Open Access The Proof of Gravity Model with Negative Exponential Land-mixed Entropy and Similar to the Method of Hyman Calibration Technology

Transcription

1 Send Orders for Reprnts to The Open Cvl Engneerng Journal, 2015, 9, Open ccess The roof of Gravty Model wth Negatve Exponental Land-mxed Entropy and Smlar to the Method of Hyman Calbraton Technology Luo Xao-Qang * rchtecture School, Chang'an Unversty, Shaanx, X'an, , Chna bstract: Mxed Land use s a common concept used n urban plannng and urban transportaton. When changng the layout of the land n urban area, essental mantenance of trp demand between local and nonlocal balance n transportaton plannng becomes a problem. On the bass of the degree of land mxng shown wth entropy functon, ths artcle makes use of maxmum entropy prncple whch deduces the double restrant gravty model wth the parameters of mxed land-use entropy. It also dscusses the calbraton of the model and problems n practcal applcaton. The research work carred out n ths artcle ndcates that trp dstrbuton of gravty model descrbed belongs to a specal knd of smple gant system, whle the mathematcal structure of parameters of land-mxed entropy model carres a negatve exponental. Ths result has certan theory value n promoton of the coordnated plannng analyss of land use and transportaton. Keywords: Gravty model, land-mxed entropy, maxmum entropy prncple, the coordnated plannng of land use and transportaton. 1. INTRODUCTION Urban transportaton demand s drectly affected by the degree of land use mxng. If the parameters are modeled by the maxmum entropy prncple whch descrbes the degree of land use mxng, and are drectly ntroduced to the gravty model, ths may play an mportant role n optmzng the land use and card traffc demand, and can also be drectly used for "hybrd land development" plannng scheme and udgment. However, at present, the study of such problems s rare. Maxmum entropy prncple establshed the theoretcal foundaton of trp dstrbuton gravty model, and s one of the mportant methods accordng to the practcal problems derved from the smulaton tme and space dstrbuton of traffc demand. Regardng the ssue, a lot of research results are present at home and abroad. Dfferent travel dstance dstrbuton studes of urban land use have formed the characterstcs usng the method of computer smulaton by gravty model, and by puttng forward the second order Erlang dstrbuton for smulatng the trp dstance dstrbuton of the cty [1, 2]. Yao Ronghuan and Wang Danha proposed a probablty torque estmaton model by usng prncple of maxmum entropy under the constrant condtons of mathematcal statstcs of the average moments, whch can be used to estmate trp dstrbuton [3, 4]. Theodore Tsekers and ntony Stathopoulos further proposed dynamc gravty model and conducted an analyss on the dfference of the sngle and double dynamc gravty model [5, 6]. Deng Mngun and Wang Te-zhong presented an mproved gravty model wth actualty OD, the nature of the land, and trp cost lnear weghtage, also wth a hgher estmaton precson than general gravty model [7]. Moreover, Xang L et al. *ddress correspondence to ths author at the rchtecture School, Chang'an Unversty, Shaanx, X'an, , Chna; Tel: ; E-mal: Luoxq37@yeah.net proposed the trp dstrbuton models wth fuzzy cost constrants and random trp generaton usng maxmum entropy prncple and presented hybrd ntellgent algorthm [8]. Ma Jng et al. brought land-mxed entropy nto the gravty model, proposng a generalzed form of unform gravty model wth nsde and outsde trp dstrbutons [9]. The above documents carry mportant reference value for the research conducted n ths paper. Ths artcle proves the double restrant gravtatonal model wth negatve exponental land-mxed entropy by usng the maxmum entropy prncple through the cardng of the concept of entropy, and gves the correspondng parameter calbraton algorthm and the applcaton example. 2. THE MOST ROBBLE DISTRIBUTION ND THE MXIMUM ENTROY RINCILE 2.1. The Most robable Dstrbuton It s unversally known that a con possesses two sdes, and f a con s contnuously tossed, the probablty for both postve and negatve becomes 0.5. Therefore, f 4 cons are thrown at a tme, followng 16 knds of results can be obtaned (Fg. (1)). Fg. (1). Four cons con experment. In the results mentoned above, f the order between the con s not consdered, and fve dfferent states are obtaned: / Bentham Open

2 1036 The Open Cvl Engneerng Journal, 2015, Volume 9 Luo Xao-Qang ll postves and ll negatves wth probablty 1/16; 1 postve and 3 negatves or 3 postves and 1 negatve: 4/16; 2 postves and 2 negatves wth probablty 6/16. It has been notced that f each con wthout nterferng wth each other, and both the probabltes of postve and negatve are equal, Every tme the four peces of flp a con for the probablty of a certan state assocated wth the state of the number of how many. So n fact, a mcroscopc (ndvdual) dstrbuton and a relatonshp between the macroscopc rsks can be establshed. In ths case, f there are 4 COINS at a tme, the most lkely state wth the largest probablty wll be of 2 postves and 2 negatves. If the number of mcro ndvduals consttutng the system gradually ncreases, ths "possble" gradually becomes the "nevtable" The Maxmum Entropy rncple To llustrate ths problem, the probablty of the state largely known as "the most probable state", wth the rest of the state beng "the general state", ntroduces a parameter ε, used to represent the rato of the most probable state probablty and general state average probablty. Let the number of COINS n the example above be as llustrated n Table 1. Table 1. Number of cons N-Cons experment's relatonshp wth the most probable state. The total number of states robablty of the most probable state n= n= n= ! (1) For a fxed n, Sn s contnuous functon of 1, 2,..., ; (2) If =1/n! =!, the correspondng S(1/n,1/n,...,1/n)! S!,!,,! s the monotone ncreasng functon of n;!!! (3) If a test s decomposed nto several successve tests, then the S weghted sum s equal to the orgnal value S; Ths moment, functon S( 1, 2,..., ) s decded as entropy, and has a unque expresson: S =!k" log x p (1) fter understandng the defnton of general concept, t can be dscussed n the system of scence Smple Gant System State Descrptons In order to more ntutvely show ths problem, the startng must be from the smplest case. Suppose there are two elements of system, the probablty of each appear respectvely as 1 and 2, havng =1. Let 1 change from 0 to 1 wth fxed step, and note that at ths tme 2 =1-1. So there are: S = 1 ln ln 2 (2) S = 1 ln 1 + (1! 1 )ln(1! 1 ) (3) When 1 =0, ths system s consttuted by 2, the system entropy s zero, and there s no uncertanty. So as 1 =1. When 1 = 2 =0.5, the possblty of two states 1 and 2 are same as 0.5. By ths tme, maxmum entropy s largest also beng the bggest uncertanty. Its functon mage s shown n Fg. (2). n= n= n= It can be seen from the above example that when there s a gradual ncrease n the quantty of the throwng COINS, the most probable state probablty gradually occupes the absolute advantage. Otherwse n a con-operated experment, t s nevtable. Ths phenomenon can be qualtatvely descrbed as: for system consstng of a large number of random elements, the actual macroscopc state contans the most mcroscopc ones. Ths prncple s often referred to as the maxmum entropy prncple. Consderng that the gas snce s composed of a large number of partcles (molecules), t must also be n complance wth the prncple of maxmum entropy. 3. ROBBILITY MESURE ENTROY ND SIM- LE GINT SYSTEM STTE DESCRITIONS 3.1. robablty Measure Entropy ssumng that a sample space X has n events of X, the probablty measure of each event s (=1,2,...,n), and there s a basc relatonshp Σ =1, 0. If there s a functon S( 1, 2,..., ), the followng condtons wll be satsfed: Fg. (2). The mages of two varables entropy functon. From the above example, t can be seen that the entropy functon s very specal. If the order of above events s not consdered as 1 and 2, the entropy functon can n fact use a sngle numercal unque dfference between system elements consttutng a state. In other words, there s one-to-one correspondence between the system entropy and ts dstrbuton structure of the elements. The entropy functon can be brefly used to descrbe a very complex smple gant system. ccordng to the defnton of system scence, "smple" of "smple gant system" refers to nteracton beng smple; "Gant" refers to a huge number of the elements of a system so each element cannot be analyzed, otherwse t makes lttle sense to the analyss of control system of macroscopc state. In lfe, there are many systems whch can be classfed as smple gant systems, such as trp dstrbuton system.

3 The roof of Gravty Model Wth Negatve Exponental Land-Mxed Entropy The Open Cvl Engneerng Journal, 2015, Volume Trp Dstrbuton as Smple Gant System Trp dstrbuton n traffc plannng s an mportant part of traffc demand analyss. The problem whch requres the man soluton s where "comes from" and where goes, whch s n a drect relatonshp n the urban spatal structure, land layout state and road traffc network (Fg. 3). Fg. (3). lannng year travel desre lne dagram n a cty. The orgnal gravty model was proposed by planner Casey (Casey, 1955), to analyze an area shoppng trp actvty between dfferent towns, havng a basc form as follows: t = kg d 2 (4) Where, t!!" s trp volume from!to ; G!; G s trp generaton volume; s trp attract volume; d s the dstance between! and ;! s the parameter. Careful comparson reveals that the model and the physcal applcaton of Newton's unversal gravtaton formula, actually s the mtaton of the formula for gravty. Gravty model later after a seres of developments has become one of the categores of the floorboard of the model. In urban transportaton plannng at present, the most common trp dstrbuton model s the double restrant gravtatonal model of mpedance functon of negatve exponent, havng the basc forms as: t = k k ' G e!!c (5) # ' k =! k e "!c ( "1 (6) $ ' k ' = # '! k G e "!c ( "1 (7) $ ' where, G s trp generaton volume;!! s trp attract volume;!!!!!" e!!c s the mpedance of the negatve exponental functon, C!!" s the mpedance, usually showed by trp dstance or tme cost,! s the parameter of mpedance functon on behalf of the end senstvty of the choce of travel tme. k, k are the teratve parameters respectvely and the ntermedate varables. In fact, the gravty model expresses only two thngs for practcal sgnfcance: n the same stuaton, the nearer the startng pont s selected, the farther s the destnaton; the opportunty of beng large trp volume s larger between the ponts of departure generaton volume and destnaton abstract volume, both of whch are large. ccordng to the prevous dscusson, t can be seen that the trp dstrbuton defned by the gravty model accords wth the defnton of smple gant system. 4. THE ROOF OF LND-MİXED ENTROY ND GRVİTY MODEL OF ENTROY 4.1. Defnton of Land-Mxed Entropy and Unon of Land-Mxed Entropy Land-Mxed Entropy Land-mxed entropy: usng entropy model, measures the degree of varous types of land use mxed wthn a certan area (Table 2). cty partton of land-mxed entropy s: n H =!K 1 " p # ln p,k,k k=1 ( ) (8) Where, H s the cty partton of land-mxed entropy, characterzaton of travel (attract) from the possblty of sze falls n ths area; k 1 s parameter related to the unts of measurement used n the varable; p,k!!,! s rato of land use type! n partton!,. Calculaton s: p,k =,k!!,k (9) "!!,m,m m Table 2. lannng year peak tme trp amount n a cty (Unt: person/h). SHN-G M YING N LONG CHENG CNG FU TN DI N JIN WEI SHNG M YING N LONG CHEN CNG FU TN DI M JIN WEI Note: s the area of land. s rato of land use type.

4 1038 The Open Cvl Engneerng Journal, 2015, Volume 9 Luo Xao-Qang,m!!,! s the area of land usng type k n traffc zone!, km 2 ; p,m!!,! s the average plot rato of land usng type! n traffc zone!, by takng the average lmt accordng to the present stuaton nvestgaton or plan. Wth ths ndex, the dstngushed combnatons of dfferent land use can be made. For example, two peces of plannng land (Fgs. 4 and 5). It s dffcult to dstngush the mxed degree hgh and low from the ntutve sense of block 1 and block 2, But from land-mxed entropy, ths can be obtaned:h1= , H2= Obvously, the degree of land mxed n block 2 s far larger than block 1.,m s the area of land use type k n traffc zone and, km 2 ; ρ,m s the average plot rato of land use type k n traffc zone and, the average lmt s taken accordng to the present stuaton nvestgaton or plan The roof of Land-Mxed Entropy Gravty Model ssumng that n a trp dstrbuton totally havng Q trps, every trp can choose OD random. So, there are n n choces for one trp. If the trp volume s q from to, t s not known how much q are present, but t can formally provde the travel choce on above average, the possblty from to s p = q /Q. H x Then, by consderng the whole trp dstrbuton entropy: ( ) =! n ) " q Q ln # q, + (. Q * + $ ' -. (12) =! 1 n Q " q ln q ( ) (13) Fg. (4). Block 1 the layout of land use (H1= ). ccordng to the maxmum entropy prncple dscussed earler, t can be known: By each of the trps randomly selected n Q, eventually on macro performance, surely one of the largest entropy s: max H x x n ( ) =! 1 Q ( ) (14) " q ln q To express n an easer way, constant 1/Q s avoded. The row and column of trps should be the same, there s:! q = (15) Fg. (5). Block 2 the layout of land use (H2= ). So far, there had been a tool whch could dstngush the mxed degree between dfferent land uses n dfferent parttons. But there are problems n usng the tool for analyss of traffc dstrbuton. Snce generaton pont and abstract pont may be the same pont n urban transportaton dvson system, called as trps n the nner zone or may be dfferent ponts known as Interval trp Unon Of Land-Mxed Entropy In order to descrbe the land mxed degree of start pont and end pont at the same tme, an ndcator unon of landmxed entropy shall be bult. For a cty traffc zone and, ndex of unon of land-mxed entropy s: n ( ) (10) H =!K 1 " p # ln p,k,k k=1 Where, H s the unon of land-mxed entropy of traffc zone and, showng the sze of the possblty of travel (attract) from fall n two areas; K 1 s parameter related to the unts of measurement used n the varable; p,k s the rato of land use type k n traffc zone and. Calculaton s: p,k =,k!!,k (11) "!!,m,m m! q = (16) Consderng that the mpedance of start and end ponts has a certan nfluence on the probablty of selecton pont actually. To assume the mpedance s c between and, so the total trp tme cost should be conserved:! q " c = Q " c = C (17) t the same tme, consderng that the characterstcs of the entropy functon are drab and addtve, so the overall land use entropy should be conserved. ssumng that the unon of land-mxed entropy of traffc zone and s h, t smlarly has:! q " h = Q " h = H (18) Formula (14) as target functon, formulae (15)-(17) as constrants, q c h beng non-negatve havng must practcal sgnfcance, and all these establshng mathematcal programmng as follows: max H x x ( ) =! n " q ln q ( ) (19)

5 The roof of Gravty Model Wth Negatve Exponental Land-Mxed Entropy The Open Cvl Engneerng Journal, 2015, Volume s.t. $ '! q = ( 20)! q = ( 21) ( )! q " c = C 22! q " h = H 23 ( ) ( ) q # 0,c # 0,h # 0 24 The result of ths plan s the trp dstrbuton whch s requred. s formulae (20) and (21) have n ndvduals (n s the number of rows or columns for trp matrx), n order to solute ths plan, the Lagrange functon can be constructed as follows: n L =!" q ln q ( ) + "! #!" q ( $ ' + "! #!" q $ ' ( +" # C! " q ) c ( $ ' # +! H!" q ) h ( $ ' α,β, θ, γ are Lagrange coeffcent. Let L dervatve wth q, and make t to 0, there s: L! = "ln q + lnq "1"! "! "!c "! h = 0 Soluton s: q = Q e e!! e!" e!#c e!$ h Change ths formula. Because: (25) (26) (27) k = Q e e!!,k ' = e!! (28) nd get: q = k k ' e!!c e!" h (29) In order to dstngush from the general double restrant gravtatonal model, formula (29) land-mxed entropy gravty model can be used lgorthm and Example In order to compare the dfference between the landmxed entropy gravty model and general double restrant gravtatonal model, formula (29) needs to be rewrtten: q = k k ' e! (!c +! h ) (30) In comparson wth the former formula (5) shown n general double restrant gravtatonal model, t can be seen that relatve to the general gravty model, the land-mxed entropy gravty model adds the descrpton of unon of land-mxed entropy, as h. It provdes drect tool to descrbe nfluence of the degree of land-mxed on trp dstrbuton arameter Calbraton Step1: Snce the number of teraton m=0, ntal value gven s:! 0 = 1 c! 0 = 1 h Step2: Snce m=m+1, land-mxed entropy gravty model s used for dstrbuton, for solvng ts average mpedance c m and average land-mxed entropy h m. Step3: Revson of estmates. If m=1, then:! 1 = c 1 c! 0! 1 = h 1 h! 0 If m>1, then: (! m = c! c m!1)! m!1! ( c! c m )! m c m! c m!1 (! m = h! h m!1)! m!1! ( h! h m )! m h m! h m!1 Step4: Gven the margn of errorε 1, ε 2. If c! c m c Then h! h m h "! 1 "! 2 s the termnatng, and output θ m, γ m. Else the process s swtched on to Step Calculaton Example ssumng that there are three traffc zones of dstrbuton system, ncludng the traffc zones n between as shown n Table 3, the trp mpedance and the n-between traffc zones are shown n Tables 4 and 5. Table 3. Observaton trp matrx (unt: m/h) Note: s the area of land. s rato of land use type.

6 1040 The Open Cvl Engneerng Journal, 2015, Volume 9 Luo Xao-Qang Table 4. Observaton trp mpedance (unt: h/m) Note: s the area of land. s rato of land use type. Table 5. Observaton unon of land-mxed entropy Note: s the area of land. s rato of land use type. ccordng to the algorthm to calculate θ=0.36, γ=0.34. t ths tme, estmate trp dstrbuton s: Table 6. Trp dstrbuton smulaton Note: s the area of land. s rato of land use type. t ths tme the average estmated error s (Table 6). CONCLUSION Ths paper dscussed the algorthm and dea of the concept of entropy used n the descrpton of degree of the landmxed, by puttng forward the dea of usng land-mxed entropy n trp dstrbuton, and through the maxmum entropy prncple that deduces the double restrant gravtatonal model. Then, t dscussed the model parameters calbraton algorthm. Through the research conducted n ths artcle, the followng results are obtaned: (1) lthough the selecton process of trp dstrbuton start and end ponts are nfluenced by many factors, but as the trp dstrbuton of the gravty model s descrbed, t s clearly a specal knd of smple gant system. (2) Land-mxed entropy ndex presents negatve exponental functon structure n the gravty model. (3) There s stll a large estmaton error n land-mxed entropy gravty model. But even so, due to the drect ntroducton of ndex of land use, t stll holds certan theoretcal value to promote the overall analyss of land use and transportaton plannng. Overall, although ths artcle dscussed the mechansm of land-mxed entropy gravty model and the parameter calbraton from two aspects of theory and practce, but based on the expermental results, the estmaton precson of the model s stll nadequate. Further research work should be carred out focusng on the mprovement of the precson of the model. CONFLICT OF INTEREST The author confrms that ths artcle content has no conflct of nterest. CKNOWLEDGEMENTS Ths work s supported by the Foundaton tem: 2014 scence and technology proect plan of housng and urban-rural development (Soft scence research proect 2014-R2-026); Shaanx provnce socal scence fund n 2014 (2014D39); The central college fund (2013G ). REFERENCES [1] S. Chen, and W. Du, Study on cty land form and traffc development model n our bg ctes, Journal of Systems Engneerng, vol. 03, pp , [2] S. Chen, Research on Urban assenger Transport System Structure and Development Strategy n Our Large Ctes, Southwest Jaotong Unversty, [3] R. Yao, Research on the Resdent Trp Dstrbuton Model Based on Maxmum Informaton Entropy, Jln Unversty, [4] R. Yao, and D. Wang, Trp dstrbuton model of nformaton entropy, Journal of Transportaton Systems Engneerng and Informaton Technology, vol. 03, pp , [5] R. Yao, and D. Wang, Inhabtant trp dstrbuton entropy model and parameter calbraton, Journal of Traffc and Transportaton Engneerng, vol. 04, pp , [6] T. Tsekers, and S. ntony, Gravty models for dynamc transport plannng: development and mplementaton n urban networks, Journal of Transport Geography, vol. 14, no. 2, pp , [7] M. Deng, and T. Wang, Research on mproved model of Resdents trp dstrbuton predcton. Journal of Transport Informaton and Safety, vol. 03, pp , [8] X. L, Z. Qn, L. Yang, and K. L. Entropy maxmzaton model for the trp dstrbuton problem wth fuzzy and random parameters. Journal of Computatonal and ppled Mathematcs, vol. 235, no. 8, pp , [9] J. Ma, G. Ln, and X. Luo, Generalzed entropy gravty model based on land-use structure entropy. Journal of Chang an Unversty (Natural Scence Edton), vol. 03, pp , Receved: February 03, 2015 Revsed: prl 03, 2015 ccepted: May 25, 2015 Luo Xao-Qang; Lcensee Bentham Open. Ths s an open access artcle lcensed under the terms of the ( whch permts unrestrcted, noncommercal use, dstrbuton and reproducton n any medum, provded the work s properly cted.