Bayesian partial pooling to reduce uncertainty in overcoring rock stress estimation

The state of in situ stress is a crucial parameter in subsurface engineering,especially for critical projects like nuclear waste repository.As one of the two ISRM suggested methods,the overcoring(OC)method is widely used to estimate the full stress tensors in rocks by independent regression analysis of the data from each OC test.However,such customary independent analysis of individual OC tests,known as no pooling,is liable to yield unreliable test-specific stress estimates due to various uncertainty sources involved in the OC method.To address this problem,a practical and no-cost solution is considered by incorporating into OC data analysis additional information implied within adjacent OC tests,which are usually available in OC measurement campaigns.Hence,this paper presents a Bayesian partial pooling(hierarchical)model for combined analysis of adjacent OC tests.We performed five case studies using OC test data made at a nuclear waste repository research site of Sweden.The results demonstrate that partial pooling of adjacent OC tests indeed allows borrowing of information across adjacent tests,and yields improved stress tensor estimates with reduced uncertainties simultaneously for all individual tests than they are independently analysed as no pooling,particularly for those unreliable no pooling stress estimates.A further model comparison shows that the partial pooling model also gives better predictive performance,and thus confirms that the information borrowed across adjacent OC tests is relevant and effective.

Local and regional flood frequency analysis based on hierarchical Bayesian model in Dongting Lake Basin,China

This study developed a hierarchical Bayesian(HB)model for local and regional flood frequency analysis in the Dongting Lake Basin,in China.The annual maximum daily flows from 15 streamflow-gauged sites in the study area were analyzed with the HB model.The generalized extreme value(GEV)distribution was selected as the extreme flood distribution,and the GEV distribution location and scale parameters were spatially modeled through a regression approach with the drainage area as a covariate.The Markov chain Monte Carlo(MCMC)method with Gibbs sampling was employed to calculate the posterior distribution in the HB model.The results showed that the proposed HB model provided satisfactory Bayesian credible intervals for flood quantiles,while the traditional delta method could not provide reliable uncertainty estimations for large flood quantiles,due to the fact that the lower confidence bounds tended to decrease as the return periods increased.Furthermore,the HB model for regional analysis allowed for a reduction in the value of some restrictive assumptions in the traditional index flood method,such as the homogeneity region assumption and the scale invariance assumption.The HB model can also provide an uncertainty band of flood quantile prediction at a poorly gauged or ungauged site,but the index flood method with L-moments does not demonstrate this uncertainty directly.Therefore,the HB model is an effective method of implementing the flexible local and regional frequency analysis scheme,and of quantifying the associated predictive uncertainty.

A Bayesian hierarchical model for the inference between metal grade with reduced variance:Case studies in porphyry Cu deposits

Ore sorting is a preconcentration technology and can dramatically reduce energy and water usage to improve the sustainability and profitability of a mining operation.In porphyry Cu deposits,Cu is the primary target,with ores usually containing secondary‘pay’metals such as Au,Mo and gangue elements such as Fe and As.Due to sensing technology limitations,secondary and deleterious materials vary in correlation type and strength with Cu but cannot be detected simultaneously via magnetic resonance(MR)ore sorting.Inferring the relationships between Cu and other elemental abundances is particularly critical for mineral processing.The variations in metal grade relationships occur due to the transition into different geological domains.This raises two questions-how to define these geological domains and how the metal grade relationship is influenced by these geological domains.In this paper,linear relationship is assumed between Cu grade and other metal grades.We applies a Bayesian hierarchical(partial-pooling)model to quantify the linear relationships between Cu,Au,and Fe grades from geochemical bore core data.The hierarchical model was compared with two other models-‘complete-pooling’model and‘nopooling’model.Mining blocks were split based on spatial domain to construct hierarchical model.Geochemical bore core data records metal grades measured from laboratory assay with spatial coordinates of sample location.Two case studies from different porphyry Cu deposits were used to evaluate the performance of the hierarchical model.Markov chain Monte Carlo(MCMC)was used to sample the posterior parameters.Our results show that the Bayesian hierarchical model dramatically reduced the posterior predictive variance for metal grades regression compared to the no-pooling model.In addition,the posterior inference in the hierarchical model is insensitive to the choice of prior.The data is wellrepresented in the posterior which indicates a robust model.The results show that the spatial domain can be successfully utilised for metal grade regression.Uncertainty in estimating the relationship between pay metals and both secondary and gangue elements is quantified and shown to be reduced with partial-pooling.Thus,the proposed Bayesian hierarchical model can offer a reliable and stable way to monitor the relationship between metal grades for ore sorting and other mineral processing options.

Bayesian meta-analysis of regional biomass factors for Quercus mongolica forests in South Korea

Indirect approaches to estimation of biomass factors are often applied to measure carbon flux in the forestry sector. An assumption underlying a country-level carbon stock estimate is the representativeness of these factors. Although intensive studies have been conducted to quantify biomass factors, each study typically covers a limited geographic area. The goal of this study was to employ a meta-analysis approach to develop regional bio- mass factors for Quercus mongolica forests in South Korea. The biomass factors of interest were biomass conversion and expansion factor （BCEF）, biomass expansion factor （BEF） and root-to-shoot ratio （RSR）. Our objectives were to select probability density functions （PDFs） that best fitted the three biomass factors and to quantify their means and uncertainties. A total of 12 scientific publications were selected as data sources based on a set of criteria. Fromthese publications we chose 52 study sites spread out across South Korea. The statistical model for the meta- analysis was a multilevel model with publication （data source） as the nesting factor specified under the Bayesian framework. Gamma, Log-normal and Weibull PDFs were evaluated. The Log-normal PDF yielded the best quanti- tative and qualitative fit for the three biomass factors. However, a poor fit of the PDF to the long right tail of observed BEF and RSR distributions was apparent. The median posterior estimates for means and 95 % credible intervals for BCEF, BEF and RSR across all 12 publica- tions were 1.016 （0.800-1.299）, 1.414 （1.304-1.560） and 0.260 （0.200-0.335）, respectively. The Log-normal PDF proved useful for estimating carbon stock of Q. mongolica forests on a regional scale and for uncertainty analysis based on Monte Carlo simulation.

Population dynamics modelling with spatial heterogeneity for yellow croaker(Larimichthys polyactis)along the coast of China

As one of the top four commercially important species in China,yellow croaker(Larimichthys polyactis)with two geographic subpopulations,has undergone profound changes during the last several decades.It is widely comprehended that understanding its population dynamics is critically important for sustainable management of this valuable fishery in China.The only two existing population dynamics models assessed the population of yellow croaker using short time-series data,without considering geographical variations.In this study,Bayesian models with and without hierarchical subpopulation structure were developed to explore the spatial heterogeneity of the population dynamics of yellow croaker from 1968 to 2015.Alternative hypotheses were constructed to test potential temporal patterns in yellow croaker’s population dynamics.Substantial variations in population dynamics characteristics among space and time were found through this study.The population growth rate was revealed to increase since the late 1980s,and the catchability increased more than twice from 1981 to 2015.The East China Sea’s subpopulation witnesses faster growth,but suffers from higher fishing pressure than that in the Bohai Sea and Yellow Sea.The global population and two subpopulations all have high risks of overfishing and being overfished according to the MSY-based reference points in recent years.More conservative management strategies with subpopulation considerations are imperative for the fishery management of yellow croaker in China.The methodology developed in this study could also be applied to the stock assessment and fishery management of other species,especially for those species with large spatial heterogeneity data.

Bayesian spatial modeling to incorporate unmeasured information at road segment levels with the INLA approach:A methodological advancement of estimating crash modification factors

Estimating safety effectiveness of roadway improvements and countermeasures,using cross-sectional models,generally requires large amounts of data such as road geometric and traffic-related characteristics at road segment levels.These models do not consider all confounding crash contributory factors such as driving culture and environmental conditions at the segment level due to a lack of readily available data.This may result in inaccurate models representing actual conditions at road segment levels,followed by erroneous estimations of safety effectiveness.To minimize the effect of not including such variables,this study develops a new methodology to estimate safety effectiveness of roadway countermeasures,based on generalized linear mixed models,assuming zeroinflated Poisson distribution for the response,and adjusting for spatial autocorrelation using the spatial random effect.The Bayesian approach,with Integrated Nested Laplace Approximation,was used to make inference on this model with computational efficiency.Results showed that incorporating a spatial random effect into the models provided better model fit than non-spatial models;hence,estimated safety effectiveness based on such models is more accurate.The proposed approach is a methodological advancement in traffic safety,which allows evaluation of safety effectiveness or roadway improvements when data are not readily available.

Combining multiple imperfect data sources for small area estimation:a Bayesian model of provincial fertility rates in Cambodia

Demographic estimation becomes a problem of small area estimation when detaileddisaggregation leads to small cell counts.The usual difficulties of small area estimation are compounded when the available data sources contain measurement errors.We present a Bayesianapproach to the problem of small area estimation with imperfect data sources.The overall modelcontains separate submodels for underlying demographic processes and for measurement processes.All unknown quantities in the model,including coverage ratios and demographic rates,are estimated jointly via Markov chain Monte Carlo methods.The approach is illustrated usingthe example of provincial fertility rates in Cambodia.

Bayesian functional enrichment analysis for the Reactome database

The first step in the analysis of high-throughput experiment results is often to identify genes orproteins with certain characteristics, such as genes being differentially expressed (DE). To gainmore insights into the underlying biology, functional enrichment analysis is then conductedto provide functional interpretation for the identified genes or proteins. The hypergeometricP value has been widely used to investigate whether genes from predefined functional terms,e.g., Reactome, are enriched in the DE genes. The hypergeometric P value has several limitations: (1) computed independently for each term, thus neglecting biological dependence;(2) subject to a size constraint that leads to the tendency of selecting less-specific terms. In this paper,a Bayesian approach is proposed to overcome these limitations by incorporating the interconnected dependence structure of biological functions in the Reactome database through a CARprior in a Bayesian hierarchical logistic model. The inference on functional enrichment is thenbased on posterior probabilities that are immune to the size constraint. This method can detectmoderate but consistent enrichment signals and identify sets of closely related and biologicallymeaningful functional terms rather than isolated terms. The performance of the Bayesian methodis demonstrated via a simulation study and a real data application.

Portfolio optimisation using constrained hierarchical bayes models

It iswell known that traditionalmean-variance optimal portfolio delivers rather erratic and unsatisfactory out-of-sample performance due to the neglect of estimation errors.Constrained solutions,such as no-short-sale-constrained and norm-constrained portfolios,can usually achieve much higher ex post Sharpe ratio.Bayesian methods have also been shown to be superior to traditional plug-in estimator by incorporating parameter uncertainty through prior distributions.In this paper,we develop an innovative method that induces priors directly on optimal portfolio weights and imposing constraints a priori in our hierarchical Bayes model.We showthat such constructed portfolios are well diversified with superior out-of-sample performance.Our proposed model is tested on a number of Fama–French industry portfolios against the na飗e diversification strategy and Chevrier and McCulloch’s(2008)economically motivated prior(EMP)strategy.On average,our model outperforms Chevrier and McCulloch’s(2008)EMP strategy by over 15%and outperform the‘1/N’strategy by over 50%.