Rapid urban growth can have undesirable impacts on agricultural lands, natural landscapes and public open spaces [1
]. In addition, urban planning, which influences the development of urban areas and the management of urban change [2
], addresses various natural and socioeconomic factors [3
]. Therefore, urban growth modelling can be an effective tool for investigating the results of different urban planning scenarios [8
Over the past two decades, many studies have focused on the modelling and dynamic simulation of urban development with the help of the spatial data analysis functionality of Geographic Information Systems (GIS), as well as statistical and artificial intelligence methods and geosimulation models such as the cellular automata (CA) (e.g., [9
However, in most of these studies, the relationships between urban development and driving forces are considered stationary. Moreover, it has been shown that the assumption of spatial stationarity in analyses of spatial relationships is unrealistic, especially for processes of land use change and urban expansion [20
Therefore, because of the ubiquity of spatial non-stationarity, it is inappropriate to assume constant values of model parameters over an entire study area [22
]. In such cases, local models can represent spatial variations more adequately than can the global models [23
Employing local models can improve the performance of dynamic simulations of land use change, and the definition of non-stationary transition rules in local CA models has led to improvements in urban growth predictions [24
Local models were first applied in analyses of clusters [29
] and hotspots [30
]. Then, they were introduced into a new field of analysis encompassing the relationships between independent and explanatory variables based on geographically weighted regression (GWR) models [22
]. Such models have been used to predict the values of spatial variables in various studies [32
], and in logistic form, these models been used for the production of urban growth probability maps [36
Machine learning algorithms, such as support vector machine (SVM) algorithms, are not as highly interpretable as statistical models. However, researchers have employed these algorithms because they can readily adapt to complex data sets and have strong nonlinear modelling capabilities [40
]. SVM has been applied to spatial data to produce probability maps for simulations of land use change (e.g., [42
In addition, the high level of efficiency of local learning algorithms, such as the k-nearest neighbours and radial basis function methods [45
], provides a basis for the local use of other learning algorithms, such as SVMs. Therefore, the development of local SVM (LSVM) models and the associated comparisons with global models have been described in the literature.
For example, Gilardi and Bengio [40
] demonstrated that local support vector regression displays a higher level of accuracy than does the global model. Considering the generalisation error, Ralaivola and d’Alché-Buc [46
] presented a methodology for training SVMs known as incremental learning, which is based on the number of nearest neighbours. Yang et al. [47
] proposed a weighted SVM model that was used to locally classify data based on the similarity between the training and testing data sets [48
Ladicky and Torr [50
] suggested a local linear SVM model that employs nonlinear manifold learning techniques. Moreover, Gu and Han [51
] compared a local clustered SVM, a combination of the k-nearest neighbours and SVM algorithms and the K-means SVM algorithm with linear and nonlinear global SVM (GSVM) models. The results showed that the clustered SVM algorithm displayed higher efficiency than the other methods. In addition, Andris et al. [52
] locally applied a linear SVM model to classify student enrolment in the United States.
A review of the research on LSVM models shows that this type of model has only been applied for data classification. The objective of this study is to develop an LSVM model that can be used to estimate the probabilities of urban expansion and to present a novel approach that integrates LSVM and GSVM models into a new model (the LGSVM model) to provide inputs for a CA and perform urban growth modelling.
This paper is structured as follows. First, the study area and the data used are introduced. Then, the LSVM and GSVM models are briefly presented. The method of calibrating the local and global models and the determination of the optimal bandwidth for the LSVM model are then described. Next, the results of the integrated models are presented and discussed, followed by the conclusions of the study.
2. Study Area
Tehran has grown extensively since the 1940s due to public and private investment and an increase in the population. In 1938, Tehran’s population was twice that of Mashhad (the second largest city in Iran); by 1966, it was 6.6 times larger [53
The population migration towards Tehran has been offset by factors such as high housing costs, but increased by factors such as the low cost of suburban properties [54
]. These factors have led to the development of an urban agglomeration around Tehran [55
]. High rates of development have occurred in some subregions in the suburbs of the Tehran metropolitan area.
For instance, one of these subregions, located in the southwestern Tehran suburbs, has developed around the Saeedi Highway. The city of Islamshahr has become the most important and largest urban centre in this subregion.
The study area investigated in this research covers the major parts of this urban region, which encompasses an area of 177 km2
and extends from 35°27′24′′ N to 35°36′05′′ N and 51°04′47′′ E to 51°18′12′′ E. The study area includes cities and towns such as Islamshahr, Golestan, Nasimshahr, Salehieh, Nasirshahr, Vavan and Qaemieh (Figure 1
illustrates the calibration results obtained with the LSVM model using the cross-validation method and based on the focal training sample. The X
-axis represents the bandwidth considering the number of nearest neighbours, and the Y
-axis shows the cross-validation error. Since the optimal bandwidth is based on the minimum value of the cross-validation error, the optimal bandwidth for the LSVM model is equivalent to 40 nearest neighbours.
The average RSS values obtained from the calibration of the GSVM model using the ten-fold cross-validation method with C
near the optimal values (C
= 1 and σ
= −4) are shown in Table 2
. Additionally, Figure 8
shows the probability maps obtained from the calibrated LSVM and GSVM models.
presents the relationship between WLocal
and the AUCs calculated from the ROCs of probability maps obtained from the global, local and integrated models. High AUC values indicate high reliabilities for the probability maps. However, although the differences in AUC values among the GSVM, LSVM and LGSVM models are low, the non-parametric statistical test [77
] shows that the AUC of the optimum LGSVM model (i.e., WLocal = 0.6
) is significantly higher than the AUCs of the GSVM and LSVM models at the 0.001 significance level (Table 4
; period: 1992–1996).
The mean values of kappa obtained from 10 simulations of urban development using the Urban-CA for the period of 1992–1996 are highlighted in Table 3
. The optimum value of α in the stochastic disturbance term is 0.01 for both the LSVM-CA and LGSVM-CA and 0.3 for the GSVM-CA. These results indicate that the accuracy of the LSVM-CA is greater than that of the GSVM-CA and that the integration of the local and global models increases the accuracy, where WLocal = 0.6
is the optimal weight. The kappa value of the optimum LGSVM-CA model increases by 18.1% and 5.3% relative to that of the GSVM-CA and LSVM-CA, respectively.
a shows the probabilities obtained from the LGSVM model, and Figure 9
b illustrates the prediction of urban development from 1992–1996 using the LGSVM-CA.
The accuracies of the urban development probabilities and the CAs for the validation period of 1996–2002 are presented in Table 5
. These predictions are based on the optimised parameters using the calibration period of 1992–1996.
The results show that the AUC of the LSVM probability map is higher than that of the GSVM probability map and that the AUC of the LGSVM probability map is higher than the AUCs of the other two maps, indicating higher accuracy. The differences in AUC values between the LGSVM and the GSVM and LSVM models are significant at the 0.001 significance level (Table 4
, period: 1996–2002).
In addition, similar to the results for the calibration period, the accuracy of the urban development prediction simulated by the LGSVM-CA is higher than the accuracies obtained with the global and local models. The kappa value of predicted urban growth using the LGSVM-CA is 1.9% to 2.9% higher than the values obtained using the LSVM-CA and GSVM-CA.
The local and global probabilities indicate that the deficiencies of one model can generally be offset by the advantages of the other model.
For example, Figure 10
a shows the map of differences between the local and global probabilities, i.e., PLocal
, around the cities of Golestan and Salehieh for the period of 1992–1996. Positive values indicate locations at which the local probability is greater than the global probability, and negative values indicate the opposite condition. Figure 10
b shows the land use in this area. At location 1, the high negative values indicate that the global probabilities are much higher than the local values despite the absence of a developed urban area at this location from 1992–1996. Thus, the LSVM model can increase the accuracy of urban prediction at this location. As illustrated in Figure 10
b, the land use at location 2 is agricultural, with no developed areas. Moreover, at this location, the probability of urban development derived from the GSVM model is low, whereas the LSVM model incorrectly estimates high probabilities. Notably, the influence of agricultural land use on urban development was considered in the GSVM model but not in the LSVM model.
Therefore, both the LSVM and GSVM models may yield low or high predictions at different locations. As a result, the integration of the LSVM and GSVM models can lead to higher accuracies than can either model alone.
In addition, a high correlation can be observed between the probability of urban development estimated by the LSVM and GSVM models and the density of urban development. To verify this finding, the kernel density function (KDF) [78
] was executed in GIS. First, developed urban cells were converted to points. Then, the KDF was implemented for these points. As shown in Figure 11
, in the KDF process, the number of points within the specified radius R is determined, and the points closer to the central point with respect to distance (r) will receive higher weights.
shows the urban development density maps produced by KDE for the periods of 1992–1996 and 1996–2002. Additionally, Table 6
presents the correlation between the density of urban development and calculated probabilities obtained by the SVM models from 1992–1996 and 1996–2002. As expected, the correlation values in the first period are relatively high and suggest that when the density of urban development is high, the probabilities estimated by the SVM models are also high.
However, Table 6
shows that the accuracy of the urban growth prediction decreases in the 1996–2002 period compared to that in the 1992–1996 period. This result is likely associated with the relationship between the density and probability. As Table 6
shows, the correlation between the density of urban development in this period and the estimated probability obtained by the GSVM, LSVM and LGSVM decreases by 14.5%, 12.5% and 11.9%, respectively. Thus, the location of urban development in the new period is not necessarily in the vicinity of the recently developed areas in the previous period, and differences exist in the density of urban development at nearby locations in the two periods.
In addition, the results of the correlation tests between probability maps of the two periods show that there is a relationship between probabilities of the calibration phase and validation phase. The correlation values are 0.718 and 0.698 for LSVM and GSVM models respectively.
In this situation, because of the temporal non-stationarity that exists for urban development at some locations, we can expect a decrease in the performance of urban growth simulations.
shows the difference between the density of urban development in the periods of 1996–2002 and 1992–1996 (i.e., KDE1996–2002
) near the cities of Golestan and Salehieh. High positive and negative values reflect the temporal non-stationarity of urban development, which may be a reason for the decrease in the prediction accuracy.
The results of this study indicate that like similar studies, the definitions of non-stationary transition rules can lead to improvements in simulation performance. Some improvements can be made by implementing a zoning approach [24
]; however, in this study, a local SVM model was developed to estimate the probability of urban development over a continuous surface to avoid discretisation issues. In addition, the improvement in simulation performance attained through integrating the local and global SVMs is another unique aspect of this study.
In this research, the number of explanatory variables used in urban development modelling was limited to only five based on the availability of local data. Generally, as additional local and historical socioeconomic data, such as population or land price data, become available, the results of urban growth modelling will be more similar to reality.
In this research, a local SVM model was designed to generate probability maps of urban development in a subregion southwest of the metropolitan area of Tehran. The model was calibrated using focal training data, and the optimum bandwidth was determined by the cross-validation method. The output is considered the basis for determining the optimum bandwidth.
For comparison, a nonlinear GSVM model was also calibrated by determining the optimal values of the parameters C and σ using the ten-fold cross-validation method. Closer investigation of the behaviours of the global and local models showed that both models have advantages and limitations in estimating the development probability. Therefore, an integrated model developed from a linear combination of the local and global SVM probabilities was also evaluated.
A comparison of the urban development probability maps produced from the local and global SVM models and the integrated models based on the AUCs for the periods of 1992–1996 and 1996–2002 revealed a higher accuracy for the LSVM model than the GSVM. Additionally, the integrated model exhibited a higher accuracy than of either of the other two models, and the LGSVM-CA outperformed the other CAs developed from global or local SVMs in the prediction of urban development for the periods of 1992–1996 and 1996–2002. The results showed that considering the temporal stationarity of urban development based on the location and area can improve simulations of urban growth.
The important contributions of this research are the development of a local SVM model for calculating the urban development probability and the integration of this model with a global SVM model to improve the accuracy of urban growth prediction. The results of the present study suggest that the integration of the LSVM and GSVM models can improve the predictive accuracy compared to that obtained using either the LSVM or GSVM model alone.
Our future work will focus on comparing the efficiencies of the LSVM and LGSVM models with the efficiency of the well-known logistic form of the GWR model in simulating land use changes and in urban development prediction.