Gravitation and Spatial Interaction

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1 Unversty College London Lectures on Urban Modellng January 2017 Gravtaton and Satal Interacton Mchael 16 January, 2017 htt:// Lectures on Urban Modellng

2 Outlne Gravtaton: he Basc Models Potental and Accessblty Satal Interacton and r Dstrbuton: Constrants on Volume & Locaton Dervaton Methods: Entroy-Maxmsng Resdental Locaton, Modal Slt he London yndall Model: Alcatons ransortaton Modellng: he Four Stage Process Next me: Modular Modellng: Couled Satal Interacton Lectures on Urban Modellng

3 Gravtaton: he Basc Models Let me begn wth satal nteracton models once agan and frst defne ey terms. We are gong to dvde our satal systems nto small zones le Census racts whch can ether be called orgns or destnatons. Orgns are notated usng the subscrt and destnatons the subscrt. Now the orgnal gravty model can be stated as PP PP ~ K 2 2 d d 2 where we defne, P, P, d, and K as trs, oulatons, dstance squared and a scalng constant Lectures on Urban Modellng

4 In fact we can generalse the model frst by notng that dstance s le n the von hunen model a measure of generalsed travel cost c and the oulatons are defned as measures of mass or actvty as orgn and destnaton actvtes O, D hen ~ O D c K O D c KO D Where s the so-called frcton of dstance arameter controllng the effect of generalsed travel cost. When s large, the effect of dstance s great and when t s small t s much less. hs gves more trase when t s small than when t s bg. c Lectures on Urban Modellng

5 In all our models, we need to estmate these arameters and ths s the rocess of calbraton. We need to choose K and so that the redcted trs are as close as ossble to the observed trs obs We can do ths n ths smlest of models by fttng a lnear regresson to the logarthmc verson of the model and when we tae logs we get log log K logc O D We fnd the arameters by mnmsng the sum of the squares (squared devatons) between the redcted obs 2 and observed trs, that s mn mn ( ) Lectures on Urban Modellng

6 Potental and Accessblty In the 1940, the astronomer John Stewart suggested that a measure of otental could be roduced from the gravty model that was an overall measure of the force of an obect on all others. He defned ths from the basc GM equaton as otental V or otental er cata v P V ~ P 2 d v V ~ P P d 2 hs s essentally accessblty or nearness and t was frst used as the bass for a smle urban model by Lectures on Urban Modellng

7 Walter Hanson n the late 1950s n a aer called How Accessblty Shaes Land Use. here he sad that the resdental develoment n a lace was a smle functon of accessblty.e. V P R ~ 2 P d In fact f total resdental develoment s R, then the equaton can be wrtten as ( V / P ) R R ( V / P ) And ths s our frst oeratonal land use model, the smlest Walter G. Hansen, W. G. (1959) How Accessblty Shaes Land Use, Journal of the Amercan Insttute of Planners, Volume 25, Pages htt://dx.do.org/ / Lectures on Urban Modellng

8 he orgnal gravty model has been used for years but n the 1960s and 1970s varous researchers cast t n a wder framewor dervng the model by settng u a seres of constrants on ts form whch showed how t mght be solved generatng consstent models. he constrants logc led to consstent accountng he generatve logc lead to analoges between utlty and entroy maxmsng and oened a door that has not been much exloted to date between entroy, energy, urban form hyscal morhology and economc structure. In artcular the economc logc s called choce theory, secfcally dscrete choce theory Lectures on Urban Modellng

9 he ey dea s to ntroduce constrants on the form that the model can tae, and these relate to secfyng what the model s able to redct. he more constrants we ntroduce on the model, the more we reduce the model s redctve ower, but the dea of constrants also relates to what we now about the system n comarson wth what we want to redct. he dea of a framewor for consstent generaton of a model s that we can then handle the constrants systematcally as we wll now show. Lectures on Urban Modellng

10 r Dstrbuton: Constrants on Volume & Locaton We must move qute qucly now so let me ntroduce the basc constrants on satal nteracton and then state varous models he constrants are usually secfed as orgn constrants and destnaton constrants as O D And we can tae our basc gravty model and mae t subect to ether or both of these constrants or not at all Lectures on Urban Modellng

11 So what we get are four ossble models Unconstraned KO D c Sngly (Orgn) Constraned so that the volume of trs at the orgns s conserved Sngly (Destnaton) Constraned so that the volume of trs at the destnatons s conserved Doubly Constraned tr volumes at orgns + destnatons are conserved he frst three are locaton models, the last s the transortaton model A B O D AO D c c B O D c Lectures on Urban Modellng

12 Now the smlest way to wor out what the constants mean s to note the constrants equatons and then add and factor the model subect to the constrants. Let us begn wth the smlest gravty model whch s KO D c hen as K ertans to the whole system f we add ths model u over and we can factor our K as K K O D c O D c O D c O D c and then and the model becomes Lectures on Urban Modellng

13 Now the sngly constraned orgn and then destnaton and the doubly constraned models follow drectly and we wll smly state there full forms notng that we need to fnd (1) (2) (3) A B AO D c B O D c 1/ A B O D c 1/ B D c AO c O D D c D c O c O c orgn - constraned destnaton constraned orgn - destnaton constraned Lectures on Urban Modellng

14 Entroy-Maxmsng and Related Measures Now we have only dealt wth constrants through consstent accountng we now need to deal wth generatve methods that lead to the same sort of accountng entroy maxmsng, nformaton mnmsng, utlty maxmsng and random utlty maxmsng, and also varous forms of nonlnear otmsaton n fact all these methods may be seen as a nd of otmsaton of an obectve functon entroy utlty and so on subect to constrants. In essence what we do s defne a functon to otmse whch s some measure of the varaton n the model and we then otmse t usng calculus subect to some basc constrants of the nd that we have been usng Frst we defne entroy as Shannon nformaton and we convert all our equatons and constrants to robabltes. Lectures on Urban Modellng

15 Shannon entroy s a measure of sread or comactness n satal systems H log We maxmse ths entroy subect to orgn and destnaton constrants or some combnaton of these but notng now that we need another constrant on travel cost whch s equvalent to energy so that we can derve a model c Cˆ Lectures on Urban Modellng

16 We thus set u the roblem as max H subect to c Cˆ log But note that the robabltes always add to 1, that s 1 From ths we get the Boltzmann-Gbbs dstrbuton for the robabltes Lectures on Urban Modellng

17 By settng u a Lagrangan whch s the method of maxmsaton, then we get or ex( c } } A O B D ) } ex( c ) Now we can generate any model n the famly of four models unconstraned, sngly-constraned (orgn or destnaton) and doubly constraned by settng the redundant constrant arameters equal to zero and smlfyng the model o derve a resdental locaton model whch s orgn constraned we now the nformaton at the orgn but want to redct the flows to the destnaton and add u these flows to redct actvty at the destnaton, we Lectures on Urban Modellng

18 We thus set u the roblem as max H subect to c Cˆ log or ex( c ) And we get D where A O ex( c ) O ex( c ex( c ) ) Lectures on Urban Modellng

19 Several thngs to note: here s no attractor value at the destnaton we would need to ut ths n as a constrant.e. a ece of nformaton to be ncororated by the model hs s a locaton model we redct actvty at the destnaton n the case of a model that redcts how many eole worng n zone O lve n zone, ths s D where the rme s the notaton for redcted Now let us ut ths model bac nto the entroy equaton and see what we get let us ut the model bac n n ts exonental form ex( c ) Lectures on Urban Modellng

20 hen what we get s H Cˆ log log ex( c ( c Cˆ What we need to note s that entroy s arttoned nto a fxed energy and free energy the fxed s the second term and the free s the frst a seres of weghted log-sums and t s often thought of a nd of accessblty. In ths case t s the sum of accessbltes, one for each orgn zone. It has strong relatons to utlty n the random utlty maxmsng verson of ths nd of model whch s central to dscrete choce theory ) / ) Lectures on Urban Modellng

21 Resdental Locaton, Modal Slt Let me llustrate n two ways how we can buld models usng ths framewor If we say that resdental locaton deends on not only travel cost but also on money avalable for housng we argue as before that he model s sngly constraned we now where eole wor and we want to fnd out where they lve so orgns are worlaces and destnatons are housng areas he model then lets us redct eole n housng We argue that eole wll trade-off money for housng aganst transort cost And we then set u the model as follows Lectures on Urban Modellng

22 hs tme usng not the robablty form but the tr actvtyvolume form, we get leads to O c A O R C R ex( R )ex( c Note that we now add a constrant on money avalable for housng (le rent) R. We can of course fnd out from ths locaton model how many eole lve n destnaton housng zones, so agan t s a dstrbuton as well as a locaton model P ) Lectures on Urban Modellng

23 We can extend ths model n lots of ways and we wll show some of these later. We also can thn about dsaggregatng the model nto dfferent transort modes let us call each mode and then set u the model so that we can redct as follows he model s sngly (orgn) constraned because we want to redcts how many eole travel from wor to home. Gven we now how many eole wor at orgns, and we want to redct what mode of transort they travel on. hen c O F C F Lectures on Urban Modellng

24 And the model can be secfed as O F ex( c F ) ex( c ) O F F ex( c ) ex( c ) Note that the mode slt s a rato of the comettve effects of each travel cost, that s ex( c ex( c ) ) In short the model s not only dstrbutng trs so that locatons comete but also that modes comete BU modes do not comete er se wth locatons Now let us see how we can buld ths model for real Lectures on Urban Modellng

25 he London yndall Model: Alcatons I already ntroduced the model last tme as an examle of an ntegrated model wth several comonents the one we wll dwell on here s the resdental locaton model whch s essentally a verson of the sngly constraned model that we have been outlnng. Here s the bloc dagram of the model stages to remnd you of what t s MoSeS Lectures on Urban Modellng

26 Essentally we have bult ths model for Greater London whch s dvded nto 633 zones the area has 7.7m oulaton and about 4.3m obs we have four modes road (car), heavy ral, lght ral and tube, and bus wal/be s a resdual mode. o fx deas let me show the extent of the area frst Lectures on Urban Modellng

27 Slash Screen for the desto verson Lectures on Urban Modellng

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29 Modes Road Bus Heavy Ral Lght Ral All rs Road: 38%; Bus: 12%: Heavy Ral: 12%: Lght Ral 19%; Other (Wal, Be, Fly): 19% Lectures on Urban Modellng

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31 Accessblty from the LUM model Many dfferent accessblty measures, 8 n all Lectures on Urban Modellng

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34 Let us run the model I need to go to my folder >> Lectures on Urban Modellng

35 Run Lectures on Urban Modellng

36 ransortaton Modellng: he Four Stage Process I should mae a bref ont about transort modellng we have ncluded transort and locaton together here but tradtonally the transort model s based on a four stage rocess that nvolves generaton, dstrbuton, modal slt and assgnment he other ssue s that n the standard transort modellng rocess, once trs are assgned to the networ, then one can assess whether the networ can tae the load ths s matchng travel demand aganst suly and f not then the model s terated to match demand to suly. hs s another generc ssue n urban modellng demand and suly and the way the maret resolves ths. Lectures on Urban Modellng

37 Juan de Dos Ortúzar, Lus G. Wllumsen 2011 Modellng ransort, 4th Edton, Wley, New Yor Lectures on Urban Modellng

38 Next me: Modular Modellng: Couled Satal Interacton Now we have a module for one nd of nteracton consder strngng these together as more than one nd of satal nteracton Classcally we mght model flows from home to wor and home to sho but there are many more and n ths sense, we can use these as buldng blocs for wder models. hs s for next tme too What we wll do s llustrate how we mght buld such a structure tang a ourney to wor model from Emloyment to Poulaton and then to Shong Lectures on Urban Modellng

39 Unversty College London Lectures on Urban Modellng January 2017 hans More Next hursday, same tme Mchael 13 January, 2017 htt:// Lectures on Urban Modellng