Global Kriging Research Articles

A new active learning surrogate model for time- and space-dependent system reliability analysis

This study introduces a novel method for time- and space-dependent system reliability analysis, integrating an active learning surrogate model with an innovative parallel updating strategy. A global Kriging model is developed to represent the signs of random samples using efficient global optimization. From a Bayesian perspective, the prediction probabilities of random sample signs within the time-space domain are calculated, and the sample with the lowest prediction probability is chosen to update the global Kriging model. The system extremum for each sample in the time-space domain is determined, and the corresponding random variables, time-space coordinates, and failure modes are selected. To further decrease iteration times, a parallel updating strategy that considers both the predicted probability and the correlation among candidate samples is proposed. Additionally, a new stopping criterion is introduced to balance accuracy and efficiency, terminating the updating process appropriately. The method's accuracy and efficiency are validated through three examples.

Multiobjective Reliability-Based Design Optimization of the Fuzzy Logic Controller for MR Damper-Based Structures

To devise an optimum and robust fuzzy logic controller for MR damper-based structures subjected to earthquake ground motions, the multiobjective reliability-based design optimization (RBDO) using the adaptive Kriging model is performed to determine the parameters of the fuzzy logic controller. The optimization problem is formulated with two objective functions, namely, the minimization of interstory drift and average control force of the concerned structure, and subjected to a probability constraint on structural dynamic responses under the effects of random structural stiffness and stochastic earthquake loadings. To reduce the computational cost of reliability assessment, a global Kriging model is constructed in an augmented space as a surrogate for computational evaluations. Subsequently, the trained metamodel combined with the nondominated sorting genetic algorithm (NSGA-II) is integrated into the framework of RBDO for solving the fuzzy logic control (FLC) optimization problem. The feasibility and effectiveness of the multiobjective RBDO in the FLC design are finally validated by conducting numerical simulations on both linear and nonlinear structures. As demonstrated in the linear case, the fuzzy logic controllers obtained from the multiobjective RBDO show more robustness than those derived from the multiobjective deterministic design optimization (DDO). In the nonlinear case, using the multiobjective DDO to prelocate a coarse safety domain can significantly reduce the number of samples for training the metamodel and facilitate the implementation of the multiobjective RBDO; in addition, the controlled structural performance with a specified fuzzy logic controller can be further improved by considering MR damper distribution optimization.

Nonmyopic Informative Path Planning Based on Global Kriging Variance Minimization

Remote, hazardous, and extreme exploration missions require robots to be equipped with on-board sensors for rich and heterogeneous information during deployment. In such tasks, path planning can directly affect the quality and quantity of the observations obtained under temporal and energetic constraints. While most informative path planners can only plan for a short horizon ahead of time (referred as to myopic planners), we propose a novel planner that is capable of planning global paths in which the nonmyopic exploration efficiency is optimized. To achieve this, a novel sampling algorithm named MPE is proposed to adaptively sample landmarks that are associated with high information capacity, in order to minimize the global Kriging variance. The traverse path for the landmarks is then obtained by the IPP-MPE algorithm for minimizing the overall traveling cost. The algorithm is flexible enough to be applied to various information acquisition tasks. The tractable computational cost allows the horizon to be long enough for scene coverage. The algorithm was deployed on a real robot for accomplishing tactile based object searching tasks, which shows superior efficiency compared to the myopic planner baseline. Last, the complexity and other theoretical analysis of the algorithm is provided.

Kriging parameter optimisation: global versus local search strategies

ABSTRACT Kriging methods require parameters to define search strategy (kriging neighbourhood). These parameters affect the precision and accuracy of its estimates. Frequently, the choice of these parameters is merely subjective. Some practitioners prioritise estimates that lead to models with a reduced smoothing effect or a regression slope as close as possible to one. However, it is prevalent to use the same kriging neighbourhood or search strategy for all blocks estimated within a stationary domain. This study presents a contribution that challenges this concept by using a block-by-block optimisation approach focused on the localised kriging parameter optimisation (LKPO) methodology. A comparative study is carried out, and some of the metrics analysed include the kriging efficiency and the slope of regression (typical in optimising methodologies in the mining industry). The results indicate that the LKPO methodology provides more accurate and precise estimates than those based on a global kriging neighbourhood.

A novel single-loop simulation method and its combination with adaptive kriging for moment-independent global sensitivity analysis

The moment-independent global sensitivity (MIGS) analysis can help decision makers to efficiently reduce the uncertainty of model output. This article proposes a novel single-loop simulation (NSLS) method for MIGS analysis. Through an interval discretization strategy, estimation of probability density functions (PDFs) in existing methods is transformed to estimation of the marginal probability of the input, the marginal probability of the output and the joint probability of the inputs–output in NSLS, which avoids the computational complexity and error resulting from PDF estimation in MIGS analysis. To drastically reduce the computational cost of NSLS to estimate the MIGS index, an NSLS-based adaptive kriging (NSLS-AK) method is subsequently developed. In NSLS-AK, a rough global kriging model is first constructed, then this kriging model is adaptively updated until the convergent condition is satisfied for accurately and efficiently predicting the responses of the samples in estimating the MIGS index.

Adaptive design of experiments for global Kriging metamodeling through cross-validation information

This paper discusses a new sequential adaptive design of experiments (DoE) approach for global Kriging metamodeling applications. The sequential implementation is established by using the current metamodel, formulated based on the existing experiments, to guide the selection of the optimal new experiment(s). The score function, defining the DoE objective, combines two components: (1) the metamodel prediction variability, expressed through the predictive variance, and (2) the metamodel bias, approximated through the leave-one-out cross validation (LOOCV) error. The latter is used as a weighting factor to extend traditional DoE approaches that focus solely on the metamodel prediction variability. Two such approaches are considered here, adopting either the integrated mean squared error or the maximum mean squared error as the basic component of the score function. The incorporation of bias information as weighting within these well-established approaches facilitates a direct extension of their respective computational workflows, making the proposed implementation attractive from computational perspective. An efficient optimization scheme for identification of the next experiment, as well as the balancing of exploration and exploitation between the two components of the score function, are also discussed. The incorporation of LOOCV weightings is shown to be highly beneficial in a total of six analytical and engineering examples. Furthermore, these examples demonstrate that for DoE approaches which use LOOCV information as weights, it is preferable to update the predictive variance to explicitly consider the impact of the new experiment, rather than relying strictly on the current metamodel variance.

Implementation of machine learning techniques into the Subset Simulation method

A hybrid reliability analysis procedure that combines the advantages of the Subset Simulation method, the application of surrogate models and clustering techniques is proposed to efficiently assess the failure probability of complex structural systems that may exhibit multiple failure modes. The proposed procedure, herein called SS-KK, integrates the Kriging surrogate modeling technique, K-means clustering algorithm into the original Subset Simulation (SS) method. The approach is found to not only improve the efficiency of the Subset Simulation method in finding the reliability of a structural system but to also help identify the important failure modes and controlling random variables. This information is important to aid engineers optimize the structural design process by focusing on the critical failures modes and design variables.The method consists of first constructing a relatively coarse global Kriging model at each subset level of SS by applying an appropriate active learning strategy. Subsequently, the initial global Kriging model is partitioned into several local Kriging models that coincide with the important failure modes identified by the K-means clustering algorithm. This partitioning, which helps identify the important failure modes, also gives a better representation of the entire failure region leading to improved estimates of the system reliability. Finally, FORM is implemented for each local Kriging to rank the importance of each identified failure mode based on its Hasofer-Lind reliability index and to use the associated design point and sensitivity coefficients to identify the most critical random variables.Several examples extracted from the available literature are analyzed to illustrate the advantages of the proposed SS-KK methodology and demonstrate its accuracy and efficiency.

Evidence‐Based Multidisciplinary Design Optimization with the Active Global Kriging Model

This article presents an approach that combines the active global Kriging method and multidisciplinary strategy to investigate the problem of evidence‐based multidisciplinary design optimization. The global Kriging model is constructed by introducing a so‐called learning function and using actively selected samples in the entire optimization space. With the Kriging model, the plausibility, Pl, of failure is obtained with evidence theory. The multidisciplinary feasible and collaborative optimization strategies of multidisciplinary design optimization are combined with the evidence‐based reliability analysis. Numerical examples are provided to illustrate the efficiency and accuracy of the proposed method. The numerical results show that the proposed algorithm is effective and valuable, which is valuable in engineering application.

Global kriging surrogate modeling for general time-variant reliability-based design optimization problems

While design optimization under uncertainty has been widely studied in the last decades, time-variant reliability-based design optimization (t-RBDO) is still an ongoing research field. The sequential and mono-level approaches show a high numerical efficiency. However, this might be to the detriment of accuracy especially in case of nonlinear performance functions and non-unique time-variant most probable failure point (MPP). A better accuracy can be obtained with the coupled approach, but this is in general computationally prohibitive. This work proposes a new t-RBDO method that overcomes the aforementioned limitations. The main idea consists in performing the time-variant reliability analysis on global kriging models that approximate the time-dependent limit state functions. These surrogates are built in an artificial augmented reliability space and an efficient adaptive enrichment strategy is developed that allows calibrating the models simultaneously. The kriging models are consequently only refined in regions that may potentially be visited by the optimizer. It is also proposed to use the same surrogates to find the deterministic design point with no extra computational cost. Using this point to launch the t-RBDO guarantees a fast convergence of the optimization algorithm. The proposed method is demonstrated on problems involving nonlinear limit state functions and non-stationary stochastic processes.

Declustering experimental variograms by global estimation with fourth order moments

The variogram is a key parameter for geostatistical estimation and simulation. Preferential sampling may bias the spatial structure and often leads to noisy and unreliable variograms. A novel technique is proposed to weight variogram pairs in order to compensate for preferential or clustered sampling . Weighting the variogram pairs by global kriging of the quadratic differences between the tail and head values gives each pair the appropriate weight, removes noise and minimizes artifacts in the experimental variogram. Moreover, variogram uncertainty could be computed by this technique. The required covariance between the pairs going into variogram calculation, is a fourth order covariance that must be calculated by second order moments. This introduces some circularity in the calculation whereby an initial variogram must be assumed before calculating how the pairs should be weighted for the experimental variogram. The methodology is assessed by synthetic and realistic examples. For synthetic example, a comparison between the traditional and declustered variograms shows that the declustered variograms are better estimates of the true underlying variograms. The realistic example also shows that the declustered sample variogram is closer to the true variogram.

Assessing Lead Contamination in Buffalo River Sediments

The Great Lakes Water Quality Agreement between Canada and the United States has identified the Buffalo River as an Area of Concern. The watershed has a long history of heavy industrial activity that contributed to its overall pollution. Sediment core data collected by the New York State Department of Environmental Conservation in 2005 were used to determine lead sediment contamination in a section of the Buffalo River. The ordinary kriging spatial interpolation technique was used to generate surface and subsurface sediment contamination estimates. Due to the meandering nature of the river, two kriging models were used to analyze surface contamination: a global kriging model and a regional kriging model, consisting of three separate sections. The results show that both the global and regional kriging models display similar interpolated surfaces and do not vary significantly. Within the sediment, lead contamination in the surface layer is lower than at the various subsurface depths. In 2011, habitat restoration efforts commenced to remediate environmental damage due to years of pollution inputs from various sources. Sediment dredging operations were initiated that are expected to be completed in 2015. The goal of these operations is to remove heavily contaminated sediments and rehabilitate the Buffalo River. The kriging results provide area-wide estimates of contamination. When compared to the dredging plan, the results indicate that additional removal of contaminated sediments may need to be considered where no dredging has occurred or is not currently planned.

Modeling multiple coal seams using signed distance functions and global kriging

An approach is developed to simultaneously model the geometry of multiple coal seams in 3-D without directly modeling surface elevations or isochore thicknesses. The approach truncates a locally refined 3-D model of estimated signed distance function values. The signed distance function measures the sign dependent distance to a coal/non-coal contact; negative in coal and positive outside. Global kriging is used to estimate the distance function at unsampled locations so that the resulting model has the correct anisotropy and is free of search artifacts. This approach is aimed at relatively large modeling volumes where the number of coal seams varies and the seams show complexities such as pinch outs and bifurcation. The technique aims to locate all of the coal boundaries with high precision through local grid refinement. The results can be used for resource/reserve evaluation.

Adaptive Neuro-Fuzzy Inference System (Anfis) Based Local Ordinary Kriging Algorithm for Scattered Data Interpolation

A new approach to the Ordinary Kriging interpolation method based on the combination of local interpolation and variogram modelling with Adaptive Neuro-Fuzzy Inference System (ANFIS) is proposed for scattered data interpolation. In this method, the experimental variogram is modelled by ANFIS and this model is used to interpolate the unknown values of specific points in a new local manner. In this local way, all the unknown points are grouped based on each reference point. The study data obtained from mathematical functions are used. The tests show that the proposed method provides better performances for all data sets in comparison to the well known and highly approved interpolation methods; Ordinary Kriging, Triangle Based Cubic and Radial Basis Function-Multiquadric. Moreover, by the proposed method the computational complexity impressively decreases compared to the global ordinary Kriging.

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In this paper, we made a simple paper feeding system which is one of MTS (media transport system) and controllers. The plant has a flexible paper and two driving rollers and two driven rollers. The control system has two conventional PID controllers. Skew angle and feeding speed of MTS deteriorate the quality of feeding system. In order to control a feeding speed and skew of feeding paper, we control rotational velocity of two driving rollers. Therefore, this controller has two inputs and two outputs as MIMO (multi-input and multi-output) system. The control inputs were the feeding speed and the skew displacement of the paper. The control outputs were the rotational velocity to each driving roller. To find appropriate PID gains of two controllers, we proposed an optimization technique. We assume the system variables and performance of a whole system as follows. PID gains of two controllers for skew and feeding speed are system variables. System performance is both skew and feeding speed. We simulates to making mathematical correlation using global Kriging interpolation. To find appropriate value of system variables, optimization method is simulation in sequence as following method. First, the optimization solver simulates with DOE (design of experiment) tables to find correlation equation of both system variable and performances. Then, the solver guesses the appropriate values and simulates if the system variables are appropriate or not. If the result of validation doesn’t satisfy the convergence and iteration tolerance, the solver makes a new Kriging models and iterates this sequence until satisfy the tolerances. 기호설명 e

Comparison of six methods for the interpolation of daily, European climate data

We compare versions of six interpolation methods for the interpolation of daily precipitation, mean, minimum and maximum temperature, and sea level pressure from station data over Europe from 1961 to 1990. The interpolation methods evaluated are global and local kriging, two versions of angular distance weighting, natural neighbor interpolation, regression, 2D and 3D thin plate splines, and conditional interpolation. We first evaluated, using station cross‐validation and several skill scores, relative skill of each method at estimating point values, looking at spatial and temporal patterns and the frequency distribution of the variables. We then compared, for precipitation, gridded area averages from the candidate interpolation methods against existing high‐resolution gridded data sets for the UK and the Alps, which are derived from a much denser network of stations. In both point and area‐average cases, differences in skill between interpolation methods at any one point are smaller than the range in skill for a single method either across the domain, or in different seasons. The main control on spatial patterns of interpolation skill is density of the station network, with topographic complexity a compounding factor. The relative skill of different methods remains relatively constant through time, despite a varying station network. Skill in interpolating extreme events is lower than for average days, but relative skill of different methods remains the same. We select global kriging as the best performing method overall, for use in the development of a daily, high‐resolution, long‐term, European data set of climate variables as part of the EU funded ENSEMBLES project.

Using spatial models and kriging techniques to optimize long‐term ground‐water monitoring networks: a case study

AbstractIn a pilot project, a spatial and temporal algorithm (geostatistical temporal–spatial or GTS) was developed for optimizing long‐term monitoring (LTM) networks. Data from two monitored ground‐water plumes were used to test the algorithm. The primary objective was to determine the degree to which sampling, laboratory analysis, and/or well construction resources could be pared without losing key statistical information concerning the plumes. Optimization of an LTM network requires an accurate assessment of both ground‐water quality over time and trends or other changes in individual monitoring wells. Changes in interpolated concentration maps over time indicate whether ground‐water quality has improved or declined. GTS separately identifies temporal and spatial redundancies. Temporal redundancy may be reduced by lengthening the time between sample collection. Spatial redundancy may be reduced by removing wells from the network which do not significantly impact assessment of ground‐water quality.Part of the temporal algorithm in GTS involves computation of a composite temporal variogram to determine the least redundant overall sampling interval. Under this measure of autocorrelation between sampling events, the lag time at which the variogram reaches a sill is the sampling interval at which same‐well measurements lack correlation and are therefore non‐redundant. The spatial algorithm assumes that well locations are redundant if nearby wells offer nearly the same statistical information about the underlying plume. A well was considered redundant if its removal did not significantly change: (i) an interpolated map of the plume; (ii) the local kriging variances in that section of the plume; and (iii) the average global kriging variance. To identify well redundancy, local kriging weights were accumulated into global weights and used to gauge each well's relative contribution to the interpolated plume map. By temporarily removing that subset of wells with the lowest global kriging weights and re‐mapping the plume, it was possible to determine how many wells could be removed without losing critical information.Test results from the Massachusetts Military Reserve (MMR) indicated that substantial savings in sampling, analysis and operational costs could be realized by utilizing GTS. Annual budgetary savings that would accrue were estimated at between 35 per cent and 5 per cent for both LTM networks under study.Copyright © 2002 John Wiley & Sons, Ltd.

Creating Field Extent Digital Elevation Models for Precision Agriculture

This paper presents a comparison of techniques for the interpolation of elevation data collected using a global positioning system. The techniques examined included global and local regression models and kriging of the residuals, global and local kriging, and the TOPOGRID tool in Arc Info. The results demonstrated the superiority of the TOPOGRID tool.

Global block kriging to estimate biomass from acoustic surveys for zooplankton in the western Aleutian Islands

Because of patchy distributions and the expense of collecting and processing net samples, zooplankton population data are often characterized by broad confidence intervals, with little detailed information on vertical and horizontal distributions. Although acoustic techniques and optical plankton counters combined with nets can supply much more detailed information on plankton distributions, application of classical statistical procedures to such data may be effected by pseudoreplication from autocorrelation in closely spaced samples. We apply two-dimensional kriging for population mapping and global estimation. The technique utilizes a computational estimator of the global block kriging mean, which can be applied to large data sets common to hydroacoustic surveys. The techniques are outlined using examples from acoustic data taken in the western Aleutian Islands. This technique has two fundamental advantages: (i) it minimizes pseudoreplication by accounting for autocorrelation, permitting rigorous statistical inferences, and (ii) it provides an effective technique for visualizing the results. These advantages may make this a useful technique for identifying changes in the size and distribution of populations.

Global block kriging to estimate biomass from acoustic surveys for zooplankton in the western Aleutian Islands

Global block kriging to estimate biomass from acoustic surveys for zooplankton in the western Aleutian Islands

Assessment of the spatial structure and biomass evaluation ofNephrops norvegicus(L.) populations in the northwestern Mediterranean by geostatistics

The spatial distribution and biomass of Nephrops norvegicus were assessed by trawling over commercial fishing grounds (“Serola”, off Barcelona, Spain) during two surveys (spring and fall 1991), using geostatistical methods. The surveys were set 6 months apart, in order to analyse possible seasonal differences. In the present surveys, Norway lobster was caught between 200 and 600 m depth, with peak abundance at about 400 m. The analysis of the structure of spatial correlation by means of semivariograms showed that densities of Nephrops norvegicus were spatially autocorrelated and lobster populations were distributed in high-density patches 6 to 9 km in diameter. No spatial segregation per biological category (size or sex) was detected. The semivariograms were consistent for all biological categories. A strong linear relationship between local mean and standard deviation (proportional effect) was modelled by the relative semivariogram. Relative experimental semivariograms were fitted to a spherical model. The shape of the semivariogram, and the spatial autocorrelation structure of Norway lobster populations, remained stable over the two surveys. The density of Nephrops norvegicus available to the experimental gear was mapped by point kriging. High-density patches of different biological categories exactly conformed and remained stable over the two surveys, showing a certain intra-annual stability. However, mean densities and overall abundance (computed by global kriging) decreased sharply in the fall survey. This was accounted for by means of knowledge on the biology of the species for the same area. The biological characteristics of Nephropspopulations in the area studied are similar to those of other Mediterranean and Atlantic populations, hence our results are not restricted to the study area. We conclude that the geostatistical analysis approach, which takes into consideration the spatial auto-correlation structure of the populations, is adequate for the direct biomass estimation and assessment of Nephrops harvestable stock.