Cell Count Method on a Network with SANET

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1 CSIS Dscusson Paper No.59 Cell Count Method on a Network wth SANET Atsuyuk Okabe* and Shno Shode** Center for Spatal Informaton Scence, Unversty of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo , Japan *atsu@ua.t.u-tokyo.ac.jp **shno@ua.t.u-tokyo.ac.jp Aug, 2003 Abstract Ths paper frst proposes a cell count method defned on a network, called the network cell count method, as an extenson of the ordnary cell count method defned on a plane. Second, the paper develops a user-frendly tool for the network cell count method n conjuncton wth a general toolbox for spatal analyss on a network, called SANET. Thrd, the paper shows an actual applcaton of the network cell count method to the dstrbuton of retal stores n Shbuya, Tokyo. Key words: cell count method, network, toolbox, software, SANET, GIS 1

2 1. Introducton Ths paper has two objectves. The frst objectve of ths paper s to propose a new cell count method, called the network cell count method, as an extenson of the ordnary cell count method (or the quadrat method). The second objectve s to develop a user-frendly tool for the network cell count method, and ncorporate t nto a toolbox called SANET, a general GIS software package for spatal analyss on a network. The cell count method s one of the most classc methods for pont pattern analyss (Baley and Gatrell, 1995; Cresse, 1993; Upton and Fngleton, 1985). Ths cell count method ordnarly makes the followng assumptons. Assumpton : Geographcal space s represented by a plane. Assumpton : The plane s homogeneous,.e., the probablty of a pont beng placed on a unt area s nvarant regardless of the locaton of the unt area on the plane. There may be some actual cases n whch these assumptons hold or approxmately hold. For example, (Matu, 1932) examned the scattered vllage (houses are scattered) on the Tonam plan, whch appears to be a farly homogeneous plane. However, when we examne the dstrbuton of features n an urbanzed area, the above assumptons are hard to accept. For example, consder the dstrbuton of beauty parlors n Shbuya, one of the sub-centers n Tokyo, whch s shown n Fgure 1. As s seen n ths fgure, the beauty parlors are located along streets, and the dstrbuton of beauty parlors seems to be strongly constraned by the street network. Ths suggests us to replace Assumpton wth the followng assumpton. Assumpton : Geographcal space s represented by a network. As a consequence of ths assumpton, we assume a heterogeneous space n the sense that the space s not sotropc. As a matter of fact, drecton s restrcted along a path of a network. However, we assume a homogeneous space n the followng sense (.e. a network verson of Assumpton ). Assumpton : The network s homogeneous n the sense that the probablty of a pont beng placed on a unt lne segment s nvarant regardless of the locaton of the unt lne segment on the network. 2

3 Obvously ths assumpton s unrealstc to deal wth the dstrbuton of features n an urbanzed area, but ths assumpton s easly relaxed n a network space through the unform network transformaton. Ths transformaton s the topc of the next secton. Fgure 1: Beauty parlors n Shbuya, Tokyo 2. Unform network transformaton A heterogeneous network, N, s represented by a densty functon defned on lnks of the network. To be explct, let D be the densty of the th lnk of the network, and L be the length of the th lnk. Then the heterogeneous network means that D vares from lnk to lnk; the homogeneous network (Assumpton ) s wrtten as D = c for = 1,, n. Note that the locaton of a pont, p, on the th lnk can be dentfed by the path length, t, from an end pont of the th lnk to the pont p along the th lnk. * Let us now consder a new network, N, by replacng the th lnk wth the new th lnk and we correspond a pont p at t on the th lnk to the pont p at s on the new th lnk n such a way that the path length, s, from an end pont of the new th lnk to the pont p along the new th lnk s gven by s = t 0 D dx. 3

4 Note that the ntegraton s done wth respect to x along the th lnk. Ths transformaton mples that the th lnk of length L s replaced wth the new th lnk of length D L. The new network obtaned from the above transformaton has a nce property. Notcng that the total quantty on the th lnk s gven by L 0 D dx = D L, and that the length of the new th lnk s D L, we obtan the densty of the new th lnk s D L /( D L ) = 1. Ths equaton hold for all lnks. Ths means that the transformed network N* s homogeneous as n Assumpton. We refer to the above transformaton as the unform network transformaton (Okabe, 2002). We can easly deal wth a heterogeneous network by transformng t nto a homogeneous network through the unform network transformaton. 3. Cell count method Let us consder a network, L, consstng of lne segments that are connected. We decompose the network L nto a set of sub-networks (called cells) satsfyng that the length of each sub-network s the same, c, and each sub-network s connected. In practce, however, ths decomposton s mpossble except for very specal cases, because the total length dvded by c s not nteger n general. PC Notcng ths fact, we consder two types of cells: proper cells, L { L1,, L n } IC mproper cells, { L L } L n+ 1,, n+ n * =, and =. The former cells are used for cell countng, but the latter cells are perpheral cells not used for cell countng. We suppose that these cells satsfy the followng condtons. () L s connected n the sense that for any par of ponts on L, there exsts a path between these ponts that s ncluded n L ( = 1,, n + n* ). 4

5 () Proper cells L,, L have the same sze,.e. 1 n L = c for = 1,, n, where L denotes the length of L. () The sze of the mproper cells L n+ 1,, L n + n* s smaller than that of the proper cells,.e. L < c for = n + 1,, n + n*. (v) The unon of the proper cells and the mproper cells cover the whole network L,.e. PC IC n n+ n* L = L L = [ L ] [ L ]. = 1 = n+ 1 (v) Cells do not ntersect each other except at boundary ponts,.e. L L = 0 for j,, j = 1,, n + n *. j We make fve remarks on ths decomposton. Frst, ths decomposton s not unque. There are many ways of decomposton satsfyng the above condtons. Second, the shape of cells s not the same, because a gven network L s not regular. Ths contrasts to the planar case where the shape s the same (usually squares) for all cells. Thrd, t s very dffcult to control the shape of cells, but the shape should be as compact as possble. Forth, the length of the mproper cells should be as small as possble. Last, mproper cells should not appear nsde a network. In the planar case, mproper cells appear only on the perphery, but n the network case, mproper cells may appear nsde a network. We call such cells hole cells. It s not easy to avod hole cells n decomposng a network, and we shall dscuss ths problem n Secton 5. The dea of the network cell count method s the same as that of the planar cell count 5

6 PC n method. Suppose that we observe m ponts dstrbuted on the network = L ], L [ = 1 and we want to test the null hypothess that the m ponts are randomly dstrbuted on PC n the network = L ] accordng to the unform dstrbuton, L [ = 1 f 1 L PC ( x) =, x L. PC We can test ths hypothess usng the goodness-of-ft test. To be explct, let observed number of cells that contans exactly j ponts, except for N k. observed number of cells that contan k or more ponts. Let N j be N k s the P j be the probablty that j ponts are placed n a cell under the above null hypothess, whch s gven by the bnomnal dstrbuton or the Posson dstrbuton for a large number of ponts. Thus we can test the null hypothess wth the test statstc k 2 χ = = 1 2 ( np N). np 4. SANET We develop a user-frendly tool for achevng the network cell count method defned above and nclude t n a general toolbox for spatal analyss on a network, called SANET (Okabe, Okunuk and Funamoto, 2002). SANET conssts of two components. The frst component s a software package for computng methods for spatal analyss on a network. We can nterface ths package wth a vewer of GIS through nput and output fles. The computaton n ths software package s ndependent of a vewer of GIS, and so we can use any vewer. The second component s an nterface wth a vewer of GIS, and ths nterface depends on a vewer of GIS. We use ArcVew8.x., and we have developed the nterface that commute data between the network computaton software package and ArcVew8.x. SANET s under development but the frst verson was released n November, The frst verson provdes the followng seven tools: Tool 1: Constructon of dataset for SANET. Tool 2: Access pont assgnment. 6

7 Tool 3: Table calculaton. Tool 4: Generaton of the network Vorono dagram (Okabe, Boots, Sughara and Chu, 2000). Tool5: Generaton of random ponts on a network (for Monte Carlo smulatons). Tool 6: Cross K-functon method (Okabe and Yamada, 2002). Tool 7: K-functon method (Okabe and Yamada, 2002; Yamada and Thll, 2002). SANET s open to non-proft users wthout charge and t can be downloaded from the SANET ste: Ths paper adds the cell count method to SANET as the eghth tool. The cell count method uses Tool 1 for convertng the network data format n the network computaton software package to data format of ArcVew8.x. It also uses Tool 2 for assgnng the representatve ponts of polygon-lke features, such as beauty parlors, to the nearest ponts on a network (whch may be regarded as entrances of the facltes). By ths assgnment, we can use the cell count method that assumes that ponts are dstrbuted over the network. 5. Computatonal method for the cell count method The computatonal method s farly complex and techncal and so we only outlne the algorthm n ths secton. In the frst step, we choose a node of a network around the center of the network. We call the node the root node. In the second step, we construct the extended shortest-path tree rooted at the root node. The extended shortest-path tree s obtaned from the ordnary shortest-path tree n the followng manner. The ordnary shortest-path tree does not cover the whole network. We call the lnks that are not ncluded n the ordnary shortest-path tree collson lnks. On each collson lnk, we can fnd such a pont, c, that the shortest-path dstance from the pont to the root node through one end node of the collson lnk s the same as the shortest-path dstance from the pont c to the root node through the other end node of the collson node. We call the pont c a collson pont. We cut the network at all collson ponts and add nodes on both cut ends. We call the resultng tree the extended shortest-path tree. Once we obtan the shortest-path tree, we can easly obtan the shortest-path from 7

8 any pont on the network to the root node. In the thrd step, we construct a tree rooted at the root node whose total length s a gven dstance. We next construct a tree rooted at each end node of the tree whose total length s a gven dstance. If we cannot construct such a tree, we leave t. We contnue ths procedure untl when we cannot construct trees. Then we obtan a set of trees that are ncluded n the network and the length of each tree s the same. We may use the resultng trees as cells. However, these cells are lkely to produce the holes cells n the network. The last step s to put out these hole cells to the perphery of the network. Ths part s too lengthy to explan, and so t s omt here. 6. An Applcaton Havng establshed the network cell count method and ts software package, let us show an actual example. The study regon s part of Shbuya, one of the sub-centers n Tokyo, and the street network s shown by colored lnes n Fgure 2. Fgure 2: Retal stores assgned to the street network n Shbuya, Tokyo (cells are ndcated by dfferent colors) Frst we decompose the street network nto cells. In Fgure 2, the cells are ndcated 8

9 by dfferent colors. The yellow lnes are mproper cells. Second, we assgn retal stores to the nearest ponts on the network (whch may be regarded as the entrances of the stores). The ponts n Fgure 2 show the assgned retal stores on the street network. Thrd, the tool counts the number of retal stores n each cell and shows the numbers of cells havng ponts as n Fgure 3. Fgure 3: The number of cells havng ponts Last we apply the goodness-of-ft test to the data shown n Fgure 4. The test rejects the null hypothess, mplyng that the retal stores are non-randomly dstrbuted n Shbuya. 7. Concludng remarks In ths paper we have proposed the network cell count method as an extenson of the ordnary cell count method defned on a plane and developed a tool for the network cell count method. Compared wth the ordnary cell count method, the proposed network cell count method has a few advantages. Frst, the network space s more natural to deal wth the dstrbuton of features n an urbanzed area than a homogeneous plane. Second, t s easy to treat heterogenety wth the unform network transformaton. For example, t s 9

10 easy to treat the dstrbuton of stores n relaton to the dstrbuton of populaton. One mght consder that the results obtaned from the ordnary cell count method are not so dfferent from those obtaned from the network cell count method. We compared these results n a few actual examples and found that ths dfference was sgnfcant. We wll further examne ths comparson usng many cases and we wll report the results n another occason. Acknowledgements We express our thanks to Yasush Asam, Yuko Sadahro and Ke-ch Okunuk for ther comments on earler drafts. Ths study was part of the outcomes of the Grant-n-ad project "Spatal Informaton Scence for Human and Socal Scences" supported by the Japanese Mnstry of Educaton and Scence. References Baley, T. and Gatrell, A. (1995) Interactve Spatal Analyss, London: Longman. Cresse, N. (1993) Statstcs for Spatal Data, New York: Wley. Matu, I. (1932) Statstcal study of the dstrbuton of scattered vllages n two regons of the Tonam plan, Toyama prefecture, Japanese Journal of Geology and Geography, 9, Okabe, A. (2002) Unform network transformaton for spatal analyss on a heterogeneous network, CSIS Dscusson Paper #51. Okabe, A., Okunuk, K. and Funamoto, S. (2002) SANET: A Toolbox for Spatal Analyss on a Network, Center for Spatal Informaton Scence, Unversty of Tokyo. Okabe, A., Boots, B., Sughara, K. and Chu, S. N. (2000) Spatal Tessellatons: Concepts and Applcatons of Vorono Dagrams, (2 nd edton), Chchester: John Wley. Okabe, A. and Yamada, I. (2001) The K-functon method on a network and ts computatonal mplementaton, Geographcal Analyss, Vol.33, No.3, pp Upton, G. and Fngleton, B. (1985) Spatal Data Analyss by Example, Chchester: Wley Yamada, I. and Thll, J-C. (2002) An Emprcal Comparson of Planar and Network K-functon Analyses, Abstract, AAG Annual Meetng, Los Angeles, CA. 10

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges