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This article was downloaded by: [Inst Nac De Pesquisas Espacia] On: 4 August 2009 Access details: Access Details: [subscription number 913173452] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Geographical Information Science Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713599799 Using neural networks and cellular automata for modelling intra-urban land-use dynamics C. M. Almeida a; J. M. Gleriani b; E. F. Castejon c; B. S. Soares-Filho d a National Institute for Space Research (INPE), Remote Sensing Division—DSR, São José dos Campos, SP, Brazil b Federal University of Viçosa (UFV), Department of Forest Engineering—DEF, Campus Universitário, s/n-36571-000, Viçosa, MG, Brazil c National Institute for Space Research (INPE), Images Processing Division-DPI, São José dos Campos, Brazil d Federal University of Minas Gerais (UFMG), Centre for Remote Sensing—CSR/IGC, Belo Horizonte, MG, Brazil Online Publication Date: 01 January 2008 To cite this Article Almeida, C. M., Gleriani, J. M., Castejon, E. F. and Soares-Filho, B. S.(2008)'Using neural networks and cellular automata for modelling intra-urban land-use dynamics',International Journal of Geographical Information Science,22:9,943 — 963 To link to this Article: DOI: 10.1080/13658810701731168 URL: http://dx.doi.org/10.1080/13658810701731168 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. International Journal of Geographical Information Science Vol. 22, No. 9, September 2008, 943–963 Research Article Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Using neural networks and cellular automata for modelling intra-urban land-use dynamics C. M. ALMEIDA*{, J. M. GLERIANI{, E. F. CASTEJON§ and B. S. SOARESFILHO" {National Institute for Space Research (INPE), Remote Sensing Division—DSR, Avenida dos Astronautas, 1758-12227-010, São José dos Campos, SP, Brazil {Federal University of Viçosa (UFV), Department of Forest Engineering—DEF, Campus Universitário, s/n-36571-000, Viçosa, MG, Brazil §National Institute for Space Research (INPE), Images Processing Division-DPI, PO Box 515, São José dos Campos, Brazil "Federal University of Minas Gerais (UFMG), Centre for Remote Sensing—CSR/IGC, Avenida Antônio Carlos, 6627-31270-900, Belo Horizonte, MG, Brazil (Received 1 June 2005; in final form 13 July 2007 ) Empirical models designed to simulate and predict urban land-use change in real situations are generally based on the utilization of statistical techniques to compute the land-use change probabilities. In contrast to these methods, artificial neural networks arise as an alternative to assess such probabilities by means of non-parametric approaches. This work introduces a simulation experiment on intra-urban land-use change in which a supervised back-propagation neural network has been employed in the parameterization of several biophysical and infrastructure variables considered in the simulation model. The spatial land-use transition probabilities estimated thereof feed a cellular automaton (CA) simulation model, based on stochastic transition rules. The model has been tested in a medium-sized town in the Midwest of São Paulo State, Piracicaba. A series of simulation outputs for the case study town in the period 1985–1999 were generated, and statistical validation tests were then conducted for the best results, based on fuzzy similarity measures. Keywords: Neural networks; Cellular automata; Urban modelling; Land-use dynamics; Fuzzy similarity measures; Town planning 1. Introduction Cellular automata (CA) models consist of a simulation environment represented by a gridded space (raster), in which a set of transition rules determine the attribute of each given cell taking into account the attributes of cells in its vicinities. These models have been very successful in view of their operationality, simplicity, and ability to embody logics—as well as mathematics-based transition rules in both theoretical and practical examples. Even in the simplest CA, complex global patterns can emerge directly from the application of local rules, and it is precisely this property of emergent complexity that makes CA so fascinating and their use so appealing. *Corresponding author. Email: almeida@dsr.inpe.br International Journal of Geographical Information Science ISSN 1365-8816 print/ISSN 1362-3087 online # 2008 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/13658810701731168 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 944 C. M. Almeida et al. The first CA models applied to urban studies were usually based on very simple methodological procedures, such as the use of neighbourhood coherence constraints (Phipps 1989) or Boolean rules (Couclelis 1985) for the transition functions. Later on, successive refinements started to be incorporated into these models, like the adoption of dynamic transition rules (Deadman et al. 1993), which could change as conditions and policies within the township under study changed. Other examples in this direction are the work of Wu (1996), who conceived transition rules to capture uncoordinated land development process based on heuristics and fuzzy sets theory, and the work of Ward et al. (1999), in which transition rules are modified in accordance with the outcomes of the optimization of economic, social, and environmental target thresholds associated with sustainable urban development. CA transition functions have also been enhanced by the incorporation of decision-support tools, including AHP, i.e. analytical hierarchy process-based techniques, which have been strongly enabled by the linkages between CA and GIS (Engelen et al. 1997). Besides supporting CA internal operations (Clarke and Gaydos 1998, Li and Yeh 2000), GIS have also been useful in implementing cellular automata devices based on proximal models of space (Takeyama and Couclelis 1997) and in articulating spatial analysis factors of micro and macro scales (Phipps and Langlois 1997). Leading theoretical progresses in the broader discipline of artificial intelligence (AI), such as expert systems, evolutionary computation and artificial neural networks have recently been included in the domain of CA simulations. Artificial neural networks (ANN) attempt to simulate human reasoning (Moore 2000) offering fault-tolerant solutions. According to Fischer and Abrahart (2000), these mechanisms are able to learn from and make decisions based on incomplete, noisy, and fuzzy information. Works associating ANN with CA models for urban analysis are still quite limited in number. Li and Yeh (2001) conducted a simulation of land-use change for a city in southern China and its immediate surroundings, using ANN embedded in a CA model upon a dual-state approach (urban/non-urban). They further refined this model dealing with multiple regional land uses (Li and Yeh 2002) and simulations for alternative development scenarios (Yeh and Li 2003). Pijanowski et al. (2002a, b) carried out forecasts of urban growth for two different regions at the margins of Lake Michigan using neural nets to assess the importance of the land-use change drivers in a so-called ‘Land Transformation Model (LTM)’, which presents the four paradigms of cellular automata according to Batty et al. (1997): (i) space constituted by an array of cells, (ii) discretization of cells states and time, (iii) local influence neighbourhoods, and (iv) universally applied transition rules. All such investigations did not scale down at the intra-urban level, inasmuch as their scope concentrated on regional (macro scale) issues. More recently, similar studies also dealt with ANN-based CA simulation models for metropolitan areas: Detroit and the Twin Cities in Minnesota in the US (Pijanowski et al. 2005) and Beijing in China (Guan and Wang 2005), always focusing their attention on the urban sprawl phenomenon, and hence, generically categorizing the model states in a binary way (urban/non-urban). In contrast to these generalized approaches, the purpose of this paper is to deal with the simulation of multiple intra-urban land uses (e.g. residential, commercial, industrial, etc.) by means of an ANN-calibrated CA model. Scaling down at the intra-urban level enables researchers and town planners to better understand the city Modelling intra-urban land-use dynamics 945 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 structure and its functioning, which hence may support more sound planning policies and actions, based on a more solid knowledge of its inner land-use dynamics. One of the first proposals towards the use of ANN for urban simulations arose already in the second half of last decade, when Clarke et al. (1997), in view of the widely acknowledged and challenging complex nature of urban systems subject to rapid growth, stated that neural network methods could be highly suitable for modelling them. Researchers in this field have come to an agreement in recent years in the sense that such non-parametric approaches could better cope with the nonlinearities and chaotic behaviour of fast-changing urban environments (Li and Yeh 2002, Yeh and Li 2003, Guan and Wang 2005), given the ANN ability to handle the uncertainties, incompleteness, overdimensionality, and multimodal behaviour of spatial data (Openshaw 1998, Fischer and Abrahart 2000). 2. Artificial neural networks ANN can be simply defined as a massively parallel distributed computational device consisting of processing units, also called neurons or nodes, which are organized in a couple of layers. The neurons are entrusted with the storage of knowledge acquired within the system, which is then rendered available for further use (Haykin 1999). A neural network usually presents one input layer, one output layer, and one or more hidden layers (or eventually none) in between. These successive layers of processing units present connections running from every unit (neuron) in one layer to every unit in the next layer. The connections are responsible for passing information throughout the network, and they are characterized by weights, which are initially set in a random way and can be positive or negative (Bishop 1995). All the neurons, except those belonging to the input layer, perform two simple processing functions—receiving the signal (activation) of the neurons in the previous layer and transmitting a new signal as the input to the next layer. Training a feed-forward neural network with supervised learning consists in propagating forward an input signal (or pattern) in the net until activation reaches the output layer. This constitutes the so-called forward propagation phase. The output of the output layer is then compared with the teaching input. The error, i.e. the difference (delta) dj between the output oj and the teaching input tj of a target output unit j is then used together with the output oi of the source unit i to compute the necessary changes in link wij. To compute the deltas of inner units for which no teaching input is available, i.e. the units of hidden layers, the deltas of the following layer (which are already computed) are retrieved. In this way, the errors (deltas) are propagated backwards, and this exact phase is called backward propagation (Rumelhart et al. 1986). The training algorithm used in this work experiment was the ‘resilient backpropagation’, which is a local adaptive learning scheme, performing supervised batch learning in multi-layer neural networks. Basically, the backtracking step of the conventional back-propagation is no longer executed, if a jump over a minimum occurred. A weight-decay term (a) is also introduced in order to reduce the output error and the size of the weights as well, which is essentially meant to improve generalization. The composite error function is as follows: X X wij 2 ð1Þ ðti {oi Þ2 z10{a E~ 946 C. M. Almeida et al. Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 where ti is the teaching input of unit i; oi is the real output of unit i; a is the weight decay term; j is an index of a successor to the current unit i with link wij from i to j. The basic principle of the resilient back-propagation is to eliminate the harmful influence of the size of the partial derivative on the weight step. In this way, only the sign of the derivative is considered to indicate the direction of the weight update (Riedmiller and Braun 1993). The size of the weight change is solely determined by a specific ‘update-value’ D(t)ij: 8 ðt Þ ðtÞ > {Dij , if LE > Lwij w0 < ð tÞ ð2Þ Dwij ~ zDðtÞ , if LE ðtÞ v0 ij > Lwij > : 0, else where LE/Lwij(t) refers to the summed gradient information over all patterns of the pattern set (‘batch learning’). The second step of the resilient back-propagation learning is to determine the new update-values D(t)ij. This is based on a signdependent adaptation process according to the equation below: 8 ðt{1Þ ðt Þ ðt{1Þ > gz
Dij , if LELwij
LE > Lwij w0 > > < ðt{1Þ ðt Þ ð tÞ Dij ~ g{
Dðijt{1Þ , if LELw
LE Lwij v0 ij > > ð3Þ > > : ðt{1Þ Dij , else where 0vg{ v1vgz where g is the learning rate, which specifies the step width of the gradient descent. The resilient back-propagation is aimed at adapting its learning process to the topology of the error function, and hence, it follows the principle of ‘batch learning’. This implies that weight-update and adaptation are performed after the gradient information of the whole pattern set is computed (Riedmiller and Braun 1993). One of the greatest advantages of ANN is their ability to generalize. This implies that a trained net could classify data from the same class as the learning data that have never been presented to it before. In real-world applications, only a small part of all possible patterns for the generation of a neural net is at hand. In order to achieve the best generalization, the data set should be split into three parts (Haykin 1999, Fischer and Abrahart 2000): N N N the training set is used to train a neural net, and its error is minimized during training; the validation set is used to determine the performance of a neural network on patterns that are not trained during learning; the test set is meant for checking the overall performance of a neural net. The learning should be stopped when the validation set error reaches its minimum. At this very point, the net is able to attain the best generalization. If learning is not stopped, overtraining occurs, and the performance of the net for the entire data set will decrease, even though the error on the training data still becomes smaller. After concluding the learning phase, the net should be finally checked with the third data set—the test set. The neural net learning process is decisive for the success of the intra-urban landuse simulation model. In some cases, depending on the study area characteristics, Modelling intra-urban land-use dynamics 947 like the observable land-use spatial configuration patterns and the respective driving forces impelling land-use dynamics, the net outcomes can become highly sensitive in face of its architecture, learning algorithm, and internal parameters. These ANN outputs concern the land-use change suitability maps, which will inform the CA model of the exact cells that are most likely to undergo changes regarding their landuse status. Although output values from a neural net cannot be directly interpreted as probabilities in the strict sense, their logic resembles a transition probabilities ranking in the particular case of this experiment, and they will be referred to as landuse change probabilities hereafter. Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 3. ANN-based CA model for the simulation of intra-urban land-use change As previously stated, ANN offer a great number of advantages for modelling complex systems, of which urban areas are a major example. Their ability to be robust and noise-resistant regardless of redundant, missing, or fuzzy data, to handle nonlinear problems, to be unconstrained by the straitjacket of mathematical formulations, and to adapt to non-normal frequency distributions (Openshaw 1998) make their use suitable for unravelling the intricacies of the relationships between site attributes and urban dynamics of growth and change. The simulation model adopted in this experiment has been calibrated by neural networks, i.e. maps of land-use change probability were generated in the Stuttgart Neural Network Simulator (SNNS) software, where the teaching inputs were the maps of land-use change, and the respective variables corresponded to maps relating to the various types of site attractiveness, characterized by biophysical and infrastructure variables. These probability maps drove an open framework CA simulation model—DINAMICA (figure 1)—developed by the Centre for Remote Sensing of the Federal University of Minas Gerais—CSR/UFMG. DINAMICA is based on eight cell Moore neighbourhoods implemented by means of two empirical land-use allocation algorithms (or transition functions): ‘expander’ and ‘patcher’. The expander function accounts for the expansion of previous patches of a certain land-use class. The patcher function, on its turn, is designed to generate new patches through a seedling mechanism. In summary, the expander function performs transitions from a state i to a state j only in the adjacent vicinities of cells with state j. And the patcher function performs transitions from a state i to a state j only in the adjacent vicinities of cells with state other than j. These two processes can be merged into the following equation: Qij ~r
ðexpander functionÞzs
ðpatcher functionÞ ð4Þ where Qij corresponds to the total amount of transitions of type ij, and r and s are respectively the percentage of transitions performed by each function, with r + s51. According to Soares-Filho et al. (2002), both transition algorithms use a stochastic selecting mechanism. The applied algorithm consists in scanning the initial land-use map to sort out the cells with the highest probabilities and then arrange them in a data array. Following this procedure, cells are selected randomly from top to bottom of the data array (the internal stochastic choosing mechanism can be loosened or tightened depending on the degree of randomization desired). In a final step, the land-use map is again scanned to perform the selected transitions. In this case, the expander function does not perform the amount of estimated changes after a fixed number of iterations; it transfers to the patcher function a residual number of transitions, so that the total number of transitions always Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 948 C. M. Almeida et al. Figure 1. Schematic data model showing the loose coupling of the ANN simulator (SNNS) and the CA model (DINAMICA). amounts to a desired value (Soares-Filho et al. 2002). The desired transitions are obtained through a simple operation of cross-tabulation in the case of common simulations, where the initial and final land-use maps are available. In the case of forecasts, the transitions are estimated by means of the Markov chain, multitemporal series or any other alike statistical model meant for predictions. The expander algorithm is expressed by the following equation: If nj w3 then P0ij ðx, yÞ~Pij ðx, yÞ, else
 P0ij ðx, yÞ~Pij ðx, yÞ
nj 4 ð5Þ where nj corresponds to the number of cells of type j occurring in a 363 window. This method ensures that the maximum P9ij will be the original Pij, whenever a cell type i is surrounded by at least 50% of type j neighbouring cells. The patcher function is meant to simulate patterns of land-use change by generating diffused patches and at the same time preventing the formation of single isolated one-cell patches. This function employs a device that searches for cells around a chosen location for a given transition. This is achieved first by selecting the core cell of the new patch, and then by selecting a specific number of cells around the core cell according to their Pij transition probabilities. The expander and patcher functions, as previously mentioned, embody an allocation mechanism which is responsible for identifying cells with the highest transition probabilities for each ij transition. This allocation device stores the cells and organizes them for later selection. In this way, each newly selected cell will build a core for a new patch or an expansion fringe, which still need to be further developed by using one of these two transition algorithms. The sizes of the new patches and the expansion fringes are set according to a log-normal probability Modelling intra-urban land-use dynamics 949 distribution, whose parameters are determined as a function of the mean size and variance of each type of patch and expansion fringe to be generated (Soares-Filho et al. 2002). 4. Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 4.1 Applications Study area and the GIS database The ANN-based CA simulation model was applied to a medium-sized city, Piracicaba, located in the Midwest of São Paulo State, at the margins of the Piracicaba River, south-east of Brazil. The city comprised a total of 198 407 inhabitants in the initial time of simulation (1985), which rose to 309 531 inhabitants in 1999. In this period, the annual population growth rate was around 1.56%, and the resulting impact in the urban area was marked by the massive expansion of existing residential areas together with a mushrooming of peripheral residential settlements, which have been mostly incorporated into the main urban agglomeration. Besides experiencing a rapid development concerning the residential use, Piracicaba also witnessed intra-urban land-use changes like the increase in industrial, institutional, and leisure areas (figure 2). The city land-use maps in 1985 and 1999 were obtained from the Piracicaba Municipal Secretariat for Town Planning. They were scanned, converted to vector format in AutoCad, and then later pre-processed using SPRING GIS (developed by the Images Processing Division of the Brazilian National Institute for Space Research—DPI-INPE). These official maps do not always correspond to the real situation, since they do not indicate informal (not legalized) settlements on the one hand, and on the other hand, they show some of the legally approved settlements that have not been built. To cope with these eventual disparities, satellite imagery has been used to update the land-use maps exclusively regarding residential settlements. Two Landsat images (WRS 220/76) were employed for this end: a TM-5 scene of 10 August 1985, and a second scene of 16 July 1999. The latest image was georeferenced by means of an official topographic chart (UTM—SAD-69) with a scale of 1:50 000, and the total average error was 1.2 pixels (with the tolerance threshold lying around 1.5–2.0 pixels). It was then used for co-registering the 1985 image, and the total error amounted to only 0.3 pixel. The geographic coordinates of the control points were later used for the registration of the city maps in vector format using SPRING. Such maps were finally superimposed on linearly enhanced colour composites of the registered images (4R_7G_1B), allowing a visual crosscheck of existent and non-existent settlements. For the purpose of simplifying the land-use maps, they were subject to generalization procedures. Similar zones were reclassified to only one category, e.g. residential zones of different densities were all reclassified to simply residential, and special use and social infrastructure zones were reclassified to institutional. Eight land-use zone categories were defined: residential, commercial, industrial, services, institutional, leisure/recreation, water bodies, and non-urban use. Districts located farther than 10 km from the main urban agglomeration were excluded from the analysis, and the traffic network and minor non-occupied areas were disregarded in the simulations. All data used in this application were resampled to 50 m650 m for a better visual adequacy of the maps (coarser resolutions would otherwise result in unpleasant jags), but also with the aim of keeping a number of cells that would enable faster Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 950 C. M. Almeida et al. Figure 2. Generalized land-use maps in Piracicaba in 1985 (left) and 1999 (right). simulations. The adopted resolution formed a grid containing 334 lines and 360 columns, with a total of 120 240 cells (30 060 ha) defining the region for simulation. 4.2 Exploratory analysis One of the first steps in the exploratory analysis is the identification of the existing types of intra-urban land-use change. A simple cross-tabulation (figure 3) between the initial and final land-use maps provides this information (table 1), besides quantifying the amount of change in terms of percentage, also called global transition rates (table 2). These rates express the likelihood of change in the study area as a whole, regardless of the influence of the driving (biophysical and Figure 3. Land use change in Piracicaba from 1985 to 1999. 951 Modelling intra-urban land-use dynamics Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Table 1. Existent land-use transitions. Notation Land-use transition NU_RES NU_IND NU_INST NU_LEIS Non-urban Non-urban Non-urban Non-urban to to to to residential industrial institutional leisure/recreation infrastructure) variables. In some cases, they can be referred to as ‘global transition probabilities’. After identifying the existent land-use transitions, the next step concerns the determination of the different sets of variables governing each of the four types of change. The variables available for the modelling analysis do not always represent the set of necessary variables able to produce ideal simulation results. In fact, intraurban land-use dynamics is subject to sudden and unforeseeable forces, like landlords’ decisions to develop certain areas in disregard of others. In other words, land-use transitions are more likely to be determined by some factors than others depending on the processes of acquisition and development undertaken by developers and consumers of land (Almeida et al. 2003). But in a general way, there is indeed a set of decisive factors for urban land-use transitions, in the sense that they substantially respond for the main drivers of such changes. And these precise factors have effectively guided the modelling experiment at issue. The several maps of biophysical and infrastructure variables have been generated on the basis of the information extracted from land-use maps, like distances to certain uses, distances to the Piracicaba River as well as distances to different categories of the traffic network, such as paved and non-paved urban or interurban roads. In all cases, the Euclidean distance has been used, and methods to initially sort out the ideal set of variables to explain a given type of land-use change were based on heuristic procedures. These procedures basically regard the visualization of distinct maps of variables (distances in grey scale) superposed on maps of land-use transition, so as to identify those that are more meaningful to explain the different types of land-use change. A map of land-use transition was made for each respective type of land-use change (NU_RES; NU_IND; NU_INST; NU_LEIS) through reclassification of the cross-tabulation map. These maps of transition indicate with different colours the areas of change and no change, and a black colour is assigned to areas not associated with the land-use change under consideration, i.e. all areas Table 2. Global transition rates for Piracicaba, 1985–1999. Land use NonU Res Comm Indust Inst Serv Water Leis/ Rec NonU Res Comm Indust Inst Serv Water bodies Leis/rec 0.8353 0 0 0 0 0 0 0 0.1501 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0.0113 0 0 1 0 0 0 0 0.0028 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0.0005 0 0 0 0 0 0 1 952 C. M. Almeida et al. Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Table 3. Definition of the land-use change driving variables. Notation Biophysical or infrastructure land use change variable dist_riv dist_com dist_ind dist_inst dist_res dist_leis int_rds main_rds Distances Distances Distances Distances Distances Distances Distances Distances to to to to to to to to rivers commercial zones small-sized industrial zones institutional zones residential zones leisure/recreation zones main interurban roads main paved and non-paved urban and interurban roads where the land-use at the initial time of simulation differs from the initial use of the transition considered. All the overlays were carried out in spring, which enables transparency in-between layers. The natural environment (soil, vegetation, relief, conservation areas) has not been regarded as a decisive driver for land-use change. In other words, natural characteristics of the physical environment, excluding the Piracicaba River, have not been considered as impedances to urban growth at a more generalized level. The city sites are relatively flat, with mild slopes, and present no constraints regarding soil, vegetation and conservation areas. All variables selected to integrate the simulation model and their respective notations are presented in table 3, and the sets of variables chosen to explain each of the four transitions are indicated in table 4. The maps of variables in grid format represent the input neurons, while the map of land-use transition consists in the net teaching input, as previously mentioned. 4.3 Training The DINAMICA model is driven by ‘n’ maps of transition probabilities which relate to each of the ‘n’ existent types of land-use transition. Given this constraint, each transition in this experiment was separately handled in the SNNS, and for each of the four types of land-use change, there was a complete process of training, validation, and testing of the respective neural net. One of the key issues in this research was the design of the neural networks architecture. Many authors have proposed mathematical formulations to meet this end, but to date they have not yet come to an agreement as to the methods to determine the optimal number of hidden layers or the ideal number of neurons in the hidden layer. It is always desirable to work with reduced sizes of nets, in which Table 4. Selection of variables determining land-use change. Variables dist_riv dist_com dist_ind dist_inst dist_res dist_leis int_rds main_rds Nu_Res Nu_Ind Nu_Inst Nu_Leis N N N N N N N N N N N 953 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Modelling intra-urban land-use dynamics fast training and good performance can be ensured. For Kavzoglu and Mather (1999), nets with more neurons or layers present the advantage to learn more complex patterns, besides being less influenced by the initial random weights (Paola and Schowengerdt 1997). But these complex neural nets, on the other hand, demand more time for training and do not generalize well with unknown data, given the excessive memorization of noise found in the training data sets (Haykin 1999). According to Kolmogorov’s theorem, any continuous function W: XnRRc can be implemented through a three-layer neural network with n neurons in the input layer, (2n + 1) neurons in the single hidden layer, and c nodes in the output layer (Wang 1994). Fletcher and Goss (1993) suggested that the ideal number of neurons in the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hidden layer would be between 2n + 1 and 2nzm, where n corresponds to the number of neurons in the input layer, and m, to the number of neurons in the output layer. Despite these divergences, there is a general consensus among researchers in the field of urban CA modelling that empiricism is a reasonable way to determine the best neural net structure for a specific problem (Li and Yeh 2001, Yeh and Li 2003, Guan and Wang 2005). All the four neural nets used in this experiment present only one hidden layer with six neurons. The neurons in the input layer correspond to the driving variables respectively available for each land-use change, and the output layer presents just a single neuron, which refers to the local land-use transition probabilities (table 5). The internal parameters of the nets were heuristically determined. The maps of land-use transition, used as teaching inputs, were converted into grids, where 0.99 was assigned to the areas of change, 0.01, to the areas of no change, and 0.000001 to the areas not concerned in the land-use change under consideration. In the face of the sigmoid nature of the activation function, absolute values of 0 and 1 were avoided in order to prevent excessively large values of weights (Kavzoglu and Mather 2003) and, hence, biases in the numerical ranking of the output nets. The grids generated in SPRING with extension SPR were converted into ASCII format for their insertion in the neural network simulator (SNNS). A special routine was created in C + + to randomly select 12 000 grid points, which accounts for nearly 10% of the total amount of pixels in the study area, and to further organize them in a array of 120 lines by 100 columns. Samples of this size were used both as the training and validation sets, and the whole area was used as the test set. As already mentioned, the learning was interrupted when the validation set error reached its minimum (figure 4). The SNNS displays three different types of error Table 5. Parameters used in the neural nets. Type of transition Training algorithm NU_RES Resilient BackProp Resilient BackProp Resilient BackProp Resilient BackProp NU_IND NU_INST NU_LEIS Maximum Initial Input Hidden Number update-value step size (D0) (Dmax) neurons neurons of cycles Weightdecay (a) 2 6 30 0.1 50 4 4 6 20 0.1 50 4 2 6 20 0.1 50 4 3 6 20 0.1 50 4 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 954 C. M. Almeida et al. Figure 4. Error decay curve for the training set (dark grey) and validation set (light grey) regarding the ‘NU_INST’ neural net. according to the number of cycles: the overall sum-of-the-squares of the output error (SSE); the overall mean square error (MSE), which refers to the average of the square of the error, namely the average of the difference between the desired response or teaching input (ti) and the actual system output (oi); and the SSE divided by the number of output units (SSE/out), which, for the kth training pattern, is: SSE=out~ X 1 ðtk {ok Þ2 no: of outputs ð6Þ The output grids were converted in thematic maps so as to allow a visual comparison between such maps of local transition probabilities (output neurons) and the respective maps of land-use transition, which correspond to the teaching inputs (figure 5). This comparison enables a final empirical evaluation of the neural nets performance. It is worth remarking that the DINAMICA model automatically sets to 0 (zero) the areas where the land use at the initial time of simulation differs from the initial use of the transition considered, in the cases when the maps of transition probabilities are generated within the DINAMICA environment through statistical methods. Since the maps of transition probabilities were produced by the SNNS, areas with high transition probabilities were somehow overestimated. But this surplus is disregarded when the DINAMICA algorithms entrusted with the cells’ land-use change scan the maps of transition, for they take into account the actual cells use according to the initial land-use map, and hence, the areas that can effectively undergo changes are considerably reduced. 5. Validation For assessing the accuracy of the CA simulation model performance, fuzzy similarity measures applied within a neighbourhood context were used. Several validation methods operating on a pixel vicinity basis have been proposed (Costanza 1989, Pontius 2002, Hagen 2003), aimed at depicting the spatial patterns similarity between a simulated and reference map, so as to relax the strictness of the pixel-bypixel agreement. The fuzzy similarity method employed in this work is a variation of 955 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Modelling intra-urban land-use dynamics Figure 5. Estimated transition probability surfaces and land-use change: 1985–1999. Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 956 C. M. Almeida et al. the fuzzy similarity metrics developed by Hagen (2003), and has been implemented in the DINAMICA model by the CSR team. Hagen’s method is based on the concept of fuzziness of location, in which the representation of a cell is influenced by the cell itself and, to a lesser extent, by the cells in its neighbourhood. Not considering fuzziness of category, the fuzzy neighbourhood vector can represent the fuzziness of location. In the fuzzy similarity validation method, a crisp vector is associated with each cell in the map. This vector has as many positions as map categories (land uses), assuming 1 for a category5i, and 0 for categories other than i. Thus, the fuzzy neighbourhood vector (Vnbhood) for each cell is given as: 2 3 mnbhood1 6m 7 6 nbhood2 7 7 Vnbhood ~6 ð7Þ .. 7 6 5 4 . mnbhoodC   mnbhood ~mcrisp i, 1
m1 , mcrisp i, 2
m2 , . . . , mcrisp i, n
mn  ð8Þ where mnbhood i represents the membership for category i within a neighbourhood of N cells (usually N5n2); mcrisp ij is the membership of category i for neighbouring cell j, assuming, as in a crisp vector, 1 for i and 0 for categories other than i (i,C); mj is the distance-based membership of neighbouring cell j, where m accounts for a distance decay function, for instance, an exponential decay (m522d/2). The selection of the most appropriate decay function and the size of the window depend on the vagueness of the data and the spatial error tolerance threshold (Hagen 2003). As the intention is to assess the model spatial fit at different resolutions, besides the exponential decay, a constant function equal to 1 inside the neighbourhood window and to 0 outside can also be applied. Equation (11) sets the category membership for the central cell, assuming that the highest contribution is found within a neighbourhood window n x n. Next, a similarity measure for a pair of maps can be obtained through a cell-by-cell fuzzy set intersection between their fuzzy and crisp vectors:    ð9Þ S(VA ,VB )~½mA,1 ,mB, 1 jMin ,mA,2 ,mB, 2 jMin ,:::::::::::,mA,i ,mB, i jMin Max where VA and VB refer to the fuzzy neighbourhood vectors for maps A and B, and mA,i and mB,i are their neighbourhood memberships for categories i,C in maps A and B, as in equation (10). According to Hagen (2003), since the similarity measure S (VA, VB) tends to overestimate the spatial fit, the two-way similarity is applied instead: Stwoway (A,B)~jS (VnbhoodA ,VcrispB ),S(VcrispA ,VnbhoodB )jMin ð10Þ The overall similarity of a pair of maps can be calculated by averaging the twoway similarity values for all map cells. However, when comparing a simulated map to the reference map (real land use in the final time of simulation), this calculation carries out an inertial similarity between them due to their areas that did not suffer any change. To avoid this problem, the CSR team introduced a modification into the overall two-way similarity method of DINAMICA, using two maps of differences, which present value 1 for the cells that underwent change, and 0 for those that did not change. In this way, each type of change is analysed separately 957 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Modelling intra-urban land-use dynamics using pairwise comparisons involving maps of differences: (i) between the initial land-use map and a simulated one and (ii) between the same initial land-use map and the reference one. This modification is able to tackle two matters. First, as it deals with only one type of change at a time, the overall two-way similarity measure can be applied to the entire map, regardless of the different number of cells per category. Second, the inherited similitude between the initial and simulated maps can be eliminated from this comparison by simply ignoring the null cells from the overall count. However, there are two ways of performing this function. One consists of counting only two-way similarity values for non-null cells in the first map of difference, and the other consists in doing the opposite. As a result, three measures of overall similarity are obtained, with the third representing the average of the two ways of counting. As random pattern maps tend to score higher due to chance depending on the manner in which the nulls are counted, it is advisable to pay close attention to the minimum overall similarity value. This method has proven to be the most comprehensive when compared with the aforementioned methods, as it yields fitness measures with the highest contrast for a series of synthetic patterns that depart from a perfect fit to a totally random pattern (Soares-Filho et al. 2004). 6. Simulations and discussion Based on the assignment of appropriate weights to the input variables, the SNNSgenerated maps of local transition probabilities drove the CA simulation model— DINAMICA. This model is also guided by internal parameters, determined empirically, and which concern the average size and variance of patches and the relative proportion of the transition algorithms (table 6). Due to the randomness of the DINAMICA transition algorithms, even though the same internal parameters are kept in different runs, different simulation results will be produced after each run of the model. In this way, the three best urban land-use simulation results for the city of Piracicaba in the period 1985–1999 are presented in figure 6. It is observed that the transition from non-urban to residential use (NU_RES) largely depends on the previous existence of residential settlements in their surroundings, because this implies the possibility of extending existing nearby infrastructure. It also depends on the available accessibility to such areas. The implementation of large institutional areas (NU_INST) occurs near peripheral roads and previously existent institutional areas, since they grow as extensions of already established institutional zones. Likewise this transition, the expansion of industrial districts (NU_IND) also requires the proximity to previously existent industrial zones and the availability of road access. This can be explained by the fact that in the industrial production process, the output of certain industries represents Table 6. DINAMICA internal parameters for the simulation of urban land-use change in Piracicaba: 1985–1999. Type of transition Average size of patches (ha) NU_RES NU_IND NU_INST NU_LEIS 300 150 75 20 Variance of patch size Proportion of Proportion of (ha) ‘expander’ ‘patcher’ 30 1 1 0 0.85 0.45 1.0 1.0 0.15 0.55 0 0 Number of iterations 10 10 10 10 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 958 C. M. Almeida et al. Figure 6. Three best simulations compared with actual land use in 1999. The central commercial zone (orange) and the services corridors (dark yellow) did not undergo any transitions during the observed time period. The new residential (dark blue) and institutional areas (light yellow) as well as the leisure and recreational zones (red) were well modelled, particularly in the first and second simulations. the input of other ones, which raises the need of rationalization and optimization of costs by the clustering of plants interrelated in the same production chain. This transition supposes as well the nearness to the labour-force supply centres (peripheral residential areas) and also a location not too far from commercial zones, since industrial activities depend on the supply of commercial goods. New leisure and recreation zones (NU_LEIS) also take place adjacent to already existent zones of this type, since they are commonly created as extensions of the latter ones. These areas are created along low and flat riverbanks, since they are floodable and hence unsuitable for sheltering other urban uses. They are also strategically located in relation to their catchment area, i.e. near central residential areas, which are those sheltering higher population densities and which in fact correspond to the demand market for leisure and recreation. It is implied by the above analyses that the landuse transitions show compliance with economic theories of urban growth and change, where there is a continuous search for the optimal location, able to ensure 959 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 Modelling intra-urban land-use dynamics competitive real estate prices, good accessibility, rationalization of transportation costs, and a strategic location in relation to supply and demand markets. The simulation accuracies were around 84.5% (table 7), which is quite acceptable. The institutional and leisure/recreational zones, respectively in light yellow and red, were well modelled in all simulations. The residential areas, in dark blue, were considerably well simulated in all of the three modelling results, especially in S1 and S2. The expansion of the industrial zone (green) located to the north was rather well modelled, but the new industrial district that arose in the south-eastern sector of the city could not be detected in any of the simulations. This may be ascribed to several concurrent reasons. First, the NU_IND map of transition probabilities generated by the SNNS does not assign high probability values to the south-eastern portion of the city given the very specific generalizing characteristic of the neural net training. Second, due to the random nature of the transition algorithms, only the areas with the highest transition probability values tend to be selected for change, and the industrial district situated south-east presents probability values ranging from medium to medium low. Third, in terms of what was exposed above and also of the adopted average size of NU_IND patches, the newly generated industrial area in the north is prone to concentrate all the amount of transitions determined by the global probabilities. Possible solutions to overcome this drawback would include the adoption of a smaller average size of patches for this transition and the generation of a greater number of simulation outputs. In a similar work carried out for the same area (Almeida 2003), in which the DINAMICA model was parameterized by the weights of evidence method, based on Bayes’ theorem, the industrial district located in the city south-eastern sector could be detected in some of the simulations. This can be explained by the fact that this Bayesian method works with discrete variables only, and the highest weights have been precisely assigned to the ranges of distances containing the two industrial districts implemented within the simulation period (1985–1999). Although the ANN and the weights of evidence simulations offered similar accuracy rates in the particular case of this experiment—85% and 86%, respectively—the former method was able to ensure a much faster operationalization of the model calibration, once it deals with continuous distances. On the other hand, in the method known as weights of evidence, the model performance is entirely dependent on the modeller’s ability to define the best distances intervals, which is generally a time-consuming procedure. 7. Final remarks and directions for future work This study has demonstrated that neural networks can be appropriately integrated with cellular automata for simulating intra-urban land-use dynamics. Cities are open and nonlinear complex systems (Yeh and Li 2003). Defining parameter values Table 7. Fuzzy similarity measures for the three best simulations of urban land-use change in Piracicaba: 1985–1999. Fuzzy similarity measures Simulations (Sn) S1 S2 S3 Window 363 Window 565 Window 767 0.839 0.845 0.834 0.842 0.848 0.836 0.845 0.850 0.838 Downloaded By: [Inst Nac De Pesquisas Espacia] At: 21:38 4 August 2009 960 C. M. Almeida et al. for assessing the relative importance of the intra-urban land-use change drivers in traditional CA models is usually done on a basis of trial-and-error approaches, which rely on the test of many possible parameter values in the search of an optimal fit (Li and Yeh 2002). These procedures are commonly computation-intensive and time-consuming, and do not always provide the best results. Neural networks, in view of their ability to model nonlinear features and handle noisy, redundant, and inaccurate spatial data, were shown to be robust and efficient for calibrating such models, thus saving time and effort in automatically determining these parameter values. The ANN-based CA simulation model has been successfully applied to a mediumsized town, Piracicaba, located in São Paulo State, south-east of Brazil. DINAMICA is endowed with a stochastic structure, which embodies unpredictability and chance in the logic of land-use change as observed in reality. And this is precisely what differs DINAMICA from other exclusively ANN-based simulation models (Yeh and Li 2003, Pijanowski et al. 2005), in which neural networks are used not only for parameterizing the model, i.e. assessing the variables weights, but also for accomplishing land-use transitions. In such models, randomization does not play a direct role in the transitions themselves. The ANN-generated maps of transition probabilities reflect the influence the set of selected variables have in defining the compatibility extent between the predicted and real land-use change areas, the latter shown in the land-use transition maps. Researchers, planners, practitioners, and consultants in the urban field are able to deal interactively with the model, so as to include or suppress one or more variables and then evaluate the resulting impacts such changes produce in the land-use change probabilities and land-use simulation maps. In this sense, the SNNS-generated maps of transition probabilities can help planners and policy makers understand the spatial driving forces and current trends of intra-urban dynamics, and hence subsidise their actions towards urban development and regulations as well as technical and social infrastructure implementation. Since knowledge can be easily learned by the model and stored for further simulation (Li and Yeh 2001), future land-use change alternatives could also be simulated on the basis of ANN-calibrated land-use transition probability maps, so as to anticipate possible urban development scenarios and orientate upcoming planning actions and policies. Neural networks present though some inherent limitations in the sense that they do not offer explicit knowledge on the process of assessing the weights of variables driving land-use change. Moreover, the user’s intervention in defining the training algorithm, the net architecture and its parameters are still important for the quality of results. The SNNS environment itself enables users to initially assign more importance to certain driving variables (input neurons) in comparison with others, according to their a priori judgements. In this way, further studies are needed to assess the responsiveness of simulations in the face of variations in the type, structure, and internal parameters of the network. Acknowledgements The authors wish to thank the Piracicaba Water Supply and Waste Water Disposal Department and the Piracicaba Planning Secretariat for providing the city maps. We are also grateful for the help and cooperation of the technical and administrative staff of the Centre for Remote Sensing of the Federal University of Minas Gerais (CSR-UFMG). This study has received financial support from the Remote Sensing Modelling intra-urban land-use dynamics 961 Division of the Brazilian National Institute for Space Research (DSR/INPE). And finally, the authors would like to thank the anonymous reviewers for their valuable contributions in improving the quality of this paper. 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