Performance analysis of empirical models for predicting rock mass deformation modulus using regression and Bayesian methods

Deformation modulus of rock mass is one of the input parameters to most rock engineering designs and constructions.The field tests for determination of deformation modulus are cumbersome,expensive and time-consuming.This has prompted the development of various regression equations to estimate deformation modulus from results of rock mass classifications,with rock mass rating(RMR)being one of the frequently used classifications.The regression equations are of different types ranging from linear to nonlinear functions like power and exponential.Bayesian method has recently been developed to incorporate regression equations into a Bayesian framework to provide better estimates of geotechnical properties.The question of whether Bayesian method improves the estimation of geotechnical properties in all circumstances remains open.Therefore,a comparative study was conducted to assess the performances of regression and Bayesian methods when they are used to characterize deformation modulus from the same set of RMR data obtained from two project sites.The study also investigated the performance of different types of regression equations in estimation of the deformation modulus.Statistics,probability distributions and prediction indicators were used to assess the performances of regression and Bayesian methods and different types of regression equations.It was found that power and exponential types of regression equations provide a better estimate than linear regression equations.In addition,it was discovered that the ability of the Bayesian method to provide better estimates of deformation modulus than regression method depends on the quality and quantity of input data as well as the type of the regression equation.

Comparative Study of Probabilistic and Least-Squares Methods for Developing Predictive Models

This article explores the comparison between the probability method and the least squares method in the design of linear predictive models. It points out that these two approaches have distinct theoretical foundations and can lead to varied or similar results in terms of precision and performance under certain assumptions. The article underlines the importance of comparing these two approaches to choose the one best suited to the context, available data and modeling objectives.

Ensemble Bayesian method for parameter distribution inference:application to reactor physics

The estimation of model parameters is an important subject in engineering.In this area of work,the prevailing approach is to estimate or calculate these as deterministic parameters.In this study,we consider the model parameters from the perspective of random variables and describe the general form of the parameter distribution inference problem.Under this framework,we propose an ensemble Bayesian method by introducing Bayesian inference and the Markov chain Monte Carlo(MCMC)method.Experiments on a finite cylindrical reactor and a 2D IAEA benchmark problem show that the proposed method converges quickly and can estimate parameters effectively,even for several correlated parameters simultaneously.Our experiments include cases of engineering software calls,demonstrating that the method can be applied to engineering,such as nuclear reactor engineering.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available,this study proposes a Bayesianmethod to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data.According to the given interval ranges of uncertainties,we determine the initial characteristic parameters of a multi-ellipsoid convex set.Moreover,to update the plausibility of characteristic parameters,a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed.Then,an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved.The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function.Several numerical examples are presented to validate the proposed Bayesian updating method.

Bayesian sequential testing for exponential life system with reliability growth

A Bayesian sequential testing method is proposed to evaluate system reliability index with reliability growth during development.The method develops a reliability growth model of repairable systems for failure censored test,and figures out the approach to determine the prior distribution of the system failure rate by applying the reliability growth model to incorporate the multistage test data collected from system development.Furthermore,the procedure for the Bayesian sequential testing is derived for the failure rate of the exponential life system,which enables the decision to terminate or continue development test.Finally,a numerical example is given to illustrate the efficiency of the proposed model and procedure.

Uncertainty analysis of strain modal parameters by Bayesian method using frequency response function

Structural strain modes are able to detect changes in local structural performance, but errors are inevitably intermixed in the measured data. In this paper, strain modal parameters are considered as random variables, and their uncertainty is analyzed by a Bayesian method based on the structural frequency response function （FRF）. The estimates of strain modal parameters with maximal posterior probability are determined. Several independent measurements of the FRF of a four-story reinforced concrete flame structural model were performed in the laboratory. The ability to identify the stiffness change in a concrete column using the strain mode was verified. It is shown that the uncertainty of the natural frequency is very small. Compared with the displacement mode shape, the variations of strain mode shapes at each point are quite different. The damping ratios are more affected by the types of test systems. Except for the case where a high order strain mode does not identify local damage, the first order strain mode can provide an exact indication of the damage location.

Improvement of X-Band Polarization Radar Melting Layer Recognition by the Bayesian Method and ITS Impact on Hydrometeor Classification

Using melting layer(ML)and non-melting layer(NML)data observed with the X-band dual linear polarization Doppler weather radar(X-POL)in Shunyi,Beijing,the reflectivity(ZH),differential reflectivity(ZDR),and correlation coefficient(CC)in the ML and NML are obtained in several stable precipitation processes.The prior probability density distributions(PDDs)of the ZH,ZDR and CC are calculated first,and then the probabilities of ZH,ZDR and CC at each radar gate are determined(PBB in the ML and PNB in the NML)by the Bayesian method.When PBB>PNB the gate belongs to the ML,and when PBB<PNB the gate belongs to the NML.The ML identification results with the Bayesian method are contrasUsing melting layer(ML)and non-melting layer(NML)data observed with the X-band dual linear polarization Doppler weather radar(X-POL)in Shunyi,Beijing,the reflectivity(ZH),differential reflectivity(ZDR),and correlation coefficient(CC)in the ML and NML are obtained in several stable precipitation processes.The prior probability density distributions(PDDs)of the ZH,ZDR and CC are calculated first,and then the probabilities of ZH,ZDR and CC at each radar gate are determined(PBB in the ML and PNB in the NML)by the Bayesian method.When PBB>PNB the gate belongs to the ML,and when PBB<PNB the gate belongs to the NML.The ML identification results with the Bayesian method are contrasted under the conditions of the independent PDDs and joint PDDs of the ZH,ZDR and CC.The results suggest that MLs can be identified effectively,although there are slight differences between the two methods.Because the values of the polarization parameters are similar in light rain and dry snow,it is difficult for the polarization radar to distinguish them.After using the Bayesian method to identify the ML,light rain and dry snow can be effectively separated with the X-POL observed data.ted under the conditions of the independent PDDs and joint PDDs of the ZH,ZDR and CC.The results suggest that MLs can be identified effectively,although there are slight differences between the two methods.Because the values of the polarization parameters are similar in light rain and dry snow,it is difficult for the polarization radar to distinguish them.After using the Bayesian method to identify the ML,light rain and dry snow can be effectively separated with the X-POL observed data.

Application of a Bayesian method to data-poor stock assessment by using Indian Ocean albacore (Thunnus alalunga) stock assessment as an example

It is widely recognized that assessments of the status of data-poor fish stocks are challenging and that Bayesian analysis is one of the methods which can be used to improve the reliability of stock assessments in data-poor situations through borrowing strength from prior information deduced from species with good-quality data or other known information. Because there is considerable uncertainty remaining in the stock assessment of albacore tuna（Thunnus alalunga） in the Indian Ocean due to the limited and low-quality data, we investigate the advantages of a Bayesian method in data-poor stock assessment by using Indian Ocean albacore stock assessment as an example. Eight Bayesian biomass dynamics models with different prior assumptions and catch data series were developed to assess the stock. The results show（1） the rationality of choice of catch data series and assumption of parameters could be enhanced by analyzing the posterior distribution of the parameters;（2） the reliability of the stock assessment could be improved by using demographic methods to construct a prior for the intrinsic rate of increase（r）. Because we can make use of more information to improve the rationality of parameter estimation and the reliability of the stock assessment compared with traditional statistical methods by incorporating any available knowledge into the informative priors and analyzing the posterior distribution based on Bayesian framework in data-poor situations, we suggest that the Bayesian method should be an alternative method to be applied in data-poor species stock assessment, such as Indian Ocean albacore.

Bayesian Study Using MCMC of Three-Parameter Frechet Distribution Based on Type-I Censored Data

Type-I censoring mechanism arises when the number of units experiencing the event is random but the total duration of the study is fixed. There are a number of mathematical approaches developed to handle this type of data. The purpose of the research was to estimate the three parameters of the Frechet distribution via the frequentist Maximum Likelihood and the Bayesian Estimators. In this paper, the maximum likelihood method (MLE) is not available of the three parameters in the closed forms;therefore, it was solved by the numerical methods. Similarly, the Bayesian estimators are implemented using Jeffreys and gamma priors with two loss functions, which are: squared error loss function and Linear Exponential Loss Function (LINEX). The parameters of the Frechet distribution via Bayesian cannot be obtained analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the three parameters is obtained via Metropolis-Hastings algorithm. Comparisons of the estimators are obtained using Mean Square Errors (MSE) to determine the best estimator of the three parameters of the Frechet distribution. The results show that the Bayesian estimation under Linear Exponential Loss Function based on Type-I censored data is a better estimator for all the parameter estimates when the value of the loss parameter is positive.

Differences in parameter estimates derived from various methods for the ORYZA(v3) Model

Parameter estimation is always a difficult issue for crop model users, and inaccurate parameter values will result in deceptive model predictions. Parameter values may vary with different inversion methods due to equifinality and differences in the estimating processes. Therefore, it is of great importance to evaluate the factors which may influence parameter estimates and to make a comparison of the current widely-used methods. In this study, three popular frequentist methods(SCE-UA, GA and PEST) and two Bayesian-based methods(GLUE and MCMC-AM) were applied to estimate nine cultivar parameters using the ORYZA(v3) Model. The results showed that there were substantial differences between the parameter estimates derived by the different methods, and they had strong effects on model predictions. The parameter estimates given by the frequentist methods were obviously sensitive to initial values, and the extent of the sensitivity varied with algorithms and objective functions. Among the frequentist methods, the SCE-UA was recommended due to the balance between stable convergence and high efficiency. All the parameter estimates remarkably improved the goodness of model-fit, and the parameter estimates derived from the Bayesian-based methods had relatively worse performance compared to the frequentist methods. In particular, the parameter estimates with the highest probability density of posterior distributions derived from the MCMC-AM method(MCMC_P_(max)) led to results equivalent to those derived from the frequentist methods, and even better in some situations. Additionally, model accuracy was greatly influenced by the values of phenology parameters in validation.

Stochastic back analysis of permeability coefficient using generalized Bayesian method

Owing to the fact that the conventional deterministic back analysis of the permeability coefficient cannot reflect the uncertainties of parameters, including the hydraulic head at the boundary, the permeability coefficient and measured hydraulic head, a stochastic back analysis taking consideration of uncertainties of parameters was performed using the generalized Bayesian method. Based on the stochastic finite element method （SFEM） for a seepage field, the variable metric algorithm and the generalized Bayesian method, formulas for stochastic back analysis of the permeability coefficient were derived. A case study of seepage analysis of a sluice foundation was performed to illustrate the proposed method. The results indicate that, with the generalized Bayesian method that considers the uncertainties of measured hydraulic head, the permeability coefficient and the hydraulic head at the boundary, both the mean and standard deviation of the permeability coefficient can be obtained and the standard deviation is less than that obtained by the conventional Bayesian method. Therefore, the present method is valid and applicable.

A Fast Bayesian Evaluation Algorithm Based on the Second Category of Maximum Likelihood

Directing at evaluation for qualifying rate in weaponry test,this article discusses firstly how field test information is flooded with lots of prior information.Then a fast Bayesian evaluation algorithm is presented based on the elaborate analysis of prior information reliability and the second category of maximum likelihood.The example demonstrates that the algorithm presented in this article is better and more robust compared with classical evaluation algorithm for safe-or-failure test and normal Bayesian method,which can make the best of prior information.

BAYESIAN PREDICTION FOR THE TWO-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON TYPE Ⅱ DOUBLY CENSORING

The two-parameter exponential distribution is proposed to be an underlying model,and prediction bounds for future observations are obtained by using Bayesian approach.Prediction intervals are derived for unobserved lifetimes in one-sample prediction and two-sample prediction based on type Ⅱ doubly censored samples.A numerical example is given to illustrate the procedures,prediction intervals are investigated via Monte Carlo method,and the accuracy of prediction intervals is presented.

Ambient noise tomography of a linear seismic array based on an improved Voronoi tessellation

Ambient noise tomography,when applied to a dense linear seismic array,has the capability to provide detailed insights into the fine velocity structures across diverse tectonic settings.The linear station arrangement naturally generates parallel and concentrated ray paths along the array trend.This unique geometry requires specific optimization of the inversion methodology and model parameterization.The Bayesian-based transdimensional inversion method,characterized by its fully non-linear nature and high degree of freedom in parameter settings,offers a powerful tool for ambient noise inversion.To effectively adapt this method to a linear array layout,we propose a modification to the Voronoi cell tessellation built in the transdimensional method.By introducing spatial priority to the Voronoi kernels,we strategically increased the density of Voronoi cells along the direction of the array.We then applied the modified approach to a linear seismic array in the North China Craton and validated its robustness through phase velocity images and resolution tests.Our improved non-uniform sampling technique in the 2-D model space accelerates convergence while simultaneously enhancing model accuracy.Compared with the conventional damped leastsquares method,the proposed algorithm revealed a shear-wave velocity map with notable low-velocity anomalies situated in the middle and lower crust beneath the borders of the Ordos block and its surrounding orogenic belt.Aligned with the crustal structures revealed by receiver function and electrical imaging,our findings indicated that the western and eastern margins of the Ordos block had experienced intensive crustal wedge deformation and re-melting,respectively.

Research on Freezing of Gait Recognition Method Based on Variational Mode Decomposition

Freezing of Gait(FOG)is the most common and disabling gait disorder in patients with Parkinson’s Disease(PD),which seriously affects the life quality and social function of patients.This paper proposes a FOG recognition method based on the Variational Mode Decomposition(VMD).Firstly,VMD instead of the traditional time-frequency analysis method to complete adaptive decomposition to the FOG signal.Secondly,to improve the accuracy and speed of the recognition algorithm,use the CART model as the base classifier and perform the feature dimension reduction.Then use the RUSBoost ensemble algorithm to solve the problem of unbalanced sample size and considerable limitations of a single classifier.Finally,the hyperparam-eters of the ensemble classifier are optimized by Bayesian optimization,and the experiment proves that the RUSBoost algorithm can complete the gait recognition task well.Compared with the Adaboost,Tomeklinks-Adaboost and ROS-Adaboost ensemble algorithms,the RUSBoost ensemble algorithm can complete the FOG recognition task more efficiently.When the maximum number of splits is 1023,and the number of base classifiers is 100,the performance of the RUSBoost ensemble algorithm can reach the best.The accuracy of the time recognition algorithm was 87.8%,the sensitivity was 89.7%,and the specificity was 87.5%.

Mixture models detect large effect QTL better than GBLUP and result in more accurate and persistent predictions

Background： Accurate evaluation of SNP effects is important for genome wide association studies and for genomic prediction. The genetic architecture of quantitative traits differs widely, with some traits exhibiting few if any quantitative trait loci （QTL） with large effects, while other traits have one or several easily detectable QTL with large effects. Methods： Body weight in broilers and egg weight in layers are two examples of traits that have QTL of large effect A commonly used method for genome wide association studies is to fit a mixture model such as BayesB that assumes some known proportion of SNP effects are zero. In contrast, the most commonly used method for genomic prediction is known as GBLUP, which involves fitting an animal model to phenotypic data with the variance-covariance or genomic relationship matrix among the animals being determined by genome wide SNP genotypes. Genotypes at each SNP are typically weighted equally in determining the genomic relationship matrix for GBLUP. We used the equivalent marker effects model formulation of GBLUP for this study. We compare these two classes of models using egg weight data collected over 8 generations from 2,324 animals genotyped with a 42 K SNP panel.Results： Using data from the first 7 generations, both BayesB and GBLUP found the largest QTL in a similar well-recognized QTL region, but this QTL was estimated to account for 24 % of genetic variation with BayesB and less than 1% with GBLUP. When predicting phenotypes in generation 8 BayesB accounted for 36 % of the phenotypic variation and GBLUP for 25 %. When using only data from any one generation, the same QTL was identified with BayesB in all but one generation but never with GBLUP. Predictions of phenotypes in generations 2 to 7 based on only 295 animals from generation 1 accounted for 10 % phenotypic variation with BayesB but only 6 % with GBLUP. Predicting phenotype using only the marker effects in the 1 Mb region that accounted for the largest effect on egg weight from generation 1 data alone accounted for almost 8 % variation using BayesB but had no predictive power with GBLUP. Conclusions： In conclusion, In the presence of large effect QTL, BayesB did a better job of QTL detection and its genomic predictions were more accurate and persistent than those from GBLUP.

Precipitation Regime Shift Associated with the Pacific Decadal Oscillation in the Maritime Continent

Recent changes in precipitation regime in South-East Asia are a subject of ongoing discussion. In this article, for the first time, evidence of a precipitation regime shift during the mid-1970s in the Northern Hemispheric part of South-East Asia is demonstrated. The detection of regime shifts is made possible by using a new comprehensive dataset of daily precipitation records (South-East Asian Climate Assessment and Dataset) and applying a novel Bayesian approach for regime shift detection. After the detected regime shift event in the mid-1970s, significant changes in precipitation distribution occurred in the Northern Hemispheric regions—Indochina Peninsula and the Philippines. More specifically, dry days became up to 10% more frequent in some regions. However, no precipitation regime shift is detected in Southern Hemisphere regions—Java and Northern Australia, were the number of observed dry days increased gradually.

Reliability Assessment for the Solenoid Valve of a High-Speed Train Braking System under Small Sample Size

Reliability assessment of the braking system in a high?speed train under small sample size and zero?failure data is veryimportant for safe operation. Traditional reliability assessment methods are only performed well under conditions of large sample size and complete failure data,which lead to large deviation under conditions of small sample size and zero?failure data. To improve this problem,a new Bayesian method is proposed. Based on the characteristics of the solenoid valve in the braking system of a high?speed train,the modified Weibull distribution is selected to describe the failure rate over the entire lifetime. Based on the assumption of a binomial distribution for the failure probability at censored time,a concave method is employed to obtain the relationships between accumulation failure prob?abilities. A numerical simulation is performed to compare the results of the proposed method with those obtained from maximum likelihood estimation,and to illustrate that the proposed Bayesian model exhibits a better accuracy for the expectation value when the sample size is less than 12. Finally,the robustness of the model is demonstrated by obtaining the reliability indicators for a numerical case involving the solenoid valve of the braking system,which shows that the change in the reliability and failure rate among the di erent hyperparameters is small. The method is provided to avoid misleading of subjective information and improve accuracy of reliability assessment under condi?tions of small sample size and zero?failure data.

MLP training in a self-organizing state space model using unscented Kalman particle filter

Many Bayesian learning approaches to the multi-layer perceptron （MLP） parameter optimization have been proposed such as the extended Kalman filter （EKF）. This paper uses the unscented Kalman particle filter （UPF） to train the MLP in a self- organizing state space （SOSS） model. This involves forming augmented state vectors consisting of all parameters （the weights of the MLP） and outputs. The UPF is used to sequentially update the true system states and high dimensional parameters that are inherent to the SOSS moder for the MLP simultaneously. Simulation results show that the new method performs better than traditional optimization methods.

Ice thickness distribution and volume estimation of Burqin Glacier No.18 in the Chinese Altay Mountains

Information on the thickness distribution and volume of glacier ice is highly important for glaciological applications;however,detailed measurements of the ice thickness of many glaciers in the Chinese Altay Mountains remain lacking.Burqin Glacier No.18 is a northeast-orientated cirque glacier located on the southern side of the Altay Mountains.This study used PulseEKKO®PRO 100A enhancement ground-penetrating radar(GPR)to survey the ice thickness and volume of Burqin Glacier No.18 in summer 2018.Together with GPR surveying,spatial distributed profiles of the GPR measurements were concurrently surveyed using the real-time kinematic(RTK)global navigation satellite system(GNSS,Unistrong E650).Besides,we used QuickBird,WorldView-2,and Landsat TM to delineate accurate boundary of the glacier for undertaking estimation of glacier ice volume.GPR measurements revealed that the basal topography of profile B1-B2 was flat,the basal topography of profile C1-C2 presented a V-type form,and the basal topography of profile D1-D2 had a typical U-type topographic feature because the bedrock near the central elevation of the glacier was relatively flat.The longitudinal profile A1-A2 showed a ladder-like distribution.Glacier ice was thin at the terminus and its thickness increased gradually from the elevation of approximately 2620 m a.s.l.along the main axis of the glacier tongue with an average value of 80(±1)m.The average ice thickness of the glacier was determined as 27(±2)m and its total ice volume was estimated at 0.031(±0.002)km3.Interpretation of remote sensing images indicated that during 1989–2016,the glacier area reduced from 1.30 to 1.17 km2(reduction of 0.37%/a)and the glacier terminus retreated at the rate of 8.48 m/a.The mean ice thickness of Burqin Glacier No.18 was less than that of the majority of other observed glaciers in China,especially those in the Qilian Mountains and Central Chinese Tianshan Mountains;this is probably attributable to differences in glacier type and climatic setting.