Acta Polytechnca Hungarca Vol. 11, No., 014 The Applcaton of Gravty Model n the Investgaton of Spatal Structure Áron Kncses, Géza Tóth Hungaran Central Statstcal Offce Kelet K. u. 5-7, 104 Budapest, Hungary E-mal: aron.kncses@ksh.hu; geza.toth@ksh.hu Abstract: In ths paper the authors wsh to ntroduce an applcaton of the gravty model through a concrete example. In ther nvestgaton the gravty model was transformed to analyse the mpact of accessblty n a way, that not only the sze of gravtatonal forces but ther drecton can also be measured. splacements were llustrated by a bdmensonal regresson. The am of ths paper to gve a new perspectve to the nvestgaton of spatal structure through a Hungaran example. Ths makes easer the transport plannng and land development modelng. In our work accessblty analyss, gravty modelng, b-dmensonal regresson calculatons and GIS vsualzaton were performed. Keywords: gravty model; b-dmensonal regresson; accessblty; spatal structure; Hungary 1 Introducton The overall goal of modellng s to smplfy realty, actual processes and nteractons and on the bass of the obtaned data to draw conclusons and make forecasts. Models based on gravtatonal analogy are the tools of spatal nteractons of classcal regonal analyses. They were frst appled n the 19 th Century [, 3, 4, 11, 19, 1, 31]. The applcaton of the geographcal gravty s confrmed by the theory of experence accordng to whch (just as n tme) the thngs that are closer to each other n space are more related than dstant thngs. Ths s called the "frst law of geography [5]. There are two basc areas of the applcaton of gravty models based on physcal analogy: the spatal flow analyss [7, 16], and the demarcaton of catchment areas [15, 17]. The potental models based on gravtatonal analogy are the most mportant groups of accessble models. In general, t can be stated that they are accessble approaches accordng to whch models show potental benefts of the regon compared to other regons where the benefts are quantfed [0]. 5
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure The use of accessble models n transport-geographcal studes s very common. However, when models are used, t s not entrely clear what s actually modelled; because of ther complexty ther nterpretatons may be dffcult [13]. It should be stressed that accessblty has no unversally accepted defnton; n emprcal studes dfferent methodologcal background ndcators are used (see [9, 10, 9, 30].The gravtatonal theory s a theory of contact, whch examnes the terrtoral nteracton between two or more ponts n a smlar way as correlatons are analysed n the law of gravtaton n physcs. Accordng to usek [5], despte the analogy, there are sgnfcant dfferences between gravty models used by socal scences and the law of gravtaton used n physcs. It s worth bearng n mnd that "the gravty model s not based on the gravtatonal law. It s a fundamental statement based on the experence of undenable statstcal character that takes nto consderaton spatal phenomena. Accordng to ths statement, phenomena nteract wth each other. The phenomena, whch are closer to each other n space, are more related than dstant phenomena [5 p. 45]. There are a number of dfferences between the law and the model. In ths study, we wsh to hghlght a new pont of vew. As a consequence of the spatal nteracton, classcal gravtatonal potental models show the magntude of potental at spatal ponts. Regardng the law of gravty n physcs, the drecton of forces cannot be evaded. In our approach each unt area s assgned an attracton drecton. That s, n the case of the gravty model (although such spaces are free of vortex) the space s characterzed usng vectors. Method The unversal gravtatonal law, Newton's gravtatonal law, states that any two pont-lke bodes mutually attract each other by a force, whose magntude s drectly proportonal to the product of the bodes weght and s nversely proportonal to the square of the dstance between them [1] (1 st formula): F m m Where γ s the proportonal factor, the gravtatonal constant (ndependent of tme and place). If r ndcates the radus vector drawn from the mass pont number to a mass pont number 1, then r/r s the unt vector drawn from 1 towards, thus, the mpact of gravtatonal force on mass pont 1 from mass pont s n (Equaton ) (Fgure 1), whch reads m1 m r F1, r r () 1 r (1) 6
Acta Polytechnca Hungarca Vol. 11, No., 014 A gravtatonal feld s set f the gradent (K) can be specfed by a drecton and magntude n each pont of the range. Snce K s a vector quantty, three numbers (two n plane) are requred to be known at each pont, for example the rght angle components of gradent K x, K y, K z, whch are functons of the ste. However, many felds, ncludng the gravtatonal feld, can be characterzed n a more smple way. They can be expressed by a sngle scalar, the so-called potental functon, nstead of three values. Fgure 1 Gravtatonal power The potental s n a smlar relaton wth gradent as work and potental energy wth force. Takng advantage of ths, gravty models are also appled n most socal scences where space s usually descrbed by a sngle scalar functon, [13], whereas n gravtatonal law vectors characterzng the space are of great mportance. The prmary reason for ths s that arthmetc operatons calculated wth numbers are easer to handle than wth vectors. Perhaps, we could say that by workng wth potentals we can avod calculaton problems n problem solvng. The potental completely characterzes the whrl-free gravty gravtatonal feld, because there s a defnte relatonshp between the feld strength and the potental: K gradu K x U ; x K y U. y In other words, the potental (as mathematcal functonal) s the negatve gradent of feld strength. Varous types of potentals and models, whch are dfferent from the ones drectly based by the gravtatonal analogy, but n ths case, force effects among space power sources are qute dfferent. In fact, these models dffer from each other snce the attractve forces reman above a predetermned lmt value and wthn set dstances. (3) 7
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure The force n a general form s: F m1 m C r (4) where C, α, β, ƴ are constants. [14]. However, how they descrbe actual power relatons between socal masses s another queston. Although potental models often characterze concentraton on focal ponts of areas and spatal structures, they fal to provde nformaton n whch drecton and wth how much force the socal attrbutes of other areas attract each delmted area unt. Thus, we attempt to use vectors n order to show the drecton where the Hungaran mcro regons tend to attract mcro regons (LAU1) n the economcal space compared to ther actual geographcal poston. Ths analyss can demonstrate the most mportant centres of attracton, or dscrepances, and the dfferences between the gravtatonal orentatons of mcro-regons can be dsplayed on a map after the evaluaton of data from 000, 005 and 010 has been performed. In the study the geometrc centres of specfc mcro-regons were the co-ordnates of Hungaran mcro-regons, whch were determned n the EOV co-ordnate system by (Geographcal Informaton System) GIS software. Our goal can be reached by usng Equaton (3) to potentals or drectly wth the help of forces. We chose the latter one. In the conventonal gravty model [1] j s the "demographc force" between and j where W and W j are the populaton sze of the settlements (regons), d j s the dstance between and j, and fnally, g s the emprcal constant (Equaton 5). W Wj g ( ) (5) j d In ths study, the W and W j weght factors represent personal ncome, whch s the base of the ncome tax n small communtes, dj s the actual dstance between and j regonal centres measured on road by a mnute (regardless of the traffc condtons and only the maxmum speed dependng on the road type s taken nto consderaton). By generalzng the aforementoned formula, we wrte the followng equaton (Equaton 6 and 7): j 8
Acta Polytechnca Hungarca Vol. 11, No., 014 j j W Wj j j C1 dj c dj W W d j (6) Where W and W j are the masses, d j s the dstance between them, c s a constant, whch s the change of the ntensty of nter-regonal relatons as a functon of dstance. As the exponent, c, ncreases and the ntensty of nter-regonal relatons becomes more senstve to dstance, ths sgnfcance of masses gradually decreases [5]. The mnus sgn express mathematcally, that the masses attract each other (see Fgure 1). Wth the extenson of the above Equaton we cannot only measure the strength of the force between the two regons, but ts drecton as well. Whle performng calculatons, t s worth dvdng the vectors nto x and y components and summarze them separately. To calculate the magntude of ths effect (vertcal and horzontal forces of components) the followng Equatons are requred (Equaton 8 and 9), whch follow from (6): X j (7) W Wj ( x ) c 1 x j (8) d j Y j W Wj (y y j) c 1 (9) d j Where x, x j, y, y j are the coordnates of the and j regons. However, f we perform the calculaton on all unt areas nvolved, we wll know n whch drecton ther forces exactly act and how strongly they affect the gven unt area. (Equaton 10) X n j1 W W c1 j d j (x x ) j Y n j1 W W c1 j d j (y y ) j (10) 9
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure It should be noted that whle n potental models, the results are modfed by the ntroducton of "self-potental", n the examnaton of forces we dsregard the ntroducton of "self- forces". Thus, t s possble to determne the magntude and drecton of force n whch other areas affect each terrtoral unty. The drecton of the vector, whch s assgned to the regon, determnes the attracton drecton of other unt areas, whle the length of the vector s proportonal to magntude of force. For the sake of mappng and llustraton, we transformed the receved forces nto shfts proportonal to them n the followng way (Equaton 11 and 1): x mod x X x x max mn k 1 X max X mn (11) y mod y Y y y max mn 1 Y max Y mn Where X mod and Y mod are the coordnates of new ponts modfed by the gravtatonal force, x and y are the coordnates of the orgnal pont set, the extreme values of whch are x max, y max, a x mn, y mn, s the force along the x and y axes, k s a constant and n ths case t s 0.5. Ths has the effect of normalzng the data magntudes. We assume that n our model the amount of nteractons between the masses s the same as n Equaton 7, and based on the superposton prncple, t can be calculated for a gven regon by Equaton 10. The new model cannot drectly be compared wth transport-geographcal data, but the results compared wth traffc data n potental models to verfy our model [13]. Our model s a knd of complement to the potental models that ensures a deeper nsght nto them. In the followng sectons of ths study we ntend to communcate some sgnfcant results of ths model. k (1) 3 Applcaton of B-mensonal Regresson The pont set obtaned by the gravtatonal calculaton (W s the populaton sze of the. mcro-regon n Hungary, d j s the dstance between and j mcro-regons), s worth comparng wth the baselne pont set, that s, wth the actual real-world geographc coordnates and examnng how the space s changed and dstorted by 10
Acta Polytechnca Hungarca Vol. 11, No., 014 the feld of force. The comparson, of course, can be done by a smple cartographc representaton, but wth such a large number of ponts, t s not really promsng good results. It s much better to use a b-dmensonal regresson. The b-dmensonal regresson s one the methods of comparng partal shapes. The comparson s possble only f one of the pont coordnates n the coordnate systems dfferng from each other s transformed to another coordnate system by an approprate rate of dsplacement, rotaton and scalng. Thus, t s possble to determne the degree of local and global smlartes of shapes as well as ther dfferences that are based on the unque and aggregated dfferences between the ponts of the shapes transformed nto a common coordnate system. The method was developed by Tobler, who publshed a study descrbng ths procedure n 1994 after the precedents of the 60s and 70s [3, 4, 6, 7, 8]. There are many examples usng ths procedure, whch are not necessarly motvated by geographc ssues [1,, 18]. As for the equaton relatng to the calculaton of the Eucldean verson, see [7, 8, 6]. Table 1 The equaton of the two dmensonal regresson of Eucldean 1. Equaton of the regresson ' A 1 1 ' B. Scale dfference 3. Rotaton 4.Calculaton of β 1 5.Calculaton of β tan 1 1 1 1 X * 1 Y ( a a)*( x x) x x ( b b )*( x x) ( b b )*( y y) ( y y) ( a a)*( y y) ( x x) ( y y) 6. Horzontal shft a x * y * 1 1 7. Vertcal Shft b x * y * 1 8. Correlaton based on error terms (a a ) (b b ) r 1 (a a) (b b) 9. Breakdown of the square sum of the dfference (a a) (b b) (a a) (b b) SST=SSR+SSE 10. Calculaton of A A ( X ) ( ) ' 1 1 Y 11. CALCULATION of B B ( X ) ( ) ' 1 Y (a a) (b b) Source: [7] and [8] based on [6 p. 14] 11
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure Where x and y are the coordnates of ndependent shapes, a and b are the coordnates of dependent shapes, and represent the coordnates of dependent shapes n the system of ndependent shapes. α 1 determnes the measure of horzontal shft, whle α determnes the measure of vertcal shft. β 1 and β are the scalar dfference and (Ф) and (Θ) determne the angle of shftng. SST s the total square sum of dfference. SSR s the square sum of dfference explaned by regresson. SSE s the square sum of dfference not explaned by the regresson (resdual). Further detals about the background of the two-dmensonal regresson can be seen n [6 pp. 14-15]. Table Two-dmensonal regresson between the gravtatonal and the geographcal space Year r α 1 α β 1 β Ф Θ 000 0,94 6304,48 017,44 0,99 0,00 0,99 0,00 005 0,94 6030,56 01,3 0,99 0,00 0,99 0,00 010 0,941 806,79 63,9 0,99 0,00 0,99 0,00 Year SST (%) SSR (%) SSE (%) 000 100,00 98,73 1,7 005 100,00 98,74 1,6 010 100,00 98,69 1,31 Source: own calculaton Our results show that there s a strong relaton between the two pont systems; the transformed verson from the orgnal pont set can be obtaned wthout usng rotaton (Θ = 0). Essental rato dfference between the two shapes was not observed. Comparng the obtaned results, t s obvous that the set of ponts behaves lke a sngle-centre md-pont smlarty, when t s dmnshed. Ths means that only the attractve force of Budapest can be determned at a natonal level. 4 Map splay and recton Analyses The aforementoned statement can be llustrated by a map dsplay of a twodmensonal regresson. The arcy program can be used n the applcaton (http://www.spatal-modellng.nfo/arcy--module-decomparason). The square grd attached onto the shape-dependent coordnate system and ts nterpolated modfed poston further generalzes the nformaton receved from the partcpatng ponts. 1
Acta Polytechnca Hungarca Vol. 11, No., 014 The arrows n Fgures and 3 show the drectons of the shfts, whle the darker shades llustrates the type of dstorton. The darker zones express the dvergent forces of the area, whch are consdered to be the most mportant gravtatonal dsplacements. The data n Table shows that the space shaped by the gravty model causes only a slght dstorton compared to the geographc space. The magntude of vertcal and horzontal dsplacements ncreased slghtly n 010. Practcally, the maps produced by arcy software verfy ths (Fgure and Fgure 3). It can be seen that the captal of Hungary s Hungary s man centre of gravty, the centre towards whch the largest power s attracted. The regonal centres lke Győr, Pécs, Szeged, ebrecen are also gravty nodes. The natonal role of regonal centres s weak. In the area of Budapest a gravty fault lne emerges. The reason for ths phenomenon s that the Hungaran captal attracts all the mcro-regons, whle very weak forces are appled to Budapest compared to ts mass. The map also llustrates that the regular force felds are the major transport corrdors, namely they are slghtly dstorted due to hghways. Between 000 and 010, the sgnfcance of green-marked gravtatonal nodes ncreased. The comparson of the two maps clearly shows the ntensfcaton of regonal dfferences. Fgure Results n 000 13
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure Fgure 3 Results n 010 Based on the obtaned results, mcro-regons can be grouped by the drecton of force appled to them by other mcro-regons. Four groups can be formed. They are as follows: South and West, North and West, North and East, South and East. Fgure 4 Results n 000 14
Acta Polytechnca Hungarca Vol. 11, No., 014 Fgure 5 Results n 010 All mcro-regons can be placed n one of the aforementoned groups. The results are shown n Fgure 4 and Fgure 5. On the maps the North-South segmentaton appears more sgnfcantly than the East-West one. Ths statement s especally true n Western Hungary where the developed mcro-regons of Gyor-Moson- Sopron and Komárom-Esztergom Countes attract other mcro- regons of Transdanuba. The effect of the east-west gradent can hardly be demonstrated because of the central locaton of Budapest and ts strong mpact on the whole country. The mpact of regonal centres s clearly seen n the areas where the drecton of forces dffers from ts envronment,.e., n the neghbourng areas where colours are dfferent from the ones n ther envronment. Accordng to the results of 000 and 010 years, stable local centres can be seen n the mcro-regons of ebrecen, Mskolc, Nyíregyháza, Szeged and Pécs. In the advanced terrtores of northwestern Hungary a smlar phenomenon, though less dstnctve, s seen when mcro-regons dfferentate from ther envronment, snce several mcro-regonal groups tend to be smlar n character n these areas. 5 Magntude of Force Per Unt Mass In a mcro-regon other regons apply dfferent forces. However, the same force strength fals to result the same mpact due to the dfference of masses. It s 15
Á. Kncses et al. The Applcaton of Gravtatonal Model n the Investgaton of Spatal Structure possble to calculate the forces actng on a unt of nternal mass by the Equaton 13: Y X F W (13) Where F s the force per unt mass, s the vertcal and horzontal forces and w s the own weght of pont. As t s shown n Fgure 6, the most sgnfcant forces compared to ther own mass are the mcro-regons located n Budapest agglomeraton, especally the Budaörs mcro-regon. Outsde the agglomeraton, Tata mcro-regon shows an outstandng magntude. The mcro-regons that represent hgher than the average value are manly located further from Budapest, generally along motorways. The long arrows around Budapest were formed due to the mpact of the motorways. Fgure 6 shows the changes between 000 and 010. The most sgnfcant changes are the outcomes of the constructon of motorways. A perfect example for ths s Tszavasvár and Komló mcro-regons, whch are located near M3 and M6 motorways. Somewhat dfferent examples are the famous Sarkad and Sárvár mcro-regons. In ther cases, the decrease of ther own weght caused a specfc power growth. There s a mcro-regon, whch emerged prmarly due to the populaton growth n ts envronment and to ts gravtatonal force growth. It s Plsvörösvár mcro-regon. Fgure 6 Forces per unt mass (populaton) as a percentage of the natonal average n 010 16
Acta Polytechnca Hungarca Vol. 11, No., 014 Conclusons Fgure 7 Changes n forces per unt mass (populaton) n 000 and n 010 In our study we made an attempt to ntroduce the potental and unexplored areas of gravty models and problems of ther nterpretaton by expandng and extendng the methodology. The forces appled were llustrated by usng the ncome tax base as weghts n mcro-regons of Hungary. On the bass of the model, the result n lne wth the experence llustrated that Budapest has no counterweght n Hungary and the local central areas are weak. However, the presence of stable local centres s detectable n ebrecen, Mskolc, Nyíregyháza, Szeged and Pécs mcro-regons. It s mportant to note that compared to ther weght, mcro-regons affected by the most sgnfcant forces are located n the broader vcnty of Budapest, manly along hghways. Most sgnfcant changes compared to unt mass are caused by hghways, but n many cases the populaton declne n mcro-regons s also a determnng factor. Acknowledgement Ths work was supported by the János Bolya Research Scholarshp of the Hungaran Academy of Scences. References [1] Budó, Á.: Expermental Physcs I. Natonal Textbook Press, Budapest, 004 [] Bhattacherjee, A.: Socal Scence Research: Prncples, Methods, and Practces. USF Tampa Bay Open Access Textbooks Collecton. Book 3, 01 17
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