Reversible Jump Markov Chain Monte Carlo Method Research Articles

Bayesian inference for order determination of double threshold variables autoregressive models

Abstract The reversible-jump Markov chain Monte Carlo (RJMCMC) algorithm can generate a jump Markov chain in the parameter space of different dimensions, and select a suitable model effectively. In this paper, when the order of the double threshold variables autoregressive (DT-AR) is unknown, the RJMCMC method is designed to identify the order of the DT-AR model in this paper. The simulation experiments and the real example show that the proposed method works well in identifying the order and estimating the parameters of the DT-AR model simultaneously.

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

Lorentzian-Model-Based Bayesian Analysis for Automated Estimation of Attenuated Resonance Spectrum

Extracting information from a signal exhibiting damped resonances is a challenging task in many practical cases due to the presence of noise and high attenuation. The interpretation of the signal relies on a model whose order (i.e., the number of resonances) is in general unknown. In this study, the signal is modeled as a sum of Lorentzian lineshapes, and a Bayesian framework is designed to simultaneously remove the baseline distortion, select the number of resonances, and recover the parameters of each lineshape including frequency, damping factor, resonance amplitude, and noise magnitude. The Bayesian problem is solved resorting to a reversible jump Markov chain Monte Carlo (RJ-MCMC) sampling scheme. The algorithm is tested on synthetic signals as well as experimental data from a resonant ultrasound spectroscopy experiment aiming to measure elastic properties. The results show that, compared to the well-known linear prediction singular value decomposition method, the RJ-MCMC method achieves a better performance with the advantages of joint model selection, high accuracy estimation, and uncertainty evaluation. We found that when the signal-to-noise-ratio is larger than 20 dB, the average relative error for frequency extraction is smaller than $\text{0.5}\%$ . Such an algorithm enables to estimate the number of resonances and extract tens of resonance parameters from a highly attenuated spectrum, which can significantly facilitate the automated processing of signals exhibiting damped resonances.

Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible Jump MCMC

This paper proposes the important issues in signal segmentation. The signal is disturbed by multiplicative noise where the number of segments is unknown. A Bayesian approach is proposed to estimate the parameter. The parameter includes the number of segments, the location of the segment, and the amplitude. The posterior distribution for the parameter does not have a simple equation so that the Bayes estimator is not easily determined. Reversible Jump Markov chain Monte Carlo (MCMC) method is adopted to overcome the problem. The Reversible Jump MCMC method creates a Markov chain whose distribution is close to the posterior distribution. The performance of the algorithm is shown by simulation data. The result of this simulation shows that the algorithm works well. As an application, the algorithm is used to segment a Synthetic Aperture Radar (SAR) signal. The advantage of this method is that the number of segments, the position of the segment change, and the amplitude are estimated simultaneously.

Random frog: An efficient reversible jump Markov Chain Monte Carlo-like approach for variable selection with applications to gene selection and disease classification

The identification of disease-relevant genes represents a challenge in microarray-based disease diagnosis where the sample size is often limited. Among established methods, reversible jump Markov Chain Monte Carlo (RJMCMC) methods have proven to be quite promising for variable selection. However, the design and application of an RJMCMC algorithm requires, for example, special criteria for prior distributions. Also, the simulation from joint posterior distributions of models is computationally extensive, and may even be mathematically intractable. These disadvantages may limit the applications of RJMCMC algorithms. Therefore, the development of algorithms that possess the advantages of RJMCMC methods and are also efficient and easy to follow for selecting disease-associated genes is required. Here we report a RJMCMC-like method, called random frog that possesses the advantages of RJMCMC methods and is much easier to implement. Using the colon and the estrogen gene expression datasets, we show that random frog is effective in identifying discriminating genes. The top 2 ranked genes for colon and estrogen are Z50753, U00968, and Y10871_at, Z22536_at, respectively. (The source codes with GNU General Public License Version 2.0 are freely available to non-commercial users at: http://code.google.com/p/randomfrog/.).

Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving

We present an application of the reversible jump Markov chain Monte Carlo (RJMCMC) method to the important problem of setting claims reserves in general insurance business for the outstanding loss liabilities. A measure of the uncertainty in these claims reserves estimates is also needed for solvency purposes. The RJMCMC method described in this paper represents an improvement over the manual processes often employed in practice. In particular, our RJMCMC method describes parameter reduction and tail factor estimation in the claims reserving process, and, moreover, it provides the full predictive distribution of the outstanding loss liabilities.

Reversible jump MCMC methods for fully automatic motion analysis in tagged MRI

Tagged magnetic resonance imaging (tMRI) is a well-known noninvasive method for studying regional heart dynamics. It offers great potential for quantitative analysis of a variety of kine(ma)tic parameters, but its clinical use has so far been limited, in part due to the lack of robustness and accuracy of existing tag tracking algorithms in dealing with low (and intrinsically time-varying) image quality. In this paper, we evaluate the performance of four frequently used concepts found in the literature (optical flow, harmonic phase (HARP) magnetic resonance imaging, active contour fitting, and non-rigid image registration) for cardiac motion analysis in 2D tMRI image sequences, using both synthetic image data (with ground truth) and real data from preclinical (small animal) and clinical (human) studies. In addition we propose a new probabilistic method for tag tracking that serves as a complementary step to existing methods. The new method is based on a Bayesian estimation framework, implemented by means of reversible jump Markov chain Monte Carlo (MCMC) methods, and combines information about the heart dynamics, the imaging process, and tag appearance. The experimental results demonstrate that the new method improves the performance of even the best of the four previous methods. Yielding higher consistency, accuracy, and intrinsic tag reliability assessment, the proposed method allows for improved analysis of cardiac motion.

Bayesian model selection in ARFIMA models

Various model selection criteria such as Akaike information criterion (AIC; Akaike, 1973), Bayesian information criterion (BIC; Akaike, 1979) and Hannan–Quinn criterion (HQC; Hannan, 1980) are used for model specification in autoregressive fractional integrated moving average (ARFIMA) models. Classical model selection criteria require to calculate both model parameters and order. This kind of approach needs much time. However, in the literature, there are proposed methods which calculate model parameters and order at the same time such as reversible jump Markov chain Monte Carlo (RJMCMC) method, Carlin and Chib (CC) method. In this paper, we proposed two new methods that are using RJMCMC method. The proposed methods are compared with classical methods by a simulation study. We obtained that our methods outperform classical methods in most cases.

Reversible Jump Markov Chain Monte Carlo for Deconvolution

To solve the problem of estimating an unknown input function to a linear time invariant system we propose an adaptive non-parametric method based on reversible jump Markov chain Monte Carlo (RJMCMC). We use piecewise polynomial functions (splines) to represent the input function. The RJMCMC algorithm allows the exploration of a large space of competing models, in our case the collection of splines corresponding to alternative positions of breakpoints, and it is based on the specification of transition probabilities between the models. RJMCMC determines: the number and the position of the breakpoints, and the coefficients determining the shape of the spline, as well as the corresponding posterior distribution of breakpoints, number of breakpoints, coefficients and arbitrary statistics of interest associated with the estimation problem. Simulation studies show that the RJMCMC method can obtain accurate reconstructions of complex input functions, and obtains better results compared with standard non-parametric deconvolution methods. Applications to real data are also reported.

Bayesian model selection and parameter estimation of nuclear emission spectra using RJMCMC

This paper addresses the general problem of estimating parameters in nuclear spectroscopy. We present a unified Bayesian formulation to tackle the various aspects of this problem. This includes deconvolution and modelling of both the peaks and background. The peaks are modelled with Gaussian or Lorentz-type functions and the background with cubic B-splines. The Bayesian model allows us to define a posterior probability in the parameter space upon which all subsequent Bayesian inference is based. Direct evaluation of this distribution or its derived features such as the conditional expectation is, unfortunately, not possible on account of the need to evaluate high-dimension integrals. As such we resort to a stochastic numerical Bayesian technique, the reversible-jump Markov-chain Monte-Carlo method. We have carried out simulations on both artificial and real data. Our results on the 1995 IAEA γ-ray test spectra shows that our program performs better than those previously reported.

Cluster modelling of disease incidence via RJMCMC methods: a comparative evaluation. Reversible jump Markov chain Monte Carlo.

The spatial modelling of small area health data has, for some time, included spatial autocorrelation as a random effect. This effect is non-specific and global and does not address the location of clusters of disease (a specific task). This paper addresses the need for specific and non-specific random effects within spatial epidemiology. In addition, individual frailty is also considered important and a computational algorithm based on reversible jump Markov chain Monte Carlo (RJMCMC) methods is described.