scholarly journals Parameter stability and consistency in an alongshore-current model determined with Markov chain Monte Carlo

2008 ◽  
Vol 10 (2) ◽  
pp. 153-162 ◽  
Author(s):  
B. G. Ruessink
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Parameter Stability ◽  
Small Uncertainty ◽  
Best Fit ◽  
Stability And Consistency

When a numerical model is to be used as a practical tool, its parameters should preferably be stable and consistent, that is, possess a small uncertainty and be time-invariant. Using data and predictions of alongshore mean currents flowing on a beach as a case study, this paper illustrates how parameter stability and consistency can be assessed using Markov chain Monte Carlo. Within a single calibration run, Markov chain Monte Carlo estimates the parameter posterior probability density function, its mode being the best-fit parameter set. Parameter stability is investigated by stepwise adding new data to a calibration run, while consistency is examined by calibrating the model on different datasets of equal length. The results for the present case study indicate that various tidal cycles with strong (say, >0.5 m/s) currents are required to obtain stable parameter estimates, and that the best-fit model parameters and the underlying posterior distribution are strongly time-varying. This inconsistent parameter behavior may reflect unresolved variability of the processes represented by the parameters, or may represent compensational behavior for temporal violations in specific model assumptions.

scholarly journals Bayesian estimation of parameters in a regional hydrological model

2002 ◽  
Vol 6 (5) ◽  
pp. 883-898 ◽  
Author(s):  
K. Engeland ◽  
L. Gottschalk
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Hydrological Model ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Parameter Uncertainties ◽  
Daily Streamflow ◽  
Likelihood Model ◽  
Simulation Errors

Abstract. This study evaluates the applicability of the distributed, process-oriented Ecomag model for prediction of daily streamflow in ungauged basins. The Ecomag model is applied as a regional model to nine catchments in the NOPEX area, using Bayesian statistics to estimate the posterior distribution of the model parameters conditioned on the observed streamflow. The distribution is calculated by Markov Chain Monte Carlo (MCMC) analysis. The Bayesian method requires formulation of a likelihood function for the parameters and three alternative formulations are used. The first is a subjectively chosen objective function that describes the goodness of fit between the simulated and observed streamflow, as defined in the GLUE framework. The second and third formulations are more statistically correct likelihood models that describe the simulation errors. The full statistical likelihood model describes the simulation errors as an AR(1) process, whereas the simple model excludes the auto-regressive part. The statistical parameters depend on the catchments and the hydrological processes and the statistical and the hydrological parameters are estimated simultaneously. The results show that the simple likelihood model gives the most robust parameter estimates. The simulation error may be explained to a large extent by the catchment characteristics and climatic conditions, so it is possible to transfer knowledge about them to ungauged catchments. The statistical models for the simulation errors indicate that structural errors in the model are more important than parameter uncertainties. Keywords: regional hydrological model, model uncertainty, Bayesian analysis, Markov Chain Monte Carlo analysis

scholarly journals Markov Chain Monte Carlo Parameter Estimation of the ODE Compartmental Cell Growth Model

2018 ◽  
Vol 13 (2) ◽  
pp. 376-391 ◽  
Author(s):  
T. Luzyanina ◽  
G. Bocharov
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Profile Likelihood ◽  
Live Cells ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Data Set ◽  
T Lymphocyte Proliferation

We use a Markov chain Monte Carlo (MCMC) method to quantify uncertainty in parameters of the heterogeneous linear compartmental model of cell population growth, described by a system of ordinary differential equations. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set specifies the following characteristics of the kinetics of human T lymphocyte proliferation assay in vitro: the total number of live cells and dead but not disintegrated cells and the number of cells divided j times. Our goal is to compare results of the MCMC analysis of the uncertainty in the best-fit parameter estimates with the ones obtained earlier, using the variance-covariance approach, the profile-likelihood based approach and the bootstrap technique. We show that the computed posterior probability density functions are Gaussian for most of the model parameters and they are close to Gaussian ones for other parameters except one. We present posterior uncertainty limits for the model solution and new observations.

scholarly journals Bayesian calibration of terrestrial ecosystem models: a study of advanced Markov chain Monte Carlo methods

2017 ◽  
Vol 14 (18) ◽  
pp. 4295-4314 ◽  
Author(s):  
Dan Lu ◽  
Daniel Ricciuto ◽  
Anthony Walker ◽  
Cosmin Safta ◽  
William Munger
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Likelihood Function ◽  
Terrestrial Ecosystem ◽  
Error Model ◽  
Model Parameters ◽  
Posterior Distributions ◽  
Ecosystem Models ◽  
Bayesian Calibration

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.

Sensitivity analysis of an urban stormwater microorganism model

2010 ◽  
Vol 62 (6) ◽  
pp. 1393-1400 ◽  
Author(s):  
D. T. McCarthy ◽  
A. Deletic ◽  
V. G. Mitchell ◽  
C. Diaper
Keyword(s):  
Sensitivity Analysis ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Analysis ◽  
Parameter Distribution ◽  
Model Parameters ◽  
Urban Stormwater ◽  
Urban Catchments ◽  
Two Parameters

This paper presents the sensitivity analysis of a newly developed model which predicts microorganism concentrations in urban stormwater (MOPUS—MicroOrganism Prediction in Urban Stormwater). The analysis used Escherichia coli data collected from four urban catchments in Melbourne, Australia. The MICA program (Model Independent Markov Chain Monte Carlo Analysis), used to conduct this analysis, applies a carefully constructed Markov Chain Monte Carlo procedure, based on the Metropolis-Hastings algorithm, to explore the model's posterior parameter distribution. It was determined that the majority of parameters in the MOPUS model were well defined, with the data from the MCMC procedure indicating that the parameters were largely independent. However, a sporadic correlation found between two parameters indicates that some improvements may be possible in the MOPUS model. This paper identifies the parameters which are the most important during model calibration; it was shown, for example, that parameters associated with the deposition of microorganisms in the catchment were more influential than those related to microorganism survival processes. These findings will help users calibrate the MOPUS model, and will help the model developer to improve the model, with efforts currently being made to reduce the number of model parameters, whilst also reducing the slight interaction identified.

scholarly journals SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

2016 ◽  
Vol III-7 ◽  
pp. 119-124 ◽  
Author(s):  
Q. H. Zhao ◽  
Y. Li ◽  
Y. Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Voronoi Tessellation ◽  
Random Variable ◽  
Model Parameters ◽  
Sar Image ◽  
Sar Images ◽  
Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

scholarly journals Inferring soil salinity in a drip irrigation system from multi-configuration EMI measurements using Adaptive Markov Chain Monte Carlo

2016 ◽  
Author(s):  
Khan Zaib Jadoon ◽  
Muhammad Umer Altaf ◽  
Matthew Francis McCabe ◽  
Ibrahim Hoteit ◽  
Nisar Muhammad ◽  
...  
Keyword(s):  
Electrical Conductivity ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Drip Irrigation ◽  
Saline Soil ◽  
Irrigation System ◽  
Earth Model ◽  
Model Parameters ◽  
Drip Irrigation System

Abstract. A substantial interpretation of electromagnetic induction (EMI) measurements requires quantifying optimal model parameters and uncertainty of a nonlinear inverse problem. For this purpose, an adaptive Bayesian Markov chain Monte Carlo (MCMC) algorithm is used to assess multi-orientation and multi-offset EMI measurements in an agriculture field with non-saline and saline soil. In the MCMC simulations, posterior distribution was computed using Bayes rule. The electromagnetic forward model based on the full solution of Maxwell's equations was used to simulate the apparent electrical conductivity measured with the configurations of EMI instrument, the CMD mini-Explorer. The model parameters and uncertainty for the three-layered earth model are investigated by using synthetic data. Our results show that in the scenario of non-saline soil, the parameters of layer thickness are not well estimated as compared to layers electrical conductivity because layer thicknesses in the model exhibits a low sensitivity to the EMI measurements, and is hence difficult to resolve. Application of the proposed MCMC based inversion to the field measurements in a drip irrigation system demonstrate that the parameters of the model can be well estimated for the saline soil as compared to the non-saline soil, and provide useful insight about parameter uncertainty for the assessment of the model outputs.

scholarly journals A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes

2021 ◽  
Vol 12 ◽  
Author(s):  
Oliver Lüdtke ◽  
Esther Ulitzsch ◽  
Alexander Robitzsch
Keyword(s):  
Monte Carlo ◽  
Factor Analysis ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Confirmatory Factor Analysis ◽  
Parameter Estimates ◽  
Penalized Maximum Likelihood ◽  
Confirmatory Factor ◽  
Bayesian Estimators

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as non-convergence or parameter estimates outside the admissible parameter space. In this article, we distinguish different Bayesian estimators that can be used to stabilize the parameter estimates of a CFA: the mode of the joint posterior distribution that is obtained from penalized maximum likelihood (PML) estimation, and the mean (EAP), median (Med), or mode (MAP) of the marginal posterior distribution that are calculated by using Markov Chain Monte Carlo (MCMC) methods. In two simulation studies, we evaluated the performance of the Bayesian estimators from a frequentist point of view. The results show that the EAP produced more accurate estimates of the latent correlation in many conditions and outperformed the other Bayesian estimators in terms of root mean squared error (RMSE). We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded, and we suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

Markov Chain Monte Carlo in Practice

2021 ◽  
Vol 9 (1) ◽  
Author(s):  
Galin L. Jones ◽  
Qian Qin
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Online Publication ◽  
Probability Distributions ◽  
Annual Review ◽  
Publication Date ◽  
Mcmc Simulation ◽  
Essential Set

Markov chain Monte Carlo (MCMC) is an essential set of tools for estimating features of probability distributions commonly encountered in modern applications. For MCMC simulation to produce reliable outcomes, it needs to generate observations representative of the target distribution, and it must be long enough so that the errors of Monte Carlo estimates are small. We review methods for assessing the reliability of the simulation effort, with an emphasis on those most useful in practically relevant settings. Both strengths and weaknesses of these methods are discussed. The methods are illustrated in several examples and in a detailed case study. Expected final online publication date for the Annual Review of Statistics, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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