Up­scalingAgent­BasedDiscrete­ChoiceTransportationModelsusingArtificialNeuralNetworksLanceE.Besaw§GraduateResearchAssistantCivilandEnvironmentalEngineeringUniversityofVermont213VoteyHall33ColchesterAve,Burlington,VT,05405Telephone:(802)656‐4595Fax:(802)656‐5838Email:lbesaw@cems.uvm.eduDonnaM.RizzoAssociateProfessorSchoolofEngineeringUniversityofVermont213VoteyHall,33ColchesterAveBurlington,VT,05405;Telephone:(802)656‐1495Fax:(802)656‐8446Email:drizzo@cems.uvm.edu

MargaretJ.EppsteinAssociateProfessorDept.ofComputerScienceUniversityofVermont327VoteyHall,33ColchesterAveBurlington,VT,05405Telephone:(802)656‐1918Fax:(802)656‐5838Email:Maggie.Eppstein@uvm.edu

MichaelB.PellonGraduateResearchAssistantDept.ofComputerScienceUniversityofVermont204FarrellHall,210ColchesterAveBurlington,VT,05405Telephone:(802)656‐4595Fax:(802)656‐5838Email:Michael.Pellon@uvm.edu

DavidK.GroverUndergraduateResearchAssistantCivilandEnvironmentalEngineeringUniversityofVermont204FarrellHall,210ColchesterAveBurlington,VT,05405Telephone:(802)656‐4595Fax:(802)656‐5838Email:David.Grover@uvm.edu

Jeffrey.S.MarshallProfessorSchoolofEngineeringUniversityofVermont231AVoteyHall,33ColchesterAveBurlington,VT,05405Telephone:(802)656‐3826Fax:(802)656‐3358Email:Jeff.Marshall@uvm.edu

§Correspondingauthor

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ABSTRACT1Agentbasedmodels(ABMs)canbeusedforsimulatingconsumertransportationdiscretechoices,2whileincorporatingtheeffectsofheterogeneousagentbehaviorsandsocial influences. However,3the application of ABMs at large‐scales may be computationally prohibitive (e.g., for millions of4agents). InanattempttoharnessthemodelingcapabilitiesofABMsatlargescales,wedevelopa5recurrent artificial neural network (ANN) to replicate nonlinear spatio‐temporal discrete choice6patternsproducedbyaspatially‐explicitABMwithsocialinfluence.ThisparticularABMhasbeen7developedtomodelconsumerdecisionmakingbetweenpurchasingaPrius‐likehybridorplug‐in8hybridelectricvehicle(PHEV)foragivengeographicregion(e.g.,cityortown).Ourgoalistoseeif9anANNtrainedatthecityscalecanoperateasa“fastfunctionapproximator”toestimatenonlinear10dynamic response functions (e.g., fleet distribution, environmental attitudes, etc.) based on city‐11wideattributes(e.g.,socio‐economicdistributions).Recurrentfeedbackconnectionswereaddedto12theANNtoleveragethetemporalhistoryandcorrelationsandimproveforecastsintime.Outputs13fromthecity‐scaleABM,runforavarietyofpopulationsizesandinitialandinputconditions,were14used to train and test the ANN. Initial results suggest the ABMmay be replaced by ANNs that15interact with each other and other agents (e.g., manufacturing agents) to investigate PHEV16penetrationatthenationalscale.17181920212223242526272829303132333435363738394041424344454647484950

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INTRODUCTIONANDMOTIVATION51Regulatoryactionsbyfederal,stateandlocalgovernmentscanplayacriticalroleininfluencingthe52transportation energy market. Due to the high degree of interdependency between various53governingmarket factors, it isdifficult topredict themarketconsequencesandsensitivity toany54givenregulatorychange.55

Discrete choice decision models are commonly used to study transportation and travel56phenomena,includingvehiclechoicebehavior(1),hybridchoicebehavior(2),andtravelmodeand57destinationchoice(3). Agentbasedmodels(ABMs)withsocial influencearecapableofmodeling58nonlinear spatio‐temporal discrete choice patterns. ABMs have been used as an alternative59methodology tomodel discrete‐choice decisions in applications like driver route choice (4) and60pedestrianwalkingbehavior(5).Incorporatingsocialinfluencesinanagent‐baseddiscretechoice61modelsmakethemmorerepresentativeofreal‐worlddecision‐makingprocess(6),butdrastically62elevatesthecomplexityofthesimplebinarychoicemodel(7).WhileABMshaveprovenusefulfor63modelingbehaviorincomplexsystems(8)anddemonstrateutilityinthefieldoftransportation(9),64they can require large amounts of computation when implementing complicated decision with65numerousagents.66

TheABM in thiswork (10)hasbeendeveloped to simulate the consumerdiscrete‐choice67decisionprocess betweenpurchasing a hybrid or plug‐in hybrid (PHEV) vehicle type for a given68demographic region (e.g., city or town),with given socioeconomic characteristics. As a proof of69concept, our ABMmodel uses synthetic data to reveal the importance of social influence on the70vehicle purchasing behavior of its agents. However, this consumer discrete‐choice model with71numerous agents requires an alternativemodeling strategy to replicate thebehavior of theABM72whenconsideringregulatorypoliciesacrossscaleslargerthanindividualtownsorcities(e.g.,larger73thanthestatelevel).74

Inthisresearch,weuseanartificialneuralnetwork(ANN)tolearnthedynamicbehaviorof75thesocially‐influencedagent‐baseddiscrete‐choicemodelandmapitsbehaviorforawiderangeof76syntheticdemographicregions(towns)andsocioeconomiccharacteristics.TheANN,operatesasa77fast functionapproximator, and requires recurrent feedback connections to allowoutputs at one78timesteptobeusedasinputsforthenexttimestep.Welooktoanswerthequestion:canasimple79ANNbeusedtoreplicate thebehaviorofacomplexagent‐basedconsumerdiscretechoicemodel80withsocialinfluence?Ifprovensuccessful,theseANNswillbesurrogatesfortheABMinanational‐81scalesimulationthatinvestigatesalternativepoliciestoinfluencePHEVfleetpenetration.82

83BACKGROUND84Artificial neural networks (ANNs) are nonparametric statistical tools that can be viewed as85universal approximators (11). They were developed as large parallel‐distributed information86processingsystemsinattempttomodelthelearningprocessesofthehumanbrain.Manydifferent87typesofANNhavebeenintroducedovertheyearsforproblemsinpatternrecognitionandfunction88approximation.ANNsspecializeinmappingnonlinearrelationshipsgivenextremelylargedatasets89(12). They have a relatively simple computational architecture, which makes them extremely90powerfulandcomputationallyefficient. TheANNknownasfeedforwardbackpropagation(FFBP)91has been used in numerous transportationmodeling studies, for example, predicting travel time92undertransienttrafficconditions(13),amongothers.However,FFBPdoeshavesomedrawbacks,93namelyitcanbecometrappedinlocalminima,requiresoptimizationofparameters(e.g.,numberof94hiddenlayersandnodes,thelearningcoefficientandmomentum)andcantakeextendedperiodsof95trainingtoconvergetoanadequatesolution.96

Inthiswork,weuseageneralizedregressionneuralnetwork(GRNN)to forecastdiscrete97consumerchoicesusingsocioeconomic,socialinfluenceandmarketconditiondescriptorsasinputs.98The GRNN has many advantages. It relaxes many of the assumptions required by traditional99parametric statistical methods (e.g., does not require an assumption of multivariate normality;100

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allowsbinaryorcategoricaldata).UnliketheFFBPnetwork, theGRNNhasone‐passtrainingand101guaranteedconvergence. Intransportationstudies,theGRNNhasbeenusedtoforecastdailytrip102flows (14),predict thehazardousnessof intersectionapproaches (15),model travelmodechoice103(16), predict CO2 fluxes (17), predict real‐time driver fatigue (18) and real‐time video traffic104modeling(19,20).105

106METHODS107AgentBasedModel108When selecting a vehicle to purchase, real‐world consumers may compare a variety of109characteristics,includingfuelefficiency,seatingandcargocapacity,safety,reliability,brand‐loyalty,110publicperception,etc.Ourmodelcurrentlyassumesthatouragentsarethesubsetofnewvehicle111consumerswhohavealreadynarrowedtheirchoicedowntoaless‐expensivePrius‐likehybridand112ahigher‐premiumPrius‐likePHEV.113

Since PHEVs are not yet available in the commercial marketplace, we based PHEV price114premiums, battery recharge requirements, electric assist ranges, and mileage with and without115electric assist, from reported specifications for the Hymotion PHEV conversion kit for the Prius116(21). The hybrid’s fuel economy is 45 mpg while the PHEV’s is 105 mpg when running on all‐117electricmode(45mpgotherwise),withanall‐electricrangeof35milesand5.5hourchargingtime118at5kWH.Weassumeotherwiseidenticalspecificationsandfeaturesbetweenthetwovehicleswith119onlygasmileageandpricepremiumdiffering.120

SeveralmajorassumptionshavebeenmadeinthedevelopmentofourABMtosimplifythe121modeledprocesses.Ourgoalwasnottoexactlymodelreal‐worldvehiclepurchasingbehavior,but122rather to grossly approximate it and investigate the impact of social influence and regulatory123policies.Here,wepresentabriefoverviewoftheABM;formoredetails,pleaserefertoPellonetal.124(10).125

Eachagentrepresentsasinglevehicleconsumer(notahousehold)anddrivesonlyonecar126(with no specification of vehicle purpose). Agents’ socioeconomic characteristics are drawn127randomly from distributions based on National Household Travel Survey (NHTS) data (22),128including annual driving distance and durations of vehicle ownership. Agents are randomly129distributed, yet clustered in space with an urban center and four suburban peripheral town‐130centers.Agentswithsimilarannualsalaryhavebeenlooselyclusteredinspaceusinga2‐Dturning131bands method (23). Several agent attributes (e.g., age, driving distance, number of years they132typicallyownavehicle, andwillingness to consideradopting thenewPHEV technology)arealso133positively or negatively correlated to salary, in varyingdegrees. The threshold forwillingness to134consider new PHEV technologywas initialized so that roughly one half of new car buyerswere135willingtoconsiderbeingPHEVearly‐adopters,consistentwithrecentsurveyresultsthatindicated136that 46% of potential consumers reported that they were some chance they might purchase a137PHEV,dependingonthepricepremium(24).138

Agents have specified social and spatial neighbors that make up their social and139geographical networks. The spatial network is defined on the physical proximity of the agents,140whilethesocialnetworkisbasedonphysicalproximityandsimilarsocioeconomiccharacteristics.141Thesenetworksaffecttheagents’decision‐makingprocess,asagentslookintheirnetworkstosee142whatvehiclesotheragentscurrentlyown;inaddition,agents’attitudescanbeinfluencedbyother143agents in their social networks. Heterogeneity in agent locations, social networks and driving144distancescausedifferentagentstobeexposedtodifferentvehiclefleets.TheproportionofPHEVs145within an agent’s observed fleet influences their willingness to consider adopting this new146technology. This “threshold” concept is found to be a very important feature in social influence147models(7;25).148

Agents stochastically decidewhen to purchase a vehicle based on their current vehicle’s149age,thenumberofyearstheyexpecttoownacarandhowmuchmoreattractiveanewvehicleis150

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comparedwith theircurrentvehicle. Onceanagentdecides topurchaseavehicle, theycompare151therelativefinancialcostsandenvironmentalbenefitstodeterminewhetherthePHEV(assuming152theirthresholdhasbeenmet)orhybridvehiclebestsuitstheirneeds,andthenpurchasethebest153vehiclefortheircircumstance.Thisprocessisrepeatedeveryyear,forallagentsoverthe15year154simulationtimeperiod.155

In addition to agents deciding to purchase vehicles, someof their internal characteristics156are updated every year. For example, their environmental attitude (or greenness) and the time157periodoverwhichtheycomputepotential fuelsavingsmaybothbeincreasedbysocial influence,158dependingontheirsocialsusceptibility. Greennessisaweightingfactorrangingbetween0and1159thatdetermineshowmuchanagentvaluestheperceivedenvironmentalbenefitsof thePHEV(in160this paper, proportion of gas saved by the PHEV relative to theHEV) vs. the perceived financial161benefitsoftheHEV(proportionofcostsavingsoftheHEVrelativetothePHEV);agreennessvalue162of0impliesthedecisionisbasedsolelyonperceivedfinancialbenefits.Inestimatingrelativecosts,163someagentsignorepotentialfuelsavingsandsimplylookatthepricepremiumofthePHEV,some164alsocomputeprojectedfuelcostsoveraperiodof1year,andotherscomputeprojectedfuelcosts165over the entire duration forwhich they anticipate owning their next car. Thus, some agents are166moreforward‐thinkingthanothers,andtheseagentscaninfluenceothersintheirsocialnetworkto167becomemoreforward‐thinking(i.e.,toconsiderprojectedfuelcostsoverlongerperiods),resulting168in interestingdynamics indiscrete‐choicebehavior.Note that if anagent’sprojected fuel savings169exceed thePHEVpricepremiumand thePHEVwillbe thusperceivedas thecheapervehicle, the170agentwill purchase thePHEV, regardless of their greenness value (assuming their thresholdhas171beenmet).See(10)formoredetails.172

Our model assumes there is no shortage of either vehicle type (i.e., no waiting period).173Thereareseveraladditionalexogenousinputs:PHEVpricepremiumaswellasfutureprojectionsof174gaspricesandcurrentnationalelectricityprices(26).175176GeneralizedRegressionNeuralNetwork177Developed as a nonlinear, non‐parametric extension ofmultiple linear regression, theGRNN is a178memory‐basednetworkcapableofestimatingcontinuousvariables(27).TheGRNNconsistsoffour179layers: input, pattern, summation, and output (Figure 1). Each layer is fully connected to the180adjacentlayersbyasetofweightsbetweenthenodes.Someoutput, ,ispredictedbasedona181setofinputvariables,x,definedbysomenon‐linearfunctionY=f(x),capturedbythetrainingdata.182Trainingdataconsistofasetofinputvectors,x,andcorrespondingoutput,Y(input‐outputpairs).183For this application, we use the algorithm to predict two output variables ( and ), which184representmediangreennessandproportionofPHEVsinagiventown,respectively.185

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186Figure 1. Architecture of a general regression neural network with recurrent feedback connections 187for two outputs variables Y1 and Y2. 188

Theuser specifies thenumberofnodes in the input layerbydeterminingwhatpredictor189variablesbestrepresentthesystembeingmodeled.Thepatternlayerhasonenodeforeachofthe190mtrainingpatterns.Theweightsontheleftsideofthepatternlayernodesstore(e.g.,aresetequal191to)theinputtrainingvectors,x.Eachnodeinthepatternlayerisconnectedtothesummationlayer192nodes,S1,S2andSG.Theweights linkingthepattern layernodeswithsummationnodesS1andS2193store themodel outputs for the two variables being predicted (e.g., , ) for all input‐output194trainingpatterns(i=1,2,…m).TheweightsfromthepatternlayernodestosummationnodeSGare195setequalto1.196

Oncetheweightsareset,theGRNNmaypredicteachofthetwooutputs.Anewinputvector197forwhich a prediction is desired,x, is presented to the pattern layer. TheEuclideandistance is198computed between the input vector and all pattern weight vectors, wi, where i=1,2,…m as:199

. The distance, , is passed to the summation layers and a prediction for200variable iscomputedas:201

,202

where isasmoothingparameterthatispivotalforestimating (thisprocessisalsoexecuted203forpredicting ). Largevaluesof smooththeregressionsurfaceandproduceestimates that204approach the sample mean; while small values produce a surface with greater chance of205discontinuity resulting in nearest neighbor estimates. Intermediate values of produce well206behaved estimates that approximate the joint probability density function of x and Y (27). The207prediction, ,isaweightedaverageofallstoredresponseobservations( , ,… ),whereeach208response isweighted exponentially according to its Euclidean distance from input vectorxi (the209samecomputationsarecomputedforoutputvariable ).210

In addition, the GRNN has been modified to allow for recurrent feedback connections211(dashedlineofFigure1).Recentpredictionsfor and ,arepassedbacktotheinputlayerand212

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usedtopredict and atthenexttimestep.Formoredetail,pleaserefertoBesawetal.(28).213TheGRNNalgorithmdescribedinthisworkwaswritteninMatLabV.7.4.0.287(R2007a).214

215ABMandGRNNimplementations216TheABMwasrununderseveralscenarios toprovidearangeof townsocioeconomicandvehicle217fleet distributions (Table 1). Prior experimentation revealed similar ABM trends when run for21810,000or1,000agents.Tosaveoncomputation,ourscenariosuse1,000agents.Thehypothetical219region of interest consisted of one larger (population of 726 agents) and 4 smaller towns220(populationsrangingfrom55to79agents)(Figure2a).Thesocioeconomicdistributionsofthese221areaswheregeneratedpseudo‐randomly,differentfromtowntotownandincludedspatialcross‐222correlation of salary and other inter‐attribute correlation. Agents in the respective towns had223different social networks based on the proximity of neighboring agents and their socioeconomic224characteristics. In addition, the ABM simulations use 10 initial random seeds to account for225potentialstochasticnoiseintheresults.226Table 1. List of ABM parameters that were varied during the generation of training, validation and 227prediction datasets. 228

Model Parameter Training & Validation Dataset Prediction Dataset

Median Social Susceptibility [0.01, 0.09, 0.33, 0.49] [0.17, 0.45] Proj. Gas Price Low, Medium, High Low-mid, Mid-High

PHEV Price Premium $5k, $10k, $15k $7k, $13k Town Identification 1, 2, 3, 4, 5 1, 2, 3, 4, 5 Region Population 1,000 1,000

Initial Random Seeds 1 to 10 1 to 10 229Forthisproof‐of‐concept, theagentthresholds forsocialsusceptibilityvariedfrom0to1,230

makingabroaddistributionofpotentialearlyadoptersandnon‐conformists.Socialsusceptibility231distributions were stochastically generated between 0 (not socially susceptible) to 1 (very232susceptible). Themedians of these distributionswere varied to generate agent populationswith233largesusceptibilityvariations(Table1andFigure2b). Agentinitialgreennessdistributionswere234also stochastically generated but remained statistically similar between scenarios. Exogenous235model inputs (i.e., projected gasprices (Figure2c) andPHEVpricepremiumwere also varied to236produce representative scenarios. U.S. Energy Information Administration (26) provided237reasonable projections of high and low gas prices (medium was the average of high and low).238Different PHEV‐premiums were used to investigate the potential impact of price incentives.239Greenness was stochastically initialized to range from 0 to 1, with an initial median value of240approximately0.17;asshownforarepresentativedistributioninFigure2d.241

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242Figure 2. (a) Spatial coordinates of agents and their annual income shown in grayscale. (b) Example 243distributions of social susceptibilities with different medians. (c) Fifteen year projections of gas 244prices (high and low data from EIA). (d) Representative distribution of agents’ initial greenness. 245

Descriptive statistics (median and interquartile range) of the ABM simulations were246computedateachtimestepforthenumerousagentdistributions.Thesewerethenaveragedover247the10randomseedsforeachtownandsimulationsetup.Thisresultedinatotalof4,320different248town simulations used to train the GRNN (comprising a 15‐year time series when PHEVs are249introduced inyear2). As is typical inANNapplications, thisdatasetwasseparated into training250andvalidationdatasets.Thetrainingdatasetcomprised3,000townsimulations(~80%)tosetthe251GRNNweights. The validation datasetwas comprised of the remaining 1,320 town simulations252(~20%)andwasusedtooptimizetheGRNNsmoothingparameter (viatrialanderror).253

GRNN inputs include many of the socioeconomic descriptors incorporated in the ABM,254includingastatisticaldescriptorsagents’age, incomeandsocialsusceptibilities,PHEVthresholds,255driving distances and car replacement age. In addition exogenous inputs include projected gas256prices, and PHEV‐price premium. All inputs were normalized such that their minimum and257maximumvalueswere0and1respectively.Outputs fromtheGRNNarestatisticaldescriptorsof258agentgreenness(median)andtheproportionofthefleetthatarePHEVSinaparticulartown.As259

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thisisarecurrentGRNN,theseoutputsarefedbackintotheGRNNandusedasinputsinthenext260timestep.261

Togeneratethepredictiondataset,anentirelynewspatialdistributionofagentsandtheir262corresponding socioeconomic characteristics were generated (Table 1). Other initial model263parameterswerechangedaswell, including:socialsusceptibility,projectedgaspricesandPHEV‐264premium. Again, descriptive statistics were computed for each town at every time step and265averaged over the 10 random seeds. This combination of parameters resulted in prediction266scenarios thatwere similar to, yet significantly different from, those uponwhich the GRNNwas267trained.268

269RESULTSANDDISCUSSION270WehavesummarizedtheaccuracyoftheGRNNpredictionsforboththevalidationandprediction271datasets(Table2)forbothPHEVpenetrationandoneofthechangingagentattributes(greenness).272Wehaveusedthecoefficientofdetermination(R2),andinsomecasestheroot‐mean‐squareerror273(RMSE)betweentheGRNNpredictionsandtheABMresultsasaccuracymetrics. Themeansand274standard deviations of R2 are computed for each town over the given number of scenarios. The275optimized smoothing parameter ( =0.004) was used to predict PHEV fleet proportion and276greennessineachofthefivetownsfor171validationand16predictionscenarios.277

278GRNNpredictionsofABMPHEVsandgreenness279Although the time horizon over which individual agents projected fuel costs for prospective280vehicleswas allowed to vary dynamically due to social influence, current implementation of the281GRNN does not feed this back as a recurrent input. Despite this known source of error, the282coefficientsofdeterminationforthevalidationdataset(Table2)showthattheGRNNwasableto283mapthegeneralbehaviorofthediscrete‐choiceABM.WhenpredictingPHEV‐fleetproportionand284greenness in the validation dataset, the GRNNwasmost accuratewhen predicting larger towns285(e.g.,towns1and3,Table2).TheGRNNpredictionswereleastaccurateintown4(PHEVaverage286andstandarddeviationofR2were0.57and0.33)wherethenumberofagentswasleast.287

TheGRNNperformedbetterwhenpredictingPHEVfleetproportionthanwhenpredicting288greennessinallvalidationscenarios.Inaddition,thereexistlargeamountsofvariabilitybetween289the coefficients of determinations computed in the validation dataset (for both PHEV and290greenness).Theseeffectsaremostlikelycausedbythelargevariationofsocialsusceptibilityinthe291trainingandvalidationdatasets(Table1).Figure2cprovidesavisualcomparisonofsomeofthese292distributions. When social susceptibilities are so drastically different, the trajectories of the293greenness,and toa lesserextentPHEV fleetproportion,areverydifferent. Itappears theremay294have been too much variation in our distributions of social susceptibility in the training and295validation datasets that limited the GRNNs ability to accurately learn all of the 3,000 scenarios.296Future experiments will add the distribution of time horizons for fuel cost projections as a297recurrentinputintotheGRNNtoseeifthisimprovesthesituation.298

TheR2’scomputed for thepredictionsdataset indicate theGRNNhassuccessfully learned299the relationships from the trainingdataset (including the influenceof social influence/networks)300andwasable toaccuratelypredictPHEV fleetproportionandgreenness. Thescenariosused for301predictionwere generated using entirely new datasets with respect to the projected gas prices,302PHEVpricepremium,spatialdistributionandsocialsusceptibility(Table1).However,wedidnot303introduce as much variation in the social susceptibility as was introduced in the training and304validationdatasets(mediansof0.17and0.45).305

As a result, for the particular scenarios presented here, the GRNN was more accurate306predicting on the prediction dataset than on the validation dataset. The estimated PHEV‐fleet307proportioninthevalidationandpredictiondatasetshadR2’sof0.8and0.97respectively(0.75and308

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0.87 for greenness). Similar to the validation dataset, we predicted PHEV‐fleet proportionmore309accurately than greenness in the prediction dataset, 0.97 and 0.87 respectively. The greater310accuraciesaremostlikelyduetothelessvariabledistributionsofsocialsusceptibilityusedinthe311predictiondataset.312Table 2. Table of summary statistics for the coefficient of determination (R2) computed for the 313validation and prediction datasets (a mean R2 of 1.0 indicates the ANN predictions perfectly match 314the ABM results). 315

Output Variable Dataset Statistic Town 1 Town

2 Town

3 Town

4 Town

5 All

Towns Number of

Agents Validation &

Prediction Mean 74 66 726 55 79 1000

Mean R2 0.89 0.83 0.88 0.57 0.83 0.80 Validation (n=171) Std. Dev. R2 0.22 0.34 0.24 0.33 0.25 0.28

Mean R2 0.97 0.98 0.99 0.94 0.96 0.97 PHEV-Fleet Proportion Prediction

(n=16) Std. Dev. R2 0.02 0.01 0.01 0.03 0.02 0.02 Mean R2 0.85 0.82 0.87 0.47 0.76 0.75 Validation

(n=171) Std. Dev. R2 0.27 0.31 0.24 0.29 0.29 0.28 Mean R2 0.83 0.74 0.97 0.89 0.91 0.87

Greenness Prediction

(n=16) Std. Dev. R2 0.09 0.06 0.01 0.04 0.01 0.04 316

Furtheranalysisintotwopredictionscenarios317To further highlight theGRNN capabilities to learn the behavior of this discrete‐choiceABM,we318have selected two scenarios (call them I and II) from our possible 16 prediction scenarios and319plotted the time‐series of PHEV proportion and greenness for towns 2, 3 and 4. The social320susceptibilitydistributionwaswithonlyparameterthatvariedbetweenscenarioIandII(Table3).321

Scenarios Iand IIdemonstrateseveraldifferentABMphenomena thatwewould likeour322GRNNtobeabletoreplicate,including:differentialratesofPHEVadoption,differentfinaladoption323proportionaswellaslinearandnon‐lineardynamicsofadoption.Theseinterestingdynamicshave324arisenoutofthedifferentdistributionsofsocialsusceptibilityusedinthesetwoscenarios.Thetime325seriesplotsprovidedbelowassumethePHEVwasbeen introduced inyear0(withyear ‐1being326ourinitialmodelconditions).327Table 3. Parameter values for the two representative scenarios I and II. 328

Parameter Scenario I Scenario II PHEV premium $13k $13k

Social Susceptibility Median 0.17 0.45 Proj. Gas Price Low-mid Low-mid

Representative Simulation Figure 3 Figure 4 329

InscenarioI,weobservethatroughly50%oftheagentsintowns2and3(Figures3aand3303b)haveadoptedPHEVs.Duetotheloweconomicstatusofagentsintown4(Figure3c),amuch331lower fraction of these agents have adopted PHEV. These three towns have different PHEV332adoption trajectories. In town 2 (Figure 3a) we see highly nonlinear behavior. There is slow333adoptionat first, followedbyan increasedrateofadoption. In town3 (Figure3b), theadoption334rate remains fairly constant over the simulation. Finally, town 4 (Figure 3c) shows a nonlinear335jumpinadoption.336

TheGRNNaccuratelypredicts the time‐seriesofPHEVadoption for towns2and3 (R2of3370.99and1,respectively).Thisisencouragingbecauseonetrajectoryisnonlinearwhiletheotheris338linear.TheGRNNdoesrelativelypoorwhenpredictingthePHEV‐fleetproportionfortown4.Asin339

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thetrainingandvalidationdataset,town4continuestobetheleastwellpredictedofthefivetowns340(Table2).However,resultsdemonstratetheGRNNhaslearnedbothlinearandnonlineardynamics341ofthisABM.342

As illustrated with the training and validation datasets, the GRNN does not predict343greenness (Figures 3d, 3e and 3f) aswells as it predicts PHEV‐fleet proportion. In Town 2, the344GRNN accurately predicts greenness (R2=0.97). However, the coefficients of determination and345RMSEshow that theGRNNdoesnotpredict greenness in towns3and4verywell. This ismost346likely due to the high variability of social susceptibility in the training dataset, upon which347greennessisprimarilybased.348

349Figure 3. Representative GRNN predictions (for scenario I in which the PHEVs are adopted by 350~50% of the consumer agents). GRNN predictions of PHEV fleet proportion versus time for (a) 351town 2, (b) town 3 and (c) town 4. (d-f) GRNN predictions of greenness for towns 2, 3 and 4 352respectively. 353

InscenarioII,weobserveagreaterproportionoftheagentsadoptingPHEVsintowns2and3543(Figures4aand4b).FewerPHEVswereadoptedbytown4(Figure4c),indirectlyduetoitslow355annualincomes(Figure2a,rightmosttown)whichwerecorrelatedwithlongervehicleownership356timesandhigherPHEVadoptionthresholds. Weseenon‐linearadoptionrates in towns2and3.357Adoption is relatively slow initially, but increases with time. In this scenario, the GRNN again358accuratelypredictsPHEV‐fleetproportion for towns2and3 (R2of0.98and1 respectively).The359GRNNgreennesspredictionsfortowns2and3aremoreaccuratethanscenarioIindicatingthatthe360GRNN may be slightly biased, as well as more accurate, when predicting using larger social361susceptibilitymedians. This could be remedied easily usingmore training scenarioswith lower362socialsusceptibilitymediansandbyallowingthetimehorizonforfuelcostprojections,whichare363alsoaffectedbysocialsusceptibility,toberecurrent.364

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365Figure 4. Representative GRNN predictions (for scenario II in which the PHEVs are adopted by 366~80% of the consumer agents). GRNN predictions of PHEV fleet proportion versus time for (a) 367town 2, (b) town 3 and (c) town 4. (d-f) GRNN predictions of greenness for towns 2, 3 and 4 368respectively. 369

ScenariosIandIIpresentagoodexamplesoftheimpactofsocialinfluenceandnetworkson370thediscrete‐choiceABM.InscenarioI,thesocialsusceptibilitydistributionwasskewedleftcausing371thegeneralpopulationtobelessinfluencedbytheirsocialnetwork(median=0.17). Conversely,372the social susceptibility distribution of scenario IIwas less skewed (median = 0.45), resulting in373agents thatweremore influenced by their social network. The PHEV‐fleet proportion curves of374scenarioII(greatersocialinfluence)hadalargernumberofPHEVsadoptedintowns2and3than375inscenarioI(roughly50%and80%respectively).Thisisduetothedifferenceincontributionof376socialinfluenceinthetwoscenarios.377

The predictions for these two scenarios show the GRNN was able to accurately predict378PHEV‐fleet proportionunder these twodifferent conditions (less andmore social susceptibility).379Thisisanimportantcontributionofthiswork;becauseaccuratepredictionsofPHEVadoptionwill380allowustousetheGRNNsasasurrogatefortheABMwhenpredictingPHEVadoptionundermany381different types of scenarios. It also demonstrates that the GRNN has been able to learn the382importanceofsocialinfluenceandnetworksgeneratedbytheABM.383

These two scenarios demonstrate the GRNN has learned and accurately predicted384importantABMphenomena,including:differentialratesofPHEVadoption,differentfinaladoption385proportion,aswellas,linearandnon‐lineardynamicsofadoption.386

387ComputationalSpeedup388Thespeedofcomputationisanotherimportantconsiderationofthiswork.Ourobjectiveistouse389theGRNNasasurrogatefortheABMforlarge‐scalesimulations(e.g.,nationscale).Tobeaviable390surrogate,theGRNNmustbecomputationallyfasterthantheABMforlarge‐scalesimulations.391

The ABM scales super linearlywith increasing number of agents (N). This is due to the392largeamountsofcomputationperformedateverytimestepforeveryagent.TheABMtakes:11.5393seconds for 1000 agents, 70 seconds for 10000 agents, and continues to rise. Example394

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computationsincludethethresholdtoconsiderpurchasingaPHEVbasedonsocialandgeographic395networks(whichscalesuperlinearlywithN).Becauseofthissuperlinearscaling,performinglarge396scale simulationswith the ABMmay not be feasible. However, as demonstrated previously, the397GRNNiscapableoflearningthenon‐lineardynamicsofthediscrete‐choiceABM.398

TheproductionoftheGRNNtrainingandvalidationdatasetsalsoscalenon‐linearlywithN,399due to the same computations discussed above. It took approximately 24 hours to produce the400training and validation dataset (4,320 scenarios) used in this proof‐of‐concept. However once401thesedatasetsaredeveloped,theGRNNtakesrelativelylittletimetotrain(~4hoursinthiswork).402OncetheGRNNistrained,ittakesverylittletimetorun(~0.4secondsforanynumberofagents),403nomatterhowlargeofatownorcityisbeingsimulated.ThisisduetothefactthattheGRNNmust404onlymarch through 15 time steps, independent of the town’s population. Thus, themajor time405requiredfortheGRNNistheproductionofthetrainingandvalidationdatasetsbytheABM.Thus,406forthe4,320scenarios(eachwith1000agents)inthissmalldemonstrationcomparabletimingsfor407theABMandANNare13.8hoursversus0.48hours,respectively.408 Thesedemonstrationtimingsweregeneratedona3GHzIntelCore2Duoprocessorwith4093GBofRAMrunningMatLabV.7.3.0.267(R2006b). 410

411412CONCLUSIONSANDFUTUREWORK413This work was originally conceived based on the premise that the GRNN can operate as a fast414functionapproximatorofadiscrete‐choiceagent‐basedtransportationmodelwithsocialnetworks415and influences. For training and validation, we produced a large dataset that exhibited spatio‐416temporal dynamics of this simplified consumer vehicle choice ABM. The inclusion of social417influence introduced through social networks, adoption thresholds and susceptibility resulted in418nonlinearagentbehavior.419

Thisproof‐of‐conceptGRNNhasprovencapableof learningthespatio‐temporaldynamics420ofthediscrete‐choiceABMwithsocialinfluence.Byincorporatingarecurrentfeedbackconnection,421the easy‐to‐train GRNNwas able to adequately replicate the behavior of ABM at the town scale.422Adding in additional recurrency for other dynamically changing attributes (in this case, time423horizonforfuelcostprojections)isexpectedtoimproveresultsevenfurther.Althoughitdoestake424significant amounts of time to generate the training and validation datasets and optimize the425GRNN’s smoothing parameter, once trained, it operates as a fast function approximator that426demonstrated tremendous speedup time relative to the ABM. The combined effects of accurate427approximation and dramatic speedup will allow us to simulate large‐scale dynamics more428computationallyefficientlythanrunningalarge‐scaleABM.429

In order to investigate potential regulatory policies that can influence the adoption of430PHEVs (e.g., price incentives, rebates) a large‐scalemodel is currently under development. This431large‐scale model incorporates additional types of agents (e.g., vehicle manufacturing and432electricityproducingagents).WiththeGRNNabletooperateasasurrogateoftheconsumerABM,433GRNNs representing neighboring townswill interactwith each other andwith other large‐scale434agents. With this framework, we can explore potential regulatory policies thatmay impact the435decisionsandbehavioroflarge‐scaleagents.436

437ACKNOWLEDGEMENTS438This work was funded in part by the United States Department of Transportation through the439UniversityofVermontTransportationResearchCenter.Wegratefullyacknowledgecomputational440resourcesandexpertiseprovidedbytheVermontAdvancedComputingCenter,supportedinpart441byNASA(NNX06AC88G).442

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