Cellular automata urban growth model calibration with genetic algorithms

2007 Urbn Remote Sensing Joint Event Cellulr utomt urbn growth model clibrtion with genetic lgorithms SHARAF AL-KHEDER, JUN WANG, JIE SHAN Geomtics Engineering School of Civil Engineering Purdue University, 550 Stdium Mll Drive, West Lfyette, IN 47907, USA Phone: +1-765-494-2168, Fx: +1-765-496-1105 shn@ecn.purdue.edu Abstrct Lst few decdes witness drmtic increse in city popultion worldwide ssocited with excessive urbniztion rtes. This rises the necessity to understnd the dynmics of urbn growth process for sustinble distribution of vilble resources. Cellulr utomt, n rtificil intelligence technique composed of pixels, sttes, neighborhood nd trnsition rules, is being widely implemented to model the urbn growth process due to its bility to fit such complex sptil nture using simple nd effective rules. The min obective of our work is to use genetic lgorithms to effectively clibrte, i.e., identify trnsition rule vlues, cellulr utomt urbn growth model tht is designed s function of multitemporl stellite imgery nd popultion density. Trnsition rules in our model identify the required neighborhood urbniztion level for test pixel to develop. Clibrtion is performed sptilly to find best rule vlues per township. Genetic lgorithms clibrtion model, through proper design of their prmeters, including obective function, initil popultion, selection, crossover nd muttion, is prepred to fit the cellulr utomt model. Genetic lgorithms strt processing the initil solution spce, through sequentil implementtion of the prmeters, to identify the best rule vlues using predefined criterion over the mximum number of itertions. Minimum obective function, representing the totl modeling errors, is used to identify the optiml rule vlues. Ech rule set is evluted in term of urbn level nd pttern mtch with relity. Clibrtion with genetic lgorithms proves to be effective in producing the optiml rule vlues in time effective mnner t n erly genertion. Proposed clibrtion lgorithm is implemented to model the historicl urbn growth of Indinpolis-IN, USA. Urbn growth results show close mtch for both urbn count nd pttern with relity. I. INTRODUCTION Much reserch efforts cn be seen in the literture towrds developing effective cellulr utomt bsed urbn growth models. A number of these models designs succeed to certin extent in modeling the urbn process. However, there remin unsolved issues to mke the developed models more relible. Clibrtion of cellulr utomt urbn growth models is mong these issues tht is still chllenge. Clibrtion in cellulr utomt urbn modeling is ment to find the best trnsition rule vlues to reproduce the sme urbn level nd pttern with reference to historicl dt [1]. Urbn cellulr utomt models re very sensitive to trnsition rules nd their prmeter vlues [2]. The difficulty in clibrting cellulr utomt rules is due to the complexity of the urbn development process [3]. Clibrtion styles in literture cn be clssified into three ctegories: visul, sttisticl, nd rtificil intelligence bsed. SLEUTH model clibrtion [4;5] is n exmple of urbn growth model clibrtion tht mkes use of visul nd sttisticl tests to identify best urbn growth prmeter vlues. Multi-criteri evlution (MCE) method [6] nd neurl networks [2] lso hve been used in previous reserch for clibrtion purposes. Common clibrtion problems of cellulr utomt re relted to the design of model itself. Most models require lrge input vribles nd composed of big set of rules which mkes the clibrtion time consuming process. Genetic lgorithms represent new clibrtion direction tht ppers recently in cellulr utomt urbn growth modeling. The erly forml strt cn be seen in the ttempt of formlizing genetic lgorithms s clibrtion tool for the SLEUTH model [7]. Further improvements re still required to mke genetic lgorithms robust technique for urbn cellulr utomt models clibrtion. Proper design of genetic lgorithms, effective setup of their prmeters, nd selection of the obective function re issues of further interest. This pper focuses on improving the clibrtion of previously designed cellulr utomt urbn model [8] through dpting genetic lgorithms. The developed cellulr utomt model is designed s function of multitemporl stellite imgery nd popultion density so tht the trnsition rules identify the required neighborhood urbniztion level for test pixel to urbnize. The min obective of the work in this pper is to use genetic lgorithms to utomte the serch method for best trnsition rule vlues of the designed cellulr utomt for relible modeling. Clibrtion is performed sptilly on township level to tke into ccount the sptil vrition in urbn dynmics where the sme trnsition rules re pplied to every township, however with different vlues. Temporl clibrtion through using historicl imges to reclibrte the model rules is lso performed. The genetic lgorithms clibrtion model is designed to fit the developed cellulr utomt urbn growth model. All the design phses of genetic lgorithms including: obective function, initil popultion preprtion nd encoding, selection process, crossover nd

2007 Urbn Remote Sensing Joint Event muttion re setup crefully to best reflect the modeling process. Initil popultion of solution strings re binry encoded with obective function being designed to represent the totl modeling errors ssocited with ech string ccording to the evlution results identified with reference to relity. Elitism nd rnk selection procedures re used to strt the production of next genertion for genetic lgorithms. After selection, crossover nd muttion opertions re implemented on the selected strings to finlize the new popultion of solutions. Finlly, the cellulr utomt is run for the new popultion to evlute their new obective function vlues. The procedure of running the genetics lgorithms with the cellulr utomt model is repeted till convergence criterion is met. The rule set tht produces the minimum obective function vlue over ll of the itertions is selected s the optiml trnsition rule for urbn modeling. Detiled nlysis of the output modeling results is performed s compred to relity. The chnge in rule vlues s function of genetic lgorithms genertion is identified nd the pttern of convergence is lso tested. II. CELLULAR AUTOMATA URBAN GROWTH MODEL A. Study Are nd Input Dt In previous reserch work, we developed cellulr utomt bsed urbn growth model [8]. The model ws tested through simulting nd predicting the historicl urbn growth of city Indinpolis, IN, USA (Fig. 1). The model uses two types of input dt: Historicl clssified stellite imges nd popultion density grids. A set of clssified stellite imges [9] over Indinpolis (1982,, nd 2003-TM) in NAD1983 UTM proection ws prepred. Seven clsses were identified in the imges, nmely: wter, rod, residentil, commercil, forest, psture, nd row crops with commercil nd residentil clsses representing the urbn clss of interest. The popultion density grids were produced using n exponentil function (1) of the distnce between the census trct centroid nd the overll city centroid. BDISTANCE POPULATION DENSITY = A e (1) Using the census trct mps t 1990 nd 2000, bsed on which the yerly chnge in model prmeters (A nd B) ws identified, the model ws used to clculte the popultion density for ech pixel throughout the yers from 1982 to 2003 to produce the input grids. B. Trnsition Rules nd Clibrition Cellulr utomt trnsition rules (Ø) of the developed model were physiclly built over the input imgery. The rules used 3x3 neighborhood, A t in (2) to identify the test pixel future stte, t+1 in (3). A t = i1, 1 1 1, 1 i1, 1, i1, + 1 + 1 1, + 1 3x3 (2) t+1 = Ø(A t ) (3) Figure 1. City of Indinpolis nd township mp, Indin, USA Trnsition rules (Ø) were designed to identify the required neighborhood urbn level for test pixel to urbnize. The following is summry of such rules: 1. IF test pixel is wter, rod OR urbn (residentil or commercil) THEN no chnge. 2. IF test pixel is non-urbn (forest, psture OR row crops) THEN it becomes urbn if its: Popultion density is equl or greter thn threshold (P i ) AND neighboring residentil pixels count is equl or greter thn threshold (R i ); or, Popultion density is equl or greter thn threshold (P i ) AND neighboring commercil pixels count is equl or greter thn threshold (C i ). where (R,C) i re integer numbers rnge from 0 to 8 (3x3 neighborhood) nd P i is rel number rnges from 0 to 3 (0.1 increment). The Clibrtion (i.e., identifying best (R,C,P) i prmeter vlues) of such rules ws performed sptilly on township level, T s (Fig. 1b) to fit the locl urbn dynmic fetures nd over time to consider the temporl urbn chnges t ech township, T t in (4). Ø clibrted = f(t s, T t, ) (4) in the clibrtion formul represents the criteri selected to find the best rule set for certin township sptil loction T s t given time epoch T t. This criterion in our model represents the totl modeling errors/mismtch between modeled output nd relity tht need to be minimized for best mtch. in (5) ws defined s function of fitness F in (6) nd totl errors E in (7) evlution mesures. Fitness nd totl errors mesure the comptibility in terms of urbn count nd pttern within ech township with respect to relity, respectively. = Abs (F-100%) + E (5) Modeled _ urbn _ count F = 100% Ground _ truth _ urbn _ count Totl _ error _ count E = 100% Totl _ count (6) (7)

2007 Urbn Remote Sensing Joint Event III. CALIBRATION WITH GENETIC ALGORITHMS Clibrting trnsition rules (Ø) to find the optiml set (R,C,P) i for ech township mong the lrge number of possible combintions (2511=9x9x31) is time consuming process. Genetic lgorithms re introduced to effectively identify such prmeters in time effective mnner. This section covers first the design phses of the genetic lgorithms clibrtion module s implemented to cellulr utomt urbn growth model. Then comprehensive nlysis is performed to evlute the modeling results. A. Design of Genetic Algorithms Clibrtion Module Genetic lgorithms erly strts come through n effort [10] to mimic the nturl process in biology of cellulr reproduction to solve problems with complex nture. Their bility s n utomtic nd effective serch method for the globl optiml solution from limited nd discontinues solution spce fvors them over other serch methods. The development phses of genetic lgorithms include obective function design, initil popultion preprtion nd encoding, selection, crossover nd muttion. tht ws defined erlier s the totl modeling errors is used s obective function to evlute the performnce for ech rule set (R,C,P) i. Rule set with minimum vlue will be selected s the optiml set. Thirty sets of (R,C,P) i re rndomly generted for ech township to represent the totl initil genetic lgorithms solution popultion. Ech (R,C,P) i set, tht is ssocited with n obective function vlue, is binry encoded to represent one string in the solution pool. R i nd C i re in the possible rnge of (0-8) integer vlues nd hve binry coding rnge of (0000 to 1000). P i continuously rnges between 0 nd 3, nd is scled to 0-30 s integer for encoding purposes, which corresponds to the binry coding rnge of (00000 to 11110). An exmple of rule string (7, 7, 1.0) is encoded s binry string (0111011100001), in which the first four digits re for R i, the second four digits for C i, nd the lst five digits for P i. At the end of this phse, totl popultion of thirty binry strings ssocited with their obective vlues for ech township is redy to be processed to produce the next genetic lgorithms genertion. In the second phse of genetic lgorithms clibrtion, rnk nd elitism selection methods re used to select the new solution popultion of thirty strings strting from the initil popultion. According to rnk selection method, ll strings re ordered bsed on their genetic lgorithms obective function vlues in scending order from minimum to mximum. The string with the lowest obective function vlue (lowest modeling error) is given rnk of 30, the second 29, etc., until the lst string, which will receive rnk of 1. The selection probbility (p i ) for ech string is clculted s rtio between its rnk (r i ) nd the rnks sum (r): p i = ri r This probbility when multiplied by the popultion size (30 strings) will identify how mny copies ech string is expected (8) to contribute in the next genertion. For exmple, string with selection probbility of 0.03988 is expected to contribute 1.1964 strings (30x0.03988) in the next popultion. This mens tht this string will reproduce one string of its type in the next genertion. Through elitism selection, the first six strings with highest rnks, or lowest obective function vlues, re copied directly to the new solution popultion. The rest of strings (i.e., 24) re selected from the old popultion ccording to their selection probbility (rnk selection method). By this step new popultion of 30 strings is produced s result of the selection process. The selected strings re processed further through the crossover opertion. Crossover in genetic lgorithms simultes the sme process in biology whereby genes from two prents meet to produce new offspring tht is mix of the prents genes. Crossover is importnt in introducing new possible solutions to explore new res of serch spce. The ssumption is lwys tht good prent strings, when crossed over, tend to produce offspring with the sme or better qulities. Strings produced s result of the selection process re mted in pirs t rndom [11]. Ech pir of strings in the crossover popultion (s n output of the selection process) is selected rndomly to be crossed over. Using single-point crossover, n integer loction k (4 in this work) between the first nd one less thn the string length (l 1) is identified s the crossover pivot point. Two new strings re produced by exchnging ll the bits between loctions k+1 nd l inclusively for the two old strings selected for crossover. As n exmple of single-point crossover, ssume tht two strings C 1,C 2 (13 bits ech s defined erlier) re selected for crossover, where k is set to be 4, the crossover will result in two new strings s follows: C = ( x, x, x, x, x, x, x, x, x, x, x, x, x ) 1 1 2 3 4 5 6 7 8 9 10 11 12 13 C = e, e, e, e, e, e, e, e, e, e, e, e, ) ( e 2 1 2 3 4 5 6 7 8 9 10 11 12 13 ' C = ( e, e, e, e, x, x, x, x, x, x, x, x, x ) 1 1 2 3 4 5 6 7 8 9 10 11 12 13 ' C = x, x, x, x, e, e, e, e, e, e, e, e, ) ( e 2 1 2 3 4 5 6 7 8 9 10 11 12 13 In our clibrtion lgorithm, the best six strings re copied directly (elitism) nd crossed over to produce new 12 strings. To complete the popultion, the best 18 strings re lso crossed over resulting in new totl popultion of 30 strings. Crossover is performed using the bove designed criteri. The lst genetic lgorithms opertion to produce the new genertion of solution spce bsed on the post crossover popultion is the muttion process. Muttion in genetic lgorithms is defined s rndom deformtion of the strings with certin probbility [12]. Muttion is introduced in genetic lgorithms to produce new formtions of strings in order to preserve genetic diversity nd to void locl optimum [12]. It simply represents the ltering of selected vrible bits in the string to enrich the serch process with new possible solution combintions representing vrious sections of the solution spce. In the clibrtion lgorithm development, the best six strings within the crossover popultion re mutted through rndom ddition of +1 or -1 to the (R,C) i prmeter vlues. The following is n exmple of string muttion: 0111011100001 (7,7,1) +1,+1 (8,8,1) 1000100000001

2007 Urbn Remote Sensing Joint Event By muttion, new genertion of genetic lgorithms representing solution spce of thirty strings is generted. Cellulr utomt model runs to evlute the new obective function vlues for the new solution pool. B. Modeling nd Evlution The genetic lgorithms opertions discussed bove re repeted recursively for totl of twenty genertions. The rule set tht produces the minimum obective function vlue over the course of the totl number of itertions (20 itertions) is selected s the optiml for ech township. City of Indinpolis urbn growth represented by the set of clssified historicl imges discussed erlier is used s test bed to test the proposed clibrtion lgorithm. Through designing n initil rndom solution spce (30 strings), the clibrtion-modeling process is implemented to model (simultion nd prediction) the historicl urbn growth of the city. Urbn growth modeling is simulted from 1982 to, where clibrtion is performed using genetic lgorithms to find the best township rule vlues. The best rules t re used to predict for short term prediction of five yers. Another clibrtion is crried out t to predict the urbn growth t 2003 for long term prediction of 11 yers. Tble I refers to prt of the numericl evlution results for simultion yer nd predicted yer ssocited with their imges t Fig. 2. TABLE I. NUMERICAL EVALUATION RESULTS Township # Fitness% Totl Error, E % Fitness % Totl Error, E Simultion Prediction % 1 148.8 22.93 101.34 19.80 2 110.0 23.21 110.27 23.45 3 96.7 26.36 85.69 32.36 4 99.2 25.42 97.64 29.48 5 113.4 23.84 92.26 25.89 6 99.3 25.10 82.33 28.89 7 99.9 30.78 87.45 34.66 8 102.2 26.81 90.82 30.40 9 100.1 28.00 97.59 31.58 10 100.3 27.59 113.25 24.20 11 101.9 26.15 111.01 21.81 12 100.1 31.38 89.59 36.65 13 85.5 22.85 87.30 26.57 14 86.6 18.00 97.33 16.51 15 101.1 11.78 113.75 7.10 16 100.2 25.93 103.42 27.17 17 90.9 24.86 83.56 28.60 18 98.7 10.89 109.63 8.11 19 100.1 16.78 99.86 16.15 20 99.4 29.42 110.35 29.47 21 118.8 26.12 75.66 27.62 22 103.6 28.00 114.58 30.57 23 105.7 24.00 76.05 28.58 24 127.2 31.15 103.02 29.73 Avg. 103.74 24.47 97.24 25.64 Rel Simulted/Predicted Figure 2. Simultion () nd prediction () imge results C. Anlysis Simultion nd prediction urbn modeling results, s shown in Tble I nd Fig. 2, show generl close mtch to relity in term of urbn count nd pttern. Tble I fitness results for both prediction nd simultion show close mtch in urbn count (close vlues to 100%) between the modeled nd rel dt with verge fitness of 103.74 (little overestimte) nd 97.24 (little underestimte), respectively. The urbn pttern mtch is lso cler in the tble where n verge totl error between 24-26% is chieved. This indictes n pproximte mtch level of 75% on pixel by pixel bsis between modeling nd relity. This is high ccurcy level compred to the results shown in literture for the urbn lnd sptil fit re tht ws only 28.15 to 44.6% [13]. The close urbn pttern mtch is lso cler in Fig. 2 where both predicted nd simulted imges hve urbn distribution similr to those shown in their corresponding rel imges. On the side of computtion time for trnsition rule clibrtion, genetic lgorithms show higher efficiency s compred to trditionl exhustive serch clibrtion method. On verge, it took genetic lgorithms six nd hlf hours of

2007 Urbn Remote Sensing Joint Event continuous CPU time to run for the twenty genertions to rech the optiml rule set per township. This is bout of time needed by the exhustive serch in this study. Looking t the fine scle (township level) regrding the chnge in obective function shows this fct s well (Fig. 3). As shown in this Figure for selected townships 7 nd 14 t different clibrtion yers (, nd 2003), the minimum obective function vlues re chieved t erly genertions (within the first 10 genertions for most townships). This indictes the bility of the proposed clibrtion lgorithm in reching n optiml solution in time effective mnner while preserving the modeling qulity s referred to relity. Township#7 townships cn be reched t n erly (<10) stge of genetic lgorithm genertions. Therefore, it is expected tht the genetic lgorithms will more significntly benefit urbn modeling problems with lrger set of input dt nd lrger solution spces. There is need to crry out clibrtion in sptil units smller thn townships to test the effect of sptil modeling unit size on the relibility of modeling with the purpose of improving the results. For this purpose, it is suggested to use census trcts tht represents the smllest sptil units bsed on which ttributes, such s popultion density, re distributed in sptil clibrtion. This will help in cpturing finer detils in the modeling process while clibrting the model over smller sptil units to reduce modeling uncertinty. GA obective function GA obective function 3 2.5 2 1.5 1 0.5 0 0.6 0.5 0.4 0.3 0.2 0.1 0 2003 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Genertion# 2003 Township#14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Genertion# Figure 3. Obective function chnge with genertions D. Conclusions nd Finl Comments As this study demonstrted, genetic lgorithms re ble to produce modeling results, both quntittively nd qulittively, close to the relity in time effective mnner. A proper selection of the initil solution popultion nd prmeters for encoding, crossover nd muttion cn enhnce the performnce while serching for the optiml rule vlues. It is shown tht the computtion time is significntly reduced from 27 hours in the trditionl exhustive serch to 6.5 hours in the cse of genetic lgorithms. Optiml rule vlues for most REFERENCES [1] F. Wu, Clibrtion of stochstic cellulr utomt: the ppliction to rurl-urbn lnd conversions,. Interntionl Journl of Geogrphicl Informtion Science, vol. 16, pp. 795 818, 2002. [2] X. Li nd A.G.O. Yeh, Neurl network-bsed cellulr utomt for simulting multiple lnd use chnges using GIS, Interntionl Journl of Geogrphicl Informtion Science, vol. 16(4), pp. 323 343, 2002. [3] M. Btty, Y. Xie, nd Z. Sun, Modelling urbn dynmics through GISbsed cellulr utomt, Computers, Environment nd Urbn Systems, vol. 23, pp. 205 233, 1999. [4] K. C. Clrke, S. Hoppen, nd L. Gydos, A self-modifying cellulr utomton model of historicl urbniztion in the Sn Frncisco By re, Environment nd plnning B, vol. 24, pp. 247 261, 1997. [5] K. C. Clrke nd J. Gydos, Loose-coupling cellulr utomton model nd GIS: long-term urbn growth prediction for Sn Frncisco nd Wshington/Bltimore, Interntionl Journl of Geogrphicl Informtion Science, vol. 12, pp. 699 714, 1998. [6] F. Wu nd C. J. Webster, Simultion of lnd development through the integrtion of cellulr utomt nd multi-criteri evlution, Environment nd Plnning B, vol. 25, pp. 103 126, 1998. [7] N. C. Goldstein, Brins vs. Brwn comprtive strtegies for the clibrtion of cellulr utomt Bsed Urbn Growth Model, 7th Interntionl Conference on GeoComputtion, Southmpton, UK, September 2003. [8] S. Alkheder nd J. Shn, Chnge detection - cellulr utomt method for urbn growth modeling Interntionl Society of Photogrmmetry nd Remote Sensing Mid-term Symposium, WG VII/5, Netherlnds, My 2006. [9] J. R. Anderson, E. E. Hrdy, J. T. Roch, nd R. E. Witmer, A lnd use nd lnd cover clssifiction system for use with remote sensor dt, USGS Professionl Pper 964, Sioux Flls, SD, USA, 1976. [10] J. H. Hollnd, Adpttion in nturl nd rtificil systems, University of Michign Press, Ann Arbor, MI, 1975. [11] D. E. Goldberg, Genetic lgorithms in serch, optimiztion, nd mchine lerning, Addison-Wesley publisher, MA., USA, 1989. [12] U. Bodenhofer, Genetic lgorithms: theory nd pplictions, Lecture notes. Johnnes Kepler University in Linz. http://www.flll.unilinz.c.t/teching/g/g-notes.pdf. 2004. [13] X. Yng, nd C. P. Lo, Modelling urbn growth nd lndscpe chnges in the Atlnt metropolitn re, Interntionl Journl of Geogrphicl Informtion Science, vol. 17, pp. 463 488. 2003.