Bayesian Nonlinear Regression Research Articles

Quantifying uncertainty in fatigue crack growth of SLM 316L through advanced predictive modeling

AbstractOptimizing structural designs is crucial today, with additive manufacturing, particularly selective laser melting, gaining prominence. Thorough mechanical characterization of new materials remains vital. This paper investigates fatigue crack growth behavior in SLM 316L specimens under cyclic loading conditions. The study addresses result uncertainties by using CT specimens aligned along three building directions per ASTM E647 standards and a constant loading ratio (R = 0.1), necessitating mean value and confidence interval predictions. Departing from linear prediction models, innovative Bootstrap Polynomial and Power Regression Models and Bayesian Nonlinear Regression Model updated posterior distribution by Markov Chain Monte Carlo are employed. These approaches leverage bootstrapping to construct confidence intervals, offering robustness and flexibility in handling non‐normal data behavior and limited sample sizes. They provide tailored fits to data curvature, revealing limitations of linear prediction models in capturing observed nonlinear behavior, enhancing reliability in additive manufacturing applications, and advancing material science and engineering.

Sorption of four antibiotics onto pristine biochar derived from macadamia nutshell

In this study, the sorption properties of ciprofloxacin, ofloxacin, sulfamethoxazole, and trimethoprim on biochar derived from macadamia nut shells were investigated. The raw biomass was pyrolyzed at 600 °C to create a highly porous material with a surface area of 392 m2 g−1. The produced biochar was found to be a valuable material for both environmental remediation and carbon sequestration due to its high carbon and oxygen content. The sorption properties of four antibiotics on the produced biochar were compared using Bayesian nonlinear regression based on second-order kinetics and the Langmuir model. The Bayesian estimation successfully compared the adsorption coefficients of the antibiotics, which can be directly visualized through graphical grammar using the probability density distribution. The results demonstrated the ability of macadamia nut shell biochar to remove antibiotics from water at neutral pH, and this material has the potential to be used for treating other emerging contaminants.

Bayesian forces identification in cable networks with small bending stiffness

The regular monitoring of cable forces is essential for ensuring the safety of cable structures both during construction and throughout their lifetime. This paper aims at developing a vibration-based identification procedure of the axial forces, bending stiffness, and, secondarily, the crossing point position of cable networks. A model constituted by two crossing stays having small bending stiffness and negligible sag effects is considered. The in-plane direct dynamic problem is solved both numerically and through a perturbation approach. The obtained results are compared to the outcomes of a finite element model for verification purposes. The theoretical studies are also supported by experimental tests performed on a real cable-stayed bridge (Haccourt bridge), which provide insights into the dynamics of the system showing that models of cables with small bending stiffness are more appropriate than taut string models. The inverse analysis based on non-linear Bayesian regression is developed and the closed-form asymptotic formulations are used to prove that the bending stiffness, the cable forces, and the crossing point position can be separately identified from a set of observed frequencies. The implemented procedure is then applied to the tested bridge as a proof of concept, showing that the proposed in-plane identification strategy provides satisfactory results.

Intrinsic regularization effect in Bayesian nonlinear regression scaled by observed data

Occam's razor is a guiding principle that models should be simple enough to describe observed data. While Bayesian model selection (BMS) embodies it by the intrinsic regularization effect (IRE), how observed data scale the IRE has not been fully understood. In the nonlinear regression with conditionally independent observations, we show that the IRE is scaled by observations' fineness, defined by the amount and quality of observed data. We introduce an observable that quantifies the IRE, referred to as the Bayes specific heat, inspired by the correspondence between statistical inference and statistical physics. We derive its scaling relation to observations' fineness. We demonstrate that the optimal model chosen by the BMS changes at critical values of observations' fineness, accompanying the IRE's variation. The changes are from choosing a coarse-grained model to a fine-grained one as observations' fineness increases. Our findings expand an understanding of BMS's typicality when observed data are insufficient.

Evaluating Electromyography and Sonomyography Sensor Fusion to Estimate Lower-Limb Kinematics Using Gaussian Process Regression

Research on robotic lower-limb assistive devices over the past decade has generated autonomous, multiple degree-of-freedom devices to augment human performance during a variety of scenarios. However, the increase in capabilities of these devices is met with an increase in the complexity of the overall control problem and requirement for an accurate and robust sensing modality for intent recognition. Due to its ability to precede changes in motion, surface electromyography (EMG) is widely studied as a peripheral sensing modality for capturing features of muscle activity as an input for control of powered assistive devices. In order to capture features that contribute to muscle contraction and joint motion beyond muscle activity of superficial muscles, researchers have introduced sonomyography, or real-time dynamic ultrasound imaging of skeletal muscle. However, the ability of these sonomyography features to continuously predict multiple lower-limb joint kinematics during widely varying ambulation tasks, and their potential as an input for powered multiple degree-of-freedom lower-limb assistive devices is unknown. The objective of this research is to evaluate surface EMG and sonomyography, as well as the fusion of features from both sensing modalities, as inputs to Gaussian process regression models for the continuous estimation of hip, knee and ankle angle and velocity during level walking, stair ascent/descent and ramp ascent/descent ambulation. Gaussian process regression is a Bayesian nonlinear regression model that has been introduced as an alternative to musculoskeletal model-based techniques. In this study, time-intensity features of sonomyography on both the anterior and posterior thigh along with time-domain features of surface EMG from eight muscles on the lower-limb were used to train and test subject-dependent and task-invariant Gaussian process regression models for the continuous estimation of hip, knee and ankle motion. Overall, anterior sonomyography sensor fusion with surface EMG significantly improved estimation of hip, knee and ankle motion for all ambulation tasks (level ground, stair and ramp ambulation) in comparison to surface EMG alone. Additionally, anterior sonomyography alone significantly improved errors at the hip and knee for most tasks compared to surface EMG. These findings help inform the implementation and integration of volitional control strategies for robotic assistive technologies.

Risk measurement of oil price based on Bayesian nonlinear quantile regression model

Oil price forecasting is one of the most challenging issues since it is noisy, non-stationary, and chaotic. In this paper, we design a Bayesian Nonlinear Quantile method consisting of a Threshold Improved model and an Adaptive MCMC model to improve predicting performance. Specifically, the threshold improve model is introduced to solve the problems caused by the completely asymmetric distribution, and the Adaptive MCMC model is used to get the optimal threshold. Besides, the two-stage framework is applied to improve traditional methods' performance, including the Indirect GARCH model and Asymmetric Slope model. The experimental results show that our approach provides a promising alternative to oil price prediction, and the framework also improve the performance of the traditional methods. The contribution of this paper is to improve the accuracy of the oil price forecasting model, and the framework applies to other energy prices as well.

Forecasting Hospitalizations Due to COVID-19 in South Dakota, USA.

Anticipating the number of hospital beds needed for patients with COVID-19 remains a challenge. Early efforts to predict hospital bed needs focused on deriving predictions from SIR models, largely at the level of countries, provinces, or states. In the USA, these models rely on data reported by state health agencies. However, predicting disease and hospitalization dynamics at the state level is complicated by geographic variation in disease parameters. In addition, it is difficult to make forecasts early in a pandemic due to minimal data. Bayesian approaches that allow models to be specified with informed prior information from areas that have already completed a disease curve can serve as prior estimates for areas that are beginning their curve. Here, a Bayesian non-linear regression (Weibull function) was used to forecast cumulative and active COVID-19 hospitalizations for SD, USA, based on data available up to 2020-07-22. As expected, early forecasts were dominated by prior information, which was derived from New York City. Importantly, hospitalization trends differed within South Dakota due to early peaks in an urban area, followed by later peaks in rural areas of the state. Combining these trends led to altered forecasts with relevant policy implications.Supplementary InformationThe online version contains supplementary material available at 10.1007/s41666-021-00094-8.

Định dạng hàm sản xuất CES bằng phương pháp hồi qui phi tuyến tính Bayes (Specifying a CES Production Function Using a Bayesian Non-linear Regression)

Vietnamese Abstract: Đa phần cac nghien cứu tại Việt Nam sử dụng ham sản xuất Cobb-Douglas va cac biến thể của no như một cong cụ trong phân tich va dự bao kinh tế. Dạng ham nay co nhược điểm la cac tiền đề của no qua cứng nhắc ma khong phu hợp với hiện thực trong nhiều trường hợp, đặc biệt trong phân tich động thai tang trưởng kinh tế. Ham CES (constant elasticity of substitution) với tiền đề linh hoạt hơn, cụ thể la hệ số co gian giữa vốn va lao động khac một, được sử dụng ngay cang phổ biến. Do vậy, nghien cứu nay được thực hiện nhằm định dạng ham CES dựa tren dữ liệu của cac doanh nghiệp phi tai chinh niem yết tren thị trường chứng khoan Việt Nam. Bằng việc sử dụng phương phap hồi quy phi tuyến tinh Bayes thong qua thuật toan lấy mẫu Random-walk Metropolis Hastings (MH), tac giả kham pha rằng, ham sản xuất được ước lượng cho cac doanh nghiệp phi tai chinh Việt Nam la dạng ham CES với hệ số co gian thay thế giữa vốn va lao động nhỏ hơn một, tức la lao động va vốn bổ sung nhau. Phat hiện nay cho thấy nền kinh tế Việt Nam co thế đối diện với đa trạng thai dừng (multiple steady-states) va cac bẫy ngheo đoi (poverty traps) nen chưa đạt được cac khả nang cho tang trưởng nội sinh (endogenous growth). English Abstract: Most studies in Vietnam use the Cobb-Douglas production function and its variants as a tool in economic analysis and forecasting. A great disadvantage of this functional form is that its propositions are extremely rigid, which is not in agreement with reality in many cases, especially in analyzing economic dynamics. The CES (constant elasticity of substitution)function with more flexible propositions, namely the elasticity of substitution between capital and labor different from unity, is used more and more popularly. Therefore, this study was conducted to specify a CES function based on data of non-financial businesses listed on Vietnam's stock market. By using a Bayesian non-linear regression through the Random-walk Metropolis Hastings (MH) sampling algorithm, the author finds that the estimated production function for Vietnamese non-financial enterprises is a CES one with the elasticity of substitution between capital and labor smaller than one. This finding shows that Vietnam's economy might face multiple steady-states and poverty traps, so it has not yet reached the possibilities of endogenous growth

Bayesian non-linear regression with spatial priors for noise reduction and error estimation in quantitative MRI with an application in T1 estimation

Purpose. To develop a method that can reduce and estimate uncertainty in quantitative MR parameter maps without the need for hand-tuning of any hyperparameters. Methods. We present an estimation method where uncertainties are reduced by incorporating information on spatial correlations between neighbouring voxels. The method is based on a Bayesian hierarchical non-linear regression model, where the parameters of interest are sampled, using Markov chain Monte Carlo (MCMC), from a high-dimensional posterior distribution with a spatial prior. The degree to which the prior affects the model is determined by an automatic hyperparameter search using an information criterion and is, therefore, free from manual user-dependent tuning. The samples obtained further provide a convenient means to obtain uncertainties in both voxels and regions. The developed method was evaluated on T 1 estimations based on the variable flip angle method. Results. The proposed method delivers noise-reduced T 1 parameter maps with associated error estimates by combining MCMC sampling, the widely applicable information criterion, and total variation-based denoising. The proposed method results in an overall decrease in estimation error when compared to conventional voxel-wise maximum likelihood estimation. However, this comes with an increased bias in some regions, predominately at tissue interfaces, as well as an increase in computational time. Conclusions. This study provides a method that generates more precise estimates compared to the conventional method, without incorporating user subjectivity, and with the added benefit of uncertainty estimation.

Macroeconomic Growth in Vietnam Transitioned to Market: An Unrestricted VES Framework

The Vietnamese economy has increased at high speed over the transformation decades; however, most recent studies on the economic growth of this country used the Cobb-Douglas or CES (Constant Elasticity of Substitution) production functions, which are unable to explore the relationship between the elasticity of capital-labour substitution and development process, and hence, are not relevant to accessing a dynamic economic system. For that reason, this study is conducted to specify an unrestricted VES (Variable Elasticity of Substitution) production function in a one-sector growth model of Vietnam, highlighted by two characteristics: successful transition from plan to market and rapid progress. The VES is given preference over the CES and the Cobb-Douglas having the elasticity of substitution between capital and labour varying with economic development. By employing a Bayesian nonlinear regression through MCMC methods, the study reported the following findings: (1) the above-unity variable elasticity of capital-labour substitution in an aggregate unrestricted VES function specified for Vietnam shows that the model generates the possibility of endogenous economic growth; (2) the capital share tends to increase, while the labour share faces a downward trend along with the development of Vietnam; (3) the VES is empirically proven through a Bayes factor test to be superior to the CES and Cobb-Douglas for analysis of the growth process of Vietnam, an emerging transition economy.

Endogenous Economic Growth: The Arrow-Romer Theory and a Test on Vietnamese Economy

The Arrow-Romer growth model helped to overcome the main drawback of the Solow-Swan model, where technical change is created exogenously, not by the firms making decisions, and formulated the conditions for endogenous growth in an economy. Nonetheless, the presentation of the Arrow-Romer model and corresponding empirical studies by the Cobb-Douglas functions hides the role of the capital-labor relationship for economic growth. A constant elasticity of substitution (CES) function, constructed by Arrow et al. (1961), allows solving this problem. So, the purpose of the current research is to test the endogenous growth of the Vietnamese economy, which has experienced a more than 30-year market-oriented reform through specifying an aggregate CES function. By applying Bayesian nonlinear regression, the research results revealed the elasticity of factor substitution (ES) lower than one. This work theoretically and empirically contributes to the endogenous growth theory in problems concerned with emerging economies. Investments in physical and human capital and technological progress are the determinants of endogenous growth. From the findings obtained, the author concludes that even though having achieved a rather impressive growth rate over more than three decades, the Vietnamese economy has not yet generated the possibility of endogenous growth, and suggests that endogenous growth can be hardly generated in emerging economies like Vietnam if important growth policies related to accumulation of physical and human capital as well as enhancement of R&D activities are not simultaneously implemented. It is indispensable to focus on substantially improving institutional quality.

How to Explain When the ES Is Lower Than One? A Bayesian Nonlinear Mixed-Effects Approach

Most studies in Vietnam use the Cobb-Douglas production function and its modifications for economic analysis. Extremely rigid presumptions are a main weak point of this functional form, particularly if the elasticity of factor substitution (ES) is equal to one, which hides the role of the ES for economic growth. The CES (constant elasticity of substitution) production function with more flexible presumptions, concretely its ES, is not unitary, and has been used more and more widely in economic investigations. So, this study is conducted to estimate the average ES through the specification of an aggregate CES function for the Vietnamese nonfinancial enterprises. By performing Bayesian nonlinear mixed-effects regression via Random-walk Metropolis Hastings (MH) algorithm, based on the data set of the listed nonfinancial enterprises of Vietnam, the author found that the CES function estimated for the researched enterprises has an ES lower than one, i.e., capital and labor are complimentary. This finding shows that Vietnamese nonfinancial enterprises can confront a downward trend of output growth.

Construction of credible intervals for nonlinear regression models with unknown error distributions

ABSTRACTThere has been continuing interest in Bayesian regressions in which no parametric assumptions are imposed on the error distribution. In this study, we consider semiparametric Bayesian nonlinear regression models. We do not impose a parametric form for the likelihood function. Instead, we treat the true density function of error terms as an infinite-dimensional nuisance parameter and estimate it nonparametrically. Thereafter, we conduct a conventional parametric Bayesian inference using MCMC methods. We derive the asymptotic properties of the resulting estimator and identify conditions for adaptive estimation, under which our two-step Bayes estimator enjoys the same asymptotic efficiency as if we knew the true density. We compare the accuracy and coverage of the adaptive Bayesian point and interval estimators to those of the maximum likelihood estimator empirically, using simulated and real data. In particular, we observe that the Bayesian inference may be superior in numerical stability for small sample sizes.

A Simplified Crossing Fiber Model in Diffusion Weighted Imaging.

Diffusion MRI (dMRI) is a vital source of imaging data for identifying anatomical connections in the living human brain that form the substrate for information transfer between brain regions. dMRI can thus play a central role toward our understanding of brain function. The quantitative modeling and analysis of dMRI data deduces the features of neural fibers at the voxel level, such as direction and density. The modeling methods that have been developed range from deterministic to probabilistic approaches. Currently, the Ball-and-Stick model serves as a widely implemented probabilistic approach in the tractography toolbox of the popular FSL software package and FreeSurfer/TRACULA software package. However, estimation of the features of neural fibers is complex under the scenario of two crossing neural fibers, which occurs in a sizeable proportion of voxels within the brain. A Bayesian non-linear regression is adopted, comprised of a mixture of multiple non-linear components. Such models can pose a difficult statistical estimation problem computationally. To make the approach of Ball-and-Stick model more feasible and accurate, we propose a simplified version of Ball-and-Stick model that reduces parameter space dimensionality. This simplified model is vastly more efficient in the terms of computation time required in estimating parameters pertaining to two crossing neural fibers through Bayesian simulation approaches. Moreover, the performance of this new model is comparable or better in terms of bias and estimation variance as compared to existing models.

Estimating the Expected Value of Sample Information across Different Sample Sizes Using Moment Matching and Nonlinear Regression

Background. The expected value of sample information (EVSI) determines the economic value of any future study with a specific design aimed at reducing uncertainty about the parameters underlying a health economic model. This has potential as a tool for trial design; the cost and value of different designs could be compared to find the trial with the greatest net benefit. However, despite recent developments, EVSI analysis can be slow, especially when optimizing over a large number of different designs. Methods. This article develops a method to reduce the computation time required to calculate the EVSI across different sample sizes. Our method extends the moment-matching approach to EVSI estimation to optimize over different sample sizes for the underlying trial while retaining a similar computational cost to a single EVSI estimate. This extension calculates the posterior variance of the net monetary benefit across alternative sample sizes and then uses Bayesian nonlinear regression to estimate the EVSI across these sample sizes. Results. A health economic model developed to assess the cost-effectiveness of interventions for chronic pain demonstrates that this EVSI calculation method is fast and accurate for realistic models. This example also highlights how different trial designs can be compared using the EVSI. Conclusion. The proposed estimation method is fast and accurate when calculating the EVSI across different sample sizes. This will allow researchers to realize the potential of using the EVSI to determine an economically optimal trial design for reducing uncertainty in health economic models. Limitations. Our method involves rerunning the health economic model, which can be more computationally expensive than some recent alternatives, especially in complex models.

A generalized Bayesian nonlinear mixed-effects regression model for zero-inflated longitudinal count data in tuberculosis trials.

In this paper, we investigate Bayesian generalized nonlinear mixed-effects (NLME) regression models for zero-inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis(TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero-inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log-transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti-TB drugs. We therefore recommend that zero-inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.

Modeling vertical gradients in water columns: A parametric autoregressive approach

AbstractVertical gradients (e.g., water temperature and density) are a measurable expression of the potential energy of a water body, which is a fundamental driver for biogeochemical processes in aquatic ecosystems. Seasonal vertical stratification is typically described by the mixed layer and thermocline depths, and these metrics are often estimated through visual assessment of graphical plots or using numerical methods. The most widely used numerical method estimates the derivative of temperature (or density) along the depth, but it is sensitive both to profile data resolution and presence of nonconforming observations. In this study, we propose a new method of modeling vertical gradients using a four‐parameter sigmoidal function, including temporal autocorrelation. The parameters were estimated through Bayesian nonlinear regression with conditional autoregressive errors. The proposed method provides a quantitative and automated way to estimate the mixed layer and thermocline depths even for data profiles with a poor resolution. It also showed good performance against high‐frequency measurement data.

Bayesian nonlinear meta regression for benefit transfer

In recent years numerous meta-regression models for benefit transfer in the context of environmental quality changes have been proposed by the academic literature and used by government agencies for policy making. We examine a set of popular specifications in terms of consistency with some basic utility-theoretic considerations, including the adding-up condition that is currently under much scrutiny by benefit transfer practitioners. We also compare these models based on econometric fit with underlying data, and ability to generate meaningful and efficient benefit transfer distributions. We find that our preferred Bayesian Nonlinear Meta-Regression Model (BNL-MRM) satisfies all theoretical requirements. Using a built-in nonlinear model search algorithm we show that it produces benefit estimates that are comparable or superior in efficiency to those flowing from better fitting, but theoretically flawed linear models that do not satisfy adding-up.

Bayesian Nonlinear Quantile Regression Approach for Longitudinal Ordinal Data

Longitudinal data with ordinal outcomes commonly arise in clinical and social studies, where the purpose of interest is usually quantile curves rather than a simple reference range. In this paper we consider Bayesian nonlinear quantile regression for longitudinal ordinal data through a latent variable. An efficient Metropolis–Hastings within Gibbs algorithm was developed for model fitting. Simulation studies and a real data example are conducted to assess the performance of the proposed method. Results show that the proposed approach performs well.

Robust Bayesian nonlinear mixed-effects modeling of time to positivity in tuberculosis trials.

Early phase 2 tuberculosis (TB) trials are conducted to characterize the early bactericidal activity (EBA) of anti-TB drugs. The EBA of anti-TB drugs has conventionally been calculated as the rate of decline in colony forming unit (CFU) count during the first 14days of treatment. The measurement of CFU count, however, is expensive and prone to contamination. Alternatively to CFU count, time to positivity (TTP), which is a potential biomarker for long-term efficacy of anti-TB drugs, can be used to characterize EBA. The current Bayesian nonlinear mixed-effects (NLME) regression model for TTP data, however, lacks robustness to gross outliers that often are present in the data. The conventional way of handling such outliers involves their identification by visual inspection and subsequent exclusion from the analysis. However, this process can be questioned because of its subjective nature. For this reason, we fitted robust versions of the Bayesian nonlinear mixed-effects regression model to a wide range of TTP datasets. The performance of the explored models was assessed through model comparison statistics and a simulation study. We conclude that fitting a robust model to TTP data obviates the need for explicit identification and subsequent "deletion" of outliers but ensures that gross outliers exert no undue influence on model fits. We recommend that the current practice of fitting conventional normal theory models be abandoned in favor of fitting robust models to TTP data.