scholarly journals Robust and fast Monte Carlo Markov Chain sampling of diffusion MRI microstructure models

2018 ◽  
Author(s):  
R.L. Harms ◽  
A. Roebroeck
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Sample Size ◽  
Diffusion Mri ◽  
Monte Carlo Markov Chain ◽  
Point Estimate ◽  
Model Parameters ◽  
Effective Sample Size ◽  
Compartment Models ◽  
Mcmc Sampling

AbstractIn diffusion MRI analysis, advances in biophysical multi-compartment modeling have gained popularity over the conventional Diffusion Tensor Imaging (DTI), because they possess greater specificity in relating the dMRI signal to underlying cellular microstructure. Biophysical multi-compartment models require parameter estimation, typically performed using either Maximum Likelihood Estimation (MLE) or using Monte Carlo Markov Chain (MCMC) sampling. Whereas MLE provides only a point estimate of the fitted model parameters, MCMC recovers the entire posterior distribution of the model parameters given the data, providing additional information such as parameter uncertainty and correlations. MCMC sampling is currently not routinely applied in dMRI microstructure modeling because it requires adjustments and tuning specific to each model, particularly in the choice of proposal distributions, burn-in length, thinning and the number of samples to store. In addition, sampling often takes at least an order of magnitude more time than non-linear optimization. Here we investigate the performance of MCMC algorithm variations over multiple popular diffusion microstructure models to see whether a single well performing variation could be applied efficiently and robustly to many models. Using an efficient GPU-based implementation, we show that run times can be removed as a prohibitive constraint for sampling of diffusion multi-compartment models. Using this implementation, we investigated the effectiveness of different adaptive MCMC algorithms, burn-in, initialization and thinning. Finally we apply the theory of Effective Sample Size to diffusion multi-compartment models as a way of determining a relatively general target for the number of samples needed to characterize parameter distributions for different models and datasets. We conclude that robust and fast sampling is achieved in most diffusion microstructure models with the Adaptive Metropolis-Within-Gibbs (AMWG) algorithm initialized with an MLE point estimate, in which case 100 to 200 samples are sufficient as a burn-in and thinning is mostly unnecessary. As a relatively general target for the number of samples, we recommend a multivariate Effective Sample Size of 2200.

Adaptive Monte-Carlo Markov chain for multivariate statistical estimation

2012 ◽  
Author(s):  
Ingrida Vaiciulyte
Keyword(s):  
Computer Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Sample Size ◽  
Test Sample ◽  
Monte Carlo Markov Chain ◽  
Likelihood Method ◽  
Estimation Of Parameters ◽  
Multivariate Statistical ◽  
T Distribution

The estimation of the multivariate skew t-distribution by the Monte-Carlo Markov Chain (MCMC) method is considered in the paper. Thus, the MCMC procedure is constructed for recurrent estimation of skew t-distribution, following the maximum likelihood method, where the Monte-Carlo sample size is regulated to ensure the convergence and to decrease the total amount of Monte-Carlo trials, required for estimation. The confidence intervals of Monte-Carlo estimators are introduced because of their asymptotic normality. The termination rule is also implemented by testing statistical hypotheses on an insignificant change of estimates in two steps of the procedure. The algorithm developed has been tested by computer simulation with test example. The test sample, following from skew t-distribution, has been simulated by computer and parameters of the skew t-distribution have been estimated by MathCAD. Next, the chi-squared criterion confirmed the hypothesis of distribution of statistics with respect to under- lying distribution function. Thus, computer simulation confirmed the applicability of the Monte-Carlo Markov chain approach with adaptively regulated sample size for estimation of parameters of the skew t- distribution with acceptable accuracy.

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 Modelling the spread of American foulbrood in honeybees

2013 ◽  
Vol 10 (88) ◽  
pp. 20130650 ◽  
Author(s):  
Samik Datta ◽  
James C. Bull ◽  
Giles E. Budge ◽  
Matt J. Keeling
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Control Strategies ◽  
Sir Model ◽  
Model Parameters ◽  
American Foulbrood ◽  
Inspection Period ◽  
Stochastic Sir ◽  
Stochastic Sir Model ◽  
The Impact

We investigate the spread of American foulbrood (AFB), a disease caused by the bacterium Paenibacillus larvae , that affects bees and can be extremely damaging to beehives. Our dataset comes from an inspection period carried out during an AFB epidemic of honeybee colonies on the island of Jersey during the summer of 2010. The data include the number of hives of honeybees, location and owner of honeybee apiaries across the island. We use a spatial SIR model with an underlying owner network to simulate the epidemic and characterize the epidemic using a Markov chain Monte Carlo (MCMC) scheme to determine model parameters and infection times (including undetected ‘occult’ infections). Likely methods of infection spread can be inferred from the analysis, with both distance- and owner-based transmissions being found to contribute to the spread of AFB. The results of the MCMC are corroborated by simulating the epidemic using a stochastic SIR model, resulting in aggregate levels of infection that are comparable to the data. We use this stochastic SIR model to simulate the impact of different control strategies on controlling the epidemic. It is found that earlier inspections result in smaller epidemics and a higher likelihood of AFB extinction.

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.

scholarly journals Locally Scaled and Stochastic Volatility Metropolis– Hastings Algorithms

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 351
Author(s):  
Wilson Tsakane Mongwe ◽  
Rendani Mbuvha ◽  
Tshilidzi Marwala
Keyword(s):  
Monte Carlo ◽  
Sample Size ◽  
Stochastic Volatility ◽  
Diffusion Processes ◽  
Predictive Performance ◽  
Jump Diffusion ◽  
Effective Sample Size ◽  
Jump Diffusion Processes ◽  
Current State ◽  
Mh Algorithm

Markov chain Monte Carlo (MCMC) techniques are usually used to infer model parameters when closed-form inference is not feasible, with one of the simplest MCMC methods being the random walk Metropolis–Hastings (MH) algorithm. The MH algorithm suffers from random walk behaviour, which results in inefficient exploration of the target posterior distribution. This method has been improved upon, with algorithms such as Metropolis Adjusted Langevin Monte Carlo (MALA) and Hamiltonian Monte Carlo being examples of popular modifications to MH. In this work, we revisit the MH algorithm to reduce the autocorrelations in the generated samples without adding significant computational time. We present the: (1) Stochastic Volatility Metropolis–Hastings (SVMH) algorithm, which is based on using a random scaling matrix in the MH algorithm, and (2) Locally Scaled Metropolis–Hastings (LSMH) algorithm, in which the scaled matrix depends on the local geometry of the target distribution. For both these algorithms, the proposal distribution is still Gaussian centred at the current state. The empirical results show that these minor additions to the MH algorithm significantly improve the effective sample rates and predictive performance over the vanilla MH method. The SVMH algorithm produces similar effective sample sizes to the LSMH method, with SVMH outperforming LSMH on an execution time normalised effective sample size basis. The performance of the proposed methods is also compared to the MALA and the current state-of-art method being the No-U-Turn sampler (NUTS). The analysis is performed using a simulation study based on Neal’s funnel and multivariate Gaussian distributions and using real world data modeled using jump diffusion processes and Bayesian logistic regression. Although both MALA and NUTS outperform the proposed algorithms on an effective sample size basis, the SVMH algorithm has similar or better predictive performance when compared to MALA and NUTS across the various targets. In addition, the SVMH algorithm outperforms the other MCMC algorithms on a normalised effective sample size basis on the jump diffusion processes datasets. These results indicate the overall usefulness of the proposed algorithms.

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