Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: a Case Study of Shenzhen in China

International Journal of Geo-Information

Article Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China

Xiaoming Man and Yanguang Chen *

Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China; [email protected] * Correspondence: [email protected]

Received: 7 October 2020; Accepted: 11 November 2020; Published: 13 November 2020

Abstract: Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in Western countries follow the common logistic function, while the fractal dimension growth curves of cities in northern China follow the quadratic logistic function. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of urban evolution and attempt to analyze the reasons. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in southern China. The urban region is divided into four subareas using ArcGIS technology, the box-counting method is adopted to extract spatial datasets, and the least squares regression method is employed to estimate fractal parameters. The results show that (1) the urban form of Shenzhen city has a clear fractal structure, but fractal dimension values of different subareas are different; (2) the fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) the peak of urban growth in Shenzhen had passed before 1986 and the fractal dimension growth is approaching its maximum capacity. It can be concluded that the urban form of Shenzhen bears characteristics of multifractals and the fractal structure has been becoming better, gradually, through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen’s urban growth are similar to those of European and American cities but differ from those of cities in northern China. This suggests that there are subtle different dynamic mechanisms of city development between northern and southern China.

Keywords: urban form and growth; fractal cities; fractal dimension growth curve; Shenzhen city

1. Introduction A study on cities begins from description and ends at understanding. To describe an urban phenomenon, we have to find its characteristic scales. Traditional mathematical methods and quantitative analysis are based on a typical scale, which is often termed characteristic length [1–3]. Unfortunately, spatial patterns of cities have no characteristic scale and cannot be effectively described by conventional measures such as length and area. At this scale, the concept of characteristic scales should be substituted with scaling ideas. Fractal geometry provides a powerful mathematical tool for scaling analysis of urban form and growth. From remote sensing images, the urban form resembles ink splashes, usually presenting a highly irregularity and self-similarity at several different scales [4,5]. It implies that it does not obey Gaussian law and traditional measures and mathematics models cannot effectively describe it [6,7]. Fractal geometry provides a proper quantitative approach in

ISPRS Int. J. Geo-Inf. 2020, 9, 0672; doi:10.3390/ijgi9110672 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2020, 9, 0672 2 of 15 this aspect [8,9]. The fractal dimension, especially multifractal parameter spectra, can be utilized to characterize spatial heterogeneity and explore the spatial complexity [10]. Ever since Mandelbrot [11] developed fractal geometry in 1983, the theory has been applied to geographical research for nearly forty years. Urban geography is one of the biggest beneficiaries of fractal ideas [12]. Since the 1980s, some pioneering studies about urban form and growth based on fractal geometry have been published. Those early research studies mainly focus on two aspects of urban form. On the one hand, they measure the fractal dimension of urban forms by using the box-counting method, area-radius scaling, or perimeter-area scaling, and then compare the fractal dimension values of different time series in the same region or different regions at the same time to realize the analysis of urban form and growth (e.g., [5,13–20]). Some of those research results were summarized by Batty and Longley [21] and Frankhauser [22] in 1994, which draw the conclusion that urban form and growth can be regarded as a fractal object and the average value of the fractal dimension is around 1.7. On the other hand, they simulate the process of urban growth based on the iterative models and the theories of chaos or self-organizing to try to find the formation mechanism or reasons for the fractal structure of urban form (e.g., [4,20,23–36]). In particular, Batty et al. [24,27,28] used the fractal model and diffusion-limited aggregation (DLA) to simulate the process of urban structure and growth, respectively. Benguigui et al. [4,29–31] used the aggregation model to simulate the urban form of Petah Tikvah and introduced the concept of leapfrogging as one of the fundamental processes that generates urban fractality. Thomas et al. [34] reported the relationships between fractal cities and self-organization. Nowadays, analysis of urban form and growth based on fractal theory and exploration of the mechanism of the emergence of fractal features of urban form have become increasingly important topics in both urban science and complexity science. Recently, there are more studies of urban form and growth based on fractal dimension as the validity indicator (e.g., [35–43]) and multifractal models (e.g., [44–51]). Compared with the simple fractal model, the calculation of multifractal models is more complex but can better reflect the heterogeneity and local characteristics of urban form distribution [45]. Encarnação et al. [52] developed the technology of fractal cartography by using the box-counting method and programming, thus enabling the visualization of fractal dimension distribution of urban form. Furthermore, a discovery was made that the fractal dimension curves of urban growth take on the squash effect and can be modeled with sigmoid functions [53]. The fractal dimension growth curves of urban forms in Europe and America satisfy the common logistic function, while those of northern Chinese cities such as Beijing meet the quadratic logistic function [6,54]. Those studies (e.g., [6,53,54]) mainly focused on theoretic modeling an appropriate time series of the fractal dimension values of a city or region and found that there was difference in the types of logistic functions between Western countries and northern China. However, the case of southern Chinese cities and some specific possible reasons for this phenomenon were seldom related. Different types of logistic functions indicate different spatial dynamics. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of development, and the possible reasons for this phenomenon. Shenzhen city in southern China is a special, notable case. It is not only a new, fast-growing city planned by the government but also an experimental city for government tests on the operation of a market economy. Nowadays, it has achieved surprising economic development and population growth and led the development of the Chinese economy since China’s reform and opening up. Taking Shenzhen as an example of southern China, we can discuss another type of fractal city in mainland China, which differs from many cities in other parts of China, except for the southeast coastal area of China. Therefore, this paper is devoted to exploring the fractal dimension growth curves of the city of Shenzhen and attempts to further analyze the reasons. We first divide Shenzhen city into four study areas and then use the box-counting method and least squares regression for extracting spatial data and estimating the fractal parameters. Then, we utilize sigmoid functions to model fractal dimension curves of urban growth in Shenzhen region. ISPRS Int. J. Geo-Inf. 2020, 9, 0672 3 of 15

Through this study, we can not only reveal the North–South differences of city development in China but also reflect the similarities and differences of urban evolution between China and the West.

2. Methods

2.1. Box-Counting Method Owing to the scale-free properties of urban form, conventional measures should be replaced by fractal parameters. The box-counting method in this paper was employed for estimating the fractal dimension of the urban form of four study regions in Shenzhen, from 1986 to 2017. With the advantages of simple understanding and operation, it has become a method widely applied by many researchers (e.g., [18,19,21,52,55]) to measure the fractal dimension in 2-dimensional images. Its basic procedure, in general, is to recursively superimpose a series of regular grids of declining box sizes over a target object and to then record the object count in each successive box, where the count records how many of the boxes are occupied by the target object. According to Benguigui et al. [18], in a 2-dimensional space, the object is covered by a grid made of squares of size ε, and the number N(ε) of squares in which a part of the object appears is counted. Then, changing the side length of boxes ε, leads to a change in non-empty boxes number, N(ε). If the object turns out to be fractal, then

D N(ε) = N1(1/ε) (1) where N1 is the proportionality coefficient, D is the urban form fractal dimension value, and the logarithmic form is ln N(ε) = ln N1 + D ln(1/ε) (2) Thus, a logarithmic plot of ln N(ε) versus ln (1/ε) yields a straight line with a slope equal to D. In this paper, the first value ε is the half-size of the box; the next value is equal to ε/4. The ith value is ε/2i; the highest value of index i is 9. The tools of create fishnet, spatial adjustment, and overlay in ArcMap10.2 were utilized for implementation the segmentation box and rotation box and obtaining the ith value of N(ε).

2.2. Fractal Dimension Growth Curve and Power Law In order to investigate the rules of urban evolution of Shenzhen city, we modelled fractal dimension values of time series of four study areas in Shenzhen by using logistic function modeling. According to Chen’s [53] elaboration, the logistic function of fractal dimension evolution can be expressed as

D D D(t) = max or D(t) = max (3) kt k(n n ) 1 + (Dmax/D0)e− 1 + Ae− − 0 where t is time order (0, 1 ... ); n is the year; n0 is the initial year; D(t) or D(n) denote the fractal dimension in the tth time or the year of n; D is the fractal dimension in the initial year; Dmax 2 0 ≤ indicates the maximum of the fractal dimension; A refers to a parameter; k is the original growth rate of fractal dimension. The parameter and variable relationships are as follows

Dmax D(t) = D(n), A = 1, t = n n0 (4) D0 − −

Equation (3) can be made into a logarithmic transform, and the result is

D D ln( max 1) = ln( max ) kt = ln A kt (5) D0 − D0 − −

Equation (5) is concerted to a log-linear equation, and the values of A and K are simple to estimate by linear regression analysis if the parameter of Dmax value is known. Here, the goodness-of-ﬁt search ISPRS Int. J. Geo-Inf. 2020, 9, 0672 4 of 15

(GOFS) parameter estimation method was selected for estimating the parameter, which has been introduced particularly by Chen [53]. Capturing the driving force factors through the mathematical model plays an important role in trying to explain the reasons of the emergence of fractal structure of urban form. An urban form is a typical complex system [56,57]. Stable urban form and growth are the result of long-term continuous interaction of various factors; those associations can be captured by the power function [58,59]. For a city system, the relationship between two measured elements representing the city, such as population, area, and GDP, usually satisfy the statistical power law. We thus captured the driving factors of urban form evolution and growth using the power law function, which is expressed as

β Dt = kXt (6) where Xt and Dt represent, in a given t year, fractal dimension value and the total number of driving factor, respectively; k and β are constants to fractal dimension. The linearized model of Equation (6) is

ln Dt = ln k + β ln Xt (7) where lnk and β are constants terms to be estimated.

2.3. Study Area and Datasets

Shenzhen (22◦270–22◦520 N, 113◦460–114◦370 E), as China’s first special economic zone, lies along the cost of the South China Sea and adjacent to Hong Kong. Before China’s reform and opening-up policy in late 1978, it was just a sleepy border town of some 30,000 inhabitants that served as a custom stop into mainland China from Hong Kong. Now, Shenzhen has become an international metropolis with a total population of 13.0366 million [60]. The mean annual temperature is around 22.4 ◦C and annual rainfall is about 1948 mm [61]. The shape characteristics of the administrative region of Shenzhen are narrow and long, and the east–west span is over 49 km, while the north–south span is only about 7 km [62]. In the southeastern part of Shenzhen is a hilly topography; in the northwestern part, it is relatively low. In this paper, four boxes were drawn as the study areas (Figure1). The first box area is the entire region of Shenzhen, which covers almost the entire built-up patch of Shenzhen. The second box area is a major center region, which mainly includes Futian District, Luohu District, and Nanshan District. The third and fourth box areas are the northwest part and northeast part, respectively. The reason for including them is because the built-up areas in these two areas visually seem to have the expansion and development trends. It is quite useful to fully assess the spatial–temporal evolution characteristics of the urban form and growth. The processing of the four boxes mainly includes box segmentation and box rotation. On the platform of ArcMap10.2, we firstly used the tool of create fishnet to generate regular grids of different scales based on the box-counting method. Secondly, we rotated the four boxes representing the four study areas with the editor tool and then matched other boxes of different scales to the four rotated boxes with the tool of spatial adjustment. In addition, the built-up areas’ data (Figure1) were extracted from Landsat TM 4 and 5 and OLI 8 images with 30-m resolution for twelve years: 1986, 1989,1992, 1995, 1998, 2001, 2003, 2006, 2010, 2013, 2015, and 2017. They were all collected from United States Geological Survey (USGS) Earth Explorer website (http://earthexplorer.usgs.gov/), with less than 10% cloud cover. In general, the methods of object-oriented supervised classification and visual interpretation post-classification are employed for extracting built-up areas’ data, and its process can be divided into three parts. First, we employed the toolbox of example-based feature-extracting workflow in ENVI 5.3 software to generate twelve periods’ land use and land cover (LULC) classification maps; here, we roughly divided land types into four categories that are water body, vegetation, bare, and built-up areas, and chose the support vector machine (SVM) classification method. Secondly, we selected the class of urban area form for each resulting vector datum in ArcMap10.2 software. Finally, and most crucially, we conducted a visual interpretation of each of the 12 periods’ built-up area maps. It included ISPRS Int. J. Geo-Inf. 2020, 9, 0672 5 of 15 removing the noisy patches by setting a threshold value and adding, removing, and modifying the misclassification of the built-up areas by using the editor tool and other toolboxes in ArcMap 10.2, GoogleISPRS Int. Earth J. Geo-Inf. and 2020 multi-period, 9, x FOR PEER false REVIEW color composite of remote sensing images. 5 of 15

Figure 1. The built-up area map in Shenzhen city from 1986 to 2017. Note: The built-up areas were Figure 1. The built-up area map in Shenzhen city from 1986 to 2017. Note: The built-up areas were extracted from Landsat TM 4 and 5 and OLI 8 images with 30-m resolution downloading from United extracted from Landsat TM 4 and 5 and OLI 8 images with 30-m resolution downloading from United States Geological Survey (USGS) Earth Explorer website (http://earthexplorer.usgs.gov/). The four boxes, States Geological Survey (USGS) Earth Explorer website (http://earthexplorer.usgs.gov/). The four 1–4, are the study areas, representing the entire region, the major business center region, the northwest boxes, 1–4, are the study areas, representing the entire region, the major business center region, the region, and the northeast region of Shenzhen, respectively. northwest region, and the northeast region of Shenzhen, respectively. 3. Results and Analyses 3. Results and Analyses 3.1. Fractal Dimension Analysis of Urban Form 3.1. Fractal Dimension Analysis of Urban Form The double logarithm linear regression based on the least square method can be employed to estimateThe fractaldouble parameters.logarithm linear Fractal regression modeling based involves on the two least types square of parameters. method can One be isemployed inferential to parametersestimate fractal and parameters. the other isFractal descriptive modeling parameters. involves two The types former of parameters. includes fractalOne is dimensioninferential andparameters proportionality and the coeotherfficient, is descriptive and the latterparameters. includes The goodness-of-fit former includes and fractal standard dimension error [63 ,and64]. Fittingproportionality results based coefficient, on the box-countingand the latter methodincludes showed goodness-of-fit that from and 1986 standard to 2017, error the four [63,64]. study Fitting areas allresults exhibit based a strong on the linear box-counting association method between showed ln N that(ε) and from ln 1986 (1/ε), to with 2017, fractal the four dimension study areas values all betweenexhibit a 1strong and 2, linear and the association goodness-of-fit betweenR2 lnvalues N(ε) areand all ln above(1/ε), with 0.98. fractal It is powerful dimension proof values that indicatesbetween that1 and the 2, urbanand the form goodness-of-fit in Shenzhen, bothR2 values its entire are andall above part regions, 0.98. It areis powerful indeed statistically proof that fractals indicates during that 1986–2017.the urban form The detailedin Shenzhen, results both of the its fractal entire dimension and part regions, estimates are of indeed the urban statistically forms of thefractals four regionsduring in1986–2017. Shenzhen The from detailed 1986 to 2017results are of presented the fractal in Tabledime1nsion. They estimates all increased of the over urban time gradually,forms of the but four the entireregions region in Shenzhen is slightly from smaller 1986 than to other2017 threeare pres subregions—theented in Table center 1. They region, all theincreased northwest over region, time andgradually, the northeast but the region.entire region is slightly smaller than other three subregions—the center region, the northwestIn fact, region, fractal and dimension the northeastD has region. now become a validity index of assessing the space-filling extent,In fact, spatial fractal complexity, dimension and D spatialhas now homogeneity become a validity or compactness index of assessing of urban the land. space-filling A larger extent, value signifiesspatial complexity, urban sprawl—urban and spatial spatialhomogeneity structure or compactness becomes more of complicatedurban land. A and larger homogeneous value signifies [65]. Inurban 2-dimension sprawl—urban digital maps,spatialD structure= 2 indicates becomes that onemore has complicated a homogeneous and spatialhomogeneous distribution [65]. ofIn the 2- objectdimension in the digital plane; themaps, other 𝐷=2 extreme indicates limitis thatD = one0, which has a suggestshomogeneous a high localspatial concentration distribution [ 4of]. Itthe is obviousobject in fromthe plane; above the results, other for extreme the past limit 40 years,is 𝐷=0 that, which the urban suggests form ina high entire local Shenzhen concentration has been [4]. in It a is obvious from above results, for the past 40 years, that the urban form in entire Shenzhen has been in a state of continuing growth and expansion in space. Meanwhile, its spatial structure has been becoming more and more complex and homogeneous over time. Considering the land-type distribution in the real world, it is inevitable that there exist other land types, such as ecological land and water bodies, in the four study areas. They may actually belong to the land types that are

ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 6 of 15

protected by the authorities and not allowed to be available for urban construction in Shenzhen. In 2017, in particular, the fractal dimension values of the four regions—the entire region, the central part, the northwest part, and the northeast part—reached their highest value, which was 1.7604, 1.8467, 1.8189, and 1.8294, respectively (Table 1). Not least because these values are close to 2, but there are probably no more other land types allowed to be available for urban development within each study area. We, thus, may speculate that Shenzhen will probably encounter the situation of urban land approaching saturation in the near future. ISPRS Int.In J. Geo-Inf.addition,2020, 9we, 0672 can obtain other further information by calculating the data in Table6 of1. 15 The average speed of urban space-filling degree in the four study areas can be calculated using (𝐷 − 𝐷)/△ 𝑡(). The calculation result values are 0.0202, 0.0133, 0.0189, and 0.0177, respectively. state of continuing growth and expansion in space. Meanwhile, its spatial structure has been becoming In terms of area, the entire region is the largest region among four study regions and its value of more and more complex and homogeneous over time. Considering the land-type distribution in the average speed of urban space-filling degree is also the largest. It suggests that Shenzhen experienced real world, it is inevitable that there exist other land types, such as ecological land and water bodies, conversion of a large number of other land types to urban land in this period, and the fastest growth in the four study areas. They may actually belong to the land types that are protected by the authorities and expansion of urban land occurred in the northwest part, followed by the northeast part and and not allowed to be available for urban construction in Shenzhen. In 2017, in particular, the fractal central part. Moreover, as shown in Figure 2, it is clear that Shenzhen city had two peaks of urban dimension values of the four regions—the entire region, the central part, the northwest part, and the sprawl during 1986–2017, which were in 1992 and in 2003, but the extent of urban expansion in 1992 northeast part—reached their highest value, which was 1.7604, 1.8467, 1.8189, and 1.8294, respectively was significantly higher than in 2003. In 1992, the northwest and northeast regions of Shenzhen had (Table1). Not least because these values are close to 2, but there are probably no more other land the same urban land-use development level; by contrast, the central region had a slower development, types allowed to be available for urban development within each study area. We, thus, may speculate but in 2003, these three subregions had basically the same urban land-use development level. that Shenzhen will probably encounter the situation of urban land approaching saturation in the near future.Table 1. Fractal dimension estimates of the urban form of four regions in Shenzhen, 1986–2017. In addition, we can obtain other further information by calculating the data in Table1. Entire Region Center Region Northwest Region Northeast Region The averageYear speed of urban𝟐 space-filling degree𝟐 in the four study𝟐 areas can be calculated𝟐 using 𝑫𝒇 𝑹 σ 𝑫𝒇 𝑹 σ 𝑫𝒇 𝑹 σ 𝑫𝒇 𝑹 σ (D2017 1986D1986 1.1352)/ t( 0.98862017 1986 0.0460). The 1.4353 calculation 0.9921 result 0.0483 values 1.2320 are 0.9914 0.0202, 0.0434 0.0133, 1.2808 0.0189, 0.9912 and 0.0177, 0.0455 − 4 − respectively.1989 1.2101 In terms 0.9908 of area, 0.0442 the entire 1.4730 region 0.9950 is 0.0395 the largest 1.3037 region 0.9947 among 0.0358 four 1.3379 study 0.9934 regions 0.0411 and its value1992 of average1.4208 speed 0.9971 of 0.0289 urban 1.6073 space-filling 0.9991 degree 0.0186 is 1.4999 also the 0.9984 largest. 0.0228 It suggests 1.5333 that 0.9982 Shenzhen 0.0245 experienced1995 conversion1.5008 0.9983 of a large 0.0235 number 1.6668 of 0.9995other land 0.0136 types 1.5739 to urban 0.9991 land in0.0175 this period, 1.5999 and 0.9985 the fastest 0.0233 1998 1.5454 0.9989 0.0193 1.6866 0.9996 0.0135 1.6174 0.9994 0.0154 1.6311 0.9987 0.0225 growth2001 and expansion1.5856 0.9992 of urban 0.0171 land 1.7145 occurred 0.9997 in the 0.0121 northwest 1.6568 part, 0.9994 followed 0.0149 by 1.6662 the northeast 0.9990 0.0200 part and central2003 part.1.6619 Moreover, 0.9995 as 0.0144 shown 1.7636 in Figure 0.99982, it is0.0103 clear that1.7315 Shenzhen 0.9997 city 0.0122 had 1.7304 two peaks 0.9993 of urban 0.0176 sprawl2006 during 1.7050 1986–2017, 0.9996 which 0.0125 were 1.7884 in 1992 0.9998 and 0.0096 in 2003, 1.7691 but the 0.9997 extent 0.0106 of urban 1.7705 expansion 0.9995 in 0.0151 1992 was significantly2010 1.7239 higher 0.9997 than 0.0117 in 2003. 1.8053 In 1992, 0.9998 the northwest 0.0106 1.7838 and 0.9997northeast 0.0110 regions 1.7898 of Shenzhen 0.9995 0.0145 had 2013 1.7401 0.9998 0.0104 1.8174 0.9998 0.0097 1.7871 0.9998 0.0103 1.8063 0.9996 0.0131 the same2015 urban 1.7489 land-use 0.9998 development 0.0098 1.8309 level; 0.9999 by contrast, 0.0082 the 1.8076 central 0.9998 region 0.0095 had a slower 1.8166 development, 0.9997 0.0124 but in 2003,2017 these1.7604 three 0.9998 subregions 0.0089 had 1.8467 basically 0.9999 the 0.0075 same 1.8189urban land-use 0.9998 0.0088 development 1.8294 level.0.9997 0.0115

Entire region Center region 0.04 0.04 0.035 0.03 0.03 0.022 0.02 0.02

0.01 0.01 0.004 0.003 Fractal dimension rate dimension growth Fractal Fractal dimension rate growth 0.00 ISPRS Int.0.00 J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 7 of 15 1985 1990 1995 2000 2005 2010 2015 2020 1985 1990 1995 2000 2005 2010 2015 2020 Year Year Northwest region Northeast region 0.04 0.04 0.033 0.033 0.03 0.03

0.02 0.02

0.01 0.01 0.004 0.004 Fractal dimension rate dimension growth Fractal Fractal dimension rate dimension growth Fractal 0.00 0.00 1985 1990 1995 2000 2005 2010 2015 2020 1985 1990 1995 2000 2005 2010 2015 2020 Year Year

Figure 2. The graph of the growth rate of fractal dimension of four regions from 1986 to 2017. Figure 2. The graph of the growth rate of fractal dimension of four regions from 1986 to 2017.

3.2. Fractal Dimension Growth Curves The results of the four regions’ fractal dimension sets to fit the logistic model by ordinary least square (OLS) estimation and goodness-of-fit search (GOFS) are shown in Figure 3. It is obvious that fractal dimension growth curves of the four regions in Shenzhen can all be very well fitted by first- order logistic function. The specific first-order logistic expressions and relevant parameters are shown in Table 2; the goodness-of-fit R2 of first-order logistic expressions for each region is very high. Meanwhile, we can also simply obtain the maximum capacity fractal dimension and predict the year of reaching maximum capacity by the logistic expression of each region. As shown in Table 2, the maximum capacity fractal dimension of space of the four study regions—entire, center, northwest, and northeast—in Shenzhen is 1.7905, 1.9, 1.86, and 1.8621, respectively, and the corresponding years of reaching above values are 2072, 2197, 2085, and 2082, respectively. It is simple to see that the center region in Shenzhen has both the largest maximum capacity fractal dimension of space and the longest time to reach the year of reaching above the value. However, connecting with the practical situation, those values, in fact, are overvalued, as within study areas, land that is completely inhospitable to humans, such as rivers and high mountains, is included.

Table 2. Four regions logistics equation information table in Shenzhen, 1986–2017.

Region Logistics Equation R2 Capacity Parameter Dmax Year 1.7905 Entire region Dtˆ()= 0.9909 1.7905 2072 + −0.1129t 1 0.5808*e 1.9000 Center region Dtˆ ()= 0.9868 1.9000 2097 + −0.0750t 1 0.3110*e 1.8600 Northwest region Dtˆ ()= 0.9819 1.8600 2085 + −0.0992t 1 0.4817*e 1.8621 Northeast region Dtˆ ()= 0.9801 1.8621 2082 + −0.1023t 1 0.4581*e

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Table 1. Fractal dimension estimates of the urban form of four regions in Shenzhen, 1986–2017.

Entire Region Center Region Northwest Region Northeast Region Year 2 2 2 2 Df R σ Df R σ Df R σ Df R σ 1986 1.1352 0.9886 0.0460 1.4353 0.9921 0.0483 1.2320 0.9914 0.0434 1.2808 0.9912 0.0455 1989 1.2101 0.9908 0.0442 1.4730 0.9950 0.0395 1.3037 0.9947 0.0358 1.3379 0.9934 0.0411 1992 1.4208 0.9971 0.0289 1.6073 0.9991 0.0186 1.4999 0.9984 0.0228 1.5333 0.9982 0.0245 1995 1.5008 0.9983 0.0235 1.6668 0.9995 0.0136 1.5739 0.9991 0.0175 1.5999 0.9985 0.0233 1998 1.5454 0.9989 0.0193 1.6866 0.9996 0.0135 1.6174 0.9994 0.0154 1.6311 0.9987 0.0225 2001 1.5856 0.9992 0.0171 1.7145 0.9997 0.0121 1.6568 0.9994 0.0149 1.6662 0.9990 0.0200 2003 1.6619 0.9995 0.0144 1.7636 0.9998 0.0103 1.7315 0.9997 0.0122 1.7304 0.9993 0.0176 2006 1.7050 0.9996 0.0125 1.7884 0.9998 0.0096 1.7691 0.9997 0.0106 1.7705 0.9995 0.0151 2010 1.7239 0.9997 0.0117 1.8053 0.9998 0.0106 1.7838 0.9997 0.0110 1.7898 0.9995 0.0145 2013 1.7401 0.9998 0.0104 1.8174 0.9998 0.0097 1.7871 0.9998 0.0103 1.8063 0.9996 0.0131 2015 1.7489 0.9998 0.0098 1.8309 0.9999 0.0082 1.8076 0.9998 0.0095 1.8166 0.9997 0.0124 2017 1.7604 0.9998 0.0089 1.8467 0.9999 0.0075 1.8189 0.9998 0.0088 1.8294 0.9997 0.0115

3.2. Fractal Dimension Growth Curves The results of the four regions’ fractal dimension sets to fit the logistic model by ordinary least square (OLS) estimation and goodness-of-fit search (GOFS) are shown in Figure3. It is obvious that fractal dimension growth curves of the four regions in Shenzhen can all be very well fitted by first-order logistic function. The specific first-order logistic expressions and relevant parameters are shown in Table2; the goodness-of-fit R2 of first-order logistic expressions for each region is very high. Meanwhile, we can also simply obtain the maximum capacity fractal dimension and predict the year of reaching maximum capacity by the logistic expression of each region. As shown in Table2, the maximum capacity fractal dimension of space of the four study regions—entire, center, northwest, and northeast—in Shenzhen is 1.7905, 1.9, 1.86, and 1.8621, respectively, and the corresponding years of reaching above values are 2072, 2197, 2085, and 2082, respectively. It is simple to see that the center region in Shenzhen has both the largest maximum capacity fractal dimension of space and the longest time to reach the year of reaching above the value. However, connecting with the practical situation, those values, in fact, are overvalued, as within study areas, land that is completely inhospitable to humans, such as rivers and high mountains, is included.

Table 2. Four regions logistics equation information table in Shenzhen, 1986–2017.

Region Logistics Equation R2 Capacity Parameter Dmax Year 1.7905 Entire region Dˆ (t) = 0.9909 1.7905 2072 1 + 0.5808 e 0.1129t 1.9000∗ − Center region Dˆ (t) = 0.9868 1.9000 2097 1 + 0.3110 e 0.0750t 1.8600∗ − Northwest region Dˆ (t) = 0.9819 1.8600 2085 1 + 0.4817 e 0.0992t 1.8621∗ − Northeast region Dˆ (t) = 0.9801 1.8621 2082 1 + 0.4581 e 0.1023t ∗ − ISPRS Int. J. Geo-Inf. 2020, 9, 0672 8 of 15 ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 8 of 15

Entire region Center region 2.0 2.0

1.8 1.8

1.6 1.6

1.4 1.4 Fractal dimension Fractal dimension

1.2 Observed value 1.2 Observed value Predicted value Predicted value

1.0 1.0 1985 1991 1997 2003 2009 2015 2021 2027 1985 1991 1997 2003 2009 2015 2021 2027 Year Year

Northwest region Northeast region 2.0 2.0

1.8 1.8

1.6 1.6

1.4 1.4 Fractal dimension Fractal dimension

1.2 Observed value 1.2 Observed value Predicted value Predicted value

1.0 1.0 1985 1991 1997 2003 2009 2015 2021 2027 1985 1991 1997 2003 2009 2015 2021 2027 Year Year

Figure 3. The logistic patterns of fractal dimension growth of the four study regions, 1986–2017. Figure 3. The logistic patterns of fractal dimension growth of the four study regions, 1986–2017. 3.3. Power Law Analysis 3.3. Power Law Analysis The results of the population size, GDP, and fractal dimension values of the entire Shenzhen from 1986The toresults 2017 of are the shown population in Figures size,4 and GDP,5. Between and fractal population dimension size, values GDP, andof the fractal entire dimension Shenzhen existfrom signiﬁcant 1986 to 2017 law are relations, shown in and Figures the values4 and 5. of Between goodness-of-ﬁt populationR2 size,are quiteGDP, high—alland fractal above dimension 0.96. 2 Theexist increase significant in fractal law relations, dimension and with the time values usually of goodness-of-fit indicates the urbanR are sprawl.quite high—all It usually above relates 0.96. to theThe factorsincrease of populationin fractal dimension size and the with level time of economicusually in developmentdicates the urban [66]. sprawl. It implies It thatusually both relates population to the sizefactors and of GDP population are the drivingsize and factors the level of of promoting economic the development urban sprawl [66]. in It Shenzhen implies that from both 1986 population to 2017. However,size and GDP the accelerating are the driving role offactors population of promoting regarding the urban urban expansion sprawl in isShenzhen much greater from than1986 GDPto 2017. in termsHowever, of their the respective accelerating power role exponents, of population which regarding indicates urban that Shenzhen expansion city is much has no greater need to than boost GDP its economicin terms of development their respective at the power expense exponents, of urban which expansion. indicates It isthat also Shenzhen worth considering city has no theneed problem to boost ofits whether economic or development not both individual at the expense behavior of urban and public expansion. policy It play is also key worth roles considering in the population the problem and economicof whether growth or not for both a city. individual It will be behavior discussed and in Sectionpublic policy4 of this play paper. key roles in the population and economic growth for a city. It will be discussed in Section 4 of this paper.

ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 9 of 15

ISPRS Int. J. J. Geo-Inf. Geo-Inf. 2020,, 9,, x 0672 FOR PEER REVIEW 9 of 15 8

8 lnD= 5.7722lnP + 3.7858 )

D 7 R² = 0.9876 lnD= 5.7722lnP + 3.7858 )

D 7 R² = 0.9876 6

6 5 Fractal dimension (ln dimension Fractal

5 Fractal dimension (ln dimension Fractal 4 0 0.1 0.2 0.3 0.4 0.5 0.6 4 Population (lnP) 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 4. Log-log plot between population (lnPopulation P) and fractal (lnP) dimension (ln D) in Shenzhen, 1986–2017. FigureSource: 4. PopulationLog-log plot data between from Shen populationzhen Statistical (lnP) and Yearbook—2019 fractal dimension. (lnD) in Shenzhen, 1986–2017. Figure 4. Log-log plot between population (ln P) and fractal dimension (ln D) in Shenzhen, 1986–2017. Source: Population data from Shenzhen Statistical Yearbook—2019. Source: Population data from Shenzhen Statistical Yearbook—2019. 8

8 lnD = 0.4094lnG - 0.5941

) R² = 0.9695

D 7 lnD = 0.4094lnG - 0.5941

) R² = 0.9695

D 7 6

6 5 Fractal dimension (ln dimension Fractal

5 Fractal dimension (ln dimension Fractal 4 12 13 14 15 16 17 18 19 20 4 GDP (lnG) 12 13 14 15 16 17 18 19 20 Figure 5. Log-log plot GDP (lnG) andGDP fractal (lnG dimension) (lnD) in Shenzhen, 1986–2017. Source:Figure Population5. Log-log dataplot fromGDP Shenzhen (lnG) and Statistical fractal dimension Yearbook—2019. (lnD) in Shenzhen, 1986–2017. Source: Population data from Shenzhen Statistical Yearbook—2019. 4. DiscussionFigure 5. Log-log plot GDP (lnG) and fractal dimension (lnD) in Shenzhen, 1986–2017. Source: Population data from Shenzhen Statistical Yearbook—2019. 4. DiscussionBy means of the calculation and analysis of fractal parameters, we can obtain a lot of new knowledge 4.about DiscussionBy the citymeans of Shenzhen. of the calculation Some knowledge and analysis can be of generalized fractal parameters, to explain the we spatio-temporal can obtain a lot evolution of new ofknowledge other cities. about Where the city city of fractals Shenzhen. are concerned, Some knowledge Shenzhen can di beffers generalized from many to explain cities in the northern spatio- ChinatemporalBy [6 ].means Itevolution is similar, of the of tocalculation other an extent, cities. and to Where cities analysis in city Europe fracof fractaltals and are American parameters, concerned, cities weShenzhen [53 can,54]. obtain This differs is a revealing lotfrom of manynew for knowledgeuscities to understand in northern about city theChina development. city [6]. of ItShenzhen. is similar, The main Some to an points knowledgeextent, of the to cities above can bein studies Europegeneralized can and be summarizedAmericanto explain cities the as followsspatio- [53,54]. (TabletemporalThis 3is). revealing First,evolution the for urban of us other to form understand cities. of Shenzhen Where city possessescitydevelopment. fractals a fractal are The concerned, structure. main points Shenzhen This of suggests the above differs that studies from the spatial manycan be citiesordersummarized in of northern this city as follows hasChina emerged [6].(Table It is by3). similar, self-organizedFirst, theto anurban extent,evolution. form to of cities Shenzhen Second, in Europe possesses diff anderent American a subareas fractal structure.cities of the [53,54]. study This Thistakesuggests onis revealing di ffthaterent the for fractal spatial us to dimension orderunderstand of this values. citycity development.has This em indicateserged by The spatial self-organized main heterogeneity points ofevolution. the ofabove Shenzhen’s Second, studies different can urban be summarizedform,subareas and of spatial theas follows study heterogeneity take (Table on 3).different suggests First, the fractal multifractal urban dimension form scalingof Shenzhen values. of city This possesses development. indicates a fractal spatial Third, structure. heterogeneity the fractal This suggestsdimensionof Shenzhen’s that values the spatialurban seem to orderform, descend of andthis from cityspatial the has center emheterogeneityerged of city by toself-organized the suggests suburbs andmultifractalevolution. exurbs. Second, Especiallyscaling different of at thecity subareasearlydevelopment. stages of the (1986–2001), studyThird, take the the onfractal fractaldifferent dimension dimension fractal valuesdimension values se ofem values. the to centerdescend This region indicates from were the spatial significantlycenter heterogeneity of city higher to the ofthansuburbs Shenzhen’s the fractal and exurbs. dimensionurban Especially form, values and at of the spatial northwestearly heterogeneitystages region (1986–2001), and suggests the northeastthe fractalmultifractal region dimension (Figure scaling values1; Tableof ofcity1 the). Thisdevelopment.center suggests region a wereThird, hidden significantly the circular fractal structure dimensionhigher than of city values the development. fractal seem dimension to descend The circularvalues from of structure thethe centernorthwest behind of city region irregular to theand suburbsurbanthe northeast forms and canexurbs. region be revealed Especially(Figure by1; Table fractalat the 1). dimensionearly This stagessugges changes (1986–2001),ts a hidden [20]. circular Fourth, the fractal thestructure goodness-of-fitdimension of city valuesdevelopment. for fractalof the centerdimensionThe circular region estimation werestructure significantly ascended behind irregularhigher over time. than urban the From frformactal 1986s dimensioncan to 2017,be revealed the valuesR square by of fractalthe value northwest dimension went region up andchanges and up theuntil[20]. northeast itFourth, was close region the togoodness-of-fit (Figure 1 (Table 1;1). Table This for 1). suggestsfractal This suggesdimension thatts the a hidden fractalestimation circular structure ascended structure of Shenzhen over of citytime. became development. From better1986 to Theand2017, better,circular the R gradually, squarestructure value through behind went self-organizedirregular up and up urban until evolution. formit wass canclose Fifth, be to revealed 1 the (Table.1). fractal by dimension Thisfractal suggests dimension growth that curvesthechanges fractal of [20].urbanstructure Fourth, form of can theShenzhen be goodness-of-fit modeled became by conventional betterfor fractal and better,dimension logistic gr function.adually, estimation through This ascended diff self-organizeders from over the time. fractal evolution. From dimension 1986 Fifth, to 2017, the R square value went up and up until it was close to 1 (Table.1). This suggests that the fractal structure of Shenzhen became better and better, gradually, through self-organized evolution. Fifth,

ISPRS Int. J. Geo-Inf. 2020, 9, 0672 10 of 15 growth curves of cities in northern China but is similar to those of the cities in Western developed countries. This, perhaps, resulted from the bottom-up urbanization process of southern cities in China, dominated by self-organized evolution, which is associated with market mechanism. Sixth, the fractal dimension values approached the capacity parameters. All the fractal dimension values in 2017 are close the maximum value, Dmax. This suggests that the urban space of Shenzhen is ﬁlled to a great degree, and there are not many remaining spaces for future development.

Table 3. The main results and ﬁndings of fractal studies and corresponding inferences or conclusions about Shenzhen city.

Results and Findings Inferences about Shenzhen Urban form bears fractal structure Spatial order emerging from self-organization Different parts bear different fractal dimension values Possible multifractal structure Fractal dimension values decay from center to edge Hidden circular structure Fractal structure become optimized by self-organized Goodness of fit ascended over time evolution Fractal dimension curves of urban growth meet Bottom-up urbanization dynamics logistic function Fractal dimension approached the maximum value Space-filling of urban development is close to its limit

The fractal dimension growth curves of urban form in the four study regions of Shenzhen can be very well modeled with first-order logistic function (Figure3), which is the same with some Western cities, such as London (UK), Tel Aviv (Israel), and Baltimore (USA) [53,54], but different from northern cities of China, such as Beijing. The biggest difference between logistic curve and quadratic logistic curve is the rate of growth before the curve reaches the maximum capital value; the logistic curve is much slower than the quadratic logistic curve (Figure6). Urban form and growth are associated with urbanization, and the process of urbanization of a region seems to impact the development of urban morphology. Urban form is one of important components of urbanization [67]. The model of fractal dimension curve of urban growth is always consistent with the urbanization curve in a country or a region [53]. Urbanization falls into two types: one is bottom-up urbanization, and the other, top-down urbanization [68]. Bottom-up urbanization is associated with market economy and chiefly dominated by the well-known “invisible hand” of free competition, while top-down urbanization is associated with command economy and is mainly dominated by the visible hand of administrative intervention [69]. Different economic mechanisms and corresponding urbanization types have their own advantages and disadvantages. Bottom-up urbanization corresponds to self-organized evolution of cities. All cities can be treated as self-organized cities [33]. However, self-organization processes of cities are influenced by the political and economic systems of a nation or a region. The fact is that China’s southeast coastal areas opened earlier and their economic development has been more strongly affected by the international community. This fact may account for the fractal dimension growth curves of Shenzhen’s urban form. Most cities in in northern China were often deeply impacted by planned economy for a long time. The planned economy in China, also known as the command economy, is generally referred to as a kind of economic system in which production, resource allocation, and consumption are planned and decided by the government in advance. Especially in the early years of China’s development, before the reformation and opening, land development could be regarded as a special product under the planned economy system and its development and utilization were all dominated by the government. Such an urban development pattern led to the rapid expansion of Chinese urban forms in a certain period of time, and time series of the fractal dimension values can be well fitted as the formal features of like Beijing quadratic logistic curve in Figure6. Until the 1990s, China began reforming and opening up, built a socialist market economy, and set up some cities as special economic zones or as the testing ground for developing the market economy. That way, the land development also gradually changed from the original government-lend mode to the mode of enterprise participation. Shenzhen, China, ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 11 of 15

formal features of like Beijing quadratic logistic curve in Figure 6. Until the 1990s, China began reforming and opening up, built a socialist market economy, and set up some cities as special economic zones or as the testing ground for developing the market economy. That way, the land ISPRSdevelopment Int. J. Geo-Inf. also2020 gradually, 9, 0672 changed from the original government-lend mode to the mode11 of 15of enterprise participation. Shenzhen, China, is one of such representative cities [70]. In spite of Shenzhen being a typical case with the characteristics of both the market economy and planned is one of such representative cities [70]. In spite of Shenzhen being a typical case with the characteristics economy, according to Table 2, it illustrates that market economy in Shenzhen has a bigger impact of both the market economy and planned economy, according to Table2, it illustrates that market than public policy, or market economy dominates the development of Shenzhen rather than public economy in Shenzhen has a bigger impact than public policy, or market economy dominates the policy. However, the role of public policy is irreplaceable in Shenzhen. It is, perhaps, precisely such development of Shenzhen rather than public policy. However, the role of public policy is irreplaceable development patterns that quickly enabled its surprising economic development and population in Shenzhen. It is, perhaps, precisely such development patterns that quickly enabled its surprising growth, becoming China's forefront of reform and opening up, leading the development of Chinese economic development and population growth, becoming China’s forefront of reform and opening up, economy. leading the development of Chinese economy.

1.10

0.90 ) t ( D

0.70

Fractal dimension dimension Fractal 0.50 Beijing-Quadratic logistic curve Shenzhen-Logistic curve

0.30 1985 1995 2005 2015 2025 2035 2045 Year t Figure 6. Two types of fractal dimension increase curves of urban form: logistic curve and quadratic Figure 6. Two types of fractal dimension increase curves of urban form: logistic curve and quadratic logistic curve. Note: The logistic curve is based on the time series of Shenzhen’s box dimension logistic curve. Note: The logistic curve is based on the time series of Shenzhen’s box dimension values, values, while the quadratic logistic curve is based on the time series of Beijing’s box dimension values. Thewhile fractal the quadratic dimension logistic values curve of Beijing is based come on from the Chentime series and Huang of Beijing’s [6]. For box comparability, dimension values. the fractal The dimensionfractal dimension values arevalues normalized of Beijing by come maximum from andChen minimum and Huang fractal [6]. dimension For comparability, values [54 the]. fractal dimension values are normalized by maximum and minimum fractal dimension values [54]. The novelty of this work lies in two aspects. One is the investigation of different subareas. ShenzhenThe novelty was divided of this into work three lies overlapped in two aspects. subareas. One is Then, the investigation we examined of fractal different structure subareas. and fractalShenzhen dimension was divided growth into of thethree entire overlapped study area suba andreas. three Then, subareas. we examined Although fractal a similar structure way wasand oncefractal used dimension by [18], growth the studied of the area entire division study ofarea this and paper three bears subareas. its characteristics. Although a similar The other way is was the modelingonce used fractal by [18], dimension the studied curves area of division urban growth of this by paper conventional bears its logistic characteristics. function. The This other results is inthe a newmodeling discovery fractal that dimension the fractal curves growth of urban of southern growth cities by conventional differs from thatlogistic of northernfunction. cities This results in China. in Thisa new discovery discovery leads that tothe a newfractal understanding growth of southern that the citi modees differs of urban from growth that of correspondsnorthern cities to thein China. mode ofThis urbanization, discovery leads and to urbanization a new understanding dynamics arethat dominated the mode of by urban the structure growth corresponds of the economic to the system. mode Theof urbanization, main shortcomings and urbanization of this studies dynamics are as are below. dominated First, the by data the beforestructure 1986 of werethe economic absent. Wesystem. only foundThe main remote shortcomings sensing images of this from studies 1986 are and as beyond.below. First, Thus, the we data cannot before identify 1986 were the time absent. in which We only the realfound peak remote of urban sensing growth images appeared. from 1986 In fact, and afterbeyond. 1986, Thus, the peak we cannot of the growthidentify rate the oftime urban in which land usethe inreal Shenzhen peak of urban had passed. growth Second,appeared the. In definition fact, after and 1986, division the peak of of study the growth area lack rate su offfi urbancient objective land use bases.in Shenzhen The principal had passed. criteria Second, of the studythe definition area and and subareas division are empiricsof study andarea research lack sufficient objective. objective Third, onlybases. the The box-counting principal criteria method of the was study used. area This and method subareas is suitable are empirics for measuring and research and estimating objective. globalThird, fractalonly the dimension. box-counting The method local fractal was used. dimension This method can be is calculated suitable for with measuring the cluster and growingestimating method, global thatfractal is, dimension. by radius-area The scalinglocal fractal [9,20 ].dimension The cluster can growing be calculated method with can the yield cluster radial growing dimension method, [71]. The radial dimension can be used to reflect urban growth from another angle of view.

5. Conclusions The scale-free spatial analysis of urban form revealed the fractal structure and evolution characteristics of Shenzhen city. This analysis is not only helpful for deeply understanding the ISPRS Int. J. Geo-Inf. 2020, 9, 0672 12 of 15 city of Shenzhen but is also useful for understanding the regularities and dynamic mechanisms of urban evolution in different regions and even in different countries. The main conclusions of this study can be drawn as follows. Firstly, the urban form of Shenzhen city has a significant fractal feature. However, the spatial distribution of urban land has heterogeneity, as different subareas of urban region show different fractal dimension values at the same time, which suggests multifractal scaling. The fractal dimension decays from the center to the edge, indicating the circular structure of urban form. Secondly, the fractal dimension curves of the urban growth of Shenzhen bear S-shape characteristics and can be modeled by the conventional logistic function, which differs from the fractal dimension curves of cities in northern China. Different logistic functions suggest different types of urbanization dynamics. The conventional logistic function indicates bottom-up urbanization, while the quadratic logistic function suggests top-down urbanization. Thirdly, the fractal structure of the urban form of Shenzhen shows a clear evolutionary process. Fractal dimension is a parameter of inference which can be used to judge fractal structure. Meanwhile, fractal structure can be evaluated by the parameter of description, i.e., the goodness of fit. The R square value of Shenzhen’s fractal modeling went up and up over time until it approached 1. This suggests the fractal structure become optimized through self-organization. Fourthly, the fractal dimension of the urban form in Shenzhen is approaching its limit, and the past urban development patterns seem to be no longer sustainable. Fractal dimension values are tending towards the maximum value, which suggests that the space-filling of Shenzhen is already near its limit, and there are few land resources available within the study area. It has to occupy precious ecological resources or water resources if it continues to expand and extend. Thus, for Shenzhen city, a new mode of development is needed in future. Finally, fractal dimension is the indicator of urban sprawl. Population plays a significant role for the urban growth of Shenzhen city rather than GDP. Shenzhen city needs no to boost its economic development at the expense of urban expansion.

Author Contributions: Xiaoming Man contributed reagents/materials/analysis tools; Yanguang Chen conceived and designed the experiments; Xiaoming Man performed the experiments; Xiaoming Man and Yanguang Chen analyzed the data; Xiaoming Man wrote the original draft; Yanguang Chen reviewed and edited the draft. All authors have read and agreed to the published version of the manuscript. Funding: This research was sponsored by the National Natural Science Foundation of China (Grant No. 41671167). Acknowledgments: We would like to thank the two anonymous reviewers whose constructive comments were helpful in improving the quality of this paper. Conﬂicts of Interest: The author declares no conﬂict of interest.

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