Lecture 4: Land Use Transportation Models:

Transcription

1 SCHOOL OF GEOGRAPHY Lecture 4: Land Use Transportaton Models: Gravtaton and Spatal Interacton, Dervaton Methods Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

2 Outlne Gravtaton: The Basc Models Trp Dstrbuton: Constrants on Volume & Locaton Dervaton Methods: Entropy Maxmsng Resdental Locaton, Modal Splt Transportaton Modellng: The Four Stage Process Modular Modellng: Coupled Spatal Interacton A Smple Example of Modularty: Lowry s Model DRAM EMPAL Style Models Demand and Supply: Market Clearng Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

3 Gravtaton: The Basc Models Ok, we can ncorporate these deas n the basc model of forces whch was frst artculated by Newton as hs second law of moton force s proportonal to mass tmes acceleraton In more conventonal terms we mght wrte the force between two bodes as F G 12 M d M 2 Or more generally F G There s a very long hstory of analoges between force and socal nteracton gong back to Newton hmself. There are many references and I wll add some to the web page M M d Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

4 But let me mmedately generalse ths and say that we need to defne many nteractons we break our system n to areas or ponts whch we defne as orgns and destnatons and And then we measure the dstance as n von Thunen not as dstance per se but as travel cost or rather generalsed cost c We also defne the mass at the orgns and destnatons as O and D and we then wrte the conventonal spatal nteracton of gravty model as T ~ P P c K P P Where K s the gravtatonal constant c Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

5 Ths s the model that has been used for years but n the 1960s and 1970s varous researchers cast t n a wder framework dervng the model by settng up a seres of constrants on ts form whch showed how t mght be solved and produced varous generatng mechansms that could generate consstent models The constrants logc led to consstent accountng The generatve logc lead to analoges between utlty and entropy maxmsng and opened a door that has not been much exploted to date between entropy, energy, urban forms physcal morphology and economc structure In partcular the economc logc rather than the energy entropy logc was called choce theory, specfcally dscrete choce Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

6 Now to ntroduce all ths, we need to defne some more terms We wll refer to the sze of volume of orgns and destnatons not as populaton P but as O and D assumng they are dfferent from one another. We wll also assume that the nverse square law on dstance or travel cost does not apply and that whenever c appears t wll be parametersed wth a value that vares whch we call We wll assume trps are as we have defned them as T but we wll also normalse trps by ther total volume T to produce probabltes T T p T T Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London Note that we use summaton extensvely n what follows

7 Trp Dstrbuton: Constrants on Volume & Locaton We must move qute quckly now so let me ntroduce the basc constrants on spatal nteracton and then state varous models The constrants are usually specfed as orgn constrants and destnaton constrants as O T D T And we can take our basc gravty model and make t subect to ether or both of these constrants or not at all Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

8 So what we get are four possble models Unconstraned T KO D c Sngly (Orgn) Constraned so that the volume of trps at the orgns s conserved T Sngly (Destnaton) Constraned so that the volume of trps at the destnatons s conserved AO D T c B O D c Doubly Constraned trp volumes at orgns + destnatons are conserved T A BO Dc The frst three are locaton models, the last s a traffc model Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

9 Ok, so what are these parameters that enable the constrants to me met well they can be very easly produced by summng each model over the relevant subscrpts e orgns or destnatons and then smply substtutng and rearrangng I wll do ths but I wll leave you to work through the algebra n your own tme and many of you wll know ths anyway. Here are the factors whch are sometmes called balancng factors K T Unconstraned O D c Orgn Constraned Destnaton Constraned Doubly Constraned A B A 1 1 D O c c 1 B D c B 1 A O c Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

10 Dervaton Methods: Entropy Maxmsng 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 entropy maxmsng, nformaton mnmsng, utlty maxmsng and random utlty maxmsng, and also varous forms of nonlnear optmsaton n fact all these methods may be seen as a knd of optmsaton of an obectve functon entropy utlty and so on subect to constrants We wll defne entropy maxmsng. Frst we defne entropy as Shannon nformaton and we convert all our equatons and constrants to probabltes. Shannon entropy s H p log p Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

11 We maxmse ths entropy subect to the prevous constrants dependent on what knd of model we seek but notng now that we need another constrant on travel cost whch s equvalent to energy so that we can derve a model We thus set up the problem as max p p H subect p to p p c Cˆ p log p But note that the probabltes always add to 1, that s p p p 1 Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

12 I am not gong to work ths through by settng up a Lagrangan and dfferentatng t and then gettng the result. There s a lot of basc algebra nvolved and all I want to show s the result For ths optmsaton the model that we get can be wrtten as p or T exp( c Tp A O B D ) exp( c ) Let us note many thngs 1. Ths s the doubly constraned model but wth an exponental of travel cost replacng the nverse power 2. We can get any of the other constraned models n the famly by droppng constrants and we can do ths drectly Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

13 3. We can begn to explore what entropy means by substtutng the probablty model nto the entropy equaton I wll reserve ths for an ad hoc semnar on entropy f you are nterested. 4. We can thnk of ths method as one n whch the most lkely model s generated gven the nformaton whch s n the constrants 5. In terms of statstcal physcs, ths model s essentally the Boltzmann Gbbs dstrbuton 6. Entropy can be seen as utlty under certan crcumstances 7. We solve the model e fnd ts parameters by solvng the entropy program whch s equvalent to solvng the maxmum lkelhood equatons 8. We can then use ths scheme to develop many dfferent knds of model where we add more and more constrants and also dsaggregate the equatons to deal wth groups Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

14 Resdental Locaton, Modal Splt Let me llustrate n two ways how we can buld models usng ths framework Frst f we say that resdental locaton depends on not only travel cost but also on money avalable for housng we can argue that 1. The model s sngly constraned we know where people work and we want to fnd out where they lve so orgns are workplaces and destnatons are housng areas 2. The model then lets us predct people n housng 3. We argue that people wll trade off money for housng aganst transport cost And we then set up the model as follows Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

15 It s T T leads T T to O c A O R C R exp( R )exp( c ) We can of course fnd out from ths locaton model how many people lve n destnaton housng zones, so t s a dstrbuton as well as a locaton model P T Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

16 In terms of modal splt we break the trps nto dfferent modes and then let the modes compete wth locatons for travellers In ths way we produce a combned modal splt locaton model. Sometmes we may want the modes to be constraned and n generatng specfc constrants on total travellers by mode, ths s equvalent to addng parameters that dstort the travel costs n fact the generc equaton can be seen as one where the travel cost or energy s modfed by the volume constrants That s p or T exp( c Tp A O B D ) exp( c ) Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

17 Transportaton Modellng: The Four Stage Process I should conclude wth sayng that the transport model s part of a four stage process that nvolves generaton, dstrbuton, modal splt and assgnment and that the spatal nteracton approach can be seen as ether applyng solely to dstrbuton and modal splt or n more ntegrated ways. To get some sense of ths process look at the book Modellng Transport, by J. Ortúzar & L. Wllumsen, Wley, 3 rd Ed, 2001 So far all our models have been demand model but the transport system s capactated and n transport modellng we need to assessng trps to the network and then fgure out f t s possble to meet theses assgnments n terms of supply f not we terate to clear the transport market The same s true of the resdental market n terms of supply and we wll develop all these deas n the next talk Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

18 Modular Modellng: Coupled Spatal Interacton Now we have a module for one knd of nteracton consder strngng these together as more than one knd of spatal nteracton Classcally we mght model flows from home to work and home to shop but there are many more and n ths sense, we can use these as buldng blocks for wder models. Ths s for next tme too What we wll now do s llustrate how we mght buld such a structure takng a ourney to work model from Employment to Populaton and then to Shoppng whch we structure as Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

19 Frst we have the ourney from work to home model as T P E F exp( c ), F exp( c ) T And then the demand from home to shop Wm exp( c m ) S m P, W exp( c ) S m S m m m m And there s a potental lnk back to employment from the retal sector E f ( S m ) m T m S m E P Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

20 A Smple Example of Modularty: Lowry s s Model Lowry s (1964) model of Pttsburgh was a model of ths nature but t also ncorporated n t or rather ts dervatves dd more formally a generatve sequence of startng wth only a porton of employment basc and then generatng the non basc that came from ths. Ths non basc set up demand for more non basc and so on untl all the non basc employment was generated, and ths sequence followed the classc multpler effect that s central to nput output models. Block dagrams of these types of model follow: they are hard to read download the book and prnt t out Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

21 From Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

22 DRAM EMPAL Style Models Essentally what we have here s the noton of smultaneous dependence.e. one actvty generates another but that other actvty generates the frst one what came frst the chcken or the egg? Stephen Putman developed an ntegrated model to predct resdental locaton DRAM and another to predct employment locaton EMPAL. In essence dfferent models are used to do each the employment model tends to be based on very dfferent factors t s a regresson lke model of key locaton factors not a flow model Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

23 Demand and Supply: Market Clearng So far most of these models have been artculated from the demand sde they are models of travel demand and locatonal demand they say nothng about supply although we dd ntroduce the noton that n smulatng trps and assgnng these to the network, we need to nvoke supply. When demand and supply are n balance, then the usual sgnal of ths s the prce that s charged. In one sense the DRAM EMPAL model confgures resdental locaton as demand and employment locaton as supply but most models tend to treat supply as beng relatvely fxed, gven, non modellable Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

24 However several models that couple more than one actvty together treat supply as beng balanced wth demand, often startng wth demand, seeng f demand s met, f not changng the bass of demand and so on untl equlbrum s ascertaned. Sometmes prces determne the sgnal of ths balance. If demand s too hgh, prce rses and demand falls untl supply s met and vce versa Most urban models do not attempt to model supply for supply sde modellng s much harder and less subect to generalsable behavour A strategy for ensurng balance s as follows for a model wth two sectors lke the one we llustrated earler Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

25 Predct work to home trps Assgn to network and check capacty Adust travel costs Predct populaton at home Check capacty Adust prces resd attractors Predct home to shop trps Assgn to network and check capacty Adust travel costs Predct retal actvty at shoppng centres Check capacty Adust prces retal attractors Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

26 The decson to nest what loop nsde what other loop s a bg ssue that makes these models non unque If the supply sde s modelled separately then the way ths s ncorporated further complcates the sequence of model operatons. In large scale ntegrated models, that we wll deal wth next tme these are crucal ssues In fact we don t have tme but there s one further structural ssue we wll deal wth when we meet next tme and ths nvolves Input Output, n partcular the Echenque Models Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

27 There s some good readng of all ths materal n Google Books n Button, K. J., Haynes, K. E., Stopher, P., and Hensher, D. A. (Edtors) (2004) Handbook of Transport Geography and Spatal Systems, Volume 5 (Handbooks n Transport), Elsever Scence, New York rt+geography+and+spatal+systems&source=bl&ots=qvggla6_ka&sg=bvokq_k5befh10nsq GcCSuSNE&hl=en&e=8CraSy1AZDosQORtrSHAQ&sa=X&o=book_result&ct=result&resnum =3&ved=0CBQQ6AEwAg#v=onepage&q&f=false Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London

28 Questons? Centre Centre for Advanced for Advanced Spatal Spatal Analyss, Unversty College London