scholarly journals Markov chain Monte Carlo based on adaptive Metropolis algorithm applied in combined model to deal with the uncertainty of weights of single models

2018 ◽  
Vol 19 (4) ◽  
pp. 1129-1136
Author(s):  
Zhenxiang Xing ◽  
Han Zhang ◽  
Yi Ji ◽  
Gong Xinglong ◽  
Qiang Fu ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Prediction Models ◽  
Metropolis Algorithm ◽  
Coefficient Of Determination ◽  
Percentage Error ◽  
Single Model ◽  
Combined Model ◽  
Study Results

Abstract Reliability and validity of model prediction play a decisive role in water resource simulation and prediction. Among many prediction models, the combined model (CM) is widely used because it can combine the prediction results of multiple single models and make full use of the information provided by various methods. CM is an effective method to improve the predictive veracity but the weight of single model estimation is the key to the CM. Previous studies take errors as the objective function to calculate the weight, and the uncertainty of the weight of the individual model cannot be considered comprehensively. In order to consider the uncertainty of the weight and to improve universal applicability of the CM, in this paper, the authors intend the Markov chain Monte Carlo based on adaptive Metropolis algorithm (AM-MCMC) to solve the weight of a single model in the CM, and obtain the probability distribution of the weight and the joint probability density of all the weight. Finally, the optimal weight combination is obtained. In order to test the validity of the established model, the author put it into the prediction of monthly groundwater level. The two single models in the CM are time series analysis model (TSAM) and grey model (GM (1,1)), respectively. The case study showed that the uncertainty characteristic of the weight in the CM can be obtained by AM-MCMC. According to the study results, CM has obtained a least average root mean square error (RMSE) of 0.85, a mean absolute percentage error (MAPE) of 8.61, and a coefficient of determination (R2) value of 0.97 for the studied forecast period.

scholarly journals An efficient interpolation technique for jump proposals in reversible-jump Markov chain Monte Carlo calculations

2015 ◽  
Vol 2 (6) ◽  
pp. 150030 ◽  
Author(s):  
W. M. Farr ◽  
I. Mandel ◽  
D. Stevens
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Theoretical Models ◽  
Reversible Jump ◽  
Single Model ◽  
Parameter Spaces ◽  
Interpolation Technique ◽  
Important Challenge ◽  
Tree Data Structure

Selection among alternative theoretical models given an observed dataset is an important challenge in many areas of physics and astronomy. Reversible-jump Markov chain Monte Carlo (RJMCMC) is an extremely powerful technique for performing Bayesian model selection, but it suffers from a fundamental difficulty and it requires jumps between model parameter spaces, but cannot efficiently explore both parameter spaces at once. Thus, a naive jump between parameter spaces is unlikely to be accepted in the Markov chain Monte Carlo (MCMC) algorithm and convergence is correspondingly slow. Here, we demonstrate an interpolation technique that uses samples from single-model MCMCs to propose intermodel jumps from an approximation to the single-model posterior of the target parameter space. The interpolation technique, based on a kD-tree data structure, is adaptive and efficient in modest dimensionality. We show that our technique leads to improved convergence over naive jumps in an RJMCMC, and compare it to other proposals in the literature to improve the convergence of RJMCMCs. We also demonstrate the use of the same interpolation technique as a way to construct efficient ‘global’ proposal distributions for single-model MCMCs without prior knowledge of the structure of the posterior distribution, and discuss improvements that permit the method to be used in higher dimensional spaces efficiently.

Statistical Approach on Grading the Student Achievement via Normal Mixture Modeling

2012 ◽  
Author(s):  
Zairul Nor Deana Md. Desa ◽  
Ismail Mohamad ◽  
Zarina Mohd. Khalid ◽  
Hanafiah Md. Zin
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Bayesian Methods ◽  
Mixture Distribution ◽  
Coefficient Of Determination ◽  
Normal Mixture ◽  
Posterior Density ◽  
Letter Grades

Kajian dijalankan untuk membanding keputusan yang didapati daripada tiga kaedah penggredan terhadap pencapaian pelajar. Kaedah konvensional yang popular adalah kaedah Skala Tegak. Pendekatan statistik yang menggunakan kaedah Sisihan Piawai dan kaedah Bayesian bersyarat dipertimbangkan untuk memberi gred. Dalam model Bayesian, dianggapkan bahawa data adalah mengikut taburan Normal Tergabung di mana setiap gred adalah dipisahkan secara berasingan oleh parameter; min dan kadar bandingan dari taburan Normal Tergabung. Masalah yang timbul adalah sukar untuk menganggarkan ketumpatan posterior bagi parameter tersebut secara analitik. Satu penyelesaiannya adalah dengan menggunakan pendekatan Markov Chain Monte Carlo iaitu melalui algoritma pensampelan Gibbs. Kaedah Skala Tegak, kaedah Sisihan Piawai dan kaedah Bayesian bersyarat diaplikasikan untuk markah mentah peperiksaan bagi dua kumpulan pelajar. Pencapaian ketiga–tiga kaedah dibandingkan melalui nilai Kehilangan Kelas Neutral, Kehilangan Kelas Tidak Tegas dan Pekali Penentuan. Didapati keputusan dari kaedah Bayesian bersyarat menunjukkan penggredan yang lebih baik berbanding kaedah Skala Tegak dan kaedah Sisihan Piawai. Kata kunci: Kaedah penggredan, pengukuran pendidikan, Skala Tegak, kaedah Sisihan Piawai, Normal Tergabung, Markov Chain Monte Carlo, pensampelan Gibbs The purpose of this study is to compare results obtained from three methods of assigning letter grades to students’ achievement. The conventional and the most popular method to assign grades is the Straight Scale method (SS). Statistical approaches which used the Standard Deviation (GC) and conditional Bayesian methods are considered to assign the grades. In the conditional Bayesian model, we assume the data to follow the Normal Mixture distribution where the grades are distinctively separated by the parameters: means and proportions of the Normal Mixture distribution. The problem lies in estimating the posterior density of the parameters which is analytically intractable. A solution to this problem is using the Markov Chain Monte Carlo approach namely Gibbs sampler algorithm. The Straight Scale, Standard Deviation and Conditional Bayesian methods are applied to the examination raw scores of two sets of students. The performances of these methods are measured using the Neutral Class Loss, Lenient Class Loss and Coefficient of Determination. The results showed that Conditional Bayesian outperformed the Conventional Methods of assigning grades. Key words: Grading methods, educational measurement, Straight Scale, Standard Deviation method, Normal Mixture, Markov Chain Monte Carlo, Gibbs sampling

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