Modelling of Spatial Data Using Semivariograms of Stationary Spatial Processes ; Erdvinių duomenų modeliavimas naudojant stacionarių erdvinių procesų semivariogramas
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Modelling of Spatial Data Using Semivariograms of Stationary Spatial Processes ; Erdvinių duomenų modeliavimas naudojant stacionarių erdvinių procesų semivariogramas


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VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Ingrida BORISENKO MODELLING OF SPATIAL DATA USING SEMIVARIOGRAMS OF STATIONARY SPATIAL PROCESSES Summary of Doctoral Dissertation Physical Sciences, Mathematics (01P) Vilnius 2009 Doctoral dissertation was prepared at Klaipėda University and Vilnius Gediminas Technical University in 2001–2009. The dissertation is defended as an external work. Scientific Consultant Prof Dr Kęstutis DUČINSKAS (Klaipėda University, Physical Sciences, Mathematics – 01P). The dissertation is being defended at the Council of Scientific Field of Mathematics at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Leonas SAULIS (Vilnius Gediminas Technical University, Physical Sciences, Mathematics – 01P). Members: Prof Dr Habil Mindaugas BLOZNELIS (Vilnius University, Physical Sciences, Mathematics – 01P), Assoc Prof Dr Vitalijus DENISOVAS (Klaipėda University, Physical Sciences, Informatics – 09P), Assoc Prof Dr Mečislavas MEILŪNAS (Vilnius Gediminas Technical University, Physical Sciences, Mathematic – 01P), Prof Dr Habil Henrikas PRAGARAUSKAS (Institute of Mathematics and Informatics, Physical Sciences, Mathematics – 01P). Opponents: Prof Dr Habil Vilijandas BAGDONAVIČIUS (Vilnius University, Physical Sciences, Mathematics – 01P), Assoc Prof Dr Marijus RADAVIČIUS (Institute of Mathematics and Informatics, Physical Sciences, Mathematics – 01P).



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Publié le 01 janvier 2010
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 SummaryofDoctoralDissertation PhysicalSciences,Mathematics(01P)


Doctoral dissertation was prepared at Klaip da University and Vilnius GediminasTechnicalUniversityin2001–2009. Thedissertationisdefendedasanexternalwork. ScientificConsultant ProfDrK stutisDU INSKAS (Klaip daUniversity,PhysicalSciences, Mathematics–01P). The dissertation is being defended at the Council o f Scientific Field of MathematicsatVilniusGediminasTechnicalUniversi ty: Chairman ProfDrHabilLeonasSAULIS (VilniusGediminasTechnicalUniversity, PhysicalSciences,Mathematics–01P).  Members: Prof Dr Habil Mindaugas BLOZNELIS  (Vilnius University, Physical Sciences,Mathematics–01P), Assoc Prof Dr Vitalijus DENISOVAS  (Klaip da University, Physical Sciences,Informatics–09P),   Assoc Prof Dr Me islavas MEIL NAS  (Vilnius Gediminas Technical University,PhysicalSciences,Mathematic–01P), Prof Dr Habil Henrikas PRAGARAUSKAS  (Institute of Mathematics andInformatics,PhysicalSciences,Mathematics–0 1P). Opponents: Prof Dr Habil Vilijandas  BAGDONAVI IUS  (Vilnius University, PhysicalSciences,Mathematics–01P), Assoc Prof Dr Marijus  RADAVI IUS  (Institute of Mathematics and Informatics,PhysicalSciences,Mathematics–01P).   The dissertation will be defended at the public mee ting of the Council of Scientific Field of Mathematics in the Senate Hall of Vilnius Gediminas TechnicalUniversityat1p.m.on6November2009.  Address:Saul tekioal.11,LT-10223Vilnius,Lithuania. Tel.:+37052744952,+37052744956;fax+3705 2700112; Thesummaryofthedoctoraldissertationwas  distributedon5October2009. A copy of the doctoral dissertation is available fo r review at the Libraries of Vilnius Gediminas Technical University (Saul tekio al. 14, LT-10223 Vilnius, Lithuania) and the Institute of Mathematics and Inf ormatics (Akademijos g. 4, LT-08663Vilnius,Lithuania). ©IngridaBorisenko,2009



 Daktarodisertacijossantrauka Fiziniaimokslai,matematika(01P)


Disertacija rengta 2001–2009 metais Klaip dos universitete ir Vilniaus Gediminotechnikosuniversitete. Disertacijaginamaeksternu. Moksliniskonsultantas prof. dr. K stutis DU INSKAS (Klaip dos universitetas, fiziniai mokslai,matematika–01P). DisertacijaginamaVilniausGediminotechnikosuniv ersitetoMatematikos mokslokryptiestaryboje: Pirmininkas prof. habil. dr. Leonas SAULIS  (Vilniaus Gedimino technikos universitetas,fiziniaimokslai,matematika–01P).  Nariai: prof.habil.dr.MindaugasBLOZNELIS (Vilniausuniversitetas,fiziniai mokslai,matematika–01P), doc. dr. Vitalijus DENISOVAS (Klaip dos universitetas, fiziniai mokslai,informatika–09P), doc. dr. Me islavas MEIL NAS (Vilniaus Gedimino technikos universitetas,fiziniaimokslai,matematika–01P),  prof. habil. dr. Henrikas PRAGARAUSKAS (Matematikos ir informatikosinstitutas,fiziniaimokslai,matemati ka–01P).  Oponentai: prof. habil. dr. Vilijandas BAGDONAVI IUS  (Vilniaus universitetas, fiziniaimokslai,matematika–01P), prof. dr. Marijus  RADAVI IUS (Matematikos ir informatikos institutas,fiziniaimokslai,matematika–01P).  DisertacijabusginamaviešameMatematikosmokslok ryptiestarybospos dyje 2009m.lapkri io6d.13val.VilniausGediminotechnikosunivers itetosenato pos dži sal je. Adresas:Saul tekioal.11,LT-10223Vilnius,Lietuva. Tel.:(85)2744952,(85)2744956;faksas(85) 2700112; el.paš Disertacijossantraukaišsiuntin ta2009m.spalio5d. Disertacij  galima perži r ti Vilniaus Gedimino technikos universiteto (Saul tekio al. 14, LT-10223 Vilnius, Lietuva) ir Matemat ikos ir informatikos instituto(Akademijosg.4,LT-08663Vilnius,Lietu va)bibliotekose. VGTUleidyklos„Technika“1662-Mmoksloliterat rosknyga. ©IngridaBorisenko,2009

Introduction Topicalityoftheproblem  Spatial statistics is one of the youngest trends in  the science of statistics. First,ithasbeenappliedin mining,duringthefi fthdecadeofthelastcentury. In fifty years after this trend of science had been  discovered, the circle of the scientists involved in it has grown drastically as well as areas of application. Also, a wide range of theoretical and practical mat erial has been issued. Nowadays, spatial statistics methods are used in: e cology, quantity geology, image processing and analysis, epidemiology, studyi ng global climate change and even cosmology. However, in Lithuania, the meth odology of spatial data analysis has been studied only from the beginning o f this Millennium. Since only few scientists (Dumbrauskas, A.; Kumetaitis, A .; Kumetaitien , A. and others) are involved, it is very important to expan d this area and develop the existing methods. Also it is essential to study the  spatial dada modelling methodsthroughlyandprovidegeneralspatialdata modellingmethodology. Inordertoapplythe methodsofspatial statistics ,it is necessaryto know the location of data in space, which is usually exp ressed in geographic coordinates.Thus,oneofthemaindistinctionsof spatialstatisticswhichmakes it different from the classical is the ability to m odel both spatial trend and spatialautocorrelation. One of the main objectives of spatial statistics is  creating a mathematical modelofspatialdata,whichcanbeusedforinterp olation(extrapolation)orfor otherpurposes.Toaccomplishdataestimationthek rigingmethodcanbeused. Under certain conditions it minimizes mean square p rediction error, that is to say, provides the best linear unbiased prediction. Depending on spatial data nature,themeanmodelofspatialprocessandother datacharacteristics,several typesofkrigingcandistinguish. Thespatialautocorrelationamongobservationsisd escribedbycovariance function or semivariogram. The last is often used i n practical researches, becauseitcoversthe widerclassofspatialproces ses.Todescribethe variaton ofspatialdatabysemivariogram,firstly,thespat ialstructureofobservationsis modelled using the graph of the empirical semivario gram. The empirical semivariogram estimator mostly is calculated by usi ng Matheron method of moments (MoM). Then one of semivariogram parametric  model is fitted. Parametricmodelisdescribedbyseveralsemivariog ramparameters . Mainof themarenugget,sillandrange. Thebestlinearunbiasedpredictorcannotbecompu tedunless  isknown, whichistypicallynot.Toevaluatesemivariogramp arameters,theleastsquares methods (ordinary, weighted and generalised least s quares methods) are often used. As stated Lahiri et al.  in “On asymptotic distribution and asymptotic

efficiency of least squares estimators of spatial v ariogram parameters“ (2002), under the general conditions the generalised least squares method (GLS) asymptoticallyeffectivebycomparingwithotherle astsquaresmethods. The least squares methods can be also used for the estimation of parameters of the mean model. Since in literature G LS method is presented through covariances functions, it is important to t ake the expression of this methodthroughsemivariogram. Should also be noted that the estimators of semivar iogram parameters by analytical method is not available. Therefore, the asymptotic analysis of estimatorsisliveissue. Another frequent task in the spatial statistic is t he determination of the optimal sampling plan. D and A criterion of samplin g design is widely used. Again, providing of expression of D and A criterion  in semivariogram form is veryimportant.  Researchobject  The research object of the dissertation is the meth odology of modelling, prediction and results analysis of spatial data gen erated by spatial processes withstationaryerrors. Aimofthework  Themainobjectiveofthesisistoproposethemeth odologyformodelling of spatial data through semivariograms and predicti on by kriging, when the spatialdataaregeneratedbyspatialprocesseswit hstationaryerrors.  Tasksofthework 1.Toreviewtheadditivemodelsofspatialprocess eswithstationaryerrors anddecribethefundamentalcharacteristicsofspat ialprocesses. 2.Toproposethemethodologyofmodellingandpred ictionofspatialdata byusingvariousparametricmodelsofmeanandstat ionarysemivariogram. 3.Toadapt the described and proposed methodology to modelling and predictionsofdepthandsedimentsdataoftheCuro nianLagoon. 4.Tocomparethespatialsamplingdesignusingvar iouscriterions. 5.To  performtheanalysisofstatisticalpropertie sofMoMestimator.  Methodologyofresearch In analytical part of work when the mathematical mo dels of spatial data andthemethodologyofmodellingandpredictionof spatialdataaredescribed, the analysis of literature and comparative analysis  are used.To summarize the steps of modelling and prediction of spatial data, the techniques of conceptual modelling and summary of the concepts are applied. For the practical

realization of modelling and prediction methodology  proposed in this work, experimentalanalysisisused. Scientificnovelty In this work the general methodology of modelling a nd prediction of spatial data through semivariograms, when the spati al data are generated by spatial  processes with stationary errors is propose d. The analysis of above mentioned methodology discovers the possibility of improving of some of the spatialstatisticalmethods. Inthisworkthefollowingfindingsareprovided: 1.  ThesemivariogramestimatorMoMisprovidedthrough spacialdesign (Laplace)matrix. 2.  The expression of generalized least squares estimat or of  is presentedthroughsemivariogram. 3.  The expressions of D and A criterion functions for parameters and estimatorsof aregiventhroughsemivariogram. 4.  Forincreasingdomainasymptotictheasymptoticaln ormalityofMoM estimator, when errors of data model are stationary  and Gausian, is proofed. 5.  Theexpressionofasymptoticvariance-covariancema trixinthecaseof exponentialisotropicsemivariogramisderived. 6.  Theproposedmethodologyofmodellingandpredictio nofspatialdata generatedbyspatialprocesseswiththestationary errorsallows: a)  to describe the variation of spatial data through s emivariograms byincludingspatialtrendandanisotropy; b)  to predict the data by ordinary, universal, median polish kriging andcokriging; c)  toapplymethods(SSE–sumofsquareoferrors,cr oss-validation method, Akaike criterion), which verificate the sel ected parametricsemivariogrammodelsandkrigingmethods . Practicalvalue  The proposed methodology of modelling and predictio n can be used in otherscientific fields(forexampleineconometric s,demographicsandothers ) toanalyseavarioustypesofspatialdata.Thethe oreticalresultscanbeusedin analysis of optimal sampling design. The practical results can be an example how to realise methodology of modelling and predict ion of spatial data by functionsinthevariouspackagesofopensourcesy stem R .

Defendedpropositions 1.  The proposed methodology of spatial data modelling through semivariograms and prediction by universal, ordinar y, median polish kriginsandcokriging. 2.  Thegivenexpressionofvariance-covariancematrix ofsemivariogram estimator (MoM) through theoretical semivariogram i n the case of stationaryGausiandatamodel. 3.  The expression of generalised least squares (GLS) e stimator through semivariogram for  is put by replacing variance-covariance matrix withmatrixofsemivariogramaccordingly. 4.  The given expressions of D and A criterion function s through semivariogramforGLSestimatorof . 5.  The asymptotical normality of MoM semivariogram est imator, when errorsofdatamodelarestationaryandGausian. 6.  The derived expression of asymptotic variance-covar iance matrix MoM semivariogram estimator for spatial sampling de sign of increasingdomainasymptoticnamedLIDA,whichdesc ribedinLahiri etal.(2002),incaseofexponentialisotropicsem ivariogram. Thescopeofthescientificwork The scientific work consists of the general charact eristic of the dissertation, 4 chapters, conclusions, list of lite rature, list of publications and addenda. The total scope of the dissertation – 127 pages, 60 figures, 20 tables and4addenda. 1.Mathematicalmodelsofspatialdata Inthefirstchapteroftheworktheadditivemodel sofspatialprocessesand theirgeneratingspatialdataareintroduced.Cress ieandothersscientistsdefine three main groups of spatial data: geostatistical, lattice and point patterns. The one of fundamental attributes of spatial data is sp atial autocorrelation: observations closer together tend to be more alike than observation further apart.Thestandardmethodologyinspatialstatisti csisessentiallybasedonthe assumptions of stationary (strictly stationarity, w eak stationarity, intrinsically stationarity)andisotropy(seesection1.2). Inthisworktheadditivespatialmodel{ Z ( s ) : s D ,where D   d }of spatialobservationatlocation s isanalyzed: Z ( s ) T f ( s ) ( s ) , (1)


where ( 0 ,..., Q ) T –vector of parameters, f ( s ) (1, f 1 ( s ),..., f Q ( s )) T – vector of explanatory variables and ( s )  is zero-mean error finite variance stationaryprocess. The spatial dependence structure of ( s )  could be expressed by covariancefunctionC( h )orbysemivariogram: ( h ) var(Z( s i 2) Z( s j )) .    (2) It is obvious that ( h ) C ( 0 ) C ( h ) , where C( 0 ) is variance 2 .  The semivariogramusedmoreoftenasitcoversthewide rclassofspatialprocesses. Sincetheclassofintrinsicstationaryprocesscon tainstheclassofsecond-order stationaryprocess,the semivariogramofa second-o rderstationaryprocesscan be constructed from the covariance function. If the  process is intrinsic but not second order stationarity, the covariance function can not be used to describe spatialcorrelationbetweenobservations.Inthisw orktheassumptionsofweak stationarityorintrinsicstationarityismostlyus ed. The semivariogram and its parameters, some semivari ogram estimators, the most common parametric models of semivariogram and advices on their practical usage are described in section 1.4. Noted , that in applications the MoM estimator is mostly used. The semivariogram est imator MoM for stacionaryspatialdatamodeliscalculatedbysuch formula: ( ^ h ) 2 N 1( h ) ( s i , s j ) N [ ( h Z ) ( s i ) Z ( s j )] 2 ,     (3) where N ( h ) arenumberofpairsinthepopulation N ( h ) , s i s j h .The semivariogramestimatorMoMmaybewritteninsuch way: ^ h ) Z 2 T NL ( h) Z ( h ,      (4) where L h ingraphtheoryisknownasLaplacematrix.Inthe spatialstatistics thismatrixcalledthespatialdesignmatrix. Depending on properties of spatial process, the emp irical semivariograms are divided into isotropic and anisotropic. The spa tial variability between two correlatedphysicalmagnitudesisdescribedbythe crosssemivariogram. 2.Methodsofspatialdatastatisticalmodellingan destimation  Predictionorestimationofspatialdataatunobser vedlocationisparticular important forgeostatisticaldata.Completesamplin gofasurfaceisimpossible andtheusermaybeinterestedinpredictingtheam ount Z atarbitrarylocation s .

The understanding of geostatistical assumptions and  methods allows choosing the best way of estimation. It looks too c onfusing for the representatives of other scientific areas the seque nce of stages of spatial data statisticalmodellingandestimation. Thus, in the second chapter the stages of spatial d ata modelling by semivariogramsandestimationbykrigingaredescri bedandtheexecutionplan is proposed. The chapter is divided into six parts.  In the section 2.1 the classification of estimation methods of spatial dat a and the common stages of geostatisticalmodellingandestimationaredescrib ed.Inthesection2.2thefirst stageofpreliminaryspatialdataanalysisandtool sforitarereviewed. In the section 2.3 the modelling by semivariogram, identification of the trend and anisotropy and types of anisotropy are an alyzed. The elimination of anisotropiesisconsideredinthesection2.3.1. The empirical semivariogram displays half the avera ge difference in Z  values of two data points as a function of the sepa ration distance between the points.Inordertocalculatetheempiricalsemivar iogram,itisneededtospecify a lag distance and a maximum number of lags. The em pirical semivariogram provides the estimates only in finite set of lags. In order to obtain estimates at arbitrary lag the empirical semivariogram must be s moothed – replaced by some parametric form. The fitting of parametric mod els to the empirical semivariogram can be done by several methods: least  squares estimation, maximum likelihood and restricted maximum likelihoo d. The least squares estimationmethodsarereviewedindetailsinthes ection2.3.3. The section 2.4 is designated for spatial data pred iction method – kriging and their types. The emphasis is put on: simple, or dinary, block (ordinary), universal,medianpolishingkrigingsandcokriging.  Further,inthesection2.5thevalidationmethods, suchascross-validation methods and Akaike criteria, of the selected semiva riogram model and/or kriging,areanalyzed. In the next section 2.6 the influence of semivariog ram parameters and modelstoestimation(prediction),theadvantagesa nddisadvantagesof kriging bycomparingwithotherpredictionmethodsarerevi ewed. The analysis of all stages of geostatistical modell ing and prediction showed that the statistical analysis of spatial dat a allows to verify the data attributes, logicality of topological information; to analyze data by classical methods of statistics (to select the data distribut ions, to detect the outliers, to analysetrendandetc.)(Fig.1). To detect the anisotropy, the semivariogram map and  directional semivariogramscanbecreated.

DATAANALYSISBYMETHODSOFCLASSICALSTATISTICS (detectionofoutliers,trend,choosingofdatadistribution) Fig.1. Thestatisticalanalysisofspatialdata
Yesexisttrend?No Do
Yes Isthedataon No regulargrid?
UsingofmedianAnalysisoftrendsurfaceand polishingtrendsubstractingfrom procedure empiricalsemivariogram
Creationofdirectional semivariogram
NoArethenugget,sillandrangesamein alldirectionalsemivariograms? Detectionand eliminationof anisotropy

Yes Creationof omnidirectional semivariogram Fittingofparametric semivariogrammodel Fig.2. Theblockschemeofsemivariogrammodellingstages

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