Modeling Urban Growth and Form with Spatial Entropy

Entropy is one of physical bases for fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using fractal dimension, we can characterize urban growth and form and describe spatial complexity. A numbers of fractal models and measurements have been proposed for urban studies. However, the precondition for fractal dimension application is to find scaling in cities. Otherwise, we can make use of entropy function and measurements. This paper is devoted to researching how to describe cities by using spatial entropy. By analogy, a pair of models can be derived and a set of measurements can be constructed to describe urban growing process and patterns. First, logistic function and Boltzmann equation can be utilized to model the entropy increase curves of urban growth. Second, a series of indexes based on spatial entropy can be used to characterize urban form. Further, multifractal dimension spectrums can be generalized to spatial entropy spectrums. Conclusions can be drawn as follows. Entropy and fractal dimension have both intersection and different spheres of application to urban research. Thus, for a given spatial measurement scale, fractal dimension can often be replaced by entropy for simplicity. The models and measurements presented in this work are significant for integrating entropy and fractal dimension into the same framework of urban spatial analysis.


Introduction
In recent years, urban research is gradually forming a new science, that is, urban science based 2 on ideas from complexity theory. It is now known that cities and regions are complex spatial systems, which cannot be modeled and analyzed by using conventional mathematical methods (Albeverio et al, 2008;Allen, 1997;Batty, 2005;Portugali, 2011;Wilson, 2000). Spatial complexity can be described with both fractal dimension and entropy (Wilson, 2000;Bar-Yam, 2004a;Bar-Yam, 2004b;Batty, 2005;Chen, 2008). If an urban phenomenon bears characteristic scales, we can characterize it by means of entropy, and if the urban phenomenon has no characteristic scale, we have to depict it by using fractal dimension. However, in practice, things are usually more complicated and more difficult. It is easy to describe a simple system or the simple aspect of a complex system. To describe a complex system, we must reveal the scaling relationships behind observed data. Sometimes, the scaling property is covered. In this case, we have to find alternative ways for scientific description.
An effective approach to making spatial analysis for cities is to make use of zonal systems (Batty, 1976;Batty and Longley, 1994;Feng, 2002;Wang and Zhou, 1999). There are three types of zonal systems. The first is natural systems of zones, the second is the artificially defined systems of zones, and the third is the results of recursive subdivision of space. Natural zonal systems come from physical evolution of geographical landscape, artificial zonal system originates from administrative or management partition (Batty, 1976;Kaye, 1989), and the recursive subdivision of space is one of means of spatial disaggregation (Batty and Longley, 1994;Goodchild and Mark, 1987). Districts and counties of cities represents artificially defined systems of zones, and box-counting method represents a way of spatial recursive subdivision. Based on the zonal systems of districts and counties, spatial entropy can be calculated for urban phenomena (Batty, 1974;Batty, 1976), and using box-counting method, we can compute both entropy and fractal dimension of urban land use Feng and Chen, 2010).
Both fractal dimension and entropy can be employed to explore spatial complexity of cities.
Fractal geometry has been adopted to research cities for a long time, and various models are built to reflect urban evolution, and various fractal measurements are presented to describe urban properties (Appleby, 1996;Ariza-Villaverde et al, 2013;Batty and Longley, 1994;Benguigui et al, 2000;Frankhauser, 1994;Frankhauser, 1998;Huang and Chen, 2018;Makse et al, 1995;Makse et al, 1998;Murcio et al, 2015;Shen, 2002;Sun and Southworth, 2013;Tannier et al, 2011;Thomas et al, 2007;Thomas et al, 2008;Thomas et al, 2010;White and Engelen, 1994). Based on different methods of fractal dimension determination, a number of fractal parameters and indexes has been 3 put forward to characterize urban form (Chen, 2020). Based on fractal dimension sets, a number of models are employed to depict urban growth. Now, the question is whether we can use spatial entropy to replace fractal dimension if the measurement of fractal dimension is limited by real conditions. This paper is devoted to deriving models and measurements based on entropy function by analogy with fractal models of cities. In Section 2, a set of mathematical models of urban growth are derived from urban fractal models, a series of urban indexes are constructed using entropy values.
In Section 3, an empirical studies are made to testify the models and measurements. In Section 4, several related questions are discussed, and finally, the discussion are concluded by summarizing the main points of this study.

Relation between entropy and fractal dimension
The starting point of this study is general entropy, based on which the generalized fractal dimension have been defined for multifractal theory. Multifractal geometry is one of important tools for exploring spatial complexity of cities. The generalized correlation dimension can also be regarded as generalized information dimension, which is on the basis of Renyi's entropy (Renyi, 1961). Based on the box-counting method, the Renyi entropy can be expressed as where q represents the order of moment (q=…,-2,-1, 0, 1, 2,…), ε denotes the linear size of boxes for spatial cover and measurement, N is the number of nonempty boxes (i=1, 2,…, N), Pi is probability (∑Pi=1), Mq refers to the qth order Renyi entropy, Cq is the corresponding generalized correlation function, and the symbol "ln" denotes natural logarithm function. If q=0, we will have M0(ε)=lnN, which represents Hartley macro state entropy; If q=1, according to the L'Hospital rule, we will have M1(ε)= -∑Pi(ε)lnPi(ε), which represents Shannon information entropy; If q=2, we will have M2(ε)= -∑lnPi(ε) 2 =lnC2(ε), which refers to the second order Renyi entropy.
The entropy function is usually employed to describe the development state and property of the systems with characteristic lengths. For a simple system or the simple aspect of a complex system, the entropy value can be uniquely determined. On the contrary, for the complex systems such as 4 fractal cities, which bears no characteristic length in many aspect, the Renyi entropy is not determinate. An effective solution to this problem is to transform the generalized entropy into generalized fractal dimension. Based on Renyi's entropy, the generalized correlation dimension is always defined as follows (Feder, 1988;Grassberger1985;Mandelbrot, 1999) where "lim" denotes the limit function in mathematical analysis. Three special fractal dimension parameters can be derived from the general definition of fractal dimension (Grassberger, 1983 This suggests that spatial correlation bears no characteristic scale, but spatial entropy has characteristic scale. According to equation (3), fractal dimension is the characteristic values of spatial entropy. According to equation (4), the fractal dimension is the scaling exponent of spatial correlation function. For a given linear size of measurement scale ε, spatial entropy is linearly proportional to fractal dimension. Therefore, under certain conditions, the models of fractal dimension increase of cities can be extended to spatial entropy increase of complex spatial systems.

Simple model: logistic function
First of all, the logistic model of spatial entropy increase of urban growth can be derived from the fractal dimension model verified by observation evidence. This model is easy to understand. If the minimum dimension Dmin=0, fractal dimension curve of urban growth can be modeled by the logistic function (Chen, 2012;Chen, 2018) where Dq(t) denotes the generalized fractal dimension of moment order q at time t, Dmax is the capacity parameter indicative of the maximum fractal dimension in the future, Dq (0) is the initial value of the fractal dimension at time t=0 in the past, r refers to the initial growth rate of fractal dimension. In the case of Dmin=0, the ratio of fractal dimension Dq(t) to the maximum dimension Dmax proved to equal the ratio of generalized entropy Mq(t) to the maximum entropy Mmax (Chen, 2020), that is, Equation (6) can be converted into the following form If t=0 as given, then equation (6) in which Mq (0) is the initial value of the generalized entropy at time t=0. Substituting equations (6) and (7) into equation (5) yields By eliminating Dmax in equation (9), we get the logistic model of spatial entropy increase of cities: where r represents the initial growth rate of generalized entropy of urban form.
Based on normalized variable of spatial entropy, the three-parameter logistic model can be simplified as a two-parameter logistic model. For equation (10), Mmin=0, thus the above logistic model can also be "normalized" and re-expressed as where Mq * (0)=Mq(0)/Mmax is the normalized result of Mq(0) when Mmin=0.The normalized spatial 6 entropy can be used to define the degree of spatial equilibrium. It proved to equal normalized fractal dimension, which is also treated as the index of space filling extent. That is, we have where Jq(t) denotes the degree of spatial equilibrium. A spatial entropy odds can be defined as in which Oq(t) refers to the spatial entropy odds, Iq(t)=Mmax-Mq(t) denotes the information gain (Batty, 1974;Batty, 1976). Using information gain Iq(t) to divided by the maximum entropy Mmax yield redundancy Zq(t)=1-Mq(t)/Mmax (Batty, 1974). Substituting equation (10) into equation (13) yields * * where Oq(0)=Mq(0)/(Mmax-1). This suggests that spatial entropy odds of urban form increases exponentially. Using equation (14), we can define a logit transform of spatial entropy.

Exquisite model: Boltzmann equation
Further, we can derive Boltzmann model of spatial entropy increase of urban growth from the corresponding fractal dimension model. In fact, logistic model can be regarded as the special case of Boltzmann equation, which has been used to describe urban growth (Benguigui et al, 2001 where Dmin denotes the lower limit of fractal dimension, Dmax is upper limit of fractal dimension, p is a scaling parameter associated with the initial growth rate r, and τ, a temporal translational parameter indicative of a critical time, when the growth rate of fractal dimension reaches its peak (Chen, 2018). The scale translation and scaling parameters can be defined respectively by τ=ln((Dmax-Dq(0))/(Dq(0)-Dmin)) p and p=1/r. For Dmin>0, the normalized fractal dimension proved 7 to equal the normalized entropy (Chen, 2020), that is min min which can be re-expressed as min min max min max min If t=0, then equation (16) can be changed to max min max min min min Inserting equations (17) and (16) into equation (15) which represent the Boltzmann equation of spatial entropy increase of urban growth. For the normalized variable, equation (20) can be re-written as which can be used to generalize multifractal spectrums to Renyi entropy spectrums.

New spatial measurements based on fractal dimension and entropy
Fractal dimension can be treated as a basic measure of urban growth, and this measure is used to replace urban area. Due to scale-dependence of urban spatial measurements, urban area cannot be 8 objectively determined, while fractal dimension is a scale-free parameter, which can be employed to substitute urban area to reflect space filling and land use extent. Based on fractal dimension of urban form, a set of urban measurements or indexes can be defined to describe city development.
The measurements based on fractal dimension are tabulated as follows (Table 1)

Dt d D t 
Ratio of used space to remained space 9 All these new measurements based on fractal dimension of cities can be generalized to spatial entropy of complex spatial systems. The space-filling ratio proved to be a normalized fractal dimension, and the normalized fractal dimension proved to equal the normalized spatial entropy of urban form (Chen, 2012;Chen, 2020). The spatial entropy reflects the land-use extent of an urban region, namely, the degrees of space-filling and spatial uniformity. The measurements based on spatial entropy are tabulated as below (Table 2).
(1) Spatial entropy range, the difference between the upper limit and lower limit of spatial entropy values, Mmax-Mmin.
(2) Space-filling index, the difference between the spatial entropy value at time t and the lower limit of spatial entropy value,  In fact, two of the above entropy indexes are widely used in various scientific fields. One is information gain, and the other is redundancy (Batty, 1974;Batty, 1976;Batty, 2010 where Ht is the H-quantity at time t, P(k, t) denotes the probability of the kth zone at time t, and where M(t) refers to Shannon information entropy. This suggests that the H-quantity is in fact information gain (Batty, 1974;Batty, 2010). That is, Ht= I(t). The H-quantity Divided by the maximum entropy yields spatial redundancy (Batty, 1974)  This suggests that the H-quantity can act as a basic measurement of urban growth and form. Please note that the measurements such as H-quantity, information gain and redundancy in literature are based on Shannon entropy (Batty, 1974;Batty, 1976;Prigogine and Stengers, 1984). In this work, they are extended to general expressions based on Renyi entropy.

Discussion
According to the theoretical derivation, we can use logistic function and Boltzmann equation to model spatial entropy increase of urban development. Based on the logistic model and Boltzmann model, we can construct a set of entropy indexes to characterize urban form. To calculate spatial entropy, we have to make use of zonal systems (Figure 1). In practice, a zonal system of cities always form a hierarchy with cascade structure (Batty, 1974;Batty, 1976). The hierarchy based on twofold cascading can be treated as spatial disaggregation (Batty and Longley, 1994). In light of ideas from fractals, we can understand the cascade structure from a new angle of view. In fact, a city can be divided into several sectors, a sector can be divided into several districts, a district can be divided into several neighborhoods, and a neighborhood can be divided into several sites (Kaye, 1989). In each part of each level, we can find residential land, commercial-industrial land, vacant 6 12km 0 0 12km 6