Standard Kriging Research Articles

The comap as a diagnostic tool for non-stationary kriging models

In this study, we demonstrate a novel use of comaps to explore spatially the performance, specification and parameterisation of a non-stationary geostatistical predictor. The comap allows the spatial investigation of the relationship between two geographically referenced variables via conditional distributions. Rather than investigating bivariate relationships in the study data, we use comaps to investigate bivariate relationships in the key outputs of a spatial predictor. In particular, we calibrate moving window kriging (MWK) models, where a local variogram is found at every target location. This predictor has often proved worthy for processes that are heterogeneous, and most standard (global variogram) kriging algorithms can be adapted in this manner. We show that the use of comaps enables a better understanding of our chosen MWK models, which in turn allows a more informed choice when selecting one MWK specification over another. As case studies, we apply four variants of MWK to two heterogeneous example data sets: (i) freshwater acidification critical load data for Great Britain and (ii) London house price data. As both of these data sets are strewn with local anomalies, three of our chosen models are robust (and novel) extensions of MWK, where at least one of which is shown to perform better than a non-robust counterpart.

Combustor Design Optimization Using Co-Kriging of Steady and Unsteady Turbulent Combustion

In the gas turbine industry, computational fluid dynamics (CFD) simulations are often used to predict and visualize the complex reacting flow dynamics, combustion environment and emissions performance of a combustor at the design stage. Given the complexity involved in obtaining accurate flow predictions and due to the expensive nature of simulations, conventional techniques for CFD based combustor design optimization are often ruled out, primarily due to the limits on available computing resources and time. The design optimization process normally requires a large number of analyses of the objective and constraint functions which necessitates a careful selection of fast, reliable and efficient computational methods for the CFD analysis and the optimization process. In this study, given a fixed computational budget, an assessment of a co-Kriging based optimization strategy against a standard Kriging based optimization strategy is presented for the design of a 2D combustor using steady and unsteady Reynolds-averaged Navier Stokes (RANS) formulation. Within the fixed computational budget, using a steady RANS formulation, the Kriging strategy successfully captures the underlying response; however with unsteady RANS the Kriging strategy fails to capture the underlying response due to the existence of a high level of noise. The co-Kriging strategy is then applied to two design problems, one using two levels of grid resolutions in a steady RANS formulation and the other using steady and unsteady RANS formulations on the same grid resolution. With the co-Kriging strategy, the multifidelity analysis is expected to find an optimum design in comparatively less time than that required using the high-fidelity model alone since less high-fidelity function calls should be required. However, using the applied computational setup for co-Kriging, the Kriging strategy beats the co-Kriging strategy under the steady RANS formulation whereas under the unsteady RANS formulation, the high level of noise stalls the co-Kriging optimization process.

Data-driven Kriging models based on FANOVA-decomposition

Kriging models have been widely used in computer experiments for the analysis of time-consuming computer codes. Based on kernels, they are flexible and can be tuned to many situations. In this paper, we construct kernels that reproduce the computer code complexity by mimicking its interaction structure. While the standard tensor-product kernel implicitly assumes that all interactions are active, the new kernels are suited for a general interaction structure, and will take advantage of the absence of interaction between some inputs. The methodology is twofold. First, the interaction structure is estimated from the data, using a first initial standard Kriging model, and represented by a so-called FANOVA graph. New FANOVA-based sensitivity indices are introduced to detect active interactions. Then this graph is used to derive the form of the kernel, and the corresponding Kriging model is estimated by maximum likelihood. The performance of the overall procedure is illustrated by several 3-dimensional and 6-dimensional simulated and real examples. A substantial improvement is observed when the computer code has a relatively high level of complexity.

Hypothesis of local stationarity in the problem of stochastic prognosis by kriging method

One of the most widely-used stochastic prognosis methods for estimating the reserves of mineral resources, namely the method of kriging, is considered. A modification of this method based on the assumption of local stationarity of the spatial variable is proposed. The modified kriging method, in combination with a morphological algorithm for detecting the subdomains of local stationarity is shown to be in good agreement with the a priori ideas on data structure. This makes it possible to decrease computational efforts for the forecasting procedure as compared to the standard kriging technique.

Combining numerical model output and particulate data using Bayesian space–time modeling

AbstractOver the past few years, Bayesian models for combining output from numerical models and air monitoring data have been applied to environmental data sets to improve spatial prediction. This paper develops a new hierarchical Bayesian model (HBM) for fine particulate matter (PM2.5) that combines U. S. EPA Federal Reference Method (FRM) PM2.5 monitoring data and Community Multi‐scale Air Quality (CMAQ) numerical model output. The model is specified in a Bayesian framework and fitted using Markov Chain Monte Carlo (MCMC) techniques. We find that the statistical model combining monitoring data and CMAQ output provides reliable information about the true underlying PM2.5 process over time and space. We base these conclusions on results of a validation exercise in which independent monitoring data were compared with predicted values from the HBM and predictions from a standard kriging model based solely on the monitoring data. Copyright © 2009 John Wiley & Sons, Ltd.

Monotonicity-Preserving Bootstrapped Kriging Metamodels for Expensive Simulations

Kriging metamodels (also called Gaussian process or spatial correlation models) approximate the Input/Output functions implied by the underlying simulation models. Such metamodels serve sensitivity analysis, especially for computationally expensive simulations. In practice, simulation analysts often know that this Input/Output function is monotonic. To obtain a Kriging metamodel that preserves this characteristic, this article uses distribution-free bootstrapping assuming each input combination is simulated several times to obtain more reliable averaged outputs. Nevertheless, these averages still show sampling variation, so the Kriging metamodel does not need to be an exact interpolator; bootstrapping gives a noninterpolating Kriging metamodel. Bootstrapping may use standard Kriging software. The method is illustrated through the popular M/M/1 model with either the mean or the 90% quantile as output; these outputs are monotonic functions of the traffic rate. The empirical results demonstrate that monotonicity-preserving bootstrapped Kriging gives higher probability of covering the true outputs, without lengthening the confidence interval.

CREDIBILITY MEASURE-BASED FUZZY MEMBERSHIP GRADE KRIGING

A fundamental problem in fuzzy analysis is that the membership function is specified subjectively. In other words, the modelers specify the membership function form and assign the values of the parameters in membership function via their working experiences. Different from its probabilistic counterpart, fuzzy mathematical theory does not provide convenient parameter estimation approach. In this paper, we first review Liu's non-classical credibility measure theory (i.e., (∨,·)-credibility measure theory) in Liu7, because the fuzzy theory initiated by Zadeh10 contains a critical weakness: non self-duality property for possibility measure. We establish a parameter estimation of the membership function in terms of maximum entropy principle on the ground of self-dual credibility measure theory. Furthermore, based on the data assimilated membership function, we can calculate membership grades on the fuzzy environmental index, using PM10 air pollution as an illustration. We treat the calculated membership grades as spatially distributed random quantity, and therefore perform the standard ordinary kriging approach for generating the predicted environmental index map, for PM10 prediction map.

Mapping Curvilinear Structures with Local Anisotropy Kriging

Interpolating geo-data with curvilinear structures using geostatistics is often disappointing. Channels, for example, become disconnected sets of lakes when interpolated from point data. In order to improve the interpolation of geological structures (e.g., curvilinear structures), we present a new form of kriging, local anisotropy kriging (LAK). Local anisotropy kriging combines a gradient algorithm from image analysis with kriging in an iterative way. After an initial standard kriging interpolation, the gradient algorithm determines the local anisotropy for each cell in the grid using a search area around the cell. Subsequently, kriging is carried out with the spatially varying anisotropy. The anisotropy calculation and subsequent kriging steps will then succeed until the result is satisfactory in the way of reproducing the curvilinear structures. Depending on the size of the search area more or less detail in the geological structures can be reproduced with LAK. Using test examples we show that LAK interpolates data with curvilinear structures more realistically than standard kriging. In a real world case, using bathymetric data of the Oosterschelde estuary, LAK also proves to be quantitatively superior to standard kriging. Absolute interpolation errors are decreased by 23%. Local anisotropy kriging only uses information from point data, which makes the method very objective, it only presents “what the data can tell.”

A multiple linear statistical model for estimating the mean maximum urban heat island

¶This study examines the spatial and quantitative influence of urban factors on the surface air temperature field of the medium-sized of Szeged, Hungary, using mobile measurements under different weather conditions in the periods of March 1999–February 2000 and April–October 2002. Efforts have been concentrated on the development of the urban heat island (UHI) in its peak development during the diurnal cycle. Tasks included: (1) determination of spatial distribution of mean maximum UHI intensity and some urban surface parameters (built-up and water surface ratios, sky view factor, building height) using the standard Kriging procedure, as well as (2) development of a statistical model in the so-called heating and non-heating seasons using the above mentioned parameters and their areal extensions. In both seasons the spatial distribution of the mean maximum UHI intensity fields had a concentric shape with some local irregularities. The intensity reaches more than 2.1 °C (heating season) and 3.1 °C (non-heating season) in the centre of the city. For both seasons statistical model equations were determined by means of stepwise multiple linear regression analysis. As the measured and calculated mean maximum UHI intensity patterns show, there is a clear connection between the spatial distribution of the urban thermal excess and the examined land-use parameters, so these parameters play an important role in the evolution of the strong UHI intensity field. From the above mentioned parameters the sky-view factor and the building height were the most determining factors which are in line with the urban surface energy balance. Therefore in the future, using our model it will be possible to predict mean maximum UHI intensity in other cities, which have land-use features similar to Szeged.

Spatial Variation of Radiocaesium Deposition in Cumbria

Over the past five decades Cumbria has been subject to radiocaesium deposition from a number of sources. In the event of a future nuclear accident, estimates of baseline radiocaesium deposition in Cumbria would be indispensable. This paper describes the application of ordinary block kriging to estimate the spatial variation of total 137 Cs and 134 Cs. This study has raised a number of issues regarding the application of kriging. Transformation of input data was necessary to improve the optimality of the standard kriging estimator. Finally, observations of 137 Cs and 134 Cs from a similar range of locations exhibit different spatial characteristics. This can be explained by differences in radioactive half-lives and differences in the deposition of the two isotopes.

A high frequency kriging approach for non‐stationary environmental processes

AbstractEmission reductions were mandated in the Clean Air Act Amendments of 1990 with the expectation that they would result in major reductions in the concentrations of atmospherically transported pollutants. The emission reductions are intended to reduce public health risks and to protect sensitive ecosystems. To determine whether the emission reductions are having the intended effect on atmospheric concentrations, monitoring data must be analyzed taking into consideration the spatial structure shown by the data. Maps of pollutant concentrations and fluxes are useful over different geopolitical boundaries, to discover when, where, and to what extent the U.S. Nation's air quality is improving or declining. Since the spatial covariance structure shown by the data changes with location, the standard kriging methodology for spatial interpolation cannot be used because it assumes stationarity of the process.We present a new methodology for spatial interpolation of non‐stationary processes. In this method the field is represented locally as a stationary isotropic random field, but the parameters of the stationary random field are allowed to vary across space. A procedure for interpolation is presented that uses an expression for the spectral density at high frequencies. New fitting algorithms are developed using spectral approaches. In cases where the data are distributed exactly or approximately on a lattice, it is argued that spectral approaches have potentially enormous computational benefits compared with maximum likelihood. The methods are extended to interpolation questions using approximate Bayesian approaches to account for parameter uncertainty. We develop applications to obtain the total loading of pollutant concentrations and fluxes over different geo‐political boundaries. Copyright © 2001 John Wiley & Sons, Ltd.