SPATIAL REGRESSION ANALYSIS USING EIGENVECTOR SPATIAL FILTERING Spatial Econometrics and Spatial Statistics SPATIAL REGRESSION ANALYSIS USING EIGENVECTOR SPATIAL FILTERING DANIEL A. GRIFFITH YONGWAN CHUN BIN LI Foreword by PIERRE LEGENDRE Series Editor GIUSEPPE ARBIA AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom ©2019ElsevierInc.Allrightsreserved Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem, withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuch astheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedical treatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluating andusinganyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuch informationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyofothers,including partiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assume anyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability, negligenceorotherwise,orfromanyuseoroperationofanymethods,products,instructions,orideas containedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN978-0-12-815043-6 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:CandiceJanco AcquisitionEditor:J.ScottBentley EditorialProjectManager:KelseyConnors ProductionProjectManager:MariaBernard CoverDesigner:MatthewLimbert TypesetbySPiGlobal,India This book is dedicated to the memory of Dr. Ruth I. Shirey, whose mentoring and support enabled Dan Griffith to co-author it; YongwanChun’sfamily,HyunjuLeeandChristopherChun;andBinLi’s wife, Maiokun. Foreword Geographers have long understood that natural phenomena contain spatial structures.Tobler’sFirstLawofGeographyprovocativelycharacterizesthis behaviorofnature:“everythingisrelatedtoeverythingelse,butnearthings aremorerelatedthandistantthings”Tobler(1970,p.236).Thisstatementis a vivid representation of the phenomenon known as spatial (auto)correla- tion. Without it, natural phenomena would be disorganized, and physical, geological, and ecological processes, among others, could not take place. On the pragmatic side, if nature did not display spatial structures, there would be no geography or geographers. Statisticianshavelongbeeninterestedinthedescriptionandquantifica- tion of spatial structures displayed by data observed in geographic space. These scientists started with a method derived from regression analysis, known as trend surface modeling. Krumbein (1956, 1959) and Grant (1957) first used this method in the earth sciences, following an earlier proposal by “Student” to describe temporal variation using a polynomial function of time (“Student” [Gosset], 1914). Intheearlydaysofspatialanalysisbypractitionersinvariousfields,spatial correlationinvariableswasviewedmerelyasanuisance:itspresenceindata made the usual tests of significance invalid when applied without correc- tions,whereasmodifiedtestsweredifficulttoimplementandhadnotbeen fully worked out by statisticians. In my own field of specialty, community ecology,keypapersappearedinLevin(1992)andLegendre(1993),arguing thatspatialstructureswereamostimportantcharacteristicofthedistribution of organisms and natural populations in ecosystems, and were worthy of study for their own sake. Trend surface analysis, which uses a polynomial function of the geo- graphiccoordinatesofstudysites,wasusedinearlyattemptsatspatialmodel- ing. This is a rather crude method: to model fine spatial structures would require a polynomial equation with more monomials than observations, which in turn would render the method useless in practice in regression. Researchers then started looking for an applicable method that would producefine-resolutionspatialmodelswithareasonablenumberofparam- eters,onethatcouldbeappliedtoirregularlyspacedstudysitesandbeused to model univariate or multivariate response data. xi xii Foreword Moran eigenvector spatial filtering: Multiple origins and convergence Amethodtomodelmultiscalespatialpatternsbasedonspatialeigenvectorsis knownasMoraneigenvectorspatialfiltering(MESF).ThisbookbyGriffith, Chun, and Li is about that methodology. Interestingly, this method was developed independently and nearly simultaneously in two different fields, statistical geography (Griffith, 1996, 2000) and quantitative community ecology(Borcard&Legendre,20021).Followingtheirfirstpaper,Borcard, Legendre, Avois-Jacquet, and Tuomisto (2004) published a series of real-world ecological applications of this method. A few years later, Dray, Legendre, and Peres-Neto (2006) formalized the theory of Moran’s eigenvector maps (MEM). Griffith’soriginalgoalwastofiltertheeffectofspatialautocorrelationout of model residuals, transferring this component to a model’s conditional mean (i.e., intercept), whereas that of Legendre and his coauthors was to explicitlymodelthemultiscalenatureofunivariateormultivariateresponse data2.TheMESFmethodofanalysiswasbasedonearlierdevelopmentsby geographerstoanalyzebinaryspatialconnection(i.e.,spatialweights)matri- ces(SWM;Garrison&Marble,1964;Gould,1967;Tinkler,1972;Griffith, 1996). The two groups quickly realized that their methods had the same algebraic bases and that their objectives were interchangeable. Researchers from these two groups jointly published a paper unifying the terminology anddefiningthefieldofspatialeigenfunctionanalysis(Griffith&Peres-Neto, 2006),whichencompassesallmethodsbasedoneigenvectorsdescribingthe spatial relationships among study sites. Subsequently, Griffith and Legendre had an opportunity to exchange notes in August 2007 during a conference organized by Academia Sinica in Taipei, where they had been invited separately and independently to present their methods. They explained to the audience that the two methods, although formally presented in different ways, were actually one and the same. 1 IhadpresentedthismethodtwoyearsearlierinakeynoteaddressdeliveredattheModel- lingComplexSystemsconferenceinMontr(cid:1)ealinJuly2000. 2 Inthefieldsofcommunityecologyandbiogeography,andcontrarytomanyproblemsin geography,mostresponsedatasetsare(highly)multivariateandnonnormal. Foreword xiii A word about the theoretical background for MESF in ecology Byaskingmetowritethis“Foreword”fortheirbook,Griffith,Chun,and Liofferedmeanopportunitytoexplainthetheoreticalbasesthatmakeecol- ogistsinterestedinspatialcorrelationanditsmodelingbyspatialeigenfunc- tions.Forthat,Ihavetogobackalittleinthehistoryofcommunityecology. Thisisthebranchofecologydevotedtothescientificstudyofrelationships among the species forming natural communities, as well as relationships between these species and their environmental conditions. In the 1990s, ecologists became aware that different kinds of generating processes could producespatialcorrelationindata.Themainmechanismsarethefollowing: (1) induced spatial dependence: the functional dependence of given response data(e.g.,species)on asetofexplanatoryvariables;thisprocessisinaction when species forming natural communities are dependent upon the envi- ronmentalconditionsinwhichtheyarefound;(2)trueautocorrelation:spatial correlation thatmay occurin multivariate databecauseof functional inter- actions among the species in a multivariate data matrix; and (3) historical dynamics: manifestations of past natural events, such as isolation by geo- graphic barriers and disturbances of various kinds (e.g., storms, forest fires, volcanic eruptions, and landslides), and anthropogenic causes, such as agri- culture,logging,mining,andconstructionsofvarioussizes;thesepastpro- cessesmayhavecausedspatialstructurestoemergeandmayhavelefttraces in present-day data that can be identified and modeled as spatial structures. Researchersinotherfieldscouldapplythesehypotheses,conceptualizations, or theory elements to the explanation of spatial structures they find in their data. ThemethodologicaldevelopmentsofMESFtodatebyquantitativegeog- raphers are described in detail in the 10 chapters of this book. Meanwhile, methodological developments of MESF continue in ecological research. Blanchet and his coauthors developed asymmetric spatial eigenvector maps inthelate2000s(Blanchet,Legendre,&Borcard,2008;Blanchet,Legendre, Maranger,Monti,&Pepin,2011),amethoddesignedtomodeltheeffectsof directionalphysicalprocesses,suchasmarineandrivercurrents,onecological communities. Gu(cid:1)enard, Legendre, Boisclair, and Bilodeau (2010) decom- posedthecorrelationbetweenvariablesintospatialscalesandthenextended their method to multivariate response data matrices (Gu(cid:1)enard & Legendre, 2018).Followinganotherresearchpath,Gu(cid:1)enard,Legendre,andPeres-Neto (2013) extended the spatial eigenvector framework to the modeling of xiv Foreword phylogenetic trees and used MESF eigenvectors to predict different types of traits and properties unobserved in rare or endangered species (Gu(cid:1)enard, Boisclair, & Legendre, 2015; Gu(cid:1)enard, von der Ohe, Walker, Lek, and Legendre (2014). Spatial eigenvectors also were used to develop a test for space–time interaction in repeated surveys (through time) of sets of sites without replication (Legendre, De Ca´ceres, & Borcard, 2010). Simultaneousdevelopmentofamethodologybyresearchersindifferent disciplinesisanindicationofitsstrength.TheMESFmethodwasdeveloped independentlybytwogroupsofresearchers,ataboutthesametime,which mayprovideusersofthemethodmoreconfidenceinit.Reviewingthevari- ety of ways MESF analysis has been applied to real-world data by geogra- phers, on the one hand, and by ecologists, on the other hand, may give users in different fields ideas for applications that they initially had not considered. Inaddition,ecologistsareinterestedinsoftwareand,inparticular,inR. TheydevelopedanRpackagedevotedtospatialandtemporalanalysiscalled adespatial (Dray et al., 2016, the first version released on CRAN), with a strong emphasis on spatial eigenfunction analysis. This software and that presented in especially Chapter 4 of this book provide a powerful toolbox for practitioners. ScientistswhoapplyMESFmethodstoanalyzedatarecognizethatspatial eigenfunctions,whichdescribeunmeasuredspatialrelationshipsamongthe sites constituting study units, may be a proxy for unmeasured explanatory variables.Thisperspectivemeans,inpractice,thatonecanusetheseeigen- functions to modelandpredictspatialstructures withoutdetailedquantita- tive knowledge of all explanatory variables affecting a set of observed data. Extensions and the future of MESF analysis Extensions of MESF to the analysis of temporal data (time series) and to space–timedataappearedinthe2010s.InthePrefaceofthisbook,Griffith, Chun, and Li mention these developments, which were the result of the workofseveralgroupsofresearchersinstatisticalgeography.Ecologistsalso extended MESF to the analysis of space–time data (Legendre & Gauthier, 2014). I would like to express my highest appreciation to Professors Griffith, Chun, and Li for the immense amount of work they have completed to produce the 10 chapters of this book. Developers and users of spatial Foreword xv eigenfunctionmethodsinallfieldsofthenaturalandsocialscienceswillread, study, and refer to this book, which constitutes the first comprehensive compendiumofspatialeigenfunctionresearch.Thisbookrepresentsanec- essary and effective step toward future development and diffusion of the method. Furthermore, seeing what new understanding will be achieved by researchers about the role of spatial structures in natural and man-made systems—researcherswhonowhaveafirmbasisofknowledgeuponwhich to base their applications of MESF and its future developments—will be exciting. Pierre Legendre Universit(cid:1)e de Montr(cid:1)eal, Montreal, QC, Canada April, 2019 References “Student”[W.S.Gosset].(1914).Theeliminationandspuriouscorrelationduetopositionin timeorspace.Biometrika,10,179–180. Blanchet,F.G.,Legendre,P.,&Borcard,D.(2008).Modellingdirectionalspatialprocesses inecologicaldata.EcologicalModelling,215,325–336. 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