Kriging Research Articles

Comparative study of HDMRs and other popular metamodeling techniques for high dimensional problems

The efficiency of optimization for the high dimensional problem has been improved by the metamodeling techniques in multidisciplinary in the past decades. In this study, comparative studies are implemented for high dimensional problems on the accuracy of four popular metamodeling methods, Kriging (KRG), radial basis function (RBF), least square support vector regression (LSSVR) and cut-high dimensional model representation (cut-HDMR) methods. Besides, HDMR methods with different basis functions are considered, including KRG-HDMR, RBF-HDMR and SVR-HDMR. Four factors that might influence the quality of metamodeling methods involving parameter interaction of problems, sample sizes, noise level and sampling strategies are considered. The results show that the LSSVR with Gaussian kernel, using Latin hypercube sampling (LHS) strategy, constructs more accurate metamodels than the KRG. The RBF with Gaussian basis function performs poor in the group. Generally, cut-HDMR methods perform much better than the other metamodeling methods when handling the function with weak parameter interaction, but not better when handling the function with strong parameter interaction.

TOPOGRAPHIC SURVEY OF AN OPEN CHANNEL IN A TIDAL FLAT

This paper describes the fundamental characteristics of an open channel in a tidal flat. In this study, the open channel was characterized using a topographic survey. Because of the limited resources available and special measurement conditions at the study area, cranes and cables were used to transport a sounding weight to the measurement point, and pressure gauges hanging on the sounding weight were used to measure the water depth. The coordinates and altitude of the surface point above the sound weight were measured using a total station. The kriging interpolation was applied to discrete geographic data to create contour lines and a topographic map of the open channel in the tidal flat. This methodology is simple, accurate, and cost-effective. Therefore, the developed methodology and theory can be applied to bathymetry measurements of river terrains with similar conditions.

On the ensemble of metamodels with multiple regional optimized weight factors

Metamodels are often used as surrogates for expensive high fidelity computational simulations (e.g., finite element analysis). Ensemble of metamodels (EM), which combines various types of individual metamodels in the form of a weighted average ensemble, is found to have improved accuracy over the individual metamodels used alone. Currently, there are mainly two kinds of EMs called as pointwise EM and average EM. The pointwise EM generally has better prediction accuracy than the average EM, but it is much more time-consuming than the average EM. In most cases, as a metamodel, EM is often used in the engineering design optimization which needs to invoke EM tens of thousands of times. Therefore, the average EM is still the most extensively used EM. To the authors’ best knowledge, the most accurate average EM is the EM with optimized weight factors proposed by Acar et al. However, the EM proposed by Acar et al. is often too “rigid” and may not have sufficient accuracy over some regions of the design space. In order to deal with this problem and further improve the prediction accuracy, a new EM with multiple regional optimized weight factors (EM-MROWF) is proposed in this study. In this new EM, the design space is divided into multiple subdomains each of which is assigned a set of optimized weight factors. This new EM was constructed by combining three typical individual metamodels, i.e., polynomial regression (PR), radial basis function (RBF), and Kriging (KRG). The proposed technique was evaluated by ten benchmark problems and two engineering application problems. The ten benchmark problems are typical mathematical functions for evaluating the approximation performance in previous studies. And, the two engineering application problems refer to the vehicular passive safety in the field of crashworthiness design. The study results showed that the EM-MROWF performed much better than the other existing average EMs as well as the three individual metamodels.

Multidisciplinary design optimization of dental implant based on finite element method and surrogate models

This study aims to propose a Multidisciplinary design optimization (MDO) approach for dental implant based on Finite element method (FEM), surrogate model and a new MDO algorithm. FEM is used to calculate the stress at the implant-bone interface first. Two surrogate models, Support vector regression (SVR) and Kriging (KRG) are built to replace FEM in the following MDO of dental implant, and their verifications indicate their accuracies. A new multidisciplinary design optimization algorithm, named as Homogenization-target-values MDO algorithm (HTV-MDO), is established and first tested by a numerical example to demonstrate its effectiveness. After that, it is applied to the MDO of dental implant based on the SVM and KRG. The results indicate that the new MDO approach proposed in this study can effectively deal with the MDO of dental implant. The stress is reduced greatly with other characteristics of dental implant (contact area and volume of implant in this study) optimizing or slightly deteriorating. This approach can be expanded to other MDO of different bio-implants.

Rigorous design optimisation for combined process of raw natural gas treating and co2 compression using surrogate models

Raw natural gas treating processes remove enormous CO2 and consume a lot of energy. The CO2 can be further compressed for storage or utilization to promote environmental protection. For the purpose of economic design of the combined process of raw natural gas treating and CO2 compression, a new superstructure combining CO2 removing and compression processes is presented. In the superstructure, the rich solvent is not regenerated in the distillation column but through a series of flash separators at different operating pressures and temperatures. The removed CO2 with varying pressures is then compressed to meet the transportation pressure separately. The optimal process structure and operating conditions are determined by minimizing the total annual cost (TAC). To obtain a good trade-off between calculation accuracy and efficiency, a surrogate based optimization framework is presented to address the NLP problem. The complex unit operation models are represented by the Kriging surrogate models, which are built from training data generated via Aspen HYSYS. The surrogate models are then incorporated into the mathematical superstructure framework for optimization. The case study results indicate that the proposed process achieves a significant TAC decrease (9 % - 21 %).

Conservative strategy-based ensemble surrogate model for optimal groundwater remediation design at DNAPLs-contaminated sites

The surrogate-based simulation-optimization techniques are frequently used for optimal groundwater remediation design. When this technique is used, surrogate errors caused by surrogate-modeling uncertainty may lead to generation of infeasible designs. In this paper, a conservative strategy that pushes the optimal design into the feasible region was used to address surrogate-modeling uncertainty. In addition, chance-constrained programming (CCP) was adopted to compare with the conservative strategy in addressing this uncertainty. Three methods, multi-gene genetic programming (MGGP), Kriging (KRG) and support vector regression (SVR), were used to construct surrogate models for a time-consuming multi-phase flow model. To improve the performance of the surrogate model, ensemble surrogates were constructed based on combinations of different stand-alone surrogate models. The results show that: (1) the surrogate-modeling uncertainty was successfully addressed by the conservative strategy, which means that this method is promising for addressing surrogate-modeling uncertainty. (2) The ensemble surrogate model that combines MGGP with KRG showed the most favorable performance, which indicates that this ensemble surrogate can utilize both stand-alone surrogate models to improve the performance of the surrogate model.

Two-stage surrogate assisted differential evolution for optimization of a non-circular drawing sequence

In this work, a two-stage surrogate assisted optimization technique is proposed for optimizing a non-circular drawing (NCD) sequence. The objective function was introduced to minimize inhomogeneity of the effective strain distribution at the cross-sections of the drawn wire which could deteriorate its delamination characteristics. The design variables introduced were die geometry and reduction of areas of the NCD sequence. Combination of Kriging (KG) models and support vector regression (SVR) which found to be an efficient technique in the literature is used as a surrogate model while a new circular sampling generation technique is proposed and applied leading to a two-stage surrogate assisted optimization strategy. The results revealed that using the proposed two-stage surrogate assisted optimization strategy can reduce surrogate model uncertainty within the same total number of training points compared to a traditional approach. The optimum results showed better effective strain homogeneity at the cross-section of the drawn wire than the original wire drawing (WD) process and NCD sequence with the same total reduction of area and less number of passes.

Chance-constrained multi-objective optimization of groundwater remediation design at DNAPLs-contaminated sites using a multi-algorithm genetically adaptive method

In this paper, a multi-algorithm genetically adaptive multi-objective (AMALGAM) method is proposed as a multi-objective optimization solver. It was implemented in the multi-objective optimization of a groundwater remediation design at sites contaminated by dense non-aqueous phase liquids. In this study, there were two objectives: minimization of the total remediation cost, and minimization of the remediation time. A non-dominated sorting genetic algorithm II (NSGA-II) was adopted to compare with the proposed method. For efficiency, the time-consuming surfactant-enhanced aquifer remediation simulation model was replaced by a surrogate model constructed by a multi-gene genetic programming (MGGP) technique. Similarly, two other surrogate modeling methods—support vector regression (SVR) and Kriging (KRG)—were employed to make comparisons with MGGP. In addition, the surrogate-modeling uncertainty was incorporated in the optimization model by chance-constrained programming (CCP). The results showed that, for the problem considered in this study, (1) the solutions obtained by AMALGAM incurred less remediation cost and required less time than those of NSGA-II, indicating that AMALGAM outperformed NSGA-II. It was additionally shown that (2) the MGGP surrogate model was more accurate than SVR and KRG; and (3) the remediation cost and time increased with the confidence level, which can enable decision makers to make a suitable choice by considering the given budget, remediation time, and reliability.

Rainfall estimate by kriging and cokrigagem in the State of Sergipe

A previsao da intensidade de chuva em determinado local nao amostrado e de grande interesse para a meteorologia e hidrologia. Esta pesquisa teve como objetivo a construcao de mapas mensais de chuvas para todo o Estado de Sergipe. Para isso foram utilizados dados de 64 estacoes meteorologicas distribuidas irregularmente por todo o estado. Para a construcao dos mapas foram utilizadas duas tecnicas de interpolacao denominadas de krigagem e cokrigagem. Para a cokrigagem a variavel secundaria utilizada foi a altitude, sendo utilizadas 162 altitudes obtidas com distribuicao geografica irregular para todo o Estado. A cokrigagem apresentou resultados mais consistentes do que a krigagem, a comparacao entre as duas tecnicas foi realizada pela validacao cruzada, em que a reta de regressao entre os valores estimados e observados, forneceu a inclinacao angular, o intercepto e o R 2 . Como parâmetros de erros ainda foram utilizados a media dos erros, o desvio padrao dos erros, a soma dos erros individuais e o erro padrao da estimativa. Para todos os meses do ano a cokrigagem apresentou a maioria dos erros com melhores resultados do que a krigagem, os unicos erros que a krigagem apresentou resultados superiores a cokrigagem foi a media dos erros e/ou o intercepto.

Comparison analysis of sampling methods to estimate regional precipitation based on the Kriging interpolation methods: A case of northwestern China

The accuracy of spatial interpolation of precipitation data is determined by the actual spatial variability of the precipitation, the interpolation method, and the distribution of observatories whose selections are particularly important. In this paper, three spatial sampling programs, including spatial random sampling, spatial stratified sampling, and spatial sandwich sampling, are used to analyze the data from meteorological stations of northwestern China. We compared the accuracy of ordinary Kriging interpolation methods on the basis of the sampling results. The error values of the regional annual precipitation interpolation based on spatial sandwich sampling, including ME (0.1513) , RMSE (95.91) , ASE (101.84) , MSE (-0.0036) , and RMSSE (1.0397) , were optimal under the premise of abundant prior knowledge. The result of spatial stratified sampling was poor, and spatial random sampling was even worse. Spatial sandwich sampling was the best sampling method, which minimized the error of regional precipitation estimation. It had a higher degree of accuracy compared with the other two methods and a wider scope of application.

Efficient VaR and CVaR Measurement via Stochastic Kriging

In this paper, stochastic kriging SK is considered as a metamodeling tool to capture risk-related properties of complex stochastic system outputs. To better assess the tail behavior of the underlyi...

Ore grade estimation of a limestone deposit in India using an Artificial Neural Network

This study describes a method used to improve ore grade estimation in a limestone deposit in India. Ore grade estimation for the limestone deposit was complicated by the complex lithological structure of the deposit. The erratic nature of the deposit and the unavailability of adequate samples for each of the lithogical units made standard geostatistical methods of capturing the spatial variation of the deposit inadequate. This paper describes an attempt to improve the ore grade estimation through the use of a feed forward neural network (NN) model. The NN model incorporated the spatial location as well as the lithological information for modeling of the ore body. The network was made up of three layers: an input, an output and a hidden layer. The input layer consisted of three spatial coordinates (x, y and z) and nine lithotypes. The output layer comprised all the grade attributes of limestone ore including silica (SiO2), alumina (Al2O3), calcium oxide (CaO) and ferrous oxide (Fe2O3). To justify the use of the NN in the deposit, a comparative evaluation between the NN method and the ordinary kriging was performed. This evaluation demonstrated that the NN model decisively outperformed the kriging model. After the superiority of the NN model had been established, it was used to predict the grades at an unknown grid location. Prior to constructing the grade maps, lithological maps of the deposit at the unknown grid were prepared. These lithological maps were generated using indicator kriging. The authors conclude by suggesting that the method described in this paper could be used for grade-control planning in ore deposits.

Automatic Mapping of Monitoring Data

This paper presents an approach, based on universal kriging, for automatic mapping of monitoring data. The performance of the mapping approach is tested on two datasets containing daily mean gamma dose rates in Germany reported by means of the national automatic monitoring network (IMIS). In the second dataset an accidental release of radioactivity in the environment was simulated in the South-Western corner of the monitored area. The approach has a tendency to smooth the actual data values, and therefore it underestimates extreme values, as seen in the second dataset. However, it is capable of identifying a release of radioactivity provided that the number of sampling locations is sufficiently high. Consequently, we believe that a combination of applying the presented mapping approach and the physical knowledge of the transport processes of radioactivity should be used to predict the extreme values.

Automated Mapping using Multilevel B-Splines

This paper describes application of multilevel B-Splines approximation (MBA) algorithm to the SIC2004 exercise as a high performance automatic mapping means in emergency situations. MBA method was compared with kriging, and various flexible MBA methods tuning controls were proposed and discussed. Outliers automatic detection and delineation techniques considered could be used to process outliers without the loss in performance. Prior data were used to adjust method parameters and discover the pattern of spatial correlation. The interpolation algorithm development did not assume any information about the phenomena in addition to values given.

Generating high spatiotemporal resolution LAI based on MODIS/GF-1 data and combined Kriging-Cressman interpolation

Abstract: Generation of high spatial and temporal resolution LAI (leaf area index) products is challenging because higher spatial resolution remotely sensed data usually have coarse temporal resolutions and vice versa. In this study, a novel method that combining Kriging interpolation and Cressman interpolation was proposed to generate high spatial and temporal resolution LAI products by fusing Moderate Resolution Imaging SpectroRadiometer (MODIS) characterized by coarse spatial resolution and high temporal resolution and Gaofen-1 (GF-1) with fine spatial resolution and coarse temporal resolution. This method was applied to the Huangpu district of Guangzhou, Guangdong, China. The results showed that compared to field observation, the predicted values of LAI had an acceptable accuracy of 73.12%. Using Moran’s I index and Kolmogorov-Smirnov tests, it was found that the MODIS data were spatially auto-correlated and characterized by normal distributions. Scaling down the 1 km×1 km spatial resolution MODIS products to a spatial resolution of 30 m×30 m using point-Kriging resulted in a precision of 79.38% compared to the results at the same spatial resolution derived from an 8 m×8 m spatial resolution GF-1 image by scaling up using block-Kriging. Moreover, the regression models that accounts for the relationship between NDVI (Normalized Difference Vegetation Index) and LAI based on MODIS data obtained the determination coefficients ranging from 0.833 to 0.870. Finally, the data fusion and interpolation of MODIS and GF-1 data using Cressman method generated high spatial and temporal resolution LAI maps, which showed reasonably spatial and temporal variability. The results imply that the proposed method is a powerful tool to create high spatial and temporal resolution LAI products. Keywords: data fusion, MODIS, GF-1, LAI, spatiotemporal resolution, spatial interpolation, remote sensing DOI: 10.3965/j.ijabe.20160905.1777 Citation: Liu Z H, Huang R G, Hu Y M, Fan S D, Feng P H. Generating high spatiotemporal resolution LAI based on MODIS/GF-1 data and combined Kriging-Cressman interpolation. Int J Agric & Biol Eng, 2016; 9(5): 120－131.

Comparative investigations on reference models for fault detection and diagnosis in centrifugal chiller systems

A successful fault detection and diagnosis (FDD) strategy for chillers is highly dependent on three core aspects, an accurate reference model, a sensitive fault detection technique and an effective fault identification method. An accurate model plays an important role in the performance of chiller FDD strategy. In this study, the accuracies of three models, MLR (multiple linear regression), KRG (Kriging) and RBF (radial basis function), are compared under various modeling criteria. The EWMA (exponentially-weighted moving average) residual control chart, which is a more powerful tool in detecting small shifts than t-statistics based method, is adopted as a sensitive fault detection technique. The most sensitive feature parameter method is improved to effectively identify faults. Seven common faults in typical centrifugal chillers are considered. Three strategies (MLR-EWMA, KRG-EWMA and RBF-EWMA), which are the combinations of three reference models with EWMA and the most sensitive feature parameter method, are validated and compared by using two sample datasets with different sizes from ASHRAE RP-1043. Through the comparisons of diagnosis performance, the most accurate chiller reference model is found and suggested to detect and diagnose faults in chiller systems. The comparison results show that the RBF-based strategy has the best diagnosis performances among the three strategies.

Spatial composition of AMSR2 soil moisture products by conditional merging technique with ground soil moisture data

The conditional merging (CM) spatial interpolation technique was applied to obtain the composite soil moisture products using the AMSR2 and in situ soil moisture for the 51 days of the summer through the late fall season of the year 2012 in Korean Peninsula. The ‘leave one out cross-validation’ analysis was conducted to assess the performance of the composite soil moisture products in estimating the soil moisture in ungagged locations. The control variable for comparison was the soil moisture products obtained by spatially interpolating the in situ soil moisture data measured at eight gage locations using the Ordinary Kriging (KR) technique. The results show that the composite soil moisture products are more accurate than the in situ only soil moisture products in estimating the soil moisture for the following cases: (1) when the spatial correlation of in situ soil moisture data is low. Such case includes when there is little rainfall and where the altitude is high (mountainous area) and (2) where the gage density is low or the area located further away from the in situ gages. For both cases, the KR method cannot use enough information due to the low spatial correlation of the in situ measurement for interpolation, while the CM method can take advantage of the satellite soil moisture measurement not affected by the spatial correlation of the in situ data.

Study of Coal Burst Source Locations in the Velenje Colliery

The Velenje coal mine (VCM) is situated on the largest Slovenian coal deposit and in one of the thickest layers of coal known in the world. The thickness of the coal layer causes problems for the efficiency of extraction, since the majority of mining operations is within the coal layer. The selected longwall coal mining method with specific geometry, increasing depth of excavations, changes in stress state and naturally given geomechanical properties of rocks induce seismic events. Induced seismic events can be caused by caving processes, blasting or bursts of coal or the surrounding rock. For 2.5D visualization, data of excavations, ash content and calorific value of coal samples, hanging wall and footwall occurrence, subsidence of the surface and coal burst source locations were collected. Data and interpolation methods available in software package Surfer®12 were statistically analyzed and a Kriging (KRG) interpolation method was chosen. As a result 2.5D visualizations of coal bursts source locations with geomechanical properties of coal samples taken at different depth in the coal seam in the VCM were made with data-visualization packages Surfer®12 and Voxler®3.

Geostatistics and exploratory factor analysis for special representation of soil chemical attributes in coffee growing

objective of this study was to evaluate the spatial dependence on chemical soil attributes by means of the geostatistical association and multivariate analysis (exploratory factor analysis), aiming to understand the behavior of the attributes in the soil, thus being able to contribute to the planning of fertilization, seeking greater productivity and sustainability of coffee growing. The research was carried in a grid of 63 sampling points, mounted on a Coffea arabica L. plantation in the cerrado region. Initially was held exploratory factor analysis that generated four factors that together accounted for 97.05% of the total variation of the data. In the geostatistical analysis it was found that the factors 1 and 3 showed spatial dependence, for which the data were interpolated by ordinary kriging (geostatistical). Already the factors 2 and 4 showed pure nugget effect (spatial independence) which held data interpolation by the inverse square of the distance method (classical statistics). The factors 1 and 3 strongly represent the acidity of soil fertility and the soil organic matter. Factors 2 and 4 represent the relationship between the attributes of the ground. The results showed that the use of multivariate techniques in combination with geostatistics can contribute to the coffee crop management in the Cerrado region.

Crashworthiness design of quenched boron steel thin-walled structures with functionally graded strength

Thin-walled structures have been widely used as energy absorbers in industries such as automobile, shipping and aerospace. Different from recent well-studied columns consisting of aluminum, this paper introduced a kind of quenched boron steel column formed by hot stamping technology, with wall strength varying along the axial direction with a specific gradient, i.e. functionally graded strength (FGS). These FGS structures are found to have higher crashworthiness in terms of peak crash force (PCF) and specific energy absorption (SEA) compared with the counterpart structures with uniform strength (US). Based on a series of numerical simulation results, it revealed that the crashing behavior of FGS columns is significantly affected by both the strength gradient and the strength in the impact end. To optimize the crashworthiness (PCF and SEA) of the FGS structures, multi-optimization based on different metamodeling techniques such as response surface method (RSM), radial basis function (RBF) neural network model, kriging (KRG) model and optimization algorithm of non-dominated sorting genetic algorithm (NSGA-II) are performed. Pareto fronts of several alternative thicknesses were obtained to provide guidance for the FGS column design and give a good insight into actual crashing engineering. It is interesting to find that the gradient exponent is taken as the main design variable when the PCF is restricted below a certain value while the parameter of steel strength in impact end will be taken as the main design variable on the contrary. The comparison of Pareto fronts between the FGS and US columns showed that the FGS columns enhance the SEA and lower the PCF concurrently. Furthermore, among the three metamodeling techniques, the RSM models are proven to be the most suitable approach for the crashworthiness optimization of FGS structure.