The Effect of Observation Scale on Urban Growth Simulation Using Particle Swarm Optimization-Based CA Models

Abstract

Cellular automata (CA) is a bottom-up self-organizing modeling tool for simulating contagion-like phenomena such as complex land-use change and urban growth. It is not known how CA modeling responds to changes in spatial observation scale when a larger-scale study area is partitioned into subregions, each with its own CA model. We examined the impact of changing observation scale on a model of urban growth at UA-Shanghai (a region within a one-hour high-speed rail distance from Shanghai) using particle swarm optimization-based CA (PSO-CA) modeling. Our models were calibrated with data from 1995 to 2005 and validated with data from 2005 to 2015 on spatial scales: (1) Regional-scale: UA-Shanghai was considered as a single study area; (2) meso-scale: UA-Shanghai was partitioned into three terrain-based subregions; and (3) city-scale: UA-Shanghai was partitioned into six cities based on administrative boundaries. All three scales yielded simulations averaging about 87% accuracy with an average Figure-of-Merit (FOM) of about 32%. Overall accuracy was reduced from calibration and validation. The regional-scale model yielded less accurate simulations as compared with the meso- and city-scales for both calibration and validation. Simulation success in different subregions is independent at the city-scale, when compared with regional- and meso-scale. Our observations indicate that observation scale is important in CA modeling and that smaller scales probably lead to more accurate simulations. We suggest smaller partitions, smaller observation scales and the construction of one CA model for each subregion to better reflect spatial variability and to produce more reliable simulations. This approach should be especially useful for large-scale areas such as huge urban agglomerations and entire nations.

1. Introduction

Cellular automata (CA) is a well-known bottom-up self-organizing model for simulation of contagion-like phenomena such as complex land-use change and urban sprawl [1]. It is also used to project future scenarios under variable conditions and strategies [2,3,4,5]. By defining land transition rules, CA models can be used to explore long-term dynamics of land-use change on regional to global scales [6,7,8,9]. These land transition rules are determined by the combined effects of the current cell state, factors driving cell state change, neighborhood, spatial and quantitative constraints, and stochasticity. These rules are usually deployed in GIS-based software to simulate changing spatial phenomena. Many studies have demonstrated that CA-based simulations are affected by spatial scale (i.e., cell size or spatial resolution) and observation scale (i.e., the spatial extent of the study area) [10,11]. CA studies of scaling effects are therefore important for accurate land-use modeling and more reliable future scenario projections.

Cell size is defined by side dimension of a square pixel [10]. Cell sizes ranging from 10 m to 1 km [12,13] have been applied to urban growth modeling. Previous studies have demonstrated complex simulation responses to changing cell size. Seemingly small changes in cell size can lead to significant impacts on transition rules and simulation results [11,14]. Some have argued that overall simulation agreement decreases as the cell size increases [15], and this has been confirmed by other modelers in modeling dynamic urban growth using multiple cell sizes [16]; but others have noted the opposite phenomenon, i.e., that overall simulation agreement increases as the cell size increases [17]. Previous studies have demonstrated the importance of spatial cell size in CA-based urban modeling, noting that the finest cell size available (e.g., 10 m) is not always the best choice [11].

The neighborhood effect is unique to CA modeling because it evaluates the impact of nearby cells on the central cell [10,11]. The neighborhood configuration represents the spatial extent (neighborhood size) and the shape that neighbors cover. Kocabas and Dragicevic [18] noted that, for the same CA models, smaller differences in neighborhood size can result in more similar outputs. However, incorrect representation of land-use transition may occur when large cell sizes are used [10]. The issue can be addressed by increasing neighborhood size, yielding a stabilizing effect [19]. Studies have shown that increased neighborhood size helps achieve better CA model outcomes. For example, Liao et al. [20] noted a distant-decaying neighborhood effect that can improve simulations of complex land-use change.

Apart from cell size and neighborhood size, observation scale also substantially impacts CA model performance. Published study areas range from city-scale to regional-scale, and from national-scale to global-scale [6,7,9,21]. The choice of observation scale often depends on data availability and on the research purpose [10]. Most modelers have focused on the simulation of land-use change at the city-scale [22,23,24]. At the regional-scale, urban agglomeration dynamics (e.g., Yangtze River Delta and Pearl River Delta) and their future scenarios have been simulated using CA models [25,26]. Urban growth simulation has also been done at national-scales [6,27] and global-scales.

While regional-scale studies are typically of interest to modelers, most workers have applied only one model across the chosen study area. This may lead to difficulty in acquiring accurate land-use patterns and driving factors, and may require huge computational capabilities [28]. To address this problem, a large region can be partitioned into subregions for more efficient modeling. Each partition may yield different land transition rules. Ke et al. [29] applied k-means clustering to partition Wuhan (China) and calibrated a CA model in each subregion to simulate the urban expansion. By considering spatial heterogeneities in each subregion, these partition-based CA models produced more accurate simulation results than a single, large-scale CA model. Despite the progress, it is not clear how the spatial observation scale relates to the effects of biophysical, socioeconomic and proximity factors on urban growth, or how CA models respond to changing spatial observation scale if one study area is partitioned using different strategies and a simulation is performed in each subregion. Therefore, an investigation of model response to changing observation scales is worthy of further examination.

This paper examines the impact of observation scale on CA modeling of urban growth. We seek to understand how CA models respond to partitioning strategies that are related to different observation scales. We examined the particle swarm optimization-based CA model (PSO-CA), an heuristic CA model proposed by Feng et al. [16]. PSO is a population-based stochastic optimization algorithm that was developed by simulating the social behavior of bird flocking. This method shares many similarities with problem-solving computation techniques such as differential evolution (DE) and genetic algorithm (GA). PSO-CA has been applied to simulate the urban growth of cities such as Tehran (Iran), Fengxian (China), Xiamen (China), and Tianjin (China) [30,31]. In another study, Feng et al. [25] compared PSO-CA with other CA methods based on generalized simulated annealing (GSA) and GA by simulating the urban growth dynamics in the Yangtze River Delta. Each application has proved the effectiveness of the PSO-CA model, particularly when comparing CA modeling based on logistic regression (LR). We applied PSO-CA to simulate urban growth in the urban agglomeration within a one-hour high-speed rail distance from Shanghai (UA-Shanghai) using three observation scales. We calibrated our PSO-CA models based on 1995–2005 urban growth and validated them using 2005–2015 urban growth, aiming to examine the response of the models to observation scale and ultimately to identify the best scale. Our research is practically useful in predicting alternative urban scenarios as well as examining the responses of these scenarios to their driving factors and issued urban planning regulations.

2. Study Area and Data

2.1. The Study Area: UA-Shanghai

UA-Shanghai lies in the hinterland of the Yangtze River Delta, and is the most economically-active resource allocation center in China [32]. Our study area has an area of about 35,000 km² (Figure 1a,b), of which more than 25% is characterized by the high slope and/or water bodies. UA-Shanghai consists of six large cities: Shanghai, Suzhou, Wuxi, Jiaxing, Huzhou, and Hangzhou (Figure 1c). In this research, Hangzhou city is part of the Hangzhou administrative area, excluding the four satellite cities Lin’an, Tonglu, Chun’an, and Jiande. According to official 2015 statistical data, the UA-Shanghai had 56 million residents with a population density of 1600 persons per km², and its gross domestic product (GDP) reached 6250 billion RMB Yuan (~930 billion USD). The huge population and vibrant economy have resulted in unprecedented rapid urbanization and demand for land resources. Classification of Landsat imagery shows a ~23% increase in urban built-up area from 1995 to 2015, reaching a total area of 9558 km² in 2015.

To test the PSO-CA models, we used three partitioning strategies based on different observation scales. The regional-scale uses UA-Shanghai as a single study area (Figure 1a); the meso-scale separates UA-Shanghai into three regions that reflect terrain (Figure 1b); while the city-scale uses administrative boundaries to divide the area into six cities (Figure 1c). At the meso-scale, the flat zone includes Shanghai and Jiaxing, the hilly zone includes Suzhou and Wuxi, and the mountainous zone includes Huzhou and Hangzhou.

2.2. Raw Data and Spatial Variables

To identify land-use patterns, we assembled Landsat images from 1995, 2005 and 2015 from the Geospatial Data Cloud and classified them using the Decision Tree Classifier in ENVI 5.2. We identified three dominant classes: Urban, non-urban, and water body. After classification, the land-use maps were resampled at 120 m (2159 rows and 2114 columns) to reduce computational load. Urban growth from 1995 to 2005 was the dependent variable used in calibrating PSO-CA.

In urban modeling, biophysical factors, socioeconomic factors, and proximity (

Table 1. Biophysical, socioeconomic and proximity factors driving UA-Shanghai urban growth.

Covariate	Abbreviation	Description	Literature Examples
Biophysical factor	Elevation	Effect of terrain conditions on urban growth	[8,24]
Socioeconomic factor	PPP	Effect of population (population per pixel) on urban growth	[30,36]
	GDP	Effect of economy on urban growth	[33,37]
Proximity disturbance	DisBuilt	Distance to the 1995 urban areas	[15,30]
	DisCenter	Distance to the city centers	[30,38]
	DisCounty	Distance to the county centers	[20,39]
	DisRoad	Distance to the main roads	[24,40]
	DisRailway	Distance to the railway	[29,41]
	DisShoreline	Distance to the shoreline	[25,42]

) are commonly taken as independent variables that drive urban growth [33,34,35]. There are many biophysical factors that affect urban growth. These include terrain drivers (e.g., elevation and slope), ecological drivers (e.g., ecological risk), environmental drivers (e.g., air pollution), and land surface parameters (e.g., urban heat islands). While the inclusion of more biophysical factors is useful to construct CA models, these factors may result in multicollinearity that reduces model accuracy. We collected GDEMV2 data from the Geospatial Data Cloud to derive elevation (Figure 2a) as a proxy for the effect of biophysical conditions on urban growth. To reflect socioeconomic aspects, we applied the rasterized population (PPP) data (Figure 2b) to represent the residential effect, and applied rasterized GDP (Figure 2c) to reflect economic impact. The PPP data were collected from WorldPop and the GDP data were collected from the Earth Observation Group. To identify proximity factors, we calculated the distance (Figure 2e–i) to 1995 urban areas, city centers, county centers, major roads, railways, and shoreline (Figure 2d–i) using a vector dataset assembled from the administrative map of the UA-Shanghai. The distance to the 1995 urban areas (DisBuilt) reflects the impact of built-up areas on the urban growth, while the other five distance-based spatial variables reflect the impact of urban infrastructure on urban growth (Table 1). Once calculated, driving factors were resampled to 120 m to match the land-use maps.

3. PSO-CA Modeling

3.1. Workflow

Our workflow (Figure 3) has four steps: (1) Data processing, which produces 1995, 2005 and 2015 land-use maps and extracts the spatial factors that drove 1995–2005 urban growth; (2) PSO training, which retrieves parameters for the transition rules using selected samples and PSO; (3) calibration, which constructs the PSO-CA model using the 1995 pattern as the starting map and establishes transition rules to simulate the 2005 urban pattern; and (4) validation, which predicts the 2015 urban pattern using the PSO-CA model. Calibration is closely related to the validation, since both compare simulation results with reference patterns [43]. We applied a cell-to-cell comparison between simulation results and reference patterns for the 1995–2005 and 2005–2015 intervals to assess and validate the model.

To examine the effects of observation scale on simulation results, we worked at three scales as shown in Figure 1. Each partition has a different spatial extent but the sum of the partitions is the same total study area. All used the same samples to calibrate the PSO-CA models. There is one regional-scale model, three meso-scale models, and six city-scale models. Each produced one simulation for 2005 and one prediction for 2015 (Figure 3).

3.2. The PSO-CA Model

PSO-CA is a hybrid method that incorporates transition rules derived by PSO into a typical CA model [16]. In CA modeling, the state of cell i at time t + 1 is determined jointly by its state at time t, interactions among its neighboring cells, specific constraint factors, and the land transition probability. Urban transition rules can be written [41,44] as:

$$Land{S}_{i,t+1}=f\left(Land{S}_{i,t},Nbr,Cst,T_p\right)$$

(1)

where$f\left(\right)$ is the CA transition function; $Land{S}_{i,t+1}$ and $Land{S}_{i,t}$ represent the states of cell i at times t + 1 and t, respectively;$Nbr$ represents the effect of the interaction among neighboring cells; $Cst$ represents the constrained conditions on urban growth; and $T_p$ represents the land transition probability retrieved based on a set of driving factors.

A unique feature of CA models is their “bottom-up” interaction among neighboring cells that can result in complex systems that mimic the real world. The neighborhood configuration ($Nbr$) therefore has a substantial effect on CA models and their simulation results. In a typical CA model, square 3 × 3, 5 × 5 and 7 × 7 pixel neighborhoods are commonly applied, where each neighboring cell has an equal opportunity to affect the state of the cell i [45,46]. We followed earlier publications and adopted a 5 × 5 neighborhood configuration. It has been noted that a spatially heterogeneous neighborhood configuration better reflects the complex interactions among neighbors, as compared with a spatially homogeneous neighborhood configuration. The spatial heterogeneity in urban systems can be represented using the urban growth gradient, derived from the gradient of land-use change. We then multiplied the gradient map with the 5 × 5 homogeneous neighborhood to produce the heterogeneous neighborhood, where each cell may yield a different effect.

Our PSO-CA model incorporates spatial and nonspatial constraints (Cst) in allocating urban cells. The spatial constraint commonly refers to broad water bodies, wetlands, and ecologically valuable lands that are institutionally undevelopable. These spatially-constrained areas persist during the model run. The nonspatial constraint refers to the total land available for development [39], which defines the total quantity of the urban land at the boundary time and helps identify optimal probability thresholds.

The land transition probability ($T_p$) is a temporally stationary element that considers the overall effect of biophysical factors (B), socioeconomic factors (S), and the proximity (P) to urban facilities. It is commonly calculated as [24,47]:

$$T_p=\frac{\exp \left(z\left(B+S+P\right)\right)}{1+\exp \left(z\left(B+S+P\right)\right)}$$

(2)

where $z\left(B+S+P\right)$ represents the collective impact of the three categories of factors described above. Typically, the impacts can be rewritten as:

$$\mathrm{z}\left(B·S·P\right)=a_0+a_1·DEM+a_2·POP+a_3·GDP+a_4·DisBuilt+a_5·DisCenter +a_6·DisCounty+a_7·DisRoad+a_8·DisRailway+a_9·DisShoreline$$

(3)

where $a_0$ is an adjustment item, and the $a_1\dots a_9$ represent the weight for each factor. Consequently, $a=\left(a_0\dots a_9\right)$ are the CA parameters in the transition rules. They are frequently retrieved using statistical methods [48] such as LR, spatial autoregressive model, and geographically weighted regression.

To address land-use and urban growth dynamics adequately, many artificial intelligence (AI) methods were applied to accurately define CA parameters [49]. Among AI methods, PSO uses a one-way sharing scheme to deliver information from a better particle (solution) to other particles [50]. The algorithm is guided by a fitness function that represents the fitting residuals using a set of selected samples. We followed earlier work [16] by using the root-mean-square error (RMSE) to reflect the model residuals. The reason for using PSO is minimization of the model RMSE, which served as the fitness function for PSO. Clearly, the fitness function links the CA transition rules with the PSO algorithm. The fitness function is:

$$Min F\left(a\right)=\sqrt{\frac{{{\displaystyle \sum }}_i^{NUM}{\left(T_p\left(a\right)-T_o\right)}^2}{NUM}}$$

(4)

where $Min F\left(a\right)$ is the fitness function to be solved, $T_p\left(a\right)$ is the predicted transition probability, $T_o$ is the observed cell state change, and $NUM$ is the total number of samples used to train the transition rules.

In PSO, we assume that there are M particles in an N-dimensional space, where each particle has two parameters: Position and velocity. Position represents the fitness value of the particle, and the velocity represents the change of position. The effect of the previous velocity on the current velocity is expressed as an inertia weight w. The particle adjusts its search direction based on its own position and that of other particles. If the fitness value of the current particle is better than its previous value, its position is considered best locally and will be shared with other neighboring particles at the next iteration. Two optimal positions in the search space are recorded at each iteration: (1) The best achieved by an individual particle; and (2) the global best achieved by the particle swarm. Each particle updates position and velocity according to these two optimal positions. The position and velocity are constrained for each particle to prevent overflying the global optimum. To avoid prematurely falling into a local optimum, PSO is improved by: (1) Defining appropriate PSO controlling parameters; and (2) integrating PSO with other search algorithms [51,52,53]. PSO can apply the Broyden-Fletcher-Goldfarb-Shanno (BFGS) search strategy to re-optimize the solutions when all particles converge locally. The algorithm stops if it encounters either the absolute convergence tolerance or the maximum number of iterations.

We performed our PSO-CA modeling using UrbanCA software developed by the first author that incorporates two PSO methods. PSO maximizes a fitness function (Figure 4a) whereas PSOv2 minimizes a fitness function (Figure 4b). The fitness functions for these two methods are therefore negatively correlated in urban CA modeling. PSO was programmed using the R-Gui package psoptim while the PSOv2 was programmed using the R-Gui package pso. Note that the controlling parameters of the left one were explicitly displayed on the graphical user interface while the parameters of PSOv2 were wrapped in the codes. Following Feng et al. [25], the absolute convergence tolerance was assigned to 1 × 10⁻¹⁰, maximum iterations were 5000, and the initial solution was assigned to zero.

3.3. Validation Methods

In the validation phase, outcomes were compared with classifications using a cell-by-cell comparison that reports the simulation successes and errors [41,54,55]. Overall accuracy is comprised of Correct Rejections and Hits, where a Correct Rejection means that observed non-urban persistence is correctly simulated as non-urban persistence, and a Hit represents an observed urban expansion being correctly simulated as urban expansion [56]. Overall accuracy is the most widely applied metric that incorporates persistence and change. In contrast, Figure-of-Merit (FOM) is the percent correctness that focuses on the change to assess a CA model, with a higher FOM indicating a stronger simulating ability of the model. We applied overall accuracy and FOM to assess and validate our PSO-CA models. FOM can be calculated as [57]:

$$\mathrm{FOM}=\frac{\mathrm{Hits}}{\mathrm{Hits}+\mathrm{Misses}+\mathrm{False}\mathrm{alarms}}$$

(5)

The simulation errors are broken down into Misses and False Alarms. Misses indicate that observed urban expansion was incorrectly simulated as non-urban persistence, and False Alarms indicate that observed non-urban persistence was incorrectly simulated as urban expansion [56].

4. Results

4.1. Transition Rules and Land Transition Probability Maps

Table 2. Cellular automata (CA) parameters retrieved by PSO at different observation scales.

Variable	Regional-scale	Meso-scale			City-scale
	UA-Shanghai	Shanghai-Jiaxing	Suzhou-Wuxi	Hangzhou-Huzhou	Shanghai	Jiaxing	Suzhou	Wuxi	Hangzhou	Huzhou
Constant	−0.14	−0.49	0.13	−0.41	−0.02	−1.90	−0.95	3.14	0.34	−1.59
DEM	−9.78	16.88	−9.20	−17.83	12.70	21.12	−24.47	−0.35	−34.82	−33.67
PPP	−1.28	−2.08	−2.10	1.07	−3.16	17.15	−3.21	−2.34	4.15	−19.96
GDP	1.03	0.99	5.46	12.17	0.16	3.29	9.98	−1.21	−0.48	48.26
DisBuilt	−16.15	−17.69	−16.60	−7.93	−15.17	−8.64	−11.73	−30.28	−6.73	−10.42
DisCenter	−0.13	0.56	−0.16	1.25	0.24	0.20	1.15	−1.84	−1.17	2.40
DisCounty	−1.76	−2.28	−1.06	−0.93	−1.21	−2.65	−1.39	−1.74	−1.40	−0.32
DisRoad	−3.25	−4.88	−2.26	−2.47	−3.48	−2.53	−0.54	−10.56	0.29	−12.73
DisRailway	−0.71	−0.31	−1.45	0.16	−1.22	0.37	−1.23	−0.07	1.15	1.65
DisShoreline	−0.08	−0.24	−0.83	−2.55	−1.78	0.17	0.33	−3.69	−2.01	−1.07

shows that a factor may have different values and even different signs for its CA parameters in different subregions, suggesting the changing effect of a driving factor across space. Except for PPP and GDP, a negative parameter reflects a promoting effect on urban growth while a positive parameter reflects an inhibitory effect. The DEM parameters are positive in Shanghai-Jiaxing at the meso-scale and Shanghai and Jiaxing at the city-scale, but negative in other regions, suggesting the possible inhibitory effect of terrain on urban growth. Note that Shanghai and Jiaxing are very flat and lie at low elevations, implying no significant DEM effect on urban growth in these regions. PPP is positively correlated with, and has a promoting effect on, urban growth in Hangzhou-Huzhou, Jiaxing and Hangzhou. PPP is negatively correlated with, and has an inhibitory effect on, urban growth elsewhere. GDP is negatively correlated with, and has an inhibitory effect on, urban growth in Wuxi and Hangzhou. GDP is positively correlated with, and has a promoting effect on, urban growth elsewhere.

DisBuilt and DisCounty are persistent promoting factors in explaining 1995–2005 urban growth. DisCenter promotes urban growth in UA-Shanghai, Suzhou-Wuxi, Wuxi and Hangzhou, but inhibits urban growth elsewhere. DisRoad promotes urban growth everywhere except Hangzhou at the city-scale. DisRailway promotes urban growth everywhere except Jiaxing, Hangzhou and Huzhou, and DisShoreline promotes urban growth everywhere except Jiaxing and Suzhou.

We merged each subregion at the meso- and city-scales to build the full study area, and produced transition probability maps at all three scales (Figure 5a–c). To compare transition probability maps, we produced difference maps (Figure 5d–f) by subtracting one probability map from another. Figure 5a–c shows significant differences in the transition probability at each scale, especially between regional- and meso-scale and between regional- and city-scale. Most transition probabilities at the regional-scale are higher than those at the meso- and city-scales (Figure 5d,e), while there are only minor differences in most areas between the meso-scale and the city-scale (Figure 5f), indicating that these two scales are spatially similar. The meso-scale has higher transition probabilities in Wuxi and Hangzhou as compared to the other two scales, whereas the city-scale has higher transition probabilities in Suzhou City center and in the satellite cities of Jiaxing and Huzhou.

4.2. The Simulation and Prediction Results

We simulated the 2005 urban patterns of all regions for each scale using PSO-CA modeling, and merged all subregions at the meso- and city-scales into the full study area. Figure 6a–c is highly similar among the three simulated urban patterns, but their local differences are significant. At all three scales, the 1995–2005 urban growth occurred primarily adjacent to the urban areas that existed in 1995. By comparison, more urban growth was simulated at the regional-scale in Shanghai (Figure 6a), the meso- and city-scales both generated more urban growth in Suzhou (Figure 6b,c), and the meso-scale produced the least urban growth in Huzhou and Jiaxing but more in Hangzhou. For further analysis, we overlaid the 1995 actual urban pattern, the 2005 actual urban pattern, and each 2005 simulated urban pattern to identify simulation successes and errors (Figure 6d–f). The overlaid maps show that the meso-scale generated more simulation successes (in red) in Shanghai (Figure 6d), the meso-scale generated more simulation successes (in red) in Hangzhou and Suzhou (Figure 6e), and the city-scale generated more simulation successes (in red) in Huzhou and Jiaxing (Figure 6f).

Table 3. Overall accuracy and FOM of 2005 simulation results for each region. For the regional-scale, the study area was partitioned into subregions to match smaller scales for assessing the model results; for the meso- and city-scales, the UA-Shanghai was composed of subregions defined in Figure 1.

Region	Overall Accuracy (%)			FOM (%)
	Regional-Scale	Meso-Scale	City-Scale	Regional-Scale	Meso-Scale	City-Scale
UA-Shanghai	86.7	86.8	86.6	31.4	31.9	31.2
Shanghai-Jiaxing	85.2	85.6	86.2	33.1	29.6	31.8
Suzhou-Wuxi	84.3	84.1	83.1	32.6	33.4	30.8
Hangzhou-Huzhou	90.4	90.4	90.1	26.5	32.6	30.9
Shanghai	82.8	82.8	83	35.9	32.6	31.3
Jiaxing	89.6	90.6	91.9	23.3	18.6	33.2
Suzhou	83.9	83.3	82.9	31.1	32.6	32
Wuxi	84.9	85.4	83.3	35	34.8	28.6
Hangzhou	86.4	84.7	86	29.3	36.1	34.9
Huzhou	93.8	95.4	93.8	20.4	20	21.5
Frequency of most accurate	3	4	3	3	5	2

lists the overall accuracy and FOM of the simulation results for each region at all scales. The two metrics show very different assessment results. Only three regions (UA-Shanghai at the meso-scale, Hangzhou-Huzhou at the meso-scale, and Jiaxing at the city-scale) yield both high overall accuracy and FOM. Due to the exclusion of persistent non-urban, FOM is considered to be more accurate in comparing PSO-CA models at each scale. Both metrics indicate that the meso-scale was the most accurate simulation in more regions, four regions for overall accuracy and five regions for FOM, as compared with the other scales. This indicates that our PSO-CA models probably produce the most accurate simulations at the meso-scale, where the regions are grouped according to their terrain conditions. For the entire UA-Shanghai study, overall accuracy and FOM both indicate that our PSO-CA models simulated the urban patterns most accurately at the meso-scale.

We predicted 2015 urban patterns using the land transition probability maps produced based on the 1995–2005 datasets. While the three predicted 2015 urban patterns are similar, visual inspection also shows significant differences (Figure 7a–c). The new 2005–2015 urban areas grew mainly near 2005 urban areas at all three scales. Comparison shows that the regional-scale predicted more urban growth at Shanghai but less at Suzhou (Figure 7a), both the meso-scale and the city-scale produced more urban growth at Suzhou (Figure 7b,c), and the meso-scale predicted the least urban growth at Jiaxing and Huzhou but more at Hangzhou. Our PSO-CA models are therefore highly similar between 1995–2005 and 2005–2015 in regard to quantity accuracy. Spatial simulation correctness and error were identified by overlaying the 2005 actual urban pattern, the 2015 actual urban pattern, and each 2015 predicted urban pattern (Figure 7d–f). The three maps in the right column show that the regional-scale yielded more prediction success in Shanghai (Figure 7d), the meso-scale yielded more prediction successes in Wuxi and Hangzhou (Figure 7e), and the city-scale yielded more prediction successes in Jiaxing, Suzhou and Huzhou (Figure 7f). This indicates that, from calibration to validation, our regional-scale PSO-CA model has good predictive ability in simulating urban growth at Shanghai, PSO-CA has good predictive ability at Hangzhou, while our city-scale PSO-CA model has good predictive ability at Huzhou.

We calculated the overall accuracies and FOMs (

Table 4. Overall accuracy and FOM of the 2015 prediction results for each region.

Region	Overall Accuracy (%)			FOM (%)
	Regional-Scale	Meso-Scale	City-Scale	Regional-Scale	Meso-Scale	City-Scale
UA-Shanghai	83.2	84.1	84.4	21.1	23.7	24.5
Shanghai-Jiaxing	82.1	84	85.4	23	21.5	25.8
Suzhou-Wuxi	80.2	80.9	80.6	22.6	26	25.3
Hangzhou-Huzhou	87.1	87.1	86.7	15.9	23	21.7
Shanghai	81.7	83.1	84.2	26	25.1	23.6
Jiaxing	82.9	85.5	87.5	16.5	13	30.1
Suzhou	78.9	79.4	79.2	21.2	24.8	26.7
Wuxi	82.2	83.4	82.8	25.2	28.4	22.5
Hangzhou	86.6	84.6	87.3	19.8	29.8	27.2
Huzhou	87.6	89.4	86.3	11.8	11.9	16.6
Frequency of most accurate	0	5	5	1	4	5

) for all the prediction results using the three maps in the right column of Figure 7. Both measures indicate that, when compared with the other two scales, the regional-scale is less accurate in predicting the 2015 urban pattern because about 70% of the overall accuracies and six-tenths of their FOMs are the lowest. The overall accuracy (Table 4) shows that the most accurate predictions in five regions were achieved at the meso- and city-scales, while the FOM (Table 4) demonstrates that the city-scale predictions were the most accurate in more regions. This suggests our PSO-CA models performed better in simulating urban growth when smaller observation scales are used.

To compare performance of the PSO-CA models at both the calibration and prediction stages, we calculated their change rates (CRate) in simulation successes as:

$$CRate=\frac{2015OA-2005OA}{2005OA}=\frac{2015\mathrm{FOM}-2005\mathrm{FOM}}{2005\mathrm{FOM}}$$

(6)

where OA is the overall accuracy. A positive CRate indicates performance improvement, while a negative CRate indicates a performance decline.

Table 5. Change rate (CRate) in performance of the PSO-CA models from calibration (1995–2005) to validation (2005–2015).

Region	Change Rate for Overall Accuracy (%)			Change Rate for FOM (%)
	Regional-Scale	Meso-Scale	City-Scale	Regional-Scale	Meso-Scale	City-Scale
UA-Shanghai	−4	−3	−3	−33	−26	−21
Shanghai-Jiaxing	−4	−2	−1	−31	−27	−19
Suzhou-Wuxi	−5	−4	−3	−31	−22	−18
Hangzhou-Huzhou	−4	−4	−4	−40	−29	−30
Shanghai	−1	0	1	−28	−23	−25
Jiaxing	−7	−6	−5	−29	−30	−9
Suzhou	−6	−5	−4	−32	−24	−17
Wuxi	−3	−2	−1	−28	−18	−21
Hangzhou	0	0	2	−32	−17	−22
Huzhou	−7	−6	−8	−42	−41	−23
Mean	−4	−3	−3	−33	−26	−21

shows that, from calibration (1995–2005) to validation (2005–2015), performance of our PSO-CA models has substantially declined, and only Shanghai and Hangzhou for the city-scale showed increased overall accuracy. The CRates of the overall accuracies fall off less rapidly when compared to the FOMs. Specifically, reductions in overall accuracy are all less than 10%, while the reduction in FOM is about 26%. This may be attributed to the less changeable persistent non-urban from calibration to validation. For overall accuracy, our PSO-CA models decreased the most in all ten regions at the regional-scale, but had the smallest decreases in nine of all ten regions at the city-scale. For FOM, our PSO-CA models decreased the most in all ten regions at the regional-scale, but had the smallest decreases in four regions under the meso-scale and six regions under the city-scale. This indicates that the PSO-CA models had better predictive power at the city-scale when partitioned by administrative boundaries, and the FOMs under the meso-scale that are grouped by terrain conditions are acceptable in regions such as Shanghai and Hangzhou.

4.3. Relationship between Simulation Accuracy and Urban Growth Rate

For the 1995–2005 calibration, overall accuracy and FOM are closely related to the urban growth rate, and the goodness-of-fit R² for overall accuracy is better than that of FOM (Figure 8). The meso-scale has the highest R², implying the strongest relation between the simulation accuracy and the terrain-based partition. In contrast, the city-scale has the lowest R² implying the weakest relation between simulation accuracy and administration-based partitioning. The change may be attributed to the fact that we applied three PSO-CA models at the meso-scale but six models at the city-scale.

Specifically, overall accuracy is negatively correlated with urban growth rate, while FOM is positively correlated with urban growth rate. At all three scales, Figure 8 shows that a lower urban growth rate (e.g., at Huzhou) is associated with a higher overall accuracy and a lower FOM; whereas a higher urban growth rate (e.g., at Shanghai) is associated with a lower overall accuracy and a higher FOM. This indicates that a region (e.g., at Huzhou) with a lower urban growth rate has more non-urban areas, leading to a higher null success (correct rejection) and hence a higher overall accuracy; however, in this area our PSO-CA models simulated less urban growth correctly, resulting in a lower FOM. In contrast, the decrease of overall accuracy in a more rapidly urbanizing area (e.g., Shanghai) led to lower null success and lower overall accuracy but higher FOM. The fitted lines (Figure 8b,e) at the meso-scale have the largest absolute slopes, indicating that the simulation performance of our PSO-CA models is more sensitive to urban growth rate at this scale. This also suggests that partitioning based on terrain (the meso-scale) had more impact on the simulations than the other two strategies in the calibration stage. In Strategies 1 and 2, four cities have FOMs lower than the average of the entire study area, while the other two cities have higher FOMs (Figure 8d,e). At the city-scale, four cities have higher FOMs but two cities have lower FOMs (Figure 8f) as compared to the entire study area. This suggests that PSO-CA modeling yields better performance when done at the smaller city-scale.

As shown by goodness-of-fit R² in Figure 9, the correlation between 2005–2015 urban growth rate and simulation accuracy significantly decreased during the prediction stage, except for overall accuracy at the regional-scale and the meso-scale. The goodness-of-fits for the overall accuracy are higher than those for the FOM, confirming the significant contribution of null success to the correlations. Shanghai produced the highest FOM at the regional-scale, Hangzhou has the highest FOM at the meso-scale, and Jiaxing yielded the highest FOM at the city-scale, where each of these cities did not yield the highest urban growth rate 2005–2015 in the corresponding scale. In contrast, at all three scales, Huzhou has the lowest FOM corresponding to the lowest urban growth rate. This suggests that the correlation between the urban growth rate and the simulation accuracy is more complex in the prediction stage than in the later calibration stage. Over the two periods, the entire study area shares a similar urban growth rate, but there is a significant spatial difference in rate across regions. While the 1995–2005 land transition probability maps cannot accurately reflect actual land transition during 2005–2015, they can reflect the performance of our PSO-CA models. It is important to note that both overall accuracy and FOM for the entire study area increased from the regional-scale to the city-scale, demonstrating better performance at city-scale.

4.4. Discussion

We used three partitioning strategies (see Figure 1) to examine the impact of observation scale on CA-based urban growth modeling. The regional-scale has no partitions, the meso-scale divided the study area into three terrain-based regions, and the city-scale partitioned the study area into six administrative city-level regions. Using the three partition strategies, we calibrated PSO-CA models using 1995–2005 urban growth in UA-Shanghai and validated these models using 2005–2015 urban growth. This produced one PSO-CA model at the regional-scale, three models at the meso-scale, and six models at the city-scale. Among the scales, the meso-scale yielded the most accurate simulations in more regions during calibration, while the city-scale yielded the most accurate predictions during validation. This means that the partitions substantially impacted model performance and that a smaller spatial observation scale may lead to more accurate simulations. The changes in model performance imply that the worst partitioning strategy for calibration (regional scale) was definitely also the worst for validation. In contrast, the best partition strategy (meso-scale) for calibration may not be optimal for validation.

During calibration, we observed a good correlation between simulation accuracy and urban growth rate (except for FOM at the city-scale), whereas during validation, simulation accuracy and urban growth rate were not correlated except for overall accuracy of the regional-scale and the meso-scale. These observations suggest that land transition probabilities constructed using calibration data more accurately reflect 1995–2005 urban growth but less accurately reflect 2005–2015 urban growth. Most importantly, simulation accuracies in different subregions are more closely related when using a single CA model (e.g., regional-scale), as compared with those using multiple CA models (e.g., city-scale).

In addition, modeling urban growth dynamics of cities of the Yangtze River Delta is of great interest to model developers and geographers [58,59,60]. These important cities include Shanghai, Hangzhou, Jiaxing, Ningbo, Nanjing, and Suzhou [38,45,61]. Methods that could be applied to construct CA models include LR, deep-belief-network, geographically weighted regression (GWR), cuckoo search, bat movement, variable weights LR, CA-Markov, PSO, GA, and GSA. These methods have overall accuracies ranging from 86% to 96% [61,62]; whereas, our PSO-CA models have achieved an averaged overall accuracy about 87% at the calibration stage and about 84% at the validation stage. Early publications cannot be directly compared with more modern research because the studies used different raw data, calibration years, and independent variables. Overall, each of these models has demonstrated simulation and prediction abilities in modeling urban growth dynamics. We believe that we have achieved better quantity agreement between simulated and actual results, leading to the ability to control the quantity in our PSO-CA models. The good performance of our PSO-CA models in quantity and allocation terms may be attributable to PSO’s “bottom-up” information-sharing method, similar to the “bottom-up” approach of CA. We have addressed model sensitivity to observation scales—an important issue that had not yet been clarified in the earlier work.

Finally, our PSO-CA models have another advantage in optimizing fitness functions with complex equality and inequality constraints. Complex fitness functions can be solved using heuristics such as GA and SA [63,64], but they cannot be solved using conventional and spatial statistical methods such as LR and GWR [40,65]. This feature of PSO facilitates urban growth modeling focusing on special conditions, and is helpful in building future urban scenarios under different conditions, providing useful suggestions to assess and adjust urban planning regulation and policy. For example, the effects of road development on urban future scenarios can be examined using our PSO-CA model by applying inequalities of roads to the PSO’s objective function. Such inequalities can make the factor more significant than the others and can make it the most influential one. The effect of the other factors on future scenarios can be examined using alternative PSO’s objective functions that are subject to similar inequalities. The inequality-based objective function can therefore lead to suitable CA models that simulate alternative urban scenarios with different patterns. Centralizing each factor using PSO could alter forecasts dramatically; for example, emphasizing the effect of eco-systems could produce urban scenarios with green settlements. However, in this research we did not perform future scenario prediction at UA-Shanghai because our objective was to examine the impact of the observation scales on urban growth modeling. Our study is an example of how to select an observation scale suitable for calibration of the best CA models. Further work should examine the optimal spatial resolution and neighborhood size corresponding to different observation scales, and study the impact of multi-observation scale on urban future scenario prediction.

5. Conclusions

CA models and their simulation results are substantially affected by partitioning of the study area and its observation scale (spatial extent). The impact of observation scale on modeling was examined by simulating urban growth in the UA-Shanghai region using PSO-CA. At three observation scales, we calibrated PSO-CA models in each subregion from 1995 to 2005 and validated these models from 2005 to 2015. All three observation scales produced reasonably accurate simulations and predictions, with an average overall accuracy about 85% and an averaged FOM of about 27%. There was a decline in simulation accuracies of our PSO-CA models from calibration and validation. In calibration and validation, the regional-scale with no partition yielded less accurate simulations when compared with smaller scale partitioning of the study area. The simulation successes in different subregions are less related to each other when applying a different PSO-CA model in each subregion, compared with a single CA model across the entire study area. The variation in simulation results was attributed to the varying effect of biophysical, socioeconomic and proximity factors at different observation scales. Future scenarios of sustainable urban growth can be generated by emphasizing eco-system drivers at different scale. In summary, (1) the choice of partition substantially impacts model performance and (2) a smaller observation scale leads to more accurate simulations.

This work improves our understanding of the impact of scale on modeling urban growth dynamics, and provides an example of the effect of choice of observation scale on urban simulations. We suggest applying partitions to retrieve smaller observation scales and constructing a single CA model in each partitioned subregion to adequately reflect spatial differences and to produce more reliable simulations. Our research is also practically helpful in projecting alternative urban scenarios and exploring the effects of driving factors and issued urban planning regulations on these scenarios. These suggestions should be particularly useful for large areas such as urban agglomerations and entire nations. Further work should examine the optimal spatial resolution and neighborhood size that corresponds to different observation scales, and should investigate the impact of multi-observation scale on urban future scenario prediction.