MRI-based brain tumor segmentation using Gaussian mixture model with reversible jump Markov chain Monte Carlo algorithm

Abstract. Atmospheric trace gas inversions often attempt to attribute fluxes to a high-dimensional grid using observations. To make this problem computationally feasible, and to reduce the degree of under-determination, some form of dimension reduction is usually performed. Here, we present an objective method for reducing the spatial dimension of the parameter space in atmospheric trace gas inversions. In addition to solving for a set of unknowns that govern emissions of a trace gas, we set out a framework that considers the number of unknowns to itself be an unknown. We rely on the well-established reversible-jump Markov chain Monte Carlo algorithm to use the data to determine the dimension of the parameter space. This framework provides a single-step process that solves for both the resolution of the inversion grid, as well as the magnitude of fluxes from this grid. Therefore, the uncertainty that surrounds the choice of aggregation is accounted for in the posterior parameter distribution. The posterior distribution of this transdimensional Markov chain provides a naturally smoothed solution, formed from an ensemble of coarser partitions of the spatial domain. We describe the form of the reversible-jump algorithm and how it may be applied to trace gas inversions. We build the system into a hierarchical Bayesian framework in which other unknown factors, such as the magnitude of the model uncertainty, can also be explored. A pseudo-data example is used to show the usefulness of this approach when compared to a subjectively chosen partitioning of a spatial domain. An inversion using real data is also shown to illustrate the scales at which the data allow for methane emissions over north-west Europe to be resolved.

Download Full-text

A reversible jump Markov chain Monte Carlo algorithm for analysis of functional neuroimages

Proceedings International Conference on Image Processing ICIP-02 ◽

10.1109/icip.2002.1038922 ◽

2003 ◽

Cited By ~ 1

Author(s):

A.S. Lukic ◽

M.N. Wernick ◽

N.P. Galatsanos ◽

Yongyi Yang ◽

S.C. Strother

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Monte Carlo Algorithm ◽

Reversible Jump

Download Full-text

Modelling heterotachy in phylogenetic inference by reversible-jump Markov chain Monte Carlo

Philosophical Transactions of the Royal Society B Biological Sciences ◽

10.1098/rstb.2008.0178 ◽

2008 ◽

Vol 363 (1512) ◽

pp. 3955-3964 ◽

Cited By ~ 52

Author(s):

Mark Pagel ◽

Andrew Meade

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Mixture Model ◽

Branch Length ◽

Monte Carlo Algorithm ◽

Morphological Data ◽

Reversible Jump ◽

Rate Variation ◽

Ancestral State

The rate at which a given site in a gene sequence alignment evolves over time may vary. This phenomenon—known as heterotachy—can bias or distort phylogenetic trees inferred from models of sequence evolution that assume rates of evolution are constant. Here, we describe a phylogenetic mixture model designed to accommodate heterotachy. The method sums the likelihood of the data at each site over more than one set of branch lengths on the same tree topology. A branch-length set that is best for one site may differ from the branch-length set that is best for some other site, thereby allowing different sites to have different rates of change throughout the tree. Because rate variation may not be present in all branches, we use a reversible-jump Markov chain Monte Carlo algorithm to identify those branches in which reliable amounts of heterotachy occur. We implement the method in combination with our ‘pattern-heterogeneity’ mixture model, applying it to simulated data and five published datasets. We find that complex evolutionary signals of heterotachy are routinely present over and above variation in the rate or pattern of evolution across sites, that the reversible-jump method requires far fewer parameters than conventional mixture models to describe it, and serves to identify the regions of the tree in which heterotachy is most pronounced. The reversible-jump procedure also removes the need for a posteriori tests of ‘significance’ such as the Akaike or Bayesian information criterion tests, or Bayes factors. Heterotachy has important consequences for the correct reconstruction of phylogenies as well as for tests of hypotheses that rely on accurate branch-length information. These include molecular clocks, analyses of tempo and mode of evolution, comparative studies and ancestral state reconstruction. The model is available from the authors' website, and can be used for the analysis of both nucleotide and morphological data.

Download Full-text

Bayesian estimation of a phenotype network structure using reversible jump Markov chain Monte Carlo

10.26686/wgtn.17012435.v1 ◽

2021 ◽

Author(s):

◽

Lisa Woods

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Network Structure ◽

Monte Carlo Algorithm ◽

Maximum Likelihood Estimates ◽

Reversible Jump ◽

Network Structures ◽

Double Step ◽

Phenotype Network

<p>In this thesis we aim to estimate the unknown phenotype network structure existing among multiple interacting quantitative traits, assuming the genetic architecture is known. We begin by taking a frequentist approach and implement a score-based greedy hill-climbing search strategy using AICc to estimate an unknown phenotype network structure. This approach was inconsistent and overfitting was common, so we then propose a Bayesian approach that extends on the reversible jump Markov chain Monte Carlo algorithm. Our approach makes use of maximum likelihood estimates in the chain, so we have an efficient sampler using well-tuned proposal distributions. The common approach is to assume uniform priors over all network structures; however, we introduce a prior on the number of edges in the phenotype network structure, which prefers simple models with fewer directed edges. We determine that the relationship between the prior penalty and the joint posterior probability of the true model is not monotonic, there is some interplay between the two. Simulation studies were carried out and our approach is also applied to a published data set. It is determined that larger trait-to-trait effects are required to recover the phenotype network structure; however, mixing is generally slow, a common occurrence with reversible jump Markov chain Monte Carlo methods. We propose the use of a double step to combine two steps that alter the phenotype network structure. This proposes larger steps than the traditional birth and death move types, possibly changing the dimension of the model by more than one. This double step helped the sampler move between different phenotype network structures in simulated data sets.</p>

Download Full-text

A reversible-jump Markov chain Monte Carlo algorithm for 1D inversion of magnetotelluric data

Computers & Geosciences ◽

10.1016/j.cageo.2018.01.011 ◽

2018 ◽

Vol 113 ◽

pp. 94-105 ◽

Cited By ~ 11

Author(s):

Eric Mandolesi ◽

Xenia Ogaya ◽

Joan Campanyà ◽

Nicola Piana Agostinetti

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Monte Carlo Algorithm ◽

Reversible Jump ◽

Magnetotelluric Data

Download Full-text

Bayesian estimation of a phenotype network structure using reversible jump Markov chain Monte Carlo

10.26686/wgtn.17012435 ◽

2021 ◽

Author(s):

◽

Lisa Woods

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Network Structure ◽

Monte Carlo Algorithm ◽

Maximum Likelihood Estimates ◽

Reversible Jump ◽

Network Structures ◽

Double Step ◽

Phenotype Network

<p>In this thesis we aim to estimate the unknown phenotype network structure existing among multiple interacting quantitative traits, assuming the genetic architecture is known. We begin by taking a frequentist approach and implement a score-based greedy hill-climbing search strategy using AICc to estimate an unknown phenotype network structure. This approach was inconsistent and overfitting was common, so we then propose a Bayesian approach that extends on the reversible jump Markov chain Monte Carlo algorithm. Our approach makes use of maximum likelihood estimates in the chain, so we have an efficient sampler using well-tuned proposal distributions. The common approach is to assume uniform priors over all network structures; however, we introduce a prior on the number of edges in the phenotype network structure, which prefers simple models with fewer directed edges. We determine that the relationship between the prior penalty and the joint posterior probability of the true model is not monotonic, there is some interplay between the two. Simulation studies were carried out and our approach is also applied to a published data set. It is determined that larger trait-to-trait effects are required to recover the phenotype network structure; however, mixing is generally slow, a common occurrence with reversible jump Markov chain Monte Carlo methods. We propose the use of a double step to combine two steps that alter the phenotype network structure. This proposes larger steps than the traditional birth and death move types, possibly changing the dimension of the model by more than one. This double step helped the sampler move between different phenotype network structures in simulated data sets.</p>

Download Full-text

A Gaussian Mixture Model as a Proposal Distribution for Efficient Markov-Chain Monte Carlo Characterization of Uncertainty in Reservoir Description and Forecasting

SPE Journal ◽

10.2118/182684-pa ◽

2019 ◽

Vol 25 (01) ◽

pp. 001-036 ◽

Cited By ~ 1

Author(s):

Xin Li ◽

Albert C. Reynolds

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Random Walk ◽

Markov Chain Monte Carlo ◽

Uncertainty Quantification ◽

Gaussian Mixture Model ◽

Gaussian Mixture ◽

Mcmc Algorithm ◽

Proposal Distribution

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.

Download Full-text

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

Applied Sciences ◽

10.3390/app11167343 ◽

2021 ◽

Vol 11 (16) ◽

pp. 7343

Author(s):

Dwi Rantini ◽

Nur Iriawan ◽

Irhamah Irhamah

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Weibull Distribution ◽

Mixture Distribution ◽

Gaussian Mixture ◽

Visual Observation ◽

Reversible Jump ◽

Leibler Divergence ◽

Gaussian Mixture Distribution

Data with a multimodal pattern can be analyzed using a mixture model. In a mixture model, the most important step is the determination of the number of mixture components, because finding the correct number of mixture components will reduce the error of the resulting model. In a Bayesian analysis, one method that can be used to determine the number of mixture components is the reversible jump Markov chain Monte Carlo (RJMCMC). The RJMCMC is used for distributions that have location and scale parameters or location-scale distribution, such as the Gaussian distribution family. In this research, we added an important step before beginning to use the RJMCMC method, namely the modification of the analyzed distribution into location-scale distribution. We called this the non-Gaussian RJMCMC (NG-RJMCMC) algorithm. The following steps are the same as for the RJMCMC. In this study, we applied it to the Weibull distribution. This will help many researchers in the field of survival analysis since most of the survival time distribution is Weibull. We transformed the Weibull distribution into a location-scale distribution, which is the extreme value (EV) type 1 (Gumbel-type for minima) distribution. Thus, for the mixture analysis, we call this EV-I mixture distribution. Based on the simulation results, we can conclude that the accuracy level is at minimum 95%. We also applied the EV-I mixture distribution and compared it with the Gaussian mixture distribution for enzyme, acidity, and galaxy datasets. Based on the Kullback–Leibler divergence (KLD) and visual observation, the EV-I mixture distribution has higher coverage than the Gaussian mixture distribution. We also applied it to our dengue hemorrhagic fever (DHF) data from eastern Surabaya, East Java, Indonesia. The estimation results show that the number of mixture components in the data is four; we also obtained the estimation results of the other parameters and labels for each observation. Based on the Kullback–Leibler divergence (KLD) and visual observation, for our data, the EV-I mixture distribution offers better coverage than the Gaussian mixture distribution.

Download Full-text