Proceedings of the 2015 Winter Simulation Conference L. Yilmaz, W. K. V. Chan, I. Moon, T. M. K. Roeder, C. Macal, and M. D. Rossetti, eds.

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1 Proceedigs of he 2015 Wier Simulaio Coferece L. Yilmaz, W. K. V. Cha, I. Moo, T. M. K. Roeder, C. Macal, ad M. D. Rossei, eds. WEAVING MULTI-AGENT MODELING AND BIG DATA FOR STOCHASTIC PROCESS INFERENCE We Dog Deparme of Compuer Sciece ad Egieerig & Isiue of Susaiable Trasporaio ad Logisics Sae Uiversiy of New Yor a Buffalo, USA ABSTRACT I his paper, we develop a sochasic process ool o ell he sories behid big daa wih age-based models. Specifically, we ideify a age-based model as a sochasic process ha geeraes he big daa, ad mae ifereces by solvig he age-based model uder he cosrai of he daa. We hope o use his ool o creae a bridge bewee hose who have access o big daa ad hose who use age-based simulaors o covey heir isigh abou hese daa. 1 INTRODUCTION Big daa have he big poeial o revoluioize boh our udersadig of complex sysems ad our way of maig daa producs. However, i is a sigifica edeavor o ell sories abou big daa wih geeraive models (age-based or sysem dyamics models) ad o impleme daa producs as opimizaio problems ivolvig big daa, because hose who wor wih big daa ad hose who wor wih models spea differe laguages. To bridge big daa ad geeraive models, we sar by ideifyig a geeraive model as a sochasic process, which geeraes ime series accordig o a well-defied probabiliy measure. We he fi he parameers of he sochasic process o he daa usig machie-learig mehods, ad reaso abou he daa by solvig he fied model uder he cosrais of hese daa. I sochasic process ermiology, he sysem sae of his geeraive model is comprised of he saes of he idividual ages, he ier-age relaioships, ad he world. Sysem sae chages over ime accordig o a sae rasiio marix/erel defied by he eves of he geeraive model, ad we are ieresed primarily i maig ifereces regardig he sochasic process by sudyig he oisy observaios abou hese sochasic process saes. However, he primary difficuly i solvig he sochasic processes defied by a age-based model fidig he probabiliies for he age-based model o geerae every possible sae sequeces sems from he explodig sae space: i a sysem wih 100 ages, ad wih each age aig wo saes, we will have sysem saes. To avoid he ecessiy of eumeraig sae combiaios, we iroduce mea-field approximaio. The esimaed probabiliy ha I cach a cold is he average probabiliy ha I cach a cold from my eighbor x (who is sifflig bu whose cold-or-o I do ow) plus he average probabiliy ha I cach a cold from my eighbor y (who appears healhy, bu whose cold-or-o I also do ow), ad so o. The fac ha my eighbor x does have a cold will o chage my esimaio, alhough i does icrease my chace of cachig a cold. This id of mea-field approximaio is employed widely i saisical physics, ad ca be a sigifica help here i dealig wih expoeially-icreasig sae space (Heses ad Zoeer 2002, Waiwrigh ad Jorda 2008). Oher poeial mehods iclude Marov chai Moe Carlo (Dog, Heller, ad Pelad 2012, Wiliso 2006) ad diffusio approximaio (Wiliso 2006). This paper herefore advocaes, for he firs ime, ha we should combie he power of big daa ad he power of model-hiig i he sochasic process framewor. Age-based modelig is a physicis s approach for modelig huma socieies whe daa are uavailable ad experimes impossible (Epsei /15/$ IEEE 713

2 Dog 2007), ad we believe ha big daa will rasform age-based modelig from speculaio io a physical sciece. This paper also offers a soluio o fi ay age-based model defied by a producio rule sysem o big ime series daa, based o mea-field approximaio. Hece, his sysem bridges modelers wih daa miers. We have bechmared our soluio o sysems of hudreds of ages, ad our bechmarig gives meaigful resuls. Wih he ool o weave muli-age modelig ad big daa for sochasic process iferece, we hope o rac real world social sysems coiuously usig big daa from social media ad Iere of Thigs o he oe had, ad muli-age models o he oher had. We ca beer udersad he diffusio of ifluece ad he formaio of ewors (Dog, Lepri, ad Pelad 2011, Dog, Heller, ad Pelad 2012) from he behavior ad ieracio daa coiuously observed by persoal mobile phoes (MIT Huma Dyamics Lab 2012), ad social physics models (Casellao, Foruao, ad Loreo 2009). We ca esimae road raffic (Dog ad Pelad 2009) i real-ime accordig o millios of raced vehicle locaios ad muli-age rasporaio simulaors ((Smih, Becma, ad Baggerly 1995, MATSim developme eam (ed.) 2007)). We ca rac aioal socio-ecoomical idicaors accordig o call deail records (Orage S. A. ) ad ciy models (Bay 2007). The res of his paper is orgaized as follows. I Secio 2, we iroduce a probabilisic producio (rule) sysem o describe he microscopic dyamics of a geeraive model, ad ideify he producio sysem as a sochasic process. I Secio 3, we derive a mea-field soluio o he geeraive model uder he cosrai of daa i oher words, give a simulaor ad oisy observaioal daa abou a process geeraed by he simulaor logic, our algorihm ifers he probabiliies o a per-age basis of all possible oucomes. I Secio 4, we give examples ad bechmar his algorihm agais oher algorihms. We summarize wha we have accomplished ad offer our speculaio abou big daa i Secio 5. 2 STOCHASTIC PROCESS INDUCED BY PRODUCTION SYSTEM We employ a producio (rule) sysem o describe muli-age dyamics. Such a producio sysem is widely used i arificial ielligece, chemisry, sysems biology, ad oher fields. A producio sysem i arificial ielligece cosiss of a worig memory ha maiais he curre sae of he world ad a rule ierpreer ha riggers differe rules (or producios) wih a well-defied mechaism, chagig he sae of he world accordig o hese riggered rules. A producio has a precodiio ad a acio whe he precodiio maches he curre sae of he world, he ierpreer ca rigger he producio, execue he acio of he producio, ad so chage he curre sae of he world. Researchers i exper sysems are ieresed i usig a deermiis algorihm ha riggers he bes rule for reachig a goal from a collecio of machig rules. I compariso, we rea a producio sysem as a sochasic process by assigig differe raes of happeig o rules, ad by defiig a ime series of sysem saes as he probabiliy of a sequece of rules firig, which geeraes he sysem-sae ime-series. We ae his approach because we are mos ieresed i fiig he producio sysem o observaios abou how people behave ad ierac, i order o solve problems ad solve for he raes. This allows us o mae predicios abou huma behavior ad ieracios usig he fied producio sysem, a effor similar o hose i research o cogiio archiecure (Aderso 1996, Newell ad Simo 1972). Algorihm: Marov process iduced by a muli-age model Ipu: iiial world saex( = 0), producios1,...,v, each producio happes wih raeh (x,c ) = c g(x ) ad chage world sae x x. Oupu: a series of imes whe producios are riggered, he IDs of he riggered producios, ad he correspodig saes brough abou by he riggered producios { i,v i,x( i ) : i} where 0 = 0 < 1 < < < +1 = T, v i {1,...,v}, x ( ) vi i x(i ) ad x() is he righ limi ad he ime series x() is lef coiuous. Procedure: 714

3 Dog Basis se curre ime o 0 = 0, se he curre sae o x( 0 ), repea he followig sep uil he curre ime i+1 > T. Iducio sample he ex reacio ime τ expoeial( h (x( i ),c )), sample he ex reacio v i+1 h / h, se curre ime o i+1 = i +τ, ad updae world sae x( i+1 ) accordig o producio v i+1. Such a model defies a sochasic process. The probabiliy for his sequece of eves { i,v i,x( i ) : i} o happe is P (v,x) = i h v i (x ) exp( i h v i (x ) ( i+1 i )) (Wiliso 2006). The ime ierval τ o he ex eve is he miimum of expoeial disribuios, ad so is iself a expoeial disribuio. We leave absrac he acios of he producios, because a iferece algorihm wih geeral producios will be more useful. For example, i he Loa-Volerra model (commoly ow as he predaor-prey model), he world is comprised of x (1) umber of predaors ad x (2) umber of prey. The prey has a ulimied food supply, ad will reproduce wih a rae proporioal o he prey populaio wihou predaio (prey 2 prey, or x (1) x (1) +1 wih rae c 1 x (1) ), bu predaio will decrease he prey populaio ad icrease he predaor populaio a a rae proporioal o he rae ha predaor ad prey mee (prey+predaor 2 predaor, or (x (1),x (2) ) (x (1) 1,x (2) +1) ). The predaor populaio will lose is members due o aural deah a a rae proporioal o he predaor populaio (predaor, or x (2) x (2) 1). By solvig he Loa-Voerra equaios ha is, fidig he emporal evoluio of he predaor populaio ad he prey populaio, cosraied by iiial ad boudary codiios we ca aswer quesios such as wheher he wo species could coexis, wheher he populaios could pass hrough cerai hresholds, ad wha he liely populaio sizes will be i he fuure. Theoreically, ay compuer program ha operaes compuer sorage hrough a se of isrucios ca describe a producio sysem wih is world sae ad rules. However, we prefer he simples model ha ca explai he pheomeo uder ispecio, because he simples model wih he fewes rules may be he mos geeralized ad applicable model for leveragig our pas experieces o gai a advaage i he fuure (Popper 1965). I fac, may elega models fid applicaios across domais. Epidemics models, for example, are applicable o oly o he sudy of epidemics hemselves, bu also o he spreadig of opiios ad iovaios (Casellao, Foruao, ad Loreo 2009). Similarly, Loa-Voerra models are applicable o ecology, bu also o ecoomics whe researchers wa o udersad he ieracios of differe idusry secors (Goodwi 1975). The sochasic process iferece ool developed i his paper aims o creae a bridge bewee big daa ad busiess isigh abou big daa, eablig daa miers o ell sories behid daa wih he logic of age-based models, ad eablig age-based modelers o mae predicios abou real-world sysems from daa. 3 MAKING INFERENCES WITH PRODUCTION SYSTEM I his secio, we derive a compuaioally-racable soluio for ay ewor dyamics model ideified by a producio sysem. I oher words, give a ewor dyamics model wih uow eve cosas {c : = 1,...,v} ad lae saes {x = (x (1),,x (C) ) : R}, we fid he mos liely esimae of he eve cosas ad lae saes from a fiie umber of observaios abou he model {y (c) : (c,)}, where he superscrip c idicaes differe ages. We firs discreize he Marov jump process io a hidde Marov process i order o mae he problem suiable for umerical evaluaio ad implemeaio o digial compuers. We he apply a mea-field approximaio such ha he lae sae of a ode evolves accordig o he average effecs of he lae saes of oher odes. Wih his approximaio i place, we give he expecaio maximizaio (EM) algorihm of he resulig hidde Marov model as a soluio o our origial problem. 715

4 3.1 Time ad Sae Discreizaio Dog We ca assume ha he lae sae x defied by a producio sysem aes a fiie umber of values i may applicaios, ad rea a producio sysem as a coiuous ime Marov chai wih rasiio rae marix (also called ifiiesimal geeraor) Q(i,j) = d dτ p(x +τ = j x = i) give by Q(x,x +τ ) = { x x+τ h (x,c ) if x x +τ 1 h (x,c ) if x x +τ. (1) These lae saes may have already ae a fiie umber of values (e.g., ifeced vs. suscepible i he SIS epidemics model). Whe he lae saes ae oly a few ieger values mos of ime (i.e., he ighess assumpio (Dudley 2002)), we se he domai of he lae saes o be hese ieger values accordig o he compuaioal precisio requireme. Whe he lae saes ae a ierval of real values mos of ime, we se he domai o be a few disjoi subiervals, agai accordig o he precisio requireme. A coiuous-ime Marov chai is relaed o a discree-ime Marov chai hrough a uiform rae parameerγ max x h (x,c ) usig eiher he uiformizaio mehod (also ow as Jese s mehod), or he radomizaio mehod (Ibe 2008). LeQ(x x + x ) = h (x,c ) be he ifiiesimal geeraor of he coiuous-ime Marov chai. The probabiliy rasiio marix of he uiformized discree-ime Marov chai is I + Q/γ, where I is he ideiy marix ad he ime of he rasiios is a Poisso process wih rae γ. The umber of rasiios bewee ime 0 ad ime is a radom variable of Poisso disribuio wih rae cosa γ ha is, he probabiliy of rasiios durig, icludig ull rasiios x x, is e γ (γ ) /!. The sae rasiio marix bewee ime 0 ad ime is herefore p(x 0 x ) = =0 (I +Q/γ) e γ (γ ) /!. Whe we le γ ad γ = 1, we ge p(x 0 x ) I + Q γ γ. The lielihood of ay sample {x,y : = } geeraed by a give ewor dyamics model is p({x,y : = } {c : = 1,...,v}) (2) observaio eves 1 x (+1) o eve x = h (x,c )) x (1 h (x, c )) 1x (+1) x.,c p(y (c) x(c) ), x( = x I he followig, we will use N o represe discree ime adh (x,c ) o represe he probabiliy for eve o happe i discree-ime hidde Marov model. 3.2 Forward-bacward Algorihm uder Mea-field Approximaio Recall ha a hidde Marov model assigs o every sequece of hidde saes{x : } ad he correspodig observaios {y : } a probabiliy p({x,y : = 1,...,T}), accordig o he sae rasiio probabiliy p(x x 1 ) ad he probabilisic observaio disribuio p(y x ). Iferrig he probabiliy disribuio of lae saes ivolves dyamic programmig o ieraively solve he forward saisics α (x ) = p(x,y 1,..., ) for = 1,...,T, he bacward saisics β (x ) = p(y +1,...,T ) for = T,...,1, ad he oe-sep saisics γ (x ) = p(x y 1,...,T ) ad wo-sep saisicsξ (x,+1 ) = p(x,+1 y 1,...,T ), ad from hese saisics we ca esimae he sae rasiio probabiliy ad he parameers of he observaio model. To avoid floaig-poi uderflow, we ofe compue / α (x ) = p(x y 1,...,,π) = α (x ) τ=1 / Z τ, β (x ) = β T (x ) τ=+1 Z τ, where 716

5 Z = p(y y 1,..., 1 ) = Dog x 1, α 1 (x 1 )p(x x 1 )p(y x ) = x 1, α 1 (x 1 )p(x x 1 )p(y x )β (x ), I a hidde Marov model ideified by a (sochasic) muli-age model or a sysem dyamics model, he sae space ca be huge, represeig he joi sae of housads o millios of ages, or housads of sysem variables. I addiio, he sae-rasiio probabiliy is ormally specified wih a limied umber of eves. I he followig, we solve he margial saisics α (c) (x (c) ), β (c) (x (c) ), γ (c) (x (c) ) ad ξ (c) (x (c),x (c) +1 ) wih mea-field approximaio he sae of a idividual ode evolves accordig o he average (mea-field) effecs of he oher odes ad esimae he eve raes accordig o he margial saisics. We adop he approximaio γ (c) (x (c) ) = α (c) (x (c) )β (c) (x (c) ), (3) p(x y 1,...,T ) = c γ (c) (x (c) ), (4) see he margial forward saisics α (c) o be cosise wih he wo-slice saisics ξ (x 1, ) for = 2,...,T, ad see he bacward saisics β (c) 1 o be cosise wih he wo-slice saisics ξ (x 1, ) for = T,...,2: fid α (c) s.. fid β (c) 1 s.. fix x (c) ξ (x 1, ) = α (c) (x (c) )β (c) (x (c) ), ξ (x 1, ) = α (c) (x (c) 1 )β(c) 1 (x(c) 1 ), fix x (c) 1 whereξ (x 1, ) = 1 α (c) 1 Z (x(c) 1 ) c c β (c) (x (c) )p(x x 1 )p(y x ). We furher le eve raes facorizable io a produc of fucios, each of which ivolves oe ode i he sysem c c g(c) (x(c) 1 ) x = x 1 + x p(x x 1 ) = :x =x 1 + x 1 (5) c c g(c) (x(c) 1 ) x = x 1 For example, he ifecio rae i he SIS epidemic model c x (1) x (2) is he rae whe oe ifeced perso is i coac wih oe suscepible perso (c) muliplied by he umber of differe ways i which x (1) suscepible perso ca be i coac wih x (2) ifecious persos. Similarly, he predaio rae i he predaor-prey model c x (1) x (2) is he rae a which oe predaor is i coac wih oe prey muliplied by he umber of differe ways ha x (1) prey ca be i coac wih x (2) predaors. We fid he margial wo-slice saisics ξ (c) (x (c) 1, ) by margializig ξ (x 1, ) over all x (c ) 1, for c c: ξ (c) (x (c) 1, ) = c c x (c ) 1, probabiliy of o sae chage (1 c (x(c) 1 ) g (c ) ξ (x 1, ) = 1 Z (6) c c 1 =x(c ) ) c x (c ) 1 =x(c ) 1 (x(c ) 1 )β(c ) (x (c ) )p(y (c ) x (c ),fix x (c) 1 )+ 717

6 Dog sae rasiio probabiliy due o eve c (x(c) 1 ) g (c ) c 1 c + x(c ) =x (c ) c 1 (x(c ) 1 )β(c ) (x (c ) )p(y (c ) x (c ) x (c ) 1 + x(c ) =x (c ),fix x (c) 1 ) where g (c ) = cosrai cosrai cosrai Z = (1 c g (c ) c c g (c ) c ) 1 g(c β (c ) p(y (c ) x (c ) ) ) 1 β(c p(y (c ) x (c ) ) 1 =x(c ) 1 + x(c ) =x (c ) ) c c, 1 (x(c ) 1 )β(c ) (x (c ) )p(y (c ) x (c ) )+ x (c ) 1 =x(c ) 1 (x(c ) 1 )β(c ) (x (c ) )p(y (c ) x (c ) ). x (c ) 1 + x(c ) =x (c ) Hece we ca mae ifereces abou age saes i a muli-age model by ieraively solvig he margial forward ad bacward saisics (Equaio 7 ad Equaio 8), ad we ca esimae eve rae cosas from he margial saisics (Equaio 9). The probabiliy of observaios uder he approximae disribuio is p = Z. α (c) (x (c) ) 1/Z probabiliy of o sae chage (1 c (x(c) 1 ) g (c ) c c 1 =x(c ) ) ( 1 c x (c ) 1 =x(c ) sae rasiio probabiliy due o eve c (x(c) 1 ) g (c ) c 1 c + x(c ) =x (c ) c β (c) 1 (x(c) 1 ) 1/Z probabiliy of o sae chage (1 c (x(c) 1 ) g (c ) c c 1 =x(c ) ( ) β (c ) x (c ) 1 + x(c ),fix x (c) 1 ( 1 1 c x (c ) 1 =x(c ) sae rasiio probabiliy due o eve c (x(c) 1 ) g (c ) ( c 1 c + x(c ) =x (c ) c c 1 ξ (c) (eve ) C c / where ξ (c) (eve ) = ξ (c) (x (c) x (c) c 1 = x(c) (c) x 1 =x(c) x (c ) 1 + x(c ) β (c ) =x (c ) ) 1c c p(y (c ),fix x (c) 1 x (c ) ) 1c c p(y (c ) )+ x (c ) ) ) 1c c β (c ) p(y (c ) x (c ) )+,fix x (c) 1 1 ) 1c c β (c ) =x (c ) cold (x(c) ) c old : x (c) = x(c) p(y (c ) x (c ),fix x (c) 1 1 ) ) c c g(c 1 ) ) c c g(c (x (c) ),, 1 + x(c ) =x (c ) 1 + x(c ) =x (c ). (7) (8) (9) 718

7 Dog 3.3 Parameer Esimaio Uder Mea-Field Approximaio I order o deermie a esimaio for he sae rasiio model (i.e., c ) ad he observaio model (p(y (c) x (c) )) ha maximizes he expeced log lielihood i he M-sep of he EM algorihm, we se he derivaives of he expeced log lielihood over he parameers o be zero ad cosider cosrais whe ecessary: Elogp c = 1 C ( ξ (c) (eve ) c c c where ξ (c) (eve ) = ξ (c) (x (c) x (c) ad c old 1 = x(c) is he old esimaio of rae cosa. (c) x 1 =x(c) ) se = 0, cold (x(c) ) c old : x (c) = x(c) 1 ) ) c c g(c 1 ) ) c c g(c (x (c) 1 + x(c ) =x (c ) x (c ) 1 + x(c ) =x (c ) I follows ha he ew esimaio of he rae cosa (c ) of rasiio is he oal umber of rasiios i he sample divided by he oal umber of imes he rasiio could happe: c 1 / ξ (c) (eve ). C c c (c) x 1 =x(c) The maximum lielihood esimaio of he parameers of he oupu models (p(y (c) x (c) )) is modeldepede. Here, we provide he soluios for he wo mos commo oupu models. Whe he observaio abou a ode aes a fiie umber of values, we ge p(y (c) x (c) ) = 1 (c) y y (c) γ(c) γ(c) (x (c) ) (x (c) ) Whe he observaio of a ode aes a ormal disribuio aroud he sae of he ode, we have (10) µ (c) (x (c) ) = σ (c) (x (c) ) = / y (c) γ (c) (x (c) ) γ (c) (x (c) ), (11) (y (c) µ (c) ) 2 γ (c) (x (c) ) / γ (c) (x (c) ). (12) 4 EXAMPLES I his secio, we mae sochasic process ifereces usig he predaor-prey sysems dyamics ad he a forage muli-age dyamics. The predaor-prey sysems dyamics are simple eough, such ha we ca ouch o he deails of ideifyig a age-based model as a sochasic process ad mae ifereces hereof, ad of how mea-field approximaio wors. As such, a forage muli-age dyamics icely demosraes how mea-field approximaio eables us o mae ifereces i big sysems. 4.1 Iferrig Predaor-Prey Dyamics The predaor-prey (Loa-Volerra) model ae from (Wiliso 2006) has hree reacios, i which predaio icreases predaor populaio by oe ad decrease prey populaio by oe a he same ime. Prey ad predaor are respecively a prey idividual ad a predaor idividual: 719

8 Dog prey 2 prey, rae = c 1 prey+predaor 2 predaor, rae = c 2 predaor, rae = c 3 Ierpreig his Loa-Volerra sysems dyamics as a discree-ime Marov chai (Secio 3.1), he ime-depede sysem-sae is he predaor ad prey populaios(x (1),x (2) ) {1,...,M 1 } {1,...,M 2 }, ad he sae rasiio erel (Equaio 13) is iduced by 3 reacios, ad a eve rae ca be esimaed as he oal eve occurrece over he umber of imes he eve could have occurred (Equaio 14). I compariso, uder mea-field approximaio (Secio 3.2), we eep rac of oly he margial sae disribuios of he predaor ad prey populaios (Equaio 15 ad Equaio 16), where he probabiliy ha he prey populaio decreases due o predaio is compued accordig o he average predaor populaio x (2) = E (2) p(x ) x(2), ad he probabiliy of predaor icrease due o predaio is compued accordig o he average prey populaio x (1) = E (1) p(x ) x(1). The a eve rae (Secio 3.3) is esimaed as he oal eve occurreces i boh populaios over he oal umber of imes he eve could have occurred i boh populaios (Equaio 17). p(x x +1 ) = (13) c 1 x (1) x +1 x = (1,0) c 2 x (1) x +1 x = ( 1,1) c 3 x (2) x +1 x = (0, 1) 1 c 1 x (1) c 2 x (1) c 3 x (2) x +1 x = (0,0) 0 oherwise p(x (1) x (1) +1 ) = (15) c 1 x (1) x (1) +1 =x(1) +1 c 2 x (1) x (1) +1 =x(1) 1 1 c 1 x (1) c 2 x (1) 0 oherwise x (1) +1 = x(1) p(x (2) x (2) +1 ) = (16) c 3 x (2) +1 =x(2) 1 c 2 x (1) +1 =x(2) +1 1 c 2 x (1) c 3 x (2) 0 oherwise +1 = x(2) c 1 = c 2 = c 3 = (,x ξ x,(x (1) +1,x (2),x x (1) ) ξ(x,x ) ( ),x ξ x,(x (1) 1,x (2) +1),x x (1) (,x ξ ) ξ(x,x ) ) 1). x,(x (1),x (2),x x (2) ξ(x,x ) (14) c 1 = c 2 = c 3 =,x (1),x (1),x (1),x (1) ξ (1) (x (1),x (1) +1) x (1) ξ (1) (x (1),x (1) ) ξ (1) (x (1),x(1) 1)+ ξ (2) (x (2),x (2),x(2) +1) x (1) ξ(1) (x (1),x(1) )x(2) + x (2),x (2) ξ(2) (x (2),x(2) )x(1) (17),x (2),x (2) ξ (2) (x (2),x (2) 1) x (2) ξ (2) (x (2),x (2) ). 720

9 Dog Wih he discree-ime Marov chais of boh he predaor-prey model ad he approximae model, we ca proceed o compare he ifereces ha use mea-field approximaio wih he exac ifereces. Mea-field approximaio is oe way o pay he price of beig able o mae sochasic process ifereces o dyamical sysems ivolvig a iracable umber of ieracig facors or ages. We se c 1 = 1, c 2 =.007, c 3 =.6, x (1) 0 = 50, x (2) 0 = 100, ad sampled a ime series of prey/predaor populaios from ime 0 o ime 40 accordig o a coiuous-ime Marov process (Secio 2). The ime series show periodiciy, radomess, ad predaor populaio followig prey populaio (Figure 1a, 1b). We perurbed he sysem saes i ime iervals [2,2.1) ad (7.7,8] wih a Gaussia radom oise of mea 0 ad a sadard deviaio of 20 (N(0,20 2 )) (resulig i oisy observaios), ad compared he exac iferece populaio rajecories bewee ime 2.1 ad ime 7.7 (Figure 1c) wih he mea-field iferece (Figure 1d). Wihi 0.5 ime ui of observaios (i.e., a imes 2.5 ad 7.5), he iferred predaor ad prey populaios usig exac forward-bac iferece are visibly correlaed. This correlaio resuls from predaio decreasig prey populaio ad icreasig predaor populaio a he same ime. The correlaio bewee populaios becomes less visible as we move away i ime from he observaios. The mea-field approximaio removes he correlaio by decouplig he predaio equaio io wo: decreasig he prey populaio due o he exisece of he predaor populaio, ad icreasig predaor due o he exisece of prey. =2.90 =2.90 predaor populaios prey predaor predaor predaor prey Time prey prey (a) phase space (b) ime series (c) exac iferece (d) mea-field iferece Figure 1: Predaor-prey dyamics (a) i phase space, (b) as ime series, (c) populaio desiy esimaio (c) from exac iferece, ad (d) mea-field iferece. 4.2 Iferrig A Forage Dyamics I his secio, we mae sochasic process ifereces usig a forage muli-age dyamics. The a forage dyamics uder discussio is from The NeLogo model As, defied by NeLogo programmig laguage (Wilesy 1997). The As world is a grid wih a a es a he ceer, hree food sources (each wih differe amous of food ad a differe disaces from he es), ad 125 as movig food from he sources o he es. The logic of idividual as is simple, bu i oeheless adds up o a ieresig collecive ielligece. A a wih food wiggles oward he es ad leaves a specified amou of chemical o grid pois alog is way. This chemical diffuses o he eigh eighborig grids ad he evaporaes. A a wihou food seses he chemical wih moderae desiy i fro of i, ad follows he chemical o a area wih high chemical coceraio. I areas wih oo low or oo high chemical coceraios, he a merely wiggles forward uil i fids food. A simulaio ru of he As model is aimaed i Figure 2a. Age-based modelers simulae he As model wihou egagig ay observaioal daa. Here, we show ha we ca raslae he a forage model i NeLogo io a discree-ime Marov chai, ad ifer 721

10 Dog how as ravel from heir origis a ime 0 o heir desiaios a ime 200, cosiderig heir ieracios hrough chemical ad food. The sysem sae is a Caresia produc of he saes of each a, which ca occupy ay of he squares, movig i ay of eigh direcios, ad be wih food or wihou; of he amou of food o each grid poi of he food sources; ad of he amou of chemical o each grid poi i he world: a sae a(a,x,y, dir, food?) {0,1}, where x { 35,...,35}, y { 35,...,35}, dir { 0 4 π,..., 7 4 π}, food {0,1} represeig a a wihou food ad a a wih food respecively, ad a {1,...,125} idexes io a a. food o grid f(x,y) {0,...,max f }, where x,y { 35,...,35} idexes io a grid poi uder ivesigaio. chemical o grid c(x,y) {0,...,max c }, where x,y { 35,...,35} idexes io a grid poi uder ivesigaio, admax c is he cu-off amou of chemical ha a grid poi ca hold (chose accordig o compuaio ime ad iferece precisio). is his he es? (x,y) {0,1}, where 0 meas ha he grid poi is ouside of he es, ad 1 i he es. The sochasic eves of he sysem iclude a a loadig food a a food source, a a aig food from a food source, a a uloadig food a he es, a a leavig food a he es, chemical diffusio, ad chemical evaporaio: a uloadig food a a wih food drops i a he es ad leaves he es, a(a,x,y, dir,1) (x,y) a(a,x,y, dir+π,0). a loadig food a a wihou food aes food a a food source ad urs bac o he es, a(a,x,y, dir,0) f(x,y) > 0 a(a,x,y, dir+π,1) f(x,y) 1. a seeig fooda(a,x,y, dir,0) f(x,y) 0 a(a,x,y, dir,1), a a wihou food wiggles owards areas wih high chemical coceraio, where (x,y ) = roud(x+ (si,cos)(dir )), ad we approximae his forward-wigglig behavior as defied i NeLogo laguage wih p(dir dir = dir±(si,cos) d) c 1 d 0 +f((x,y)+(cos,si)dir ) γ. a headig o he esa(a,x,y, dir,1) o (x,y) a(a,x,y, dir,1), a a wih food wiggles owards he es. chemical diffusio c(x,y) c(x+ x,y+ y) (1 δ) c(x,y) c(x+ x,y+ y)+δ c(x,y), where x, y {0,±1}. Nex, we decompose he sysem sae io a saes, food saes, ad chemical saes, ad assume mea-field ieracios amog he as, food, ad chemical i order o circumve racig a asroomical umber of saes i he sae space (( ) 125 max f maxc ). The amou of food o each grid poi is a caegorical disribuio aig values from 1,...,max f. The amou of chemical o each grid poi is a caegorical disribuio aig values from 1,...,max c. Each a ca be a ay of he grid pois, movig i ay of he eigh headigs, wih or wihou food. The average amou of food o be ae by a a a a grid poi is he average amou of food a he grid poi imes he probabiliy ha a a wihou food is o his grid poi. The average amou of chemical o be lef by a a a a grid poi is he probabiliy ha he a wih food is o his grid poi. The chemical diffuses ad evaporaes accordig o a compoud Poisso process. Wih he mea-field iferece algorihm, we ca ifer how as i heir iiial posiios a ime 0 could move o heir posiios a ime 200 based o heir ieracios wih boh chemical ad food (Figure 2a), ad o how oher simulaios could move he as from he same origis o he same desiaios wih he same umber of seps. We compare a locaios i a simulaio wih our iferece from he firs frame ad fial frames i Figure 2b, where differe colors i he hea map represe he x 12 perceiles, where x {0,..., 12}. This aimaio demosraes visually ha our approximae iferece has capured he dyamics of he model. To assess he accuracy of he approximae iferece o he As model, we idepedely sampled he ime series of as, chemical, ad food from heir respecive hidde Marov processes uder he mea-field approximaio wih he forward filerig bacward samplig (FFBS) algorihm, ad used he Meropolis-Hasigs algorihm o accep or rejec samples accordig o he exac probabiliy of he A model. We se bur-i period ad hiig ierval heurisically, ad evaluaed covergece by comparig various saisics of idepede Meropolis-Hasigs rus. We foud he probabiliy of a sample pah uder he mea-field approximaio o be a good esimaio of he exac probabiliy iduced by he As world (R 2 = 0.99,p < 10 6 ), ad he margial oe-slice saisics o be a good esimaio of hose compued 722

11 Dog =127 = (a) sample (b) iferece Figure 2: Olie versio shows a simulaio ru of he As model (lef) ad he sochasic iferece of a rajecories from he ed pois (righ). Pried versio shows a saes (lef) ad iferred a locaios (righ) a ime 127. from he Meropolis-Hasigs sample (R 2 = 0.95,p < 10 3 ). Hece, we believe he approximae iferece developed i his paper has good accuracy. The ime/space complexiy of he approximae algorihm scales liearly wih he umber of ages or sysem variables, ad for each age or sysem variable he complexiy scales liearly wih he umber of lae sae (because he sae rasiio marix iduced by producios is sparse). The approximae forward-bacward algorihm could ge oiceable improveme afer 4 ieraios, ad he approximae forward-bacward + parameer esimaio algorihm could ge oiceable improveme afer 20 ieraios for a wide rage of iiial cofiguraios. Hece he approximaio based o greedy local projecio is scalable o much larger daa se. While his example maes a sochasic iferece abou model-geeraed daa assumig he model is correc, here is o resricio i maig sochasic ifereces abou real-world daa wih compeig age-based models. Such a combiaio of model ad daa helps daa scieiss o ell sories abou daa ad o ideify a daa busiess as a opimizaio problem o daa, ad also helps modelers o bechmar compeig models agais real-world daa ad o mae predicios abou real-world daa wih heir models. For example, i could be ieresig o combie radiioal age-based road-raffic models wih peabye vehicle racig daa ad mae predicios abou eves ha have ever previously occurred. Or, oe could combie meme-diffusio models wih real-world mobile daa ad web surfig daa o predic he ex fad, or combie ciy growh models wih demographics o predic ecoomic reds. 5 CONCLUSIONS AND DISCUSSIONS I his paper, we gave a mea-field iferece algorihm o approximae a coiuous-ime coupled Marov process defied by muli-age models. Such iferece algorihm is useful i bridgig ogeher big daa ad he world of geeraive models describable wih rule-based producio sysems. Saisics such as soppig imes abou a idividual chai ca be esimaed from fiig Marov models if we employ srucured 723

12 Dog variaioal approximaio. Such variaioal algorihms decouple he ieracig chais, ad so cao be used o esimae saisics abou he ieracios amog chais; however, i such siuaios esimaios ca be based o Moe Carlo algorihms. REFERENCES Aderso, J. R The archiecure of cogiio. Mahwah, N.J.: Lawrece Erlbaum Associaes. Bay, M Ciies ad Complexiy: Udersadig Ciies Wih Cellular Auomaa, Age-Based Models, ad Fracals. Mi Press. Casellao, C., S. Foruao, ad V. Loreo Saisical physics of social dyamics. Reviews of Moder Physics 81 (2): Dog, W., K. A. Heller, ad A. Pelad Modelig Ifecio wih Muli-age Dyamics. I SBP, edied by S. J. Yag, A. M. Greeberg, ad M. R. Edsley, Volume 7227 of Lecure Noes i Compuer Sciece, : Spriger. Dog, W., B. Lepri, ad A. Pelad Modelig he co-evoluio of behaviors ad social relaioships usig mobile phoe daa. I MUM, edied by Q. Dai, R. Jai, X. Ji, ad M. Kraz, : ACM. Dog, W., ad A. Pelad A Newor Aalysis of Road Traffic wih Vehicle Tracig Daa. Dudley, R. M Real aalysis ad probabiliy, 2d ed. Cambridge Uiversiy Press. Epsei, J. M. M Geeraive Social Sciece: Sudies i Age-Based Compuaioal Modelig (Priceo Sudies i Complexiy). Priceo Uiversiy Press. Goodwi, R. M Socialism, capialism ad ecoomic growh, Chaper A growh cycle. Cambridge Uiversiy Press. Heses, T., ad O. Zoeer Expecaio Propogaio for Approximae Iferece i Dyamic Bayesia Newors. I UAI, edied by A. Darwiche ad N. Friedma, : Morga Kaufma. Ibe, O. C Marov processes for sochasic modelig. Academic Press. MATSim developme eam (ed.) MATSIM-T: Aims, approach ad implemeaio. Techical repor, IVT, ETH Zürich, Zürich. MIT Huma Dyamics Lab hp://realiycommos.mi.edu Realiy Commos. Newell, A., ad H. A. Simo Huma problem solvig. Eglewood Cliffs, N.J.: Preice-Hall. Orage S. A. hp:// Daa for Developme. Popper, K Cojecures ad refuaios: he growh of scieific owledge. Lodo: Rouledge ad Kega Paul. Smih, L., R. Becma, ad K. Baggerly TRANSIMS: Trasporaio aalysis ad simulaio sysem. Techical repor, Los Alamos Naioal Lab., NM (Uied Saes). Waiwrigh, M. J., ad M. I. Jorda Graphical Models, Expoeial Families, ad Variaioal Iferece. Foudaios ad Treds i Machie Learig 1 (1-2): Uri Wilesy NeLogo As model. hp://ccl.orhweser.edu/elogo/models/as. Ceer for Coeced Learig ad Compuer-Based Modelig, Norhweser Uiversiy, Evaso, IL. Wiliso, D. J Sochasic modellig for sysems biology. Boca Rao, FL: Taylor & Fracis. AUTHOR BIOGRAPHIES We Dog is a Assisa Professor of Compuer Sciece ad Egieerig a he Sae Uiversiy of New Yor a Buffalo wih a joi appoime i he Isiue of Susaiable Trasporaio ad Logisics. He focuses o modelig huma ieracio dyamics wih sochasic process heory hrough combiig he power of big daa ad he logic/reasoig power of age-based models, o solve our socieies mos challegig problems such as rasporaio susaiabiliy ad efficiecy. We Dog holds a Ph.D. i Media Ars ad Scieces from Massachuses Isiue of Techology. His address is wedog@buffalo.edu. 724