en-wikipedia-org.pdf

7/28/2019 en-wikipedia-org.pdf

1/6

Kriging

Example of one-dimensional data interpolation by kriging,with confidence intervals. Squares indicate the location ofthe data. The kriging interpolation is in red. The confidenceintervals are in green.

From Wikipedia, the free encyclopedia

Kriging is a geostatistical estimator that infers the value of a random field at an unobserved location (e.g. elevation as a function ofgeographic coordinates) from samples (see spatial analysis and computer experiments).

The theory behind interpolating orextrapolatingby kriging was developed by theFrenchmathematician Georges Matheron based on the Master's thesis ofDanieG. Krige, the pioneering plotter of distance-weighted average gold grades at theWitwatersrand reef complex in South Africa. The English verb is to krige and themost common noun is kriging; both are often pronounced with a hard "g",following the pronunciation of the name "Krige".

Contents [hide]

1 Geostatistical estimation1.1 Related terms and techniques

1.2 Geostatistical estimator

1.3 Linear estimation

2 Methods2.1 Ordinary kriging

2.1.1 Kriging system

2.2 Simple kriging2.2.1 Kriging system

2.2.2 Estimation

2.3 Properties

3 Applications3.1 Design and analysis of computer experiments

4 See also

5 References5.1 Books

5.2 Historical references

Geostatistical estimation [edit]

Related terms and techniques [edit]

Kriging is based on the idea that the value at an unknown point should be the average of the known values at its neighbors; weighted bythe neighbors' distance to the unknown point. The method is mathematically closely related to regression analysis. Both theories derive abest linear unbiased estimator, based on assumptions on covariances, make use ofGauss-Markov theorem to prove independence of theestimate and error, and make use of very similar formulae. They are nevertheless useful in different frameworks: kriging is made forestimation of a single realization of a random field, while regression models are based on multiple observations of a multivariate dataset.

In the statistical community the same technique is also known as Gaussian process regression, Kolmogorov Wiener prediction, orbestlinear unbiased prediction.

The kriging estimation may also be seen as a spline in a reproducing kernel Hilbert space, with reproducing kernel given by thecovariance function.[1] The difference with the classical kriging approach is provided by the interpretation: while the spline is motivated bya minimum norm interpolation based on a Hilbert space structure, kriging is motivated by an expected squared prediction error based ona stochastic model.

Kriging withpolynomial trend surfaces is mathematically identical to generalized least squares polynomial curve fitting.

Kriging can also be understood as a form ofBayesian inference.[2] Kriging starts with a priordistribution overfunctions. This prior takesthe form of a Gaussian process: samples from a function will benormally distributed, where the covariance between any two samples

is the covariance function (orkernel) of the Gaussian process evaluated at the spatial location of two points. A set of values is thenobserved, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location, by combining theGaussian prior with a Gaussian likelihood function for each of the observed values. The resulting posteriordistribution is also Gaussian,with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from theprior.

Geostatistical estimator [edit]

In geostatistical models, sampled data is interpreted as a result of a random process. The fact that this models incorporate uncertainty inits conceptualization doesn't mean that the phenomenon - the forest, the aquifer, the mineral deposit - has resulted from a randomprocess, but solely allows to build a methodological basis for the spatial inference of quantities in unobserved locations and to thequantification of the uncertainty associated with the estimator.

A stochastic process is simply, in the context of this model, a way to approach the set of data collected from the samples. The first stepin geostatistical modulation is the creation of a random process that best describes the set of experimental observed data.[3]

A value spatially located at (generic denomination of a set ofgeographic coordinates) is interpreted as a realization of therandom variable . In the space , where the set of samples is dispersed, exists realizations of the random variables

, correlated between themselves.

The set of random variables, constitutes a random function of which only one realization is known - the set of experimental data.

With only one realization of each random variable it's theoretically impossible to determine any statistical parameter of the individualvariables or the function.

Read EditArticle Talk Search

Main page

Contents

Featured contentCurrent events

Random article

Donate to Wikipedia

Wikimedia Shop

Interaction

Help

About Wikipedia

Community portal

Recent changes

Contact Wikipedia

Toolbox

Print/export

Languages

Deutsch

Espaol

Euskara

Franais

Hrvatski

Bahasa Indonesia

Italiano

Magyar

Norsk bokml

Polski

Portugus

Romn

Basa Sunda

Ting VitEdit links

Create account Log in

converted by Web2PDFConvert.com
http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=3http://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Kernel_(set_theory)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Prior_probability_distributionhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Generalized_least_squareshttp://en.wikipedia.org/wiki/Generalized_least_squareshttp://en.wikipedia.org/wiki/Curve_fittinghttp://en.wikipedia.org/wiki/Spline_(mathematics)http://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_spacehttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=1http://en.wikipedia.org/wiki/Spatial_analysis#Samplinghttp://en.wikipedia.org/wiki/Computer_experimenthttp://en.wikipedia.org/w/index.php?title=Kriging&action=edithttp://en.wikipedia.org/wiki/Best_linear_unbiased_estimatorhttp://vi.wikipedia.org/wiki/Kriginghttp://ro.wikipedia.org/wiki/Kriginghttp://pl.wikipedia.org/wiki/Kriginghttp://hr.wikipedia.org/wiki/Kriginghttp://es.wikipedia.org/wiki/Krigeajehttp://en.wikipedia.org/wiki/Special:RecentChangeshttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://shop.wikimedia.org/http://en.wikipedia.org/wiki/Special:Randomhttp://en.wikipedia.org/wiki/Portal:Featured_contenthttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://www.wikidata.org/wiki/Q225926#sitelinks-wikipediahttp://vi.wikipedia.org/wiki/Kriginghttp://su.wikipedia.org/wiki/Kriginghttp://ro.wikipedia.org/wiki/Kriginghttp://pt.wikipedia.org/wiki/Kriginghttp://pl.wikipedia.org/wiki/Kriginghttp://no.wikipedia.org/wiki/Kriginghttp://hu.wikipedia.org/wiki/Krigel%C3%A9shttp://it.wikipedia.org/wiki/Kriginghttp://id.wikipedia.org/wiki/Kriginghttp://hr.wikipedia.org/wiki/Kriginghttp://fr.wikipedia.org/wiki/Krigeagehttp://eu.wikipedia.org/wiki/Krigekuntzahttp://es.wikipedia.org/wiki/Krigeajehttp://de.wikipedia.org/wiki/Kriginghttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://en.wikipedia.org/wiki/Special:RecentChangeshttp://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Help:Contentshttp://shop.wikimedia.org/http://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=enhttp://en.wikipedia.org/wiki/Special:Randomhttp://en.wikipedia.org/wiki/Portal:Current_eventshttp://en.wikipedia.org/wiki/Portal:Featured_contenthttp://en.wikipedia.org/wiki/Portal:Contentshttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/w/index.php?title=Kriging&action=edithttp://en.wikipedia.org/wiki/Talk:Kriginghttp://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Kriginghttp://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Kriging&type=signuphttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Geographic_coordinate_systemhttp://en.wikipedia.org/wiki/Stochastic_processhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=3http://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Kernel_(set_theory)http://en.wikipedia.org/wiki/Covariancehttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Prior_probability_distributionhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Curve_fittinghttp://en.wikipedia.org/wiki/Generalized_least_squareshttp://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_spacehttp://en.wikipedia.org/wiki/Spline_(mathematics)http://en.wikipedia.org/wiki/Gaussian_processhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Gauss-Markov_theoremhttp://en.wikipedia.org/wiki/Covariancehttp://en.wikipedia.org/wiki/Best_linear_unbiased_estimatorhttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=1http://en.wikipedia.org/wiki/Hard_and_soft_ghttp://en.wikipedia.org/wiki/South_Africahttp://en.wikipedia.org/wiki/Witwatersrandhttp://en.wikipedia.org/wiki/Danie_G._Krigehttp://en.wikipedia.org/wiki/Georges_Matheronhttp://en.wikipedia.org/wiki/Extrapolationhttp://en.wikipedia.org/wiki/Interpolationhttp://en.wikipedia.org/wiki/File:Example_krig.pnghttp://en.wikipedia.org/wiki/File:Example_krig.pnghttp://en.wikipedia.org/wiki/Computer_experimenthttp://en.wikipedia.org/wiki/Spatial_analysis#Samplinghttp://en.wikipedia.org/wiki/Geographic_coordinate_systemhttp://en.wikipedia.org/wiki/Random_fieldhttp://en.wikipedia.org/wiki/Geostatistics


2/6

The proposed solution in the geostatistical formalism consists in assuming various degrees of stationarity in the random function, inorder to make possible the inference of some statistic values.[4]

For instance, if a workgroup of scientists assumes appropriate, based on the homogeneity of samples in area where the variable is

distributed, the hypothesis that the first moment is stationary (i.e. all random variables have the same mean), than, they are implying thatthe mean can be estimated by the arithmetic mean of sampled values. Judging an hypothesis like this as appropriate is the same asconsidering that sample values are sufficiently homogeneous to validate that representativity.

The hypothesis of stationarity related to the second moment is defined in the following way: the correlation between two random variablessolely depends on the spatial distance that separates them and is independent of its location:

where

This hypothesis allows to infer those two measures - the variogram and the covariogram - based on the samples:

where

Linear estimation [edit]

Spatial inference, or estimation, of a quantity , at an unobserved location , is calculated from a linear combination of

the observed values and weights :

The weights are intended to summarize two extremely important procedures in a spatial inference process:

reflect the structural "proximity" of samples to the estimation location,

at the same time, they should have a desegregation effect, in order to avoid bias caused by eventual sample clusters

When calculating the weights , there are two objectives in the geostatistical formalism: unbias and minimal variance of estimation.If the cloud of real values is plotted against the estimated values , the criterion for global unbias, intrinsic stationarityor

wide sense stationarity of the field, implies that the mean of the estimations must be equal to mean of the real values.

The second criterion says that the mean of the squared deviations must be minimal, what means that when the

cloud of estimated values versus the cloud real values is more disperse, more imprecise the estimator is.

Methods [edit]

Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods forcalculating the weights can be deducted, i.e different types of kriging apply. Classical methods are:

Ordinary krigingassumes stationarity of the first moment of all random variables: , where

is unknown.

Simple krigingassumes a known stationary mean: , where is known.

Universal krigingassumes a general polynomial trend model, such as linear trend model .

IRFk-krigingassumes to be an unknown polynomial in .

Indicator kriginguses indicator functions instead of the process itself, in order to estimate transition probabilities.

Multiple-indicator krigingis a version of indicator kriging working with a family of indicators. However, MIK has fallen out of favour as aninterpolation technique in recent years. This is due to some inherent difficulties related to operation and model validation. Conditionalsimulation is fast becoming the accepted replacement technique in this case.

Disjunctive krigingis a nonlinear generalisation of kriging.

Lognormalkriginginterpolates positive data by means oflogarithms.

Ordinary kriging [edit]

The unknown value is interpreted as a random variable located in , as well as the values of neighboors samples

. The estimator is also interpreted as a random variable located in , a result of the linear

combination of variables.

In order to deduce the kriging system for the assumptions of the model, the following error committed while estimating in is

declared:

http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=6http://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Log-normal_distributionhttp://en.wikipedia.org/wiki/Multiple-indicator_kriginghttp://en.wikipedia.org/wiki/Indicator_functionhttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Moment_(mathematics)#Meanhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=5http://en.wikipedia.org/wiki/Stationary_processhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Covariogram&action=edit&redlink=1http://en.wikipedia.org/wiki/Variogramhttp://en.wikipedia.org/wiki/Moment_(mathematics)#Variancehttp://en.wikipedia.org/wiki/Moment_(mathematics)#Mean


3/6

The two quality criteria referred previously can now be expressed in terms of the mean and variance of the new random variable :

Unbias

Since the random function is stationary, , the following constraint is observed:

In order to ensure that the model is unbiased, the sum of weights needs to be one.

Minimal Variance: minimize

Two estimators can have , but the dispersion around their mean determines the difference between the quality of estimators.

* seecovariance matrix for a detailed explanation

* where the literals stand for

.

Once defined the covariance model orvariogram, or , valid in all field of analysis of , than we can write an expression

for the estimation variance of any estimator in function of the covariance between the samples and the covariances between the samplesand the point to estimate:

Some conclusions can be asserted from this expressions. The variance of estimation:

is not quantifiable to any linear estimator, once the stationarity of the mean and of the spatial covariances, or variograms, are

assumed.grows with the covariance between samples , i.e. to the same distance to the estimating point, if the samples are

proximal to each other, than the clustering effect, or informational redundancy, is bigger, so the estimation is worst. This conclusion isvalid to any value of the weights: a preferential grouping of samples is always worst, which means that for the same number ofsamples the estimation variance grows with the relative weight of the sample clusters.

grows when the covariance between the samples and the point to estimate decreases. This means that, when the samples are morefar away from , the worst is the estimation.

grows with the a priorivariance of the variable . When the variable is less disperse, the variance is lower in any point of

the area .

does not depend on the values of the samples. This means that the same spatial configuration (with the same geometrical relationsbetween samples and the point to estimate) always reproduces the same estimation variance in any part of the area . This way, the

variance does not measures the uncertainty of estimation produced by the local variable.

Kriging system [edit]

Solving this optimization problem (see Lagrange multipliers) results in the kriging system:

the additional parameter is a Lagrange multiplierused in the minimization of the kriging error to honor the unbiasedness

condition.

Typical workflow1. First of all, a variogram model needs to be defined, . This needs to take into account concepts such as: range, sill,

anisotropy, which do not belong to the scope of this article.

2. With this function, can now be calculated with the previous equation.

3. Finally, the estimation and error by Ordinary Kriging is given by:

http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://en.wikipedia.org/wiki/Variogramhttp://en.wikipedia.org/wiki/Lagrange_multiplierhttp://en.wikipedia.org/wiki/Lagrange_multipliershttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=7http://en.wikipedia.org/wiki/Variogramhttp://en.wikipedia.org/wiki/Covariance_matrix#As_a_linear_operator


4/6

Using NumPy in Python it can be implemented as:

import numpy as np

def Variogram(xi, xj): """ arguments: two locations

returns: float with value of variogram"""(...)

def OrdinaryKriging(locations, values, x0):

n =len(values)

X = np.array([(lon,lat) for lon,lat in locations])Z = np.matrix(np.concatenate((np.array(values), np.array([0.])))).T

#prepares matrixes with Variogram model__k = [[Variogram(X[row],X[col]) for col inrange(n)]+[1.] for row inrange(n)]__k += [[1. for col inrange(n)]]+[[0.]]K = np.matrix(__k)M = np.matrix([[Variogram(x[i], x0)] for i inrange(n)]+[[1.]])

#processes kriging systemW = K.I*M

returnfloat(W.T*Z)

Simple kriging [edit]

this section is very poor and needs to be improved

Simple kriging is mathematically the simplest, but the least general. It assumes the expectation of the random field to be known, andrelies on a covariance function. However, in most applications neither the expectation nor the covariance are known beforehand.

The practical assumptions for the application ofsimple krigingare:

wide sense stationarity of the field.The expectation is zero everywhere: .

Known covariance function

Kriging system [edit]

Thekriging weights ofsimple kriginghave no unbiasedness condition and are given by the simple kriging equation system:

This is analogous to a linear regression of on the other .

Estimation [edit]

The interpolation by simple kriging is given by:

The kriging error is given by:

which leads to the generalised least squares version of the Gauss-Markov theorem (Chiles & Delfiner 1999, p. 159):

Properties [edit]

this section needs revision. Incorrect or confusing text should be removed.

(Cressie 1993, Chiles&Delfiner 1999, Wackernagel 1995)

The kriging estimation is unbiased:

The kriging estimation honors the actually observed value: (assuming no measurement error is incurred)

The kriging estimation is the best linear unbiased estimatorof if the assumptions hold. However (e.g. Cressie 1993):

As with any method: If the assumptions do not hold, kriging might be bad.

There might be better nonlinear and/or biased methods.No properties are guaranteed, when the wrong variogram is used. However typically still a 'good' interpolation is achieved.

http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://en.wikipedia.org/wiki/Best_linear_unbiased_estimatorhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=11http://en.wikipedia.org/wiki/Gauss-Markov_theoremhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=9http://en.wikipedia.org/wiki/Covariance_functionhttp://en.wikipedia.org/wiki/Stationary_processhttp://en.wikipedia.org/wiki/Covariance_functionhttp://en.wikipedia.org/wiki/Random_fieldhttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=8http://en.wikipedia.org/wiki/Python_(programming_language)http://en.wikipedia.org/wiki/NumPy


5/6

Wikimedia Commons has mediarelated to: Kriging

Best is not necessarily good: e.g. In case of no spatial dependence the kriging interpolation is only as good as the arithmeticmean.

Kriging provides as a measure of precision. However this measure relies on the correctness of the variogram.

Applications [edit]

this section is very poor and needs to be improved

Although kriging was developed originally for applications in geostatistics, it is a general method of statistical interpolation that can beapplied within any discipline to sampled data from random fields that satisfy the appropriate mathematical assumptions.

To date kriging has been used in a variety of disciplines, including the following:Environmental science[5]

Hydrogeology[6][7][8]

Mining[9][10]

Natural resources[11][12]

Remote sensing[13]

Real estate appraisal[14][15]

and many others.

Design and analysis of computer experiments [edit]

Another very important and rapidly growing field of application, in engineering, is the interpolation of data coming out as responsevariables of deterministic computer simulations,[16] e.g. finite element method (FEM) simulations. In this case, kriging is used as ametamodeling tool, i.e. a black box model built over a designed set of computer experiments. In many practical engineering problems,such as the design of a metal forming process, a single FEM simulation might be several hours or even a few days long. It is therefore

more efficient to design and run a limited number of computer simulations, and then use a kriging interpolator to rapidly predict theresponse in any other design point. Kriging is therefore used very often as a so-called surrogate model, implemented inside optimizationroutines.[17]

See also [edit]

Bayes linear statisticsGaussian processMultiple-indicator kriging

Spatial dependenceVariogram

Multivariate interpolation

References [edit]

1. ^ Grace Wahba (1990). Spline Models for Observational Data59. SIAM. p. 162.2. ^ Will iams, Christopher K.I. (1998). "Prediction with Gaussian processes: From linear regression to linear prediction and beyond". In M. I.

Jordan.Learning in graphical models. MIT Press. pp. 599612.3. ^ Soares 2006, p.184. ^ Matheron G. 19785. ^ Hanefi Bayraktar and F. Sezer. Turalioglu (2005) "AKriging-based approach for locating a sampling sitein the assessment of air

quality, SERRA, 19 (4), 301-305doi:10.1007/s00477-005-0234-86. ^ Chiles, J.-P. and P. Delfiner (1999) Geostatistics, Modeling Spatial Uncertainty, Wiley Series in Probability and statistics.7. ^ Zimmerman, D.A. et al. (1998) "Acomparison of seven geostatistically based inverse approaches to estimate transmissivies for

modelling advective transport by groundwater flow", Water Resources Research, 34 (6), 1273-14138. ^ Tonkin M.J. Larson (2002) "Kriging Water Levels with a Regional-Linear and Point Logarithmic Drift",Ground Water, 33 (1), 338-353,9. ^ Journel, A.G. and C.J. Huijbregts (1978) Mining Geostatistics, Academic Press London

10. ^ Andrew Richmond (2003) "Financially Efficient Ore Selection Incorporating Grade Uncertainty", Mathematical Geology, 35 (2), 195-21511. ^ Goovaerts (1997) Geostatistics for natural resource evaluation, OUP. ISBN 0-19-511538-412. ^ X. Emery (2005) "Simple and Ordinary Kriging Multigaussian Kriging for Estimating recoverarble Reserves", Mathematical Geology, 37

(3), 295-31)

13. ^ A. Stein, F. van der Meer, B. Gorte (Eds.) (2002) Spatial Statistics for remote sensing. Springer. ISBN 0-7923-5978-X14. ^ Barris, J. (2008)An expert system for appraisal by the method of comparison. PhD Thesis, UPC, Barcelona15. ^ Barris, J. and Garcia Almirall ,P.(2010)A density function of the appraisal value., UPC, Barcelona16. ^ Sacks, J. and Welch, W.J. and Mitchell, T.J. and Wynn, H.P. (1989). Design and Analysis of Computer Experiments4 (4). Statistical

Science. pp. 409435.17. ^ Strano, M. (2008). "Atechnique for FEM optimization under reliability constraint of process variables in sheet metal forming". International

Journal of Material Forming1: 1320. doi:10.1007/s12289-008-0001-8 . edit

Books [edit]

Abramowitz, M., and Stegun, I. (1972), Handbook of Mathematical Functions, Dover Publications, New York.Banerjee, S., Carlin, B.P. and Gelfand, A.E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRCPress, Taylor and Francis Group.Chiles, J.-P. and P. Delfiner (1999) Geostatistics, Modeling Spatial uncertainty, Wiley Series in Probability and statistics.

Cressie, N (1993) Statistics for spatial data, Wiley, New YorkDavid, M (1988) Handbook of Applied Advanced Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing

Deutsch, C.V., and Journel, A. G. (1992), GSLIB - Geostatistical Software Library and User's Guide, Oxford University Press, NewYork, 338 pp.Goovaerts, P. (1997) Geostatistics for Natural Resources Evaluation, Oxford University Press, New York ISBN 0-19-511538-4

Isaaks, E. H., and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 pp.Journel, A. G. and C. J. Huijbregts (1978) Mining Geostatistics, Academic Press London

Journel, A. G. (1989), Fundamentals of Geostatistics in Five Lessons, American Geophysical Union, Washington D.C.Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 3.7.4. Interpolation by Kriging" , Numerical Recipes: The

http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://apps.nrbook.com/empanel/index.html?pg=144http://en.wikipedia.org/wiki/Special:BookSources/0195115384http://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=16http://en.wikipedia.org/w/index.php?title=Template:Cite_doi/10.1007.2Fs12289-008-0001-8&action=edit&editintro=Template:Cite_doi/editintro2http://dx.doi.org/10.1007%2Fs12289-008-0001-8http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Special:BookSources/079235978Xhttp://en.wikipedia.org/wiki/Special:BookSources/0195115384http://dx.doi.org/10.1007%2Fs00477-005-0234-8http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=15http://en.wikipedia.org/wiki/Multivariate_interpolationhttp://en.wikipedia.org/wiki/Variogramhttp://en.wikipedia.org/wiki/Spatial_dependencehttp://en.wikipedia.org/wiki/Multiple-indicator_kriginghttp://en.wikipedia.org/wiki/Gaussian_processhttp://en.wikipedia.org/wiki/Bayes_linear_statisticshttp://commons.wikimedia.org/wiki/Category:Kriginghttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=14http://en.wikipedia.org/wiki/Optimizationhttp://en.wikipedia.org/wiki/Surrogate_modelhttp://en.wikipedia.org/wiki/Metal_forminghttp://en.wikipedia.org/wiki/Computer_experimenthttp://en.wikipedia.org/wiki/Metamodelinghttp://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=13http://en.wikipedia.org/wiki/Real_estate_appraisalhttp://en.wikipedia.org/wiki/Remote_sensinghttp://en.wikipedia.org/wiki/Natural_resourcehttp://en.wikipedia.org/wiki/Mininghttp://en.wikipedia.org/wiki/Hydrogeologyhttp://en.wikipedia.org/wiki/Environmental_sciencehttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=12


6/6

Privacy policy About Wikipedia Disclaimers Contact Wikipedia Mobile view

This page was last modified on 7 July 2013 at 08:23.

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and PrivacyPolicy.Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

Art of Scientific Computing(3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8. Also, "Section 15.9. GaussianProcess Regression" .Soares, A. (2000), Geoestatstica para as Cincias da Terra e do Ambiente, IST Press, Lisbon, ISBN 972-8469-46-2

Stein, M. L. (1999), Statistical Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York.Wackernagel, H. (1995) Multivariate Geostatistics - An Introduction with Applications, Springer Berlin

Historical references [edit]

1. Agterberg, F P, Geomathematics, Mathematical Background and Geo-Science Applications, Elsevier Scientific PublishingCompany, Amsterdam, 1974

2. Cressie, N. A. C., The Origins of Kriging, Mathematical Geology, v. 22, pp 239252, 19903. Krige, D.G,A statistical approach to some mine valuations and allied problems at the Witwatersrand, Master's thesis of the

University of Witwatersrand, 19514. Link, R F and Koch, G S, Experimental Designs and Trend-Surface Analsysis, Geostatistics, A colloquium, Plenum Press, New

York, 1970

5. Matheron, G., "Principles of geostatistics", Economic Geology, 58, pp 12461266, 19636. Matheron, G., "The intrinsic random functions, and their applications",Adv. Appl. Prob., 5, pp 439468, 19737. Merriam, D F, Editor, Geostatistics, a colloquium, Plenum Press, New York, 1970

Categories: Geostatistics Interpolation Multivariate interpolation
http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://www.mediawiki.org/http://wikimediafoundation.org/http://en.m.wikipedia.org/wiki/Kriginghttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://en.wikipedia.org/wiki/Wikipedia:General_disclaimerhttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://wikimediafoundation.org/wiki/Privacy_policyhttp://www.wikimediafoundation.org/http://wikimediafoundation.org/wiki/Privacy_policyhttp://wikimediafoundation.org/wiki/Terms_of_Usehttp://en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_Licensehttp://en.wikipedia.org/wiki/Category:Multivariate_interpolationhttp://en.wikipedia.org/wiki/Category:Interpolationhttp://en.wikipedia.org/wiki/Category:Geostatisticshttp://en.wikipedia.org/wiki/Help:Categorieshttp://en.wikipedia.org/w/index.php?title=Kriging&action=edit&section=17http://en.wikipedia.org/wiki/Special:BookSources/9728469462http://apps.nrbook.com/empanel/index.html?pg=836http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-88068-8http://en.wikipedia.org/wiki/International_Standard_Book_Number

Documents

en-wikipedia-org.pdf