PathDependenceandTheValidationofAgent-basedSpatialModelsofLandUseDANIELG.BROWN,SCOTTPAGE,RICKRIOLO,MOIRAZELLNERandWILLIAMRANDInthispaper,weidentifytwodistinctnotionsofaccuracyofland-usemodelsandhighlightatensionbetweenthem.Amodelcanhavepredictiveaccuracy:itspredictedland-usepatterncanbehighlycorrelatedwiththeactualland-usepattern.Amodelcanalsohaveprocessaccuracy:theprocessbywhichlocationsorland-usepatternsaredeterminedcanbeconsistentwithrealworldprocesses.Tobalancethesetwopotentiallyconflictingmotivations,weintroducetheconceptoftheinvariantregion,i.e.,theareawhereland-usetypeisalmostcertain,andthuspathindependent;andthevariantregion,i.e.,theareawherelandusedependsonaparticularseriesofevents,andisthuspathdependent.Wedemonstrateourmethodsusinganagent-basedland-usemodelandusingmulti-temporalland-usedatacollectedforWashtenawCounty,Michigan,USA.Theresultsindicatethat,usingthemethodswedescribe,researcherscanimprovetheirabilitytocommunicatehowwelltheirmodelperforms,thesituationsorinstancesinwhichitdoesnotperformwell,andthecasesinwhichitisrelativelyunlikelytopredictwellbecauseofeitherpathdependenceorstochasticuncertainty.Keywords:Agent-basedmodeling;Land-usechange;Urbansprawl;Modelvalidation;Complexsystems1.IntroductionTheriseofmodelsthatrepresentthefunctioningofcomplexadaptivesystemshasledtoanincreasedawarenessofthepossibilityforpathdependencyandmultipleequilibriaineconomicandecologicalsystemsingeneralandspatialland-usesystemsinparticular(AtkinsonandOelson1996,Wilson2000,Balmann2001).Pathdependencearisesfromnegativeandpositivefeedbacks.Negativefeedbacksintheformofspatialdis-amenitiesruleoutsomepatternsofdevelopmentandpositivefeedbacksfromroadsandotherinfrastructureandfromservicecentersreinforceexistingpaths(Arthur1988,Arthur1989).Thus,asmallrandomcomponentinlocationdecisionscanleadtolargedeviationsinsettlementpatternswhichcouldnotresultwerethosefeedbacksnotpresent(AtkinsonandOleson1996).Concurrentwiththisawarenessoftheunpredictabilityofsettlementpatternshasbeenanincreasedavailabilityofspatialdatawithingeographicinformationsystems(GIS).Thishasledtogreateremphasisonthevalidationofspatialland-usemodels(Costanza1989,Pontius2000,2002,Koketal.2001).Thesetwoscientificadvances,onetheoreticalandoneempirical,haveleadtotwocontradictoryimpulsesinland-usemodeling:thedesireforincreasedaccuracyofpredictionandtherecognitionofunpredictabilityintheprocess.Thispaperaddressesthebalancebetweenthesetwoimpulses:thedesireforaccuracyofpredictionandaccuracyofprocess.Accuracyofpredictionreferstotheresemblanceofmodeloutputtodataabouttheenvironmentsandregionstheyaremeanttodescribe,usuallymeasuredaseitheraggregatesimilarityandspatialsimilarity.Aggregatesimilarityreferstosimilaritiesinstatisticsthatdescribethemappedpatternoflandusesuchasthedistributionsofsizesofdevelopedclusters,thefunctionalrelationshipbetweendistancetocitycenteranddensity(BattyandLongley1994,Makseetal.1998;Anderssonetal.2002,Randetal.2003),orlandscapepatternmetricsdevelopedwithinthelandscapeecologyliterature(e.g.,McGarigalandMarks1995)tomeasurethedegreeoffragmentationinthelandscape(ParkerandMeretsky2004).Spatialsimilarityreferstothedegreeofmatchbetweenland-usemapsandasinglerunorsummaryofmultiplerunsofaland-usemodel.Themostcommonapproachesbuildonthebasicerrormatrixapproach(Congalton1991),bywhichagreementcanbesummarizedusingthekappastatisti...
