at the treatment of data uncertainty in Markov chain Monte Carlo inversion and other probabilistic methods." Geophysical Journal International 222, no. 1 (May 6, 2020): 388–405. http://dx.doi.org/10.1093/gji/ggaa168.
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SUMMARY In probabilistic Bayesian inversions, data uncertainty is a crucial parameter for quantifying the uncertainties and correlations of the resulting model parameters or, in transdimensional approaches, even the complexity of the model. However, in many geophysical inference problems it is poorly known. Therefore, it is common practice to allow the data uncertainty itself to be a parameter to be determined. Although in principle any arbitrary uncertainty distribution can be assumed, Gaussian distributions whose standard deviation is then the unknown parameter to be estimated are the usual choice. In this special case, the paper demonstrates that a simple analytical integration is sufficient to marginalise out this uncertainty parameter, reducing the complexity of the model space without compromising the accuracy of the posterior model probability distribution. However, it is well known that the distribution of geophysical measurement errors, although superficially similar to a Gaussian distribution, typically contains more frequent samples along the tail of the distribution, so-called outliers. In linearized inversions these are often removed in subsequent iterations based on some threshold criterion, but in Markov chain Monte Carlo (McMC) inversions this approach is not possible as they rely on the likelihood ratios, which cannot be formed if the number of data points varies between the steps of the Markov chain. The flexibility to define the data error probability distribution in McMC can be exploited in order to account for this pattern of uncertainties in a natural way, without having to make arbitrary choices regarding residual thresholds. In particular, we can regard the data uncertainty distribution as a mixture between a Gaussian distribution, which represent valid measurements with some measurement error, and a uniform distribution, which represents invalid measurements. The relative balance between them is an unknown parameter to be estimated alongside the standard deviation of the Gauss distribution. For each data point, the algorithm can then assign a probability to be an outlier, and the influence of each data point will be effectively downgraded according to its probability to be an outlier. Furthermore, this assignment can change as the McMC search is exploring different parts of the model space. The approach is demonstrated with both synthetic and real tomography examples. In a synthetic test, the proposed mixed measurement error distribution allows recovery of the underlying model even in the presence of 6 per cent outliers, which completely destroy the ability of a regular McMC or linear search to provide a meaningful image. Applied to an actual ambient noise tomography study based on automatically picked dispersion curves, the resulting model is shown to be much more consistent for different data sets, which differ in the applied quality criteria, while retaining the ability to recover strong anomalies in selected parts of the model.
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Noh, S. J., Y. Tachikawa, M. Shiiba, and S. Kim. "Applying sequential Monte Carlo methods into a distributed hydrologic model: lagged particle filtering approach with regularization." Hydrology and Earth System Sciences Discussions 8, no. 2 (April 4, 2011): 3383–420. http://dx.doi.org/10.5194/hessd-8-3383-2011.
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Abstract. Applications of data assimilation techniques have been widely used to improve hydrologic prediction. Among various data assimilation techniques, sequential Monte Carlo (SMC) methods, known as "particle filters", provide the capability to handle non-linear and non-Gaussian state-space models. In this paper, we propose an improved particle filtering approach to consider different response time of internal state variables in a hydrologic model. The proposed method adopts a lagged filtering approach to aggregate model response until uncertainty of each hydrologic process is propagated. The regularization with an additional move step based on Markov chain Monte Carlo (MCMC) is also implemented to preserve sample diversity under the lagged filtering approach. A distributed hydrologic model, WEP is implemented for the sequential data assimilation through the updating of state variables. Particle filtering is parallelized and implemented in the multi-core computing environment via open message passing interface (MPI). We compare performance results of particle filters in terms of model efficiency, predictive QQ plots and particle diversity. The improvement of model efficiency and the preservation of particle diversity are found in the lagged regularized particle filter.
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Cai, Rong-Gen, and Tao Yang. "Standard sirens and dark sector with Gaussian process." EPJ Web of Conferences 168 (2018): 01008. http://dx.doi.org/10.1051/epjconf/201816801008.
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The gravitational waves from compact binary systems are viewed as a standard siren to probe the evolution of the universe. This paper summarizes the potential and ability to use the gravitational waves to constrain the cosmological parameters and the dark sector interaction in the Gaussian process methodology. After briefly introducing the method to reconstruct the dark sector interaction by the Gaussian process, the concept of standard sirens and the analysis of reconstructing the dark sector interaction with LISA are outlined. Furthermore, we estimate the constraint ability of the gravitational waves on cosmological parameters with ET. The numerical methods we use are Gaussian process and the Markov-Chain Monte-Carlo. Finally, we also forecast the improvements of the abilities to constrain the cosmological parameters with ET and LISA combined with the Planck.
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Cui, Kai. "Semiparametric Gaussian Variance-Mean Mixtures for Heavy-Tailed and Skewed Data." ISRN Probability and Statistics 2012 (December 23, 2012): 1–18. http://dx.doi.org/10.5402/2012/345784.
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There is a need for new classes of flexible multivariate distributions that can capture heavy tails and skewness without being so flexible as to fully incur the curse of dimensionality intrinsic to nonparametric density estimation. We focus on the family of Gaussian variance-mean mixtures, which have received limited attention in multivariate settings beyond simple special cases. By using a Bayesian semiparametric approach, we allow the data to infer about the unknown mixing distribution. Properties are considered and an approach to posterior computation is developed relying on Markov chain Monte Carlo. The methods are evaluated through simulation studies and applied to a variety of applications, illustrating their flexible performance in characterizing heavy tails, tail dependence, and skewness.
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Chin, T. M., M. J. Turmon, J. B. Jewell, and M. Ghil. "An Ensemble-Based Smoother with Retrospectively Updated Weights for Highly Nonlinear Systems." Monthly Weather Review 135, no. 1 (January 1, 2007): 186–202. http://dx.doi.org/10.1175/mwr3353.1.
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Abstract Monte Carlo computational methods have been introduced into data assimilation for nonlinear systems in order to alleviate the computational burden of updating and propagating the full probability distribution. By propagating an ensemble of representative states, algorithms like the ensemble Kalman filter (EnKF) and the resampled particle filter (RPF) rely on the existing modeling infrastructure to approximate the distribution based on the evolution of this ensemble. This work presents an ensemble-based smoother that is applicable to the Monte Carlo filtering schemes like EnKF and RPF. At the minor cost of retrospectively updating a set of weights for ensemble members, this smoother has demonstrated superior capabilities in state tracking for two highly nonlinear problems: the double-well potential and trivariate Lorenz systems. The algorithm does not require retrospective adaptation of the ensemble members themselves, and it is thus suited to a streaming operational mode. The accuracy of the proposed backward-update scheme in estimating non-Gaussian distributions is evaluated by comparison to the more accurate estimates provided by a Markov chain Monte Carlo algorithm.
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Aleardi, Mattia, Fabio Ciabarri, and Timur Gukov. "A two-step inversion approach for seismic-reservoir characterization and a comparison with a single-loop Markov-chain Monte Carlo algorithm." GEOPHYSICS 83, no. 3 (May 1, 2018): R227—R244. http://dx.doi.org/10.1190/geo2017-0387.1.
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We have evaluated a two-step Bayesian algorithm for seismic-reservoir characterization, which, thanks to some simplifying assumptions, is computationally very efficient. The applicability and reliability of this method are assessed by comparison with a more sophisticated and computer-intensive Markov-chain Monte Carlo (MCMC) algorithm, which in a single loop directly estimates petrophysical properties and lithofluid facies from prestack data. The two-step method first combines a linear rock-physics model (RPM) with the analytical solution of a linearized amplitude versus angle (AVA) inversion, to directly estimate the petrophysical properties, and related uncertainties, from prestack data under the assumptions of a Gaussian prior model and weak elastic contrasts at the reflecting interface. In particular, we use an empirical, linear RPM, properly calibrated for the investigated area, to reparameterize the linear time-continuous P-wave reflectivity equation in terms of petrophysical contrasts instead of elastic constants. In the second step, a downward 1D Markov-chain prior model is used to infer the lithofluid classes from the outcomes of the first step. The single-loop (SL) MCMC algorithm uses a convolutional forward modeling based on the exact Zoeppritz equations, and it adopts a nonlinear RPM. Moreover, it assumes a more realistic Gaussian mixture distribution for the petrophysical properties. Both approaches are applied on an onshore 3D seismic data set for the characterization of a gas-bearing, clastic reservoir. Notwithstanding the differences in the forward-model parameterization, in the considered RPM, and in the assumed a priori probability density functions, the two methods yield maximum a posteriori solutions that are consistent with well-log data, although the Gaussian mixture assumption adopted by the SL method slightly improves the description of the multimodal behavior of the petrophysical parameters. However, in the considered reservoir, the main difference between the two approaches remains the very different computational times, the SL method being much more computationally intensive than the two-step approach.
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Wu, Chen, Li, and Peng. "Acoustic Impedance Inversion Using Gaussian Metropolis–Hastings Sampling with Data Driving." Energies 12, no. 14 (July 17, 2019): 2744. http://dx.doi.org/10.3390/en12142744.
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The Markov chain Monte Carlo (MCMC) method based on Metropolis–Hastings (MH) sampling is a popular approach in solving seismic acoustic impedance (AI) inversion problem, as it can improve the inversion resolution by statistical prior information. However, the sampling function of the traditional MH sampling is a fixed parameter distribution. The parameter ignores the statistical information of AI that expands sampling range and reduces the inversion efficiency and resolution. To reduce the sampling range and improve the efficiency, we apply the statistical information of AI to the sampling function and build a Gaussian MH sampling with data driving (GMHDD) approach to the sampling function. Moreover, combining GMHDD and MCMC, we propose a novel Bayesian AI inversion method based on GMHDD. Finally, we use the Marmousi2 data and field data to test the proposed method based on GMHDD and other methods based on traditional MH. The results reveal that the proposed method can improve the efficiency and resolution of impedance inversion than other methods.
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Maniatis, G., N. Demiris, A. Kranis, G. Banos, and A. Kominakis. "Comparison of inference methods of genetic parameters with an application to body weight in broilers." Archives Animal Breeding 58, no. 2 (July 27, 2015): 277–86. http://dx.doi.org/10.5194/aab-58-277-2015.
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Abstract. REML (restricted maximum likelihood) has become the standard method of variance component estimation in animal breeding. Inference in Bayesian animal models is typically based upon Markov chain Monte Carlo (MCMC) methods, which are generally flexible but time-consuming. Recently, a new Bayesian computational method, integrated nested Laplace approximation (INLA), has been introduced for making fast non-sampling-based Bayesian inference for hierarchical latent Gaussian models. This paper is concerned with the comparison of estimates provided by three representative programs (ASReml, WinBUGS and the R package AnimalINLA) of the corresponding methods (REML, MCMC and INLA), with a view to their applicability for the typical animal breeder. Gaussian and binary as well as simulated data were used to assess the relative efficiency of the methods. Analysis of 2319 records of body weight at 35 days of age from a broiler line suggested a purely additive animal model, in which the heritability estimates ranged from 0.31 to 0.34 for the Gaussian trait and from 0.19 to 0.36 for the binary trait, depending on the estimation method. Although in need of further development, AnimalINLA seems a fast program for Bayesian modeling, particularly suitable for the inference of Gaussian traits, while WinBUGS appeared to successfully accommodate a complicated structure between the random effects. However, ASReml remains the best practical choice for the serious animal breeder.
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Noh, S. J., Y. Tachikawa, M. Shiiba, and S. Kim. "Applying sequential Monte Carlo methods into a distributed hydrologic model: lagged particle filtering approach with regularization." Hydrology and Earth System Sciences 15, no. 10 (October 25, 2011): 3237–51. http://dx.doi.org/10.5194/hess-15-3237-2011.
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Abstract. Data assimilation techniques have received growing attention due to their capability to improve prediction. Among various data assimilation techniques, sequential Monte Carlo (SMC) methods, known as "particle filters", are a Bayesian learning process that has the capability to handle non-linear and non-Gaussian state-space models. In this paper, we propose an improved particle filtering approach to consider different response times of internal state variables in a hydrologic model. The proposed method adopts a lagged filtering approach to aggregate model response until the uncertainty of each hydrologic process is propagated. The regularization with an additional move step based on the Markov chain Monte Carlo (MCMC) methods is also implemented to preserve sample diversity under the lagged filtering approach. A distributed hydrologic model, water and energy transfer processes (WEP), is implemented for the sequential data assimilation through the updating of state variables. The lagged regularized particle filter (LRPF) and the sequential importance resampling (SIR) particle filter are implemented for hindcasting of streamflow at the Katsura catchment, Japan. Control state variables for filtering are soil moisture content and overland flow. Streamflow measurements are used for data assimilation. LRPF shows consistent forecasts regardless of the process noise assumption, while SIR has different values of optimal process noise and shows sensitive variation of confidential intervals, depending on the process noise. Improvement of LRPF forecasts compared to SIR is particularly found for rapidly varied high flows due to preservation of sample diversity from the kernel, even if particle impoverishment takes place.
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Beneš, Viktor, Karel Bodlák, Jesper Møller, and Rasmus Waagepetersen. "A CASE STUDY ON POINT PROCESS MODELLING IN DISEASE MAPPING." Image Analysis & Stereology 23, no. 3 (May 3, 2011): 159. http://dx.doi.org/10.5566/ias.v24.p159-168.
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We consider a data set of locations where people in Central Bohemia have been infected by tick-borne encephalitis (TBE), and where population census data and covariates concerning vegetation and altitude are available. The aims are to estimate the risk map of the disease and to study the dependence of the risk on the covariates. Instead of using the common area level approaches we base the analysis on a Bayesian approach for a log Gaussian Cox point process with covariates. Posterior characteristics for a discretized version of the log Gaussian Cox process are computed using Markov chain Monte Carlo methods. A particular problem which is thoroughly discussed is to determine a model for the background population density. The risk map shows a clear dependency with the population intensity models and the basic model which is adopted for the population intensity determines what covariates influence the risk of TBE. Model validation is based on the posterior predictive distribution of various summary statistics.
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Li, Xin, and Albert C. Reynolds. "A Gaussian Mixture Model as a Proposal Distribution for Efficient Markov-Chain Monte Carlo Characterization of Uncertainty in Reservoir Description and Forecasting." SPE Journal 25, no. 01 (September 23, 2019): 001–36. http://dx.doi.org/10.2118/182684-pa.
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Summary Generating an estimate of uncertainty in production forecasts has become nearly standard in the oil industry, but is often performed with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability-density function (PDF) of reservoir-model parameters conditional to dynamic data and then sampling this PDF correctly. Although Markov-chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target PDF that is known up to a normalizing constant, in reservoir-engineering applications, researchers have found that it might require extraordinarily long chains containing millions to hundreds of millions of states to obtain a correct characterization of the target PDF. When the target PDF has a single mode or has multiple modes concentrated in a small region, it might be possible to implement a proposal distribution dependent on a random walk so that the resulting MCMC algorithm derived from the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior PDF with a computationally feasible chain length. However, for a high-dimensional multimodal PDF with modes separated by large regions of low or zero probability, characterizing the PDF with MCMC using a random walk is not computationally feasible. Although methods such as population MCMC exist for characterizing a multimodal PDF, their computational cost generally makes the application of these algorithms far too costly for field application. In this paper, we design a new proposal distribution using a Gaussian mixture PDF for use in MCMC where the posterior PDF can be multimodal with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model (GMM) to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with that of random-walk MCMC and is also compared with that of population MCMC for a target PDF that is multimodal.
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Ruchi, Sangeetika, Svetlana Dubinkina, and Jana de Wiljes. "Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation." Nonlinear Processes in Geophysics 28, no. 1 (January 15, 2021): 23–41. http://dx.doi.org/10.5194/npg-28-23-2021.
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Abstract. Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark.
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Girolami, Mark, and Simon Rogers. "Variational Bayesian Multinomial Probit Regression with Gaussian Process Priors." Neural Computation 18, no. 8 (August 2006): 1790–817. http://dx.doi.org/10.1162/neco.2006.18.8.1790.
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It is well known in the statistics literature that augmenting binary and polychotomous response models with gaussian latent variables enables exact Bayesian analysis via Gibbs sampling from the parameter posterior. By adopting such a data augmentation strategy, dispensing with priors over regression coefficients in favor of gaussian process (GP) priors over functions, and employing variational approximations to the full posterior, we obtain efficient computational methods for GP classification in the multiclass setting.1 The model augmentation with additional latent variables ensures full a posteriori class coupling while retaining the simple a priori independent GP covariance structure from which sparse approximations, such as multiclass informative vector machines (IVM), emerge in a natural and straightforward manner. This is the first time that a fully variational Bayesian treatment for multiclass GP classification has been developed without having to resort to additional explicit approximations to the nongaussian likelihood term. Empirical comparisons with exact analysis use Markov Chain Monte Carlo (MCMC) and Laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full MCMC and it is shown to be more accurate than the Laplace approximation.
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Madan, Hennadii, Franjo Pernuš, and Žiga Špiclin. "Reference-free error estimation for multiple measurement methods." Statistical Methods in Medical Research 28, no. 7 (January 31, 2018): 2196–209. http://dx.doi.org/10.1177/0962280217754231.
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We present a computational framework to select the most accurate and precise method of measurement of a certain quantity, when there is no access to the true value of the measurand. A typical use case is when several image analysis methods are applied to measure the value of a particular quantitative imaging biomarker from the same images. The accuracy of each measurement method is characterized by systematic error (bias), which is modeled as a polynomial in true values of measurand, and the precision as random error modeled with a Gaussian random variable. In contrast to previous works, the random errors are modeled jointly across all methods, thereby enabling the framework to analyze measurement methods based on similar principles, which may have correlated random errors. Furthermore, the posterior distribution of the error model parameters is estimated from samples obtained by Markov chain Monte-Carlo and analyzed to estimate the parameter values and the unknown true values of the measurand. The framework was validated on six synthetic and one clinical dataset containing measurements of total lesion load, a biomarker of neurodegenerative diseases, which was obtained with four automatic methods by analyzing brain magnetic resonance images. The estimates of bias and random error were in a good agreement with the corresponding least squares regression estimates against a reference.
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Feng, Xia, Timothy DelSole, and Paul Houser. "Comparison of Seasonal Potential Predictability of Precipitation." Journal of Climate 27, no. 11 (May 29, 2014): 4094–110. http://dx.doi.org/10.1175/jcli-d-13-00489.1.
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Abstract Three methods for estimating potential seasonal predictability of precipitation from a single realization of daily data are assessed. The estimation methods include a first-order Markov chain model proposed by Katz (KZ), and an analysis of covariance (ANOCOVA) method and a bootstrap method proposed by the authors. The assessment is based on Monte Carlo experiments, ensemble atmospheric general circulation model (AGCM) simulations, and observation-based data. For AGCM time series, ANOCOVA produces the most accurate estimates of weather noise variance, despite the fact that it makes the most unrealistic assumptions about precipitation (in particular, it assumes precipitation is generated by a Gaussian autoregressive model). The KZ method significantly underestimates noise variance unless the autocorrelation of precipitation amounts on consecutive wet days is taken into account. Both AGCM and observation-based data reveal that the fraction of potentially predictable variance is greatest in the tropics, smallest in the extratropics, and undergoes a strong seasonal variation. The three methods give consistent estimates of potential predictability for 67% of the globe.
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Lu, Rong, Jennifer L. Miskimins, and Mikhail Zhizhin. "Learning from Nighttime Observations of Gas Flaring in North Dakota for Better Decision and Policy Making." Remote Sensing 13, no. 5 (March 3, 2021): 941. http://dx.doi.org/10.3390/rs13050941.
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In today’s oil industry, companies frequently flare the produced natural gas from oil wells. The flaring activities are extensive in some regions including North Dakota. Besides company-reported data, which are compiled by the North Dakota Industrial Commission, flaring statistics such as count and volume can be estimated via Visible Infrared Imaging Radiometer Suite nighttime observations. Following data gathering and preprocessing, Bayesian machine learning implemented with Markov chain Monte Carlo methods is performed to tackle two tasks: flaring time series analysis and distribution approximation. They help further understanding of the flaring profiles and reporting qualities, which are important for decision/policy making. First, although fraught with measurement and estimation errors, the time series provide insights into flaring approaches and characteristics. Gaussian processes are successful in inferring the latent flaring trends. Second, distribution approximation is achieved by unsupervised learning. The negative binomial and Gaussian mixture models are utilized to describe the distributions of field flare count and volume, respectively. Finally, a nearest-neighbor-based approach for company level flared volume allocation is developed. Potential discrepancies are spotted between the company reported and the remotely sensed flaring profiles.
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Gangopadhyay, A., and W. C. Gau. "Bayesian Nonparametric Approach to Credibility Modelling." Annals of Actuarial Science 2, no. 1 (March 2007): 91–114. http://dx.doi.org/10.1017/s1748499500000270.
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ABSTRACTCurrent methods in credibility theory often rely on parametric models, e.g., a linear function of past experience. During the last decade, the existence of high speed computers and statistical software packages allowed the introduction of more sophisticated and flexible modelling strategies. In recent years, some of these techniques, which made use of the Markov Chain Monte Carlo (MCMC) approach to modelling, have been incorporated in credibility theory. However, very few of these methods made use of additional covariate information related to risk, or collection of risks; and at the same time account for the correlated structure in the data. In this paper, we consider a Bayesian nonparametric approach to the problem of risk modelling. The model incorporates past and present observations related to risk, as well as relevant covariate information. This Bayesian modelling is carried out by sampling from a multivariate Gaussian prior, where the covariance structure is based on a thin-plate spline. The model uses the MCMC technique to compute the predictive distribution of future claims based on available data.
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Vehtari, Aki, and Jouko Lampinen. "Bayesian Model Assessment and Comparison Using Cross-Validation Predictive Densities." Neural Computation 14, no. 10 (October 1, 2002): 2439–68. http://dx.doi.org/10.1162/08997660260293292.
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In this work, we discuss practical methods for the assessment, comparison, and selection of complex hierarchical Bayesian models. A natural way to assess the goodness of the model is to estimate its future predictive capability by estimating expected utilities. Instead of just making a point estimate, it is important to obtain the distribution of the expected utility estimate because it describes the uncertainty in the estimate. The distributions of the expected utility estimates can also be used to compare models, for example, by computing the probability of one model having a better expected utility than some other model. We propose an approach using cross-validation predictive densities to obtain expected utility estimates and Bayesian bootstrap to obtain samples from their distributions. We also discuss the probabilistic assumptions made and properties of two practical cross-validation methods, importance sampling and k-fold cross-validation. As illustrative examples, we use multilayer perceptron neural networks and gaussian processes with Markov chain Monte Carlo sampling in one toy problem and two challenging real-world problems.
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Miller, S. M., A. M. Michalak, and P. J. Levi. "Atmospheric inverse modeling with known physical bounds: an example from trace gas emissions." Geoscientific Model Development 7, no. 1 (February 13, 2014): 303–15. http://dx.doi.org/10.5194/gmd-7-303-2014.
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Abstract. Many inverse problems in the atmospheric sciences involve parameters with known physical constraints. Examples include nonnegativity (e.g., emissions of some urban air pollutants) or upward limits implied by reaction or solubility constants. However, probabilistic inverse modeling approaches based on Gaussian assumptions cannot incorporate such bounds and thus often produce unrealistic results. The atmospheric literature lacks consensus on the best means to overcome this problem, and existing atmospheric studies rely on a limited number of the possible methods with little examination of the relative merits of each. This paper investigates the applicability of several approaches to bounded inverse problems. A common method of data transformations is found to unrealistically skew estimates for the examined example application. The method of Lagrange multipliers and two Markov chain Monte Carlo (MCMC) methods yield more realistic and accurate results. In general, the examined MCMC approaches produce the most realistic result but can require substantial computational time. Lagrange multipliers offer an appealing option for large, computationally intensive problems when exact uncertainty bounds are less central to the analysis. A synthetic data inversion of US anthropogenic methane emissions illustrates the strengths and weaknesses of each approach.
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Potthast, Roland, Anne Walter, and Andreas Rhodin. "A Localized Adaptive Particle Filter within an Operational NWP Framework." Monthly Weather Review 147, no. 1 (January 2019): 345–62. http://dx.doi.org/10.1175/mwr-d-18-0028.1.
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Particle filters are well known in statistics. They have a long tradition in the framework of ensemble data assimilation (EDA) as well as Markov chain Monte Carlo (MCMC) methods. A key challenge today is to employ such methods in a high-dimensional environment, since the naïve application of the classical particle filter usually leads to filter divergence or filter collapse when applied within the very high dimension of many practical assimilation problems (known as the curse of dimensionality). The goal of this work is to develop a localized adaptive particle filter (LAPF), which follows closely the idea of the classical MCMC or bootstrap-type particle filter, but overcomes the problems of collapse and divergence based on localization in the spirit of the local ensemble transform Kalman filter (LETKF) and adaptivity with an adaptive Gaussian resampling or rejuvenation scheme in ensemble space. The particle filter has been implemented in the data assimilation system for the global forecast model ICON at Deutscher Wetterdienst (DWD). We carry out simulations over a period of 1 month with a global horizontal resolution of 52 km and 90 layers. With four variables analyzed per grid point, this leads to 6.6 × 106 degrees of freedom. The LAPF can be run stably and shows a reasonable performance. We compare its scores to the operational setup of the ICON LETKF.
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Mendoza, Alberto, Lassi Roininen, Mark Girolami, Jere Heikkinen, and Heikki Haario. "Statistical methods to enable practical on-site tomographic imaging of whole-core samples." GEOPHYSICS 84, no. 3 (May 1, 2019): D89—D100. http://dx.doi.org/10.1190/geo2018-0436.1.
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Statistical methods enable the use of portable industrial scanners with sparse measurements, suitable for fast on-site whole-core X-ray computed tomography (CT), as opposed to conventional (medical) devices that use dense measurements. This approach accelerates an informed first-stage general assessment of core samples. To that end, this novel industrial tomographic measurement principle is feasible for rock-sample imaging, in conjunction with suitable forms of priors in Bayesian inversion algorithms. Gaussian, Cauchy, and total variation priors yield different inversion characteristics for similar material combinations. An evaluation of the inversion performance in rock samples considers, in a discrete form, conditional mean estimators, via Markov Chain Monte Carlo algorithms with noise-contaminated measurements. Additionally, further assessment indicates that this statistical approach better characterizes the attenuation contrast of rock materials, compared with simultaneous iterative reconstruction techniques. Benchmarking includes X-ray CT from numerical simulations of synthetic and measurement-based whole-core samples. To this end, we consider tomographic measurements of fine- to medium-grained sandstone core samples, with igneous-rich pebbles from the Miocene, off the Shimokita Peninsula in Japan, and fractured welded tuff from Big Bend National Park, Texas. Bayesian inversion results confirm that with only 16 radiograms, natural fractures with aperture of less than 2 mm wide are detectable. Additionally, reconstructed images found approximately spherical concretions of 6 mm diameter. To achieve similar results, filtered back projection techniques require hundreds of radiograms, only possible with conventional laboratory scanners.
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Arjas, Arttu, Andreas Hauptmann, and Mikko J. Sillanpää. "Estimation of dynamic SNP-heritability with Bayesian Gaussian process models." Bioinformatics 36, no. 12 (March 18, 2020): 3795–802. http://dx.doi.org/10.1093/bioinformatics/btaa199.
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Abstract Motivation Improved DNA technology has made it practical to estimate single-nucleotide polymorphism (SNP)-heritability among distantly related individuals with unknown relationships. For growth- and development-related traits, it is meaningful to base SNP-heritability estimation on longitudinal data due to the time-dependency of the process. However, only few statistical methods have been developed so far for estimating dynamic SNP-heritability and quantifying its full uncertainty. Results We introduce a completely tuning-free Bayesian Gaussian process (GP)-based approach for estimating dynamic variance components and heritability as their function. For parameter estimation, we use a modern Markov Chain Monte Carlo method which allows full uncertainty quantification. Several datasets are analysed and our results clearly illustrate that the 95% credible intervals of the proposed joint estimation method (which ‘borrows strength’ from adjacent time points) are significantly narrower than of a two-stage baseline method that first estimates the variance components at each time point independently and then performs smoothing. We compare the method with a random regression model using MTG2 and BLUPF90 software and quantitative measures indicate superior performance of our method. Results are presented for simulated and real data with up to 1000 time points. Finally, we demonstrate scalability of the proposed method for simulated data with tens of thousands of individuals. Availability and implementation The C++ implementation dynBGP and simulated data are available in GitHub: https://github.com/aarjas/dynBGP. The programmes can be run in R. Real datasets are available in QTL archive: https://phenome.jax.org/centers/QTLA. Supplementary information Supplementary data are available at Bioinformatics online.
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Pfreundschuh, Simon, Patrick Eriksson, David Duncan, Bengt Rydberg, Nina Håkansson, and Anke Thoss. "A neural network approach to estimating a posteriori distributions of Bayesian retrieval problems." Atmospheric Measurement Techniques 11, no. 8 (August 9, 2018): 4627–43. http://dx.doi.org/10.5194/amt-11-4627-2018.
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Abstract. A neural-network-based method, quantile regression neural networks (QRNNs), is proposed as a novel approach to estimating the a posteriori distribution of Bayesian remote sensing retrievals. The advantage of QRNNs over conventional neural network retrievals is that they learn to predict not only a single retrieval value but also the associated, case-specific uncertainties. In this study, the retrieval performance of QRNNs is characterized and compared to that of other state-of-the-art retrieval methods. A synthetic retrieval scenario is presented and used as a validation case for the application of QRNNs to Bayesian retrieval problems. The QRNN retrieval performance is evaluated against Markov chain Monte Carlo simulation and another Bayesian method based on Monte Carlo integration over a retrieval database. The scenario is also used to investigate how different hyperparameter configurations and training set sizes affect the retrieval performance. In the second part of the study, QRNNs are applied to the retrieval of cloud top pressure from observations by the Moderate Resolution Imaging Spectroradiometer (MODIS). It is shown that QRNNs are not only capable of achieving similar accuracy to standard neural network retrievals but also provide statistically consistent uncertainty estimates for non-Gaussian retrieval errors. The results presented in this work show that QRNNs are able to combine the flexibility and computational efficiency of the machine learning approach with the theoretically sound handling of uncertainties of the Bayesian framework. Together with this article, a Python implementation of QRNNs is released through a public repository to make the method available to the scientific community.
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Lenoir, Guillaume, and Michel Crucifix. "A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis." Nonlinear Processes in Geophysics 25, no. 1 (March 5, 2018): 175–200. http://dx.doi.org/10.5194/npg-25-175-2018.
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Abstract. Geophysical time series are sometimes sampled irregularly along the time axis. The situation is particularly frequent in palaeoclimatology. Yet, there is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. Here we provide such a framework. To this end, we define the scalogram as the continuous-wavelet-transform equivalent of the extended Lomb–Scargle periodogram defined in Part 1 of this study (Lenoir and Crucifix, 2018). The signal being analysed is modelled as the sum of a locally periodic component in the time–frequency plane, a polynomial trend, and a background noise. The mother wavelet adopted here is the Morlet wavelet classically used in geophysical applications. The background noise model is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, which is more general than the traditional Gaussian white and red noise processes. The scalogram is smoothed by averaging over neighbouring times in order to reduce its variance. The Shannon–Nyquist exclusion zone is however defined as the area corrupted by local aliasing issues. The local amplitude in the time–frequency plane is then estimated with least-squares methods. We also derive an approximate formula linking the squared amplitude and the scalogram. Based on this property, we define a new analysis tool: the weighted smoothed scalogram, which we recommend for most analyses. The estimated signal amplitude also gives access to band and ridge filtering. Finally, we design a test of significance for the weighted smoothed scalogram against the stationary Gaussian CARMA background noise, and provide algorithms for computing confidence levels, either analytically or with Monte Carlo Markov chain methods. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.
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Hassan, Masoud M. "A Fully Bayesian Logistic Regression Model for Classification of ZADA Diabetes Dataset." Science Journal of University of Zakho 8, no. 3 (September 30, 2020): 105–11. http://dx.doi.org/10.25271/sjuoz.2020.8.3.707.
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Classification of diabetes data with existing data mining and machine learning algorithms is challenging and the predictions are not always accurate. We aim to build a model that effectively addresses these challenges (misclassification) and can accurately diagnose and classify diabetes. In this study, we investigated the use of Bayesian Logistic Regression (BLR) for mining such data to diagnose and classify various diabetes conditions. This approach is fully Bayesian suited for automating Markov Chain Monte Carlo (MCMC) simulation. Using Bayesian methods in analysing medical data is useful because of the rich hierarchical models, uncertainty quantification, and prior information they provide. The analysis was done on a real medical dataset created for 909 patients in Zakho city with a binary class label and seven independent variables. Three different prior distributions (Gaussian, Laplace and Cauchy) were investigated for our proposed model implemented by MCMC. The performance and behaviour of the Bayesian approach were illustrated and compared with the traditional classification algorithms on this dataset using 10-fold cross-validation. Experimental results show overall that classification under BLR with informative Gaussian priors performed better in terms of various accuracy metrics. It provides an accuracy of 92.53%, a recall of 94.85%, a precision of 91.42% and an F1 score of 93.11%. Experimental results suggest that it is worthwhile to explore the application of BLR to predictive modelling tasks in medical studies using informative prior distributions.
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Pollard, David, Won Chang, Murali Haran, Patrick Applegate, and Robert DeConto. "Large ensemble modeling of the last deglacial retreat of the West Antarctic Ice Sheet: comparison of simple and advanced statistical techniques." Geoscientific Model Development 9, no. 5 (May 4, 2016): 1697–723. http://dx.doi.org/10.5194/gmd-9-1697-2016.
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Abstract. A 3-D hybrid ice-sheet model is applied to the last deglacial retreat of the West Antarctic Ice Sheet over the last ∼ 20 000 yr. A large ensemble of 625 model runs is used to calibrate the model to modern and geologic data, including reconstructed grounding lines, relative sea-level records, elevation–age data and uplift rates, with an aggregate score computed for each run that measures overall model–data misfit. Two types of statistical methods are used to analyze the large-ensemble results: simple averaging weighted by the aggregate score, and more advanced Bayesian techniques involving Gaussian process-based emulation and calibration, and Markov chain Monte Carlo. The analyses provide sea-level-rise envelopes with well-defined parametric uncertainty bounds, but the simple averaging method only provides robust results with full-factorial parameter sampling in the large ensemble. Results for best-fit parameter ranges and envelopes of equivalent sea-level rise with the simple averaging method agree well with the more advanced techniques. Best-fit parameter ranges confirm earlier values expected from prior model tuning, including large basal sliding coefficients on modern ocean beds.
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Miller, S. M., A. M. Michalak, and P. J. Levi. "Atmospheric inverse modeling with known physical bounds: an example from trace gas emissions." Geoscientific Model Development Discussions 6, no. 3 (September 6, 2013): 4531–62. http://dx.doi.org/10.5194/gmdd-6-4531-2013.
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Abstract. Many inverse problems in the atmospheric sciences involve parameters with known physical constraints. Examples include non-negativity (e.g., emissions of some urban air pollutants) or upward limits implied by reaction or solubility constants. However, probabilistic inverse modeling approaches based on Gaussian assumptions cannot incorporate such bounds and thus often produce unrealistic results. The atmospheric literature lacks consensus on the best means to overcome this problem, and existing atmospheric studies rely on a limited number of the possible methods with little examination of the relative merits of each. This paper investigates the applicability of several approaches to bounded inverse problems and is also the first application of Markov chain Monte Carlo (MCMC) to estimation of atmospheric trace gas fluxes. The approaches discussed here are broadly applicable. A common method of data transformations is found to unrealistically skew estimates for the examined example application. The method of Lagrange multipliers and two MCMC methods yield more realistic and accurate results. In general, the examined MCMC approaches produce the most realistic result but can require substantial computational time. Lagrange multipliers offer an appealing alternative for large, computationally intensive problems when exact uncertainty bounds are less central to the analysis. A synthetic data inversion of US anthropogenic methane emissions illustrates the strengths and weaknesses of each approach.
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Chen, Jinsong, G. Michael Hoversten, Kerry Key, Gregg Nordquist, and William Cumming. "Stochastic inversion of magnetotelluric data using a sharp boundary parameterization and application to a geothermal site." GEOPHYSICS 77, no. 4 (July 1, 2012): E265—E279. http://dx.doi.org/10.1190/geo2011-0430.1.
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We developed a Bayesian model to invert magnetotelluric (MT) data using a 2D sharp boundary parameterization. We divided the 2D cross section into layers and considered the locations of interfaces and resistivity of the regions formed by the interfaces as random variables. We assumed that those variables are independent in the vertical direction and dependent along the lateral direction, whose spatial dependence is described by either pairwise difference or multivariate Gaussian priors. We used a parallel, adaptive finite-element algorithm to rapidly forward simulate frequency-domain MT responses of the 2D resistivity structure and used Markov chain Monte Carlo methods to draw many samples from the joint posterior probability distribution. We applied the Bayesian model to a synthetic case that mimics a geothermal exploration scenario. Our results demonstrated that the developed method is effective in estimating the resistivity and depths to interfaces and in quantifying uncertainty on the estimates. We also applied the developed method to the field MT data collected from the Darajat geothermal site in Indonesia. We compared our inversion results with those obtained from a deterministic inversion of 3D MT data; they are consistent even if the two inversion methods are very different and the amount of information used for inversion is different.
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Blatter, Daniel, Anandaroop Ray, and Kerry Key. "Two-dimensional Bayesian inversion of magnetotelluric data using trans-dimensional Gaussian processes." Geophysical Journal International 226, no. 1 (March 25, 2021): 548–63. http://dx.doi.org/10.1093/gji/ggab110.
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SUMMARY Bayesian inversion of electromagnetic data produces crucial uncertainty information on inferred subsurface resistivity. Due to their high computational cost, however, Bayesian inverse methods have largely been restricted to computationally expedient 1-D resistivity models. In this study, we successfully demonstrate, for the first time, a fully 2-D, trans-dimensional Bayesian inversion of magnetotelluric (MT) data. We render this problem tractable from a computational standpoint by using a stochastic interpolation algorithm known as a Gaussian process (GP) to achieve a parsimonious parametrization of the model vis-a-vis the dense parameter grids used in numerical forward modelling codes. The GP links a trans-dimensional, parallel tempered Markov chain Monte Carlo sampler, which explores the parsimonious model space, to MARE2DEM, an adaptive finite element forward solver. MARE2DEM computes the model response using a dense parameter mesh with resistivity assigned via the GP model. We demonstrate the new trans-dimensional GP sampler by inverting both synthetic and field MT data for 2-D models of electrical resistivity, with the field data example converging within 10 d on 148 cores, a non-negligible but tractable computational cost. For a field data inversion, our algorithm achieves a parameter reduction of over 32× compared to the fixed parameter grid used for the MARE2DEM regularized inversion. Resistivity probability distributions computed from the ensemble of models produced by the inversion yield credible intervals and interquartile plots that quantitatively show the non-linear 2-D uncertainty in model structure. This uncertainty could then be propagated to other physical properties that impact resistivity including bulk composition, porosity and pore-fluid content.
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Ghofrani, Faeze, Qing He, Reza Mohammadi, Abhishek Pathak, and Amjad Aref. "Bayesian Survival Approach to Analyzing the Risk of Recurrent Rail Defects." Transportation Research Record: Journal of the Transportation Research Board 2673, no. 7 (May 2, 2019): 281–93. http://dx.doi.org/10.1177/0361198119844241.
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This paper develops a Bayesian framework to explore the impact of different factors and to predict the risk of recurrence of rail defects, based upon datasets collected from a US Class I railroad between 2011 and 2016. To this end, this study constructs a parametric Weibull baseline hazard function and a proportional hazard (PH) model under a Gaussian frailty approach. The analysis is performed using Markov chain Monte Carlo simulation methods and the fit of the model is checked using a Cox–Snell residual plot. The results of the model show that the recurrence of a defect is correlated with different factors such as the type of rail defect, the location of the defect, train speed limit, the number of geometry defects in the last three years, and the weight of the rail. First, unlike the ordinary PH model in which the occurrence times of rail defects at the same location are assumed to be independent, a PH model under frailty induces the correlation between times to the recurrence of rail defects for the same segment, which is essential in the case of recurrent events. Second, considering Gaussian frailties is useful for exploring the influence of unobserved covariates in the model. Third, integrating a Bayesian framework for the parameters of the Weibull baseline hazard function as well as other parameters provides greater flexibility to the model. Fourth, the findings are useful for responsive maintenance planning, capital planning, and even preventive maintenance planning.
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Fang, Zhilong, Curt Da Silva, Rachel Kuske, and Felix J. Herrmann. "Uncertainty quantification for inverse problems with weak partial-differential-equation constraints." GEOPHYSICS 83, no. 6 (November 1, 2018): R629—R647. http://dx.doi.org/10.1190/geo2017-0824.1.
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In statistical inverse problems, the objective is a complete statistical description of unknown parameters from noisy observations to quantify uncertainties in unknown parameters. We consider inverse problems with partial-differential-equation (PDE) constraints, which are applicable to many seismic problems. Bayesian inference is one of the most widely used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of parameters. Typically when formulating the posterior distribution, the PDE constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo (MCMC) methods are known to converge slowly for realistically sized problems. To overcome these drawbacks, we relax the PDE constraints by introducing an auxiliary variable, which allows for Gaussian errors in the PDE and yields a bilinear posterior distribution with weak PDE constraints that is more amenable to uncertainty quantification because of its special structure. We determine that for a particular range of variance choices for the PDE misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Because it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for problems with more than [Formula: see text] unknowns, we have developed a method to implicitly construct it, which enables efficient sampling. We apply this framework to 2D seismic inverse problems with 1800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities with those produced by conventional MCMC-type methods while requiring far fewer PDE solves, which are the main computational bottlenecks in these problems.
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Pollard, D., W. Chang, M. Haran, P. Applegate, and R. DeConto. "Large ensemble modeling of last deglacial retreat of the West Antarctic Ice Sheet: comparison of simple and advanced statistical techniques." Geoscientific Model Development Discussions 8, no. 11 (November 12, 2015): 9925–63. http://dx.doi.org/10.5194/gmdd-8-9925-2015.
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Abstract. A 3-D hybrid ice-sheet model is applied to the last deglacial retreat of the West Antarctic Ice Sheet over the last ~ 20 000 years. A large ensemble of 625 model runs is used to calibrate the model to modern and geologic data, including reconstructed grounding lines, relative sea-level records, elevation-age data and uplift rates, with an aggregate score computed for each run that measures overall model-data misfit. Two types of statistical methods are used to analyze the large-ensemble results: simple averaging weighted by the aggregate score, and more advanced Bayesian techniques involving Gaussian process-based emulation and calibration, and Markov chain Monte Carlo. Results for best-fit parameter ranges and envelopes of equivalent sea-level rise with the simple averaging method agree quite well with the more advanced techniques, but only for a large ensemble with full factorial parameter sampling. Best-fit parameter ranges confirm earlier values expected from prior model tuning, including large basal sliding coefficients on modern ocean beds. Each run is extended 5000 years into the "future" with idealized ramped climate warming. In the majority of runs with reasonable scores, this produces grounding-line retreat deep into the West Antarctic interior, and the analysis provides sea-level-rise envelopes with well defined parametric uncertainty bounds.
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Ju, Feng, Ru An, and Yaxing Sun. "Immune Evolution Particle Filter for Soil Moisture Data Assimilation." Water 11, no. 2 (January 26, 2019): 211. http://dx.doi.org/10.3390/w11020211.
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Data assimilation (DA) has been widely used in land surface models (LSM) to improve model state estimates. Among various DA methods, the particle filter (PF) with Markov chain Monte Carlo (MCMC) has become increasingly popular for estimating the states of the nonlinear and non-Gaussian LSMs. However, the standard PF always suffers from the particle impoverishment problem, characterized by loss of particle diversity. To solve this problem, an immune evolution particle filter with MCMC simulation inspired by the biological immune system, entitled IEPFM, is proposed for DA in this paper. The merit of this approach is in imitating the antibody diversity preservation mechanism to further improve particle diversity, thus increasing the accuracy of estimates. Furthermore, the immune memory function refers to promise particle evolution process towards optimal estimates. Effectiveness of the proposed approach is demonstrated by the numerical simulation experiment using a highly nonlinear atmospheric model. Finally, IEPFM is applied to a soil moisture (SM) assimilation experiment, which assimilates in situ observations into the Variable Infiltration Capacity (VIC) model to estimate SM in the MaQu network region of the Tibetan Plateau. Both synthetic and real case experiments demonstrate that IEPFM mitigates particle impoverishment and provides more accurate assimilation results compared with other popular DA algorithms.
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Posselt, Derek J., and Gerald G. Mace. "MCMC-Based Assessment of the Error Characteristics of a Surface-Based Combined Radar–Passive Microwave Cloud Property Retrieval." Journal of Applied Meteorology and Climatology 53, no. 8 (August 2014): 2034–57. http://dx.doi.org/10.1175/jamc-d-13-0237.1.
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AbstractCollocated active and passive remote sensing measurements collected at U.S. Department of Energy Atmospheric Radiation Measurement Program sites enable simultaneous retrieval of cloud and precipitation properties and air motion. Previous studies indicate the parameters of a bimodal cloud particle size distribution can be effectively constrained using a combination of passive microwave radiometer and radar observations; however, aspects of the particle size distribution and particle shape are typically assumed to be known. In addition, many retrievals assume the observation and retrieval error statistics have Gaussian distributions and use least squares minimization techniques to find a solution. In truth, the retrieval error characteristics are largely unknown. Markov chain Monte Carlo (MCMC) methods can be used to produce a robust estimate of the probability distribution of a retrieved quantity that is nonlinearly related to the measurements and that has non-Gaussian error statistics. In this work, an MCMC algorithm is used to explore the error characteristics of cloud property retrievals from surface-based W-band radar and low-frequency microwave radiometer observations for a case of orographic snowfall. In this particular case, it is found that a combination of passive microwave radiometer measurements with radar reflectivity and Doppler velocity is sufficient to constrain the liquid and ice particle size distributions, but only if the width parameter of the assumed gamma particle size distribution and mass–dimensional relationships are specified. If the width parameter and mass–dimensional relationships are allowed to vary realistically, a unique retrieval of the liquid and ice particle size distribution for this orographic snowfall case is rendered far more problematic.
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Lenoir, Guillaume, and Michel Crucifix. "A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis." Nonlinear Processes in Geophysics 25, no. 1 (March 5, 2018): 145–73. http://dx.doi.org/10.5194/npg-25-145-2018.
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Abstract. We develop a general framework for the frequency analysis of irregularly sampled time series. It is based on the Lomb–Scargle periodogram, but extended to algebraic operators accounting for the presence of a polynomial trend in the model for the data, in addition to a periodic component and a background noise. Special care is devoted to the correlation between the trend and the periodic component. This new periodogram is then cast into the Welch overlapping segment averaging (WOSA) method in order to reduce its variance. We also design a test of significance for the WOSA periodogram, against the background noise. The model for the background noise is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, more general than the classical Gaussian white or red noise processes. CARMA parameters are estimated following a Bayesian framework. We provide algorithms that compute the confidence levels for the WOSA periodogram and fully take into account the uncertainty in the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters provides analytical confidence levels for the WOSA periodogram, which are more accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below some threshold for the number of data points, less costly in computing time. We then estimate the amplitude of the periodic component with least-squares methods, and derive an approximate proportionality between the squared amplitude and the periodogram. This proportionality leads to a new extension for the periodogram: the weighted WOSA periodogram, which we recommend for most frequency analyses with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalise and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. They also constitute the starting point for an extension to the continuous wavelet transform developed in a companion article (Lenoir and Crucifix, 2018). All the methods presented in this paper are available to the reader in the Python package WAVEPAL.
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Heydenreich, Sven, Benjamin Brück, and Joachim Harnois-Déraps. "Persistent homology in cosmic shear: Constraining parameters with topological data analysis." Astronomy & Astrophysics 648 (April 2021): A74. http://dx.doi.org/10.1051/0004-6361/202039048.
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In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of N-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on S8 ≡ σ8Ωm/0.3 that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg2 of Euclid-like simulations, we are able to improve the constraints on S8 and Ωm by 19% and 12%, respectively, while breaking some of the degeneracy between S8 and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.
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Martikainen, J., K. Muinonen, A. Penttilä, A. Cellino, and X. B. Wang. "Asteroid absolute magnitudes and phase curve parameters from Gaia photometry." Astronomy & Astrophysics 649 (May 2021): A98. http://dx.doi.org/10.1051/0004-6361/202039796.
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Aims. We perform light curve inversion for 491 asteroids to retrieve phase curve parameters, rotation periods, pole longitudes and latitudes, and convex and triaxial ellipsoid shapes by using the sparse photometric observations from Gaia Data Release 2 and the dense ground-based observations from the DAMIT database. We develop a method for the derivation of reference absolute magnitudes and phase curves from the Gaia data, allowing for comparative studies involving hundreds of asteroids. Methods. For both general convex shapes and ellipsoid shapes, we computed least-squares solutions using either the Levenberg-Marquardt optimization algorithm or the Nelder-Mead downhill simplex method. Virtual observations were generated by adding Gaussian random errors to the observations, and, later on, a Markov chain Monte Carlo method was applied to sample the spin, shape, and scattering parameters. Absolute magnitude and phase curve retrieval was developed for the reference geometry of equatorial illumination and observations based on model magnitudes averaged over rotational phase. Results. The derived photometric slope values showed wide variations within each assumed Tholen class. The computed Gaia G-band absolute magnitudes matched notably well with the V-band absolute magnitudes retrieved from the Jet Propulsion Laboratory Small-Body Database. Finally, the reference phase curves were well fitted with the H, G1, G2 phase function. The resulting G1, G2 distribution differed, in an intriguing way, from the G1, G2 distribution that is based on the phase curves corresponding to light curve brightness maxima.
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Fernández Martínez, Juan Luis, Tapan Mukerji, Esperanza García Gonzalo, and Amit Suman. "Reservoir characterization and inversion uncertainty via a family of particle swarm optimizers." GEOPHYSICS 77, no. 1 (January 2012): M1—M16. http://dx.doi.org/10.1190/geo2011-0041.1.
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History matching provides to reservoir engineers an improved spatial distribution of physical properties to be used in forecasting the reservoir response for field management. The ill-posed character of the history-matching problem yields nonuniqueness and numerical instabilities that increase with the reservoir complexity. These features might cause local optimization methods to provide unpredictable results not being able to discriminate among the multiple models that fit the observed data (production history). Also, the high dimensionality of the inverse problem impedes estimation of uncertainties using classical Markov-chain Monte Carlo methods. We attenuated the ill-conditioned character of this history-matching inverse problem by reducing the model complexity using a spatial principal component basis and by combining as observables flow production measurements and time-lapse seismic crosswell tomographic images. Additionally the inverse problem was solved in a stochastic framework. For this purpose, we used a family of particle swarm optimization (PSO) optimizers that have been deduced from a physical analogy of the swarm system. For a synthetic sand-and-shale reservoir, we analyzed the performance of the different PSO optimizers, both in terms of exploration and convergence rate for two different reservoir models with different complexity and under the presence of different levels of white Gaussian noise added to the synthetic observed data. We demonstrated that PSO optimizers have a very good convergence rate for this example, and provide in addition, approximate measures of uncertainty around the optimum facies model. The PSO algorithms are robust in presence of noise, which is always the case for real data.
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Basson, Abigail R., Fabio Cominelli, and Alexander Rodriguez-Palacios. "‘Statistical Irreproducibility’ Does Not Improve with Larger Sample Size: How to Quantify and Address Disease Data Multimodality in Human and Animal Research." Journal of Personalized Medicine 11, no. 3 (March 23, 2021): 234. http://dx.doi.org/10.3390/jpm11030234.
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Poor study reproducibility is a concern in translational research. As a solution, it is recommended to increase sample size (N), i.e., add more subjects to experiments. The goal of this study was to examine/visualize data multimodality (data with >1 data peak/mode) as cause of study irreproducibility. To emulate the repetition of studies and random sampling of study subjects, we first used various simulation methods of random number generation based on preclinical published disease outcome data from human gut microbiota-transplantation rodent studies (e.g., intestinal inflammation and univariate/continuous). We first used unimodal distributions (one-mode, Gaussian, and binomial) to generate random numbers. We showed that increasing N does not reproducibly identify statistical differences when group comparisons are repeatedly simulated. We then used multimodal distributions (>1-modes and Markov chain Monte Carlo methods of random sampling) to simulate similar multimodal datasets A and B (t-test-p = 0.95; N = 100,000), and confirmed that increasing N does not improve the ‘reproducibility of statistical results or direction of the effects’. Data visualization with violin plots of categorical random data simulations with five-integer categories/five-groups illustrated how multimodality leads to irreproducibility. Re-analysis of data from a human clinical trial that used maltodextrin as dietary placebo illustrated multimodal responses between human groups, and after placebo consumption. In conclusion, increasing N does not necessarily ensure reproducible statistical findings across repeated simulations due to randomness and multimodality. Herein, we clarify how to quantify, visualize and address disease data multimodality in research. Data visualization could facilitate study designs focused on disease subtypes/modes to help understand person–person differences and personalized medicine.
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Ruchi, Sangeetika, and Svetlana Dubinkina. "Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling." Nonlinear Processes in Geophysics 25, no. 4 (November 6, 2018): 731–46. http://dx.doi.org/10.5194/npg-25-731-2018.
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Abstract. Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable Markov chain Monte Carlo (MCMC) methods are computationally expensive. Sequential ensemble methods such as ensemble Kalman filters and particle filters provide a favorable alternative. However, ensemble Kalman filter has an assumption of Gaussianity. Ensemble transform particle filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimations in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ ensemble transform particle filter (ETPF) and ensemble transform Kalman filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of particular interest for subsurface reservoir modeling as it allows us to parameterize permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (one and five) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble, while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen–Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands knowledge of which mode to truncate.
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Athreya, K. B., Mohan Delampady, and T. Krishnan. "Markov Chain Monte Carlo methods." Resonance 8, no. 4 (April 2003): 17–26. http://dx.doi.org/10.1007/bf02883528.
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