Kriging Predictor Research Articles

A dynamic sample augmentation Kriging metamodeling method for industrial residue hydrogenation process

AbstractResidue hydrogenation process (RHP) plays an important role in efficient utilization of heavy oil resources. The high‐fidelity model of RHP is so complex that its optimization cost is expensive and even intractable. Furthermore, in the actual industrial processes, due to the low sampling frequencies of some sensors, only a few sampling points can be obtained with some missing values. The insufficient samples can directly affect the performance of the final model. Therefore, a new adaptive dynamic sampling (ADS) method for Kriging metamodeling is proposed. Based on a small number of valid industrial sample points, the proposed method adaptively obtains key information sampling points and selectively adds key information sampling points to update the model. First, the difference maximization strategy and the outlier distance strategy are proposed to obtain new sampling points with both global and regionalized information. Then, the new key points are used to iteratively update the Kriging predictor to obtain satisfactory accuracy. Finally, the calculation result of benchmark cases and two actual experiments validate the effectiveness of the proposed method for constructing surrogate models of complex industrial processes.

System reliability analysis based on dependent Kriging predictions and parallel learning strategy

Reliability analysis of a complex system is challenging because of complex failure regions and frequent requirement of time-consuming simulations. To address these problems, combining adaptive surrogate models with Monte Carlo simulation has received considerable attention in recent years. The core of existing adaptive methods is the construction of an effective learning function as the guideline to select new training samples. In this paper, a new learning function with a parallel processing strategy is proposed for selecting new training samples for complex systems. It combines dependent Kriging predictions and parallel learning strategy to further improve the computational efficiency. Using the proposed parallel learning strategy for system reliability problems, one or several new training samples can be selected at each iteration to refine the constructed surrogate models. This causes the total number of iterations to decrease. Compared with existing adaptive Kriging-based system methods, the proposed method offers the following advantages: (1) it is capable of parallel processing, i.e., multiple training samples can be selected at each iteration for refinement to reduce the overall computational time, (2) it is easy to implement for complex systems regardless of their structure, and (3) it is generally more effective than most existing methods. Three numerical examples are investigated to demonstrate the proposed method, and the results show that it has high applicability and accuracy for complex reliability problems.

Statistical Tests for Cross-Validation of Kriging Models

Kriging or Gaussian process models are popular metamodels (surrogate models or emulators) of simulation models; these metamodels give predictors for input combinations that are not simulated. To validate these metamodels for computationally expensive simulation models, the analysts often apply computationally efficient cross-validation. In this paper, we derive new statistical tests for so-called leave-one-out cross-validation. Graphically, we present these tests as scatterplots augmented with confidence intervals that use the estimated variances of the Kriging predictors. To estimate the true variances of these predictors, we might use bootstrapping. Like other statistical tests, our tests—with or without bootstrapping—have type I and type II error probabilities; to estimate these probabilities, we use Monte Carlo experiments. We also use such experiments to investigate statistical convergence. To illustrate the application of our tests, we use (i) an example with two inputs and (ii) the popular borehole example with eight inputs. Summary of Contribution: Simulation models are very popular in operations research (OR) and are also known as computer simulations or computer experiments. A popular topic is design and analysis of computer experiments. This paper focuses on Kriging methods and cross-validation methods applied to simulation models; these methods and models are often applied in OR. More specifically, the paper provides the following; (1) the basic variant of a new statistical test for leave-one–out cross-validation; (2) a bootstrap method for the estimation of the true variance of the Kriging predictor; and (3) Monte Carlo experiments for the evaluation of the consistency of the Kriging predictor, the convergence of the Studentized prediction error to the standard normal variable, and the convergence of the expected experimentwise type I error rate to the prespecified nominal value. The new statistical test is illustrated through examples, including the popular borehole model.

An adaptive method fusing the kriging model and multimodal importance sampling for profust reliability analysis

Profust reliability analysis is investigated in this article. The main aim is to address problems with very small failure probabilities. First, a single-layer multimodal importance sampling (MMIS) method based on the most probable points (MPPs) in the fuzzy failure region is proposed. Secondly, to reduce function calls, an adaptive method fusing kriging model and MMIS is developed. In this method, a surrogate fuzzy failure region (SFFR) based on the kriging prediction information is defined. The MPPs in the SFFR are explored, the important samples are generated, the optimal training point is chosen, and the kriging model and the SFFR are updated. The learning process is executed iteratively until the SFFR converges to the true one. To find all MPPs and maintain unbiasedness, an intelligent multimodal optimization method based on multi-objective optimization is introduced. Four case studies are used to verify the advantages of the proposed method.

Copula-based multiple indicator kriging for non-Gaussian random fields

In spatial statistics, the kriging predictor is the best linear predictor at unsampled locations, but not the optimal predictor for non-Gaussian processes. In this paper, we introduce a copula-based multiple indicator kriging model for the analysis of non-Gaussian spatial data by thresholding the spatial observations at a given set of quantile values. The proposed copula model allows for flexible marginal distributions while modeling the spatial dependence via copulas. We show that the covariances required by kriging have a direct link to the chosen copula function. We then develop a semiparametric estimation procedure. The proposed method provides the entire predictive distribution function at a new location, and thus allows for both point and interval predictions. The proposed method demonstrates better predictive performance than the commonly used variogram approach and Gaussian kriging in the simulation studies. We illustrate our methods on precipitation data in Spain during November 2019, and heavy metal dataset in topsoil along the river Meuse, and obtain probability exceedance maps.

GIS-Based Approach to Spatio-Temporal Interpolation of Atmospheric CO2 Concentrations in Limited Monitoring Dataset

Understanding the magnitude and distribution of the mixes of the near-ground carbon dioxide (CO2) components spatially (related to the surface characteristics) and temporally (over seasonal timescales) is critical to evaluating present and future climate impacts. Thus, the application of in situ measurement approaches, combined with the spatial interpolation methods, will help to explore variations in source contribution to the total CO2 mixing ratios in the urban atmosphere. This study presents the spatial characteristic and temporal trend of atmospheric CO2 levels observed within the city of Wroclaw, Poland for the July 2017–August 2018 period. The seasonal variability of atmospheric CO2 around the city was directly measured at the selected sites using flask sampling with a Picarro G2201-I Cavity Ring-Down Spectroscopy (CRDS) technique. The current work aimed at determining the accuracy of the interpolation techniques and adjusting the interpolation parameters for estimating the magnitude of CO2 time series/seasonal variability in terms of limited observations during the vegetation and non-vegetation periods. The objective was to evaluate how different interpolation methods will affect the assessment of air pollutant levels in the urban environment and identify the optimal sampling strategy. The study discusses the schemes for optimization of the interpolation results that may be adopted in areas where no observations are available, which is based on the kriging error predictions for an appropriate spatial density of measurement locations. Finally, the interpolation results were extended regarding the average prediction bias by exploring additional experimental configurations and introducing the limitation of the future sampling strategy on the seasonal representation of the CO2 levels in the urban area.

Improved Regression Kriging Prediction of the Spatial Distribution of the Soil Cadmium by Integrating Natural and Human Factors

Mastering the spatial distribution of heavy metals in the soil plays an important supporting role in the scientific formulation of soil pollution risk management and control strategies. Few factors were considered and multiple collinearity between parallel variables existed,resulting in low prediction accuracy. OK (common Kriging method), NRK (regressive Kriging method based on natural factors only), and NARK (regressive Kriging based on natural-human factors)were used to simulate the spatial distribution of soil Cd by selecting 23 natural-artificial influencing factors. The prediction accuracy was evaluated based on an empirical study of the area around Shaoguan Qujiang smelter. The results showed that the above-standard rate of soil cadmium in this area reached 85.93%, and the effect on the spatial heterogeneity of soil cadmium was shown as air emissions from smelters > air emissions from steel plants > pH > organic matter > Euclidean distance to road > Euclidean distance to river. The root-mean-square error and average absolute error of NARK's prediction results for soil cadmium were 26.86% and 30.56% lower than that of the OK method, respectively. The model determination coefficient R2 increased from 0.78 to 0.88. Compared with that of NRK, it was reduced by 24.15% and 24.23% and R2 increased from 0.81 to 0.88. The NRK combining natural and human factors significantly improved the simulation accuracy of the spatial distribution of soil cadmium, and the addition of human factors as auxiliary variables, especially atmospheric point source pollution emissions, greatly contributed to the improvement of the model accuracy.

Thermal reliability-based design optimization using Kriging model of PCM based pin fin heat sink

This paper deals with the thermal reliability optimization of a pin fin heat sink (HS) filled with phase change material (PCM) in order to study the heat transfer performance for cooling of electronic devices. Aluminium is taken to carry out the pin fin heat sink with a constant volume fraction of 9% by choosing 72 pin fins. Paraffin wax is used as PCM inside the heat sink to retain the heat produced by the electronic device. A reliability-based design optimization (RBDO) is addressed to determine the optimum thermal configuration of the pin fin heat sink that satisfies the desired reliability. The presented methodology is summarized by the combination of the 3D finite element simulation (FES), the Kriging substitution model and the optimization procedure. The surrogate model is built and validated using two metrics such as error predictions and cross validation (CV). A comparison of the number of training points shows that the Kriging predictor with 100 Latin hypercube sampling (LHS) points provides a better approximation of the original model. Thermal optimization results show that the time taken by the latent heating phase of the optimal design is higher than that obtained by the baseline model. With the reliability optimization, the maximum temperature of the optimal design was reduced by 30%.

Incident cardiovascular disease and particulate matter air pollution in South Korea using a population-based and nationwide cohort of 0.2 million adults

BackgroundWhile many studies reported the association between long-term exposure to particulate matter air pollution (PM) and cardiovascular disease (CVD), few studies focused on incidence with relatively high-dose exposure using a nationwide cohort. This study aimed to investigate the association between long-term exposure to PM10 and PM2.5 and incidence of CVD in a nationwide and population-based cohort in South Korea where the annual average concentration of PM2.5 is above 20 μg/m3.MethodsWe selected 196,167 adults in the National Health Insurance Service-National Sample Cohort (NHIS-NSC) constructed based on the entire South Korean population. Incidence of four CVD subtypes including ischemic heart disease (IHD), myocardial infarction, heart failure, and stroke, and total CVD including all four was identified as the first diagnosis for 2007–2015. To assess individual exposures, we used annually-updated district-level residential addresses and district-specific PM concentrations predicted by a previously developed universal kriging prediction model. We computed individual-level long-term PM concentrations for four exposure windows: previous 1, 3, and 5 year(s) and 5 years before baseline. We applied time-dependent Cox proportional hazards models to estimate hazard ratios (HRs) of incident CVDs per 10 μg/m3 increase in PM10 and PM2.5 after adjusting for individual- and area-level characteristics.ResultsDuring 1,578,846 person-year, there were 33,580 cases of total incident CVD. Average PM10 and PM2.5 concentrations for the previous 5 years were 52.3 and 28.1 μg/m3, respectively. A 10 μg/m3 increase in PM2.5 exposed for the previous 5 years was associated with 4 and 10% increases in the incidence of total CVD (95% confidence interval: 0–9%) and IHD (4–16%), respectively. HRs tended to be higher with earlier exposure for IHD and more recent exposure for stroke. The estimated shape of the concentration-response relationship showed non-linear patterns. We did not find evidence of the association for PM10.ConclusionsUsing a population-based nationwide cohort exposed to relatively high PM concentration, this study confirmed the association between PM2.5 and CVD incidence that was reported in previous studies mostly with low-dose environments. The magnitude and the shape of the association were generally consistent with previous findings.

A kriging method for fuzzy spatial data

This paper extends the ordinary kriging prediction method and proposes a trend removal method for observed spatial data represented as LR-fuzzy numbers. To this end, a covariance concept was developed and discussed in the fuzzy domain. Then, a notion of semi-variogram and its empirical estimator were proposed. The proposed semi-variogram exhibited all of the characteristics of a typical semi-variogram in the fuzzy domain. A non-parametric kernel-based method was proposed to remove the trend across the fuzzy data. Some common goodness-of-fit criteria were employed to examine the performance of the proposed kriging prediction method. The proposed method was then put to test using a simulated set of fuzzy data. In order to demonstrate the applicability of this approach, it was applied to a set of pH data in the fuzzy domain to investigate the quality of groundwater. The results proved the potentials of the proposed method for the fuzzy spatial data encountered in real applications.

System Reliability Analysis With Autocorrelated Kriging Predictions

AbstractWhen limit-state functions are highly nonlinear, traditional reliability methods, such as the first-order and second-order reliability methods, are not accurate. Monte Carlo simulation (MCS), on the other hand, is accurate if a sufficient sample size is used but is computationally intensive. This research proposes a new system reliability method that combines MCS and the Kriging method with improved accuracy and efficiency. Accurate surrogate models are created for limit-state functions with minimal variance in the estimate of the system reliability, thereby producing high accuracy for the system reliability prediction. Instead of employing global optimization, this method uses MCS samples from which training points for the surrogate models are selected. By considering the autocorrelation of a surrogate model, this method captures the more accurate contribution of each MCS sample to the uncertainty in the estimate of the serial system reliability and therefore chooses training points efficiently. Good accuracy and efficiency are demonstrated by four examples.

A new local update-based method for reliability-based design optimization

Reliability-based design optimization (RBDO) is often associated with computationally expensive computer simulations in practical engineering applications, in which both objective and constraint functions may be implicit and time-consuming to evaluate. To improve the efficiency of RBDO, metamodels are often employed to substitute the implicit and expensive functions. In this paper, a new local update-based method is proposed for RBDO under the approximation of both objective and constraint functions. In this method, Kriging metamodels are employed to replace both objective and constraint functions, and sequential optimization and reliability assessment (SORA) is adopted to perform RBDO. Meanwhile, a local update method is developed for the approximation of objective and constraint functions, termed as LUOC. In LUOC, two local update strategies are proposed. For the constraint functions, one local update strategy judges the active constraints and then sequentially refines their Kriging metamodels in the local regions around the most probable target points obtained by SORA. For the objective function, the other local update strategy based on expected improvement is developed to refine the Kriging metamodel in the local region around the SORA solution with considering the Kriging prediction variance. Three examples including a battery design application are presented to test the accuracy and efficiency of the proposed method for RBDO.

A fast Bayesian inference scheme for identification of local structural properties of layered composites based on wave and finite element-assisted metamodeling strategy and ultrasound measurements

Reliable verification and evaluation of the mechanical properties of a layered composite ensemble are critical for industrially relevant applications, however it still remains an open engineering challenge. In this study, a fast Bayesian inference scheme based on multi-frequency single shot measurements of wave propagation characteristics is developed to overcome the limitations of ill-conditioning and non-uniqueness associated with the conventional approaches. A Transitional Markov chain Monte Carlo (TMCMC) algorithm is employed for the sampling process. A Wave and Finite Element (WFE)-assisted metamodeling scheme in lieu of expensive-to-evaluate explicit FE analysis is proposed to cope with the high computational cost involved in TMCMC sampling. For this, the Kriging predictor providing a surrogate mapping between the probability spaces of the model predictions for the wave characteristics and the mechanical properties in the likelihood evaluations is established based on the training outputs computed using a WFE forward solver, coupling periodic structure theory to conventional FE. The valuable uncertainty information of the prediction variance introduced by the use of a surrogate model is also properly taken into account when estimating the parameters’ posterior probability distribution by TMCMC. A numerical study as well as an experimental study are conducted to verify the computational efficiency and accuracy of the proposed methodology. Results show that the TMCMC algorithm in conjunction with the WFE forward solver-aided metamodeling can sample the posterior Probability Density Function (PDF) of the updated parameters at a very reasonable cost. This approach is capable of quantifying the uncertainties of recovered independent characteristics for each layer of the composite structure under investigation through fast and inexpensive experimental measurements on localized portions of the structure.

Atakrig: An R package for multivariate area-to-area and area-to-point kriging predictions

Geostatistical interpolation methods are used in diverse disciplines, such as environmental science, ecology, and hydrology. With the increasing availability of areal spatial data, area-to-area and area-to-point interpolations have great application potential. In this study, based on the variogram deconvolution algorithm proposed by Goovaerts (2008), an open-source area-to-area kriging package atakrig is developed in the R environment. In atakrig, point-scale variogram and cross-variogram can be automatically deconvoluted from spatial areal samples. It provides a general framework for area-to-area and area-to-point ordinary kriging and cokriging. Two applications show that the package works well in river runoff prediction and missing data interpolation for remote sensing aerosol optical depth. The package can be deployed on different operating systems and computer hardware platforms.

EEK-SYS: System reliability analysis through estimation error-guided adaptive Kriging approximation of multiple limit state surfaces

In order to approximate the multiple limit state functions for different failure events, the active learning Kriging model proposed for component reliability analysis has been extended to system reliability analysis. Meanwhile, many efficient sampling strategies have been applied to reduce the high computational burden. However, these strategies meet a challenge in wasting some training points and terminating the training process inappropriately, since they do not directly relate to the estimation error of system failure probability. To address the challenge, this work proposes an estimation error-guided adaptive Kriging method. As Kriging prediction may be inaccurate before being well trained, the predicted system failure probability may deviate from the true result. To quantify this estimation error, the true number of failure points is approximated by adding the number of predicted failure points and the number of wrongly classified points. Since it is impossible to learn the exact number of wrongly classified points, its confidence interval is derived based on the probability of making wrong state classification. Subsequently, the refinement of Kriging is achieved by using the probability to identify new points and using the estimation error to determine the termination, which has been demonstrated by three different cases.

Prediction for Big Data Through Kriging: Small Sequential and One-Shot Designs

Kriging—or Gaussian process (GP) modeling—is an interpolation method assuming that the outputs (responses) are more correlated, as the inputs (explanatory or independent variables) are closer. Such a GP has unknown (hyper)parameters that are usually estimated through the maximum-likelihood method. Big data, however, make it problematic to compute these estimated parameters, and the corresponding Kriging predictor and its predictor variance. To solve this problem, some authors select a relatively small subset from the big set of previously observed “old” data. These selection methods are sequential, and they depend on the variance of the Kriging predictor; this variance requires a specific Kriging model and the estimation of its parameters. The resulting designs turn out to be “local”; i.e., most selected old input combinations are concentrated around the new combination to be predicted. We develop a simpler one-shot (fixed-sample, non-sequential) design; i.e., from the big data set we select a small subset with the nearest neighbors of the new combination. To compare our designs and the sequential designs empirically, we use the squared prediction errors, in several numerical experiments. These experiments show that our design may yield reasonable performance.

Bayesian Finite Element Model Updating of a Long-Span Suspension Bridge Utilizing Hybrid Monte Carlo Simulation and Kriging Predictor

Bayesian model updating technique has been widely investigated and utilized in the field of finite element model (FEM) updating for its advantages in system uncertainty quantification. Most existing studies focus on numerical and experimental models. More studies on large-scale civil infrastructures based on field monitoring are still required. A case study on Bayesian FEM updating of the Runyang Suspension Bridge (RSB), a long-span suspension bridge with a main span of 1,490 m, is carried out in this paper. The Bayesian updating method is utilized to update the initial FEM of RSB, aiming to make the numerical modal properties match the field monitoring results. Two stochastic sampling algorithms, i.e., the Metropolis-Hastings (MH) algorithm and the Hybrid Monte Carlo (HMC) algorithm, are respectively investigated to show their advantages and limitations in Bayesian updating. Subsequently, based on the experimentalsamples generated by the Latin hypercube sampling algorithm, a Kriging predictor is established as a surrogate model to reduce the computational burden of model updating. Results show that the HMC algorithm could guarantee much higher acceptance rate of the sampled chain than the MH algorithm especially when the updating step size is large. In addition, combined with the Kriging predictor, Bayesian model updating method could serve as an effective and efficient tool to calibrate the FEM of large-scale civil infrastructures.

Estimation of in-situ maximum horizontal principal stress magnitudes from borehole breakout data using machine learning

This paper presents a technique for in-situ horizontal stress prediction in vertical boreholes based on borehole breakout data using a machine learning-based meta-modelling technique, i.e. Kriging. This model is generated based on the deterministic mean function and a stationary Gaussian process function derived from the 106 data points from published experimental studies. To examine the reliability of the model, a leave-one-out cross-validation process is conducted against the published experimental results. The Kriging prediction gives an average error of 10.59% on maximum horizontal principal stress (σH). A total of 23 field data were also extracted from both literature and mine site A with stress measurements conducted at the similar depth. The model yields an average prediction error of 8.4% in comparison with field stress measurement results on σH, which is remarkable considering its simplicity, reliability and low cost. This study limits the scope to σH estimation only due to the nature of the experimental setup, in which minimum horizontal principal stress (σh) values are often kept constant while changing σH. This results in a lot of repetition σh values in training data and it makes the prediction of σh unreliable. By collecting a broader range of data on breakout geometries, stress magnitudes and the rock strength, it is expected that the accuracy and the parameter coverage of this technique can be further improved in practical conditions.

Kriging Predictor for Facial Emotion Recognition Using Numerical Proximities of Human Emotions

Emotion recognition from facial expressions has gained much interest over the last few decades. In the literature, the common approach, used for facial emotion recognition (FER), consists of these steps: image pre-processing, face detection, facial feature extraction, and facial expression classification (recognition). We have developed a method for FER that is absolutely different from this common approach. Our method is based on the dimensional model of emotions as well as on using the kriging predictor of Fractional Brownian Vector Field. The classification problem, related to the recognition of facial emotions, is formulated and solved. The relationship of different emotions is estimated by expert psychologists by putting different emotions as the points on the plane. The goal is to get an estimate of a new picture emotion on the plane by kriging and determine which emotion, identified by psychologists, is the closest one. Seven basic emotions (Joy, Sadness, Surprise, Disgust, Anger, Fear, and Neutral) have been chosen. The accuracy of classification into seven classes has been obtained approximately 50%, if we make a decision on the basis of the closest basic emotion. It has been ascertained that the kriging predictor is suitable for facial emotion recognition in the case of small sets of pictures. More sophisticated classification strategies may increase the accuracy, when grouping of the basic emotions is applied.

Improved Mapping of Soil Heavy Metals Using a Vis-NIR Spectroscopy Index in an Agricultural Area of Eastern China

Visible and near-infrared reflectance (Vis-NIR) spectroscopy can provide low-cost and high-density data for mapping various soil properties. However, a weak correlation between the spectra and measurements of soil heavy metals makes spectroscopy difficult to use in predicting incipient risk areas. In this study, we introduce a new spectral index (SI) based on Vis-NIR spectra and use it as a covariate in ordinary cokriging (OCK) to improve the mapping of soil heavy metals. The SI was defined from the highest covariance between spectra and heavy metal content in the partial least squares regression (PLSR) model. The proposed mapping approach was compared with an ordinary kriging (OK) predictor that uses only soil heavy metal data and an OCK predictor that uses soil organic matter (SOM) and Fe as covariates. To this end, a total of 100 topsoil (0-20 cm) samples were collected in an agricultural area near Longkou City, and the contents of As, Pb and Zn in the soil were determined. The results showed that OCK with the SI provided better results in terms of unbiasedness and accuracy compared to other comparative methods. Additionally, we explored the SI through simple strategies based on spectral analysis and correlation statistics and found that the SI synthesized most of the soil properties affected by heavy metals and was not limited to Fe and SOM. In summary, the SI method is cost-effective for improving soil heavy metal mapping and can be applied to other areas.