# A Wave-Spectrum Analysis of Urban Population Density: Entropy, Fractal, and Spatial Localization

Authors: Yanguang Chen
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

The method of spectral analysis is employed to research the spatial dynamics of urban population distribution. First of all, the negative exponential model is derived in a new way by using an entropy-maximizing idea. Then an approximate scaling relation between wave number and spectral density is derived by Fourier transform of the negative exponential model. The theoretical results suggest the locality of urban population activities. So the principle of entropy maximization can be utilized to interpret the locality and localization of urban morphology. The wave-spectrum model is applied to the city in the real world, Hangzhou, China, and spectral exponents can give the dimension values of the fractal lines of urban population profiles. The changing trend of the fractal dimension does reflect the localization of urban population growth and diffusion. This research on spatial dynamics of urban evolvement is significant for modeling spatial complexity and simulating spatial complication of city systems by cellular automata.


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## Body

## 1. Introduction

The greatest shortcoming of the human race is our inability to understand the exponential function.Albert A. Bartlett


Simulating the spatial dynamics of urban population is an
interesting but a difficult project. Urban population density can be modeled by
two types of functions: one is the exponential function known as Clark’s law [1], the other is the power function proposed by Smeed [2].
Geographers used to employ the exponential model to characterize population
density of monocentric cities. However, Smeed’s
model has been favored since fractal cities came to front (see, e.g., [3–5]). In fact, if we reject Smeed’s model, we will be unable to
interpret the law of allometric growth on urban area and population in theory.
On the other hand, if we avoid Clark’s model,
we will not be able to describe many cities’ population density empirically.
Geographers are often placed in a dilemma when dealing with spatial dynamics of
urban evolution.In fact, the exponential
function implies translational symmetry, while the power function denotes
dilation symmetry or scaling symmetry; the exponential function implies
simplicity and randomness, while the power function indicates complexity and
structure [6, 7]. In fractal geometry, two exponential functions can often
construct a power function, while a power function can always be decomposed into two
exponential functions [8]. It
is difficult for us to understand the exponential function, and it is
especially difficult to understand the relation between the exponential distribution
and the power-law distribution. A conjecture is that
exponential law and power law represent, respectively, two modes of urban
evolvement which supplement each other.Population is one of the two central variables which can be
employed to explore the dynamics of cities [9]. However, the underlying rationale of
intraurban population growth and diffusion is still a question pending further
discussion. Clark’s law on urban density can
provide a window for us to apprehend the dynamics of urban morphology from the
angle of view of population. The negative exponential distribution seems
to mean nonfractal
structure of urban population, but it can be associated with fractal structure by the Fourier
transform. In order to probe the mysteries of fractal cities and the related spatial
dynamics, we must research the essentials of negative exponential distribution.In this paper,
the exponential model of urban density will be explored by using the wave-spectral
analysis. The significance of studying the classical model is in three aspects.
The first is to reveal the locality and localization of urban population evolution,
which is very important for simulating spatial complexity of cities through
computers. The second is to find a new approach to evaluating a kind of fractal dimension of
urban form, which differs from but can make upbox dimension and radial
dimension. The third is to understand spatial
complexity of urban evolvement in the new perspective. The study of complexity
concerns emergence of fractals, localization, strange attractor, symmetry
breaking, and so on [10]. Fractal structure
and localization can be brought to light to some extent from the negative
exponential distribution by means of spectral analysis.The rest of this paper is structured as follows. Section2 presents a new
derivation of the negative exponential model of urban population density by the entropy-maximizing principle,
which is actually one of the fundamental reasons of fractal cities [11]. Based on the
exponential function, an approximate power-law relation between wave number and
spectral density is derived by Fourier transform. Section 3 provides an
empirical analysis, including spectral analysis, correlogram analysis,
and information entropy analysis, by applying the theoretical
models to the city of Hangzhou, China. The computations lend support to the theoretical inferences given in Section 2. In Section 4, the differences
and relationships between the negative exponential distribution and the inverse
power-law distribution are discussed to distinguish the concept of locality from that of action at a distance.


Simulating the spatial dynamics of urban population is an
interesting but a difficult project. Urban population density can be modeled by
two types of functions: one is the exponential function known as Clark’s law [1], the other is the power function proposed by Smeed [2].
Geographers used to employ the exponential model to characterize population
density of monocentric cities. However, Smeed’s
model has been favored since fractal cities came to front (see, e.g., [3–5]). In fact, if we reject Smeed’s model, we will be unable to
interpret the law of allometric growth on urban area and population in theory.
On the other hand, if we avoid Clark’s model,
we will not be able to describe many cities’ population density empirically.
Geographers are often placed in a dilemma when dealing with spatial dynamics of
urban evolution.In fact, the exponential
function implies translational symmetry, while the power function denotes
dilation symmetry or scaling symmetry; the exponential function implies
simplicity and randomness, while the power function indicates complexity and
structure [6, 7]. In fractal geometry, two exponential functions can often
construct a power function, while a power function can always be decomposed into two
exponential functions [8]. It
is difficult for us to understand the exponential function, and it is
especially difficult to understand the relation between the exponential distribution
and the power-law distribution. A conjecture is that
exponential law and power law represent, respectively, two modes of urban
evolvement which supplement each other.Population is one of the two central variables which can be
employed to explore the dynamics of cities [9]. However, the underlying rationale of
intraurban population growth and diffusion is still a question pending further
discussion. Clark’s law on urban density can
provide a window for us to apprehend the dynamics of urban morphology from the
angle of view of population. The negative exponential distribution seems
to mean nonfractal
structure of urban population, but it can be associated with fractal structure by the Fourier
transform. In order to probe the mysteries of fractal cities and the related spatial
dynamics, we must research the essentials of negative exponential distribution.In this paper,
the exponential model of urban density will be explored by using the wave-spectral
analysis. The significance of studying the classical model is in three aspects.
The first is to reveal the locality and localization of urban population evolution,
which is very important for simulating spatial complexity of cities through
computers. The second is to find a new approach to evaluating a kind of fractal dimension of
urban form, which differs from but can make upbox dimension and radial
dimension. The third is to understand spatial
complexity of urban evolvement in the new perspective. The study of complexity
concerns emergence of fractals, localization, strange attractor, symmetry
breaking, and so on [10]. Fractal structure
and localization can be brought to light to some extent from the negative
exponential distribution by means of spectral analysis.The rest of this paper is structured as follows. Section2 presents a new
derivation of the negative exponential model of urban population density by the entropy-maximizing principle,
which is actually one of the fundamental reasons of fractal cities [11]. Based on the
exponential function, an approximate power-law relation between wave number and
spectral density is derived by Fourier transform. Section 3 provides an
empirical analysis, including spectral analysis, correlogram analysis,
and information entropy analysis, by applying the theoretical
models to the city of Hangzhou, China. The computations lend support to the theoretical inferences given in Section 2. In Section 4, the differences
and relationships between the negative exponential distribution and the inverse
power-law distribution are discussed to distinguish the concept of locality from that of action at a distance.


## 2. Mathematical Models

### 2.1. New Derivation of Clark’s
Law

A power law indicating fractal
structure of urban systems can be decomposed into two exponential laws [8], and
the exponential laws can be derived by using the entropy-maximizing method [12]. This
suggests that fractal structure can be interpreted with the principle of
entropy maximization, and exponential function is an important bridge between
entropy and power law. On the other hand, as complex spatial system, an urban
phenomenon can be modeled with different mathematical expressions under
different conditions. Urban population density can be described by a number of
functions, among which the negative exponential function is always valid in
empirical analysis [1, 13, 14]. Since that the exponential
law can connect entropy maximization and fractal, we are naturally interested
in the cause and effect of the urban exponential distribution. It will be shown
that Clark’s law comes between entropy-maximizing
process and special fractal structure.As a theoretical
study, this paper is focused on a monocentric city, and all the data analyses
are based on the idea from statistical average. In this instance, the growth of
cities is often regarded as a process of spatiotemporal diffusion [15], which can be abstracted as the
following partial differential equation (2.1)∂ρ(x,y,t)∂t=K[∂2ρ(x,y,t)∂x2+∂2ρ(x,y,t)∂y2]−aρ(x,y,t), where a denotes growth/decay
coefficient or transfer coefficient, K is called “diffusivity” or diffusion coefficient, and x and y refer to two directions of spatial
diffusion. For an isotropic diffusion, one direction
(say, x) has no difference from the
other direction (say, y), we have x=y=r, where r represents the distance from the center of city (where r=0).
In order words, we can substitute one-dimension diffusion process for
two-dimension process to analyze the isotropic city systems. Now, if ρ does not change with time, namely, if ∂ρ/∂t=0,
then (2.1) reduces to the common differential equation characterizing one-dimension
diffusion such as d2ρ(r)/dr2−aρ(r)/K=0 (the initial condition is ρ|r=0=ρ0, while the boundary condition ρ|r→∞=0), whose
solution is just the exponential function known as Clark’s
law (see Appendix A). This suggests that the
exponential law in fact reflects an instantaneous equilibrium of urban
population diffusion.Assuming that
population densityρ(r) at distance r from the city center
declines monotonically, Clark [1] proposed an empirical model that can be
written as (2.2)ρ(r)=ρ0exp(−br)=ρ0exp(−rr0), where ρ0 is a constant of proportionality
which is supposed to equal the central density, that is, ρ0=ρ(0),b denotes a rate at which the effect of distance attenuates,
and r0=1/b refers to a characteristic radius of urban
population distribution. Thus we have r0=K/a.
Clark [1] fitted the log transform of (2.1) to more than 20 cities by using linear
regression. The results form the solid empirical foundation of the negative
exponential law of monocentric urban density.In the real world, urban growth is often not isotropic, but
in an average sense, we can regard an anisotropic process as an isotropic
process. Just based on this idea, Clark’s law
is propounded. An urban population density function is actually defined in
one-dimension space but it includes information of two-dimension space. Generally
speaking, an exponential distribution function can be derived from the entropy maximization principle. Bussiere and Snickers [16]
once showed that Clark’s model could be derived from Wilson’s
[17] spatial interaction models (see also Wilson [18]), which is based on the entropy-maximizing
principle. In fact, under ideal conditions, Clark’s
model can be derived in a very simple way from several geographical assumptions
by using entropy-maximizing methods. Now, in
order to reveal the physical essence of exponential distribution of urban
population, a new derivation of (2.2) is
given in this subsection. The mathematical deduction is more graceful and compendious
than previous derivation presented by Bussiere and Snickers [16],
and the process is helpful for exploring the spatial dynamics of urban
evolution.Suppose that the
total population in the urban field of a monocentric city isPt, and the urban growth is considered
to be a continuous process in time and space. An urban field is defined as a
bounding circle based on the center of the urban cluster, marked by the maximum
radius R which contains the whole
cluster [3, page 340]. Imagining
that the urban map has been
digitalized with low resolution, we can “string” n+1pixels, which
may be called “cells,”
by drawing a straight line or a radial from the center of the city to the boundary
(see Figure 1). Further, suppose the population
in the ith cell along the “line” is ρi(i=0,1,2,…,n), and the whole population
along the line is P. The variable ρi has
dual attributes. On the one hand, it denotes the population size within the ith
cell along the line, and on the other, it represents just the average
population density of the ith ring comprising a number of cells.Figure 1
A sketch map of urban field with rings and cells.Since Clark’s law is just
the solution to the one-dimension diffusion equation, we can examine
one-dimension population distribution based on the idea from statistical
average. The postulates of this study can be summarized as follows. (1) A
monocentric city has no strict boundary because of scaling invariance of urban
form. (2) Population is dense enough in urban field. The next step is to
find the functional relationship between densityρi and
distance r. For this purpose, the
entropy-maximizing method is employed. The
number of states of the population distributed in all the cells along the radial, W, can be expressed as an ordered division problem (2.3)W=(Pρ0,ρ1,…,ρn)=P!∏i=0n+1ρi!. We use the state in one-dimension
urban space to represent the state in two-dimension urban space in average
sense. Then the entropy of population distribution profile, He,
is given by (2.4)He=lnW=lnP!−∑i=0n+1lnρi!. Suppose that the entropy approaches to maximization. We can
define an objective function such as (2.5)MaxHe=lnW. According to our assumptions stated above, the objective of
city evolution is subject to two constraint
conditions as follows: (2.6)∑i=0nρi=P,(2.7)ρ0+∑i=1n2πiρi=Pt. Equation (2.6) can be understood easily, but (2.7) need be made clear. Here ρ0 refers to
the population number in the center of the city, and 2πiρi to the population number in the ith
ring that is measured with a circle of cells. In fact, if we measure the distance by the size of the cells,
namely, take the diameter of cells as length unit, then i is just the
distance from the centroid of the ith cell to the center of the city.
That is, when the cells
are very very small, the ordinal number i can represent the radius of
the ith ring, and 2πi is the corresponding perimeter.In this case, if
we can find a mathematical expression to describe the relationship betweenρi and i, the problem will be solved immediately. Thus our question can be turned
into the process of finding conditional extremum because that the value of
entropy depends on the density of spatial distribution of urban population. A
Lagrangian function is constructed as (2.8)L(ρi)=lnP!−∑ilnρi!+λ1(P−∑iρi)+λ2(Pt−ρ0−∑i2πiρi), where λ is the Lagrangian multiplier
(LM). Theoretically ρi and P are both large
enough in terms of our postulates. According to the well-known Stirling’s formula N!=(2π)1/2NN+1/2e−N,
we have an approximate relation, ∂lnN!/∂N=lnN, where N is considerably big. So, differentiating (2.8)
partially with respect to ρi yields (2.9)∂L(ρi)∂ρi=−lnρi−λ1−2πλ2i. Considering the condition of extremum ∂L(ρi)/∂ρi=0,
we have (2.10)ρi=e−λ1e−2πλ2i.In theory, we
can improve the resolution of digital map unlimitedly, and thus, the
cell/pixels become infinitesimal. That is to say, for simplicity, the discrete
distance variable represented byi can be replaced with a continuous one
represented by r for the time
being, that is, i→r,ρi→ρ(r).
Inserted with (2.10), the discrete (2.7) in which r is used as a substitute for i can be rewritten as an integration
expression (2.11)∫0R2πrρ(r)dr=e−λ1∫0R2πre−2πλ2rdr=Pt, where R is the radius of urban field and it can be
defined by R=F/2, here F is the Feret’s diameter [3], (see Kaye [19]). Equation (2.11) is the continuous
expression replacing (2.7). In keeping with the first postulate, R is large enough. Using
integration by decomposition and taking l’Hospital’s rule into account, we
can find the solution of (2.11) such as (see Appendix B) (2.12)e−λ1=2πλ22Pt. Substituting (2.12) into (2.10) yields (2.13)ρ(r)=2πλ22Pte−2πλ2r. If r=0
as given, then (2.13) collapses to (2.14)ρ(0)=2πλ22Pt=ρ0. The characteristic radius of the city, r0, can be defined by (2.15)ρ0=e−λ1=Pt2πr02. Inserting (2.15) into (2.14) gives (2.16)2πλ2=b=1r0. Substituting (2.14) and (2.16)
into (2.13) immediately yields Clark’s
law, that is, (2.2). Further,
inserting (2.2)
into (2.6), we
can derive r0=P/ρ0. The maximum of entropy can be proved to be Hmax=elnr0, where e is the base of the natural system of
logarithms, having a numerical value of 2.7183 approximately.Entropy maximization suggests the most probable
distribution on some conditions. The negative exponential distribution of urban
population density is not inevitable, but it is the most probable state for at least the
monocentric city. This kind of distribution suggests a special fractal profile,
which can be brought to light by Fourier analysis and scaling wave-spectrum
relation.


### 2.2. Wave-Spectrum Function of Urban Density

The negative exponential model of urban density is in essence
a special spatial correlation function. A power-law relation between wave
number and spectrum density can be derived from the exponential function.
Considering the interaction between the cells along the radial in Figure1, we can construct a density-density correlation
function as follows: (2.17)C(r)=∫−∞∞ρ(x)ρ(x+r)dx=2ρ02∫0∞e−2x/r0−r/r0dx, where ρ(x) denotes the population density of
cell X at distance x from the city
center, ρ(x+r) refers to the population
density of another cell at distance r from X. Given x=0, it follows that one cell becomes the center of the
city, and the spatial correlation function collapses to an exponential function (2.18)C(r)=ρ(0)ρ(r)=ρ02e−r/r0. If the data are so normalized that ρ0=1, we
have C(r)=ρ(r), and thus
(2.18) is equivalent to (2.2). In this case,
Clark’s law is just a special density-density
correlation function, which indicates spatial correlating action between the
city center and the location at distance r from the center. The distance parameter, r0,
is relative to the spatial correlation length. A larger value of the
characteristic radius (r0)
suggests a longer correlation distance.Note that the autocorrelation
function and the energy spectrum can be converted to each other through Fourier’s
cosine transform:(2.19)S(k)=∫−∞∞C(r)e−j2πkrdr=2∫0∞C(r)cos2πkrdr, where j=−1 is the unit of complex number, k denotes
wave number, that is, the reciprocal of the wavelength, S(k)
represents corresponding energy spectral density.
The concept of energy
spectrum comes from engineering mathematics. The product of Fourier
transform of a function and its conjugate bears an analogy with the mathematical
form of energy in physics [20]. In
the light of the symmetry of correlation function, the Fourier transform of (2.18) can be given in the form (2.20)F(k)=ρ02∫−∞∞e−r/r0e−j2πkrdr=2r0ρ021+j2πkr0. As 2r0ρ02 is large enough, we have (2.21)[12r0ρ02]2→0. Thus the energy spectral density can be gained according to
energy integral such as [21] (2.22)S(k)=|F(k)|2=(2r0ρ02)21+r02(2πk)2≈1(πk)2/(ρ04)∝k−2. In practice, the length of sample path, L, is generally limited, therefore the wave spectrum density W(k)=|G(k)|2/L is always employed to substitute for the energy spectrum density S(k).
Then (2.22) can be rewritten as [22] (2.23)W(k)∝k−2. Equation (2.23) is an approximate expression
based on ideal conditions, and it can be generalized to the following scaling
relation: (2.24)W(k)∝k−β, where β is called “spectral
exponent” which usually ranges from 0 to 3. When β value is near 1, (2.24)
indicates what is called 1/β noise (see, e.g., [23, 24]). In fact, the spectral
exponent is associated with a fractal dimension of urban population profiles.For a time series or spatial
series, if the relation between spectral density and frequency or wave number
follows the scaling law defined by (2.24), a
fractal structure can be revealed. It has been demonstrated that, for dE=1 dimension variables, the connection between β and D is given by [25–27] (2.25)D=dE+3−β2=5−β2=2−H, where dE refers to the dimension of Euclidean space. Accordingly, β=5−2D, where D is the fractal dimension of urban population profiles (dE<D<dE+1), and H denotes
the Hurst exponent (0≤H≤1). Further, the autocorrelation coefficients
of the rate of changes can be derived from the fractional Brownian motion as in [25] (2.26)CΔ(r)=〈−ρ(r−1)ρ(r+1)〉〈ρ(r)2〉=22H−1−1. This is a special density-density
correlation function, which can be understood by means of the knowledge of time
series analysis. Many methods of analyzing times series, including
autocorrelation analysis, autoregression analysis, and spectral analysis, can
be employed to deal with spatial series [28]. If D=1.5
or β=2, then we have H=1/2, and thus CΔ=0. In this
case, the ith cells act directly on
and only on the (i±1)th cells, and do not act on the (i±2)th cells or more. If D<1.5 or β>2,
then we have H>1/2, and
thus CΔ>0. In
this case, for u>1, the ith cell can act directly on the (i±u)th
cells positively. If D>1.5 or β<2, then we have H<1/2, and thus CΔ<0. In
this case, the ith cell will act
directly on the (i±u)th
cells negatively.For the negative exponential function of urban population
density, the expected result of spectral exponent isβ≈2, thus the fractal dimension is D≈3/2=1.5, and the
Hurst exponent is H=2−D≈0.5, which gives the
autocorrelation coefficient CΔ≈0. That suggests
a spatial locality of city systems. In physics, the principle of locality coming from Einstein
[29] is that distant objects cannot have
direct influence on each another. In other words, an object is influenced
directly only by its immediate surroundings. The fact of spatial
autocorrelation coefficient CΔ→0 implies that
a population cell tends to interact only on the immediate cells.


## 2.1. New Derivation of Clark’s
Law

A power law indicating fractal
structure of urban systems can be decomposed into two exponential laws [8], and
the exponential laws can be derived by using the entropy-maximizing method [12]. This
suggests that fractal structure can be interpreted with the principle of
entropy maximization, and exponential function is an important bridge between
entropy and power law. On the other hand, as complex spatial system, an urban
phenomenon can be modeled with different mathematical expressions under
different conditions. Urban population density can be described by a number of
functions, among which the negative exponential function is always valid in
empirical analysis [1, 13, 14]. Since that the exponential
law can connect entropy maximization and fractal, we are naturally interested
in the cause and effect of the urban exponential distribution. It will be shown
that Clark’s law comes between entropy-maximizing
process and special fractal structure.As a theoretical
study, this paper is focused on a monocentric city, and all the data analyses
are based on the idea from statistical average. In this instance, the growth of
cities is often regarded as a process of spatiotemporal diffusion [15], which can be abstracted as the
following partial differential equation (2.1)∂ρ(x,y,t)∂t=K[∂2ρ(x,y,t)∂x2+∂2ρ(x,y,t)∂y2]−aρ(x,y,t), where a denotes growth/decay
coefficient or transfer coefficient, K is called “diffusivity” or diffusion coefficient, and x and y refer to two directions of spatial
diffusion. For an isotropic diffusion, one direction
(say, x) has no difference from the
other direction (say, y), we have x=y=r, where r represents the distance from the center of city (where r=0).
In order words, we can substitute one-dimension diffusion process for
two-dimension process to analyze the isotropic city systems. Now, if ρ does not change with time, namely, if ∂ρ/∂t=0,
then (2.1) reduces to the common differential equation characterizing one-dimension
diffusion such as d2ρ(r)/dr2−aρ(r)/K=0 (the initial condition is ρ|r=0=ρ0, while the boundary condition ρ|r→∞=0), whose
solution is just the exponential function known as Clark’s
law (see Appendix A). This suggests that the
exponential law in fact reflects an instantaneous equilibrium of urban
population diffusion.Assuming that
population densityρ(r) at distance r from the city center
declines monotonically, Clark [1] proposed an empirical model that can be
written as (2.2)ρ(r)=ρ0exp(−br)=ρ0exp(−rr0), where ρ0 is a constant of proportionality
which is supposed to equal the central density, that is, ρ0=ρ(0),b denotes a rate at which the effect of distance attenuates,
and r0=1/b refers to a characteristic radius of urban
population distribution. Thus we have r0=K/a.
Clark [1] fitted the log transform of (2.1) to more than 20 cities by using linear
regression. The results form the solid empirical foundation of the negative
exponential law of monocentric urban density.In the real world, urban growth is often not isotropic, but
in an average sense, we can regard an anisotropic process as an isotropic
process. Just based on this idea, Clark’s law
is propounded. An urban population density function is actually defined in
one-dimension space but it includes information of two-dimension space. Generally
speaking, an exponential distribution function can be derived from the entropy maximization principle. Bussiere and Snickers [16]
once showed that Clark’s model could be derived from Wilson’s
[17] spatial interaction models (see also Wilson [18]), which is based on the entropy-maximizing
principle. In fact, under ideal conditions, Clark’s
model can be derived in a very simple way from several geographical assumptions
by using entropy-maximizing methods. Now, in
order to reveal the physical essence of exponential distribution of urban
population, a new derivation of (2.2) is
given in this subsection. The mathematical deduction is more graceful and compendious
than previous derivation presented by Bussiere and Snickers [16],
and the process is helpful for exploring the spatial dynamics of urban
evolution.Suppose that the
total population in the urban field of a monocentric city isPt, and the urban growth is considered
to be a continuous process in time and space. An urban field is defined as a
bounding circle based on the center of the urban cluster, marked by the maximum
radius R which contains the whole
cluster [3, page 340]. Imagining
that the urban map has been
digitalized with low resolution, we can “string” n+1pixels, which
may be called “cells,”
by drawing a straight line or a radial from the center of the city to the boundary
(see Figure 1). Further, suppose the population
in the ith cell along the “line” is ρi(i=0,1,2,…,n), and the whole population
along the line is P. The variable ρi has
dual attributes. On the one hand, it denotes the population size within the ith
cell along the line, and on the other, it represents just the average
population density of the ith ring comprising a number of cells.Figure 1
A sketch map of urban field with rings and cells.Since Clark’s law is just
the solution to the one-dimension diffusion equation, we can examine
one-dimension population distribution based on the idea from statistical
average. The postulates of this study can be summarized as follows. (1) A
monocentric city has no strict boundary because of scaling invariance of urban
form. (2) Population is dense enough in urban field. The next step is to
find the functional relationship between densityρi and
distance r. For this purpose, the
entropy-maximizing method is employed. The
number of states of the population distributed in all the cells along the radial, W, can be expressed as an ordered division problem (2.3)W=(Pρ0,ρ1,…,ρn)=P!∏i=0n+1ρi!. We use the state in one-dimension
urban space to represent the state in two-dimension urban space in average
sense. Then the entropy of population distribution profile, He,
is given by (2.4)He=lnW=lnP!−∑i=0n+1lnρi!. Suppose that the entropy approaches to maximization. We can
define an objective function such as (2.5)MaxHe=lnW. According to our assumptions stated above, the objective of
city evolution is subject to two constraint
conditions as follows: (2.6)∑i=0nρi=P,(2.7)ρ0+∑i=1n2πiρi=Pt. Equation (2.6) can be understood easily, but (2.7) need be made clear. Here ρ0 refers to
the population number in the center of the city, and 2πiρi to the population number in the ith
ring that is measured with a circle of cells. In fact, if we measure the distance by the size of the cells,
namely, take the diameter of cells as length unit, then i is just the
distance from the centroid of the ith cell to the center of the city.
That is, when the cells
are very very small, the ordinal number i can represent the radius of
the ith ring, and 2πi is the corresponding perimeter.In this case, if
we can find a mathematical expression to describe the relationship betweenρi and i, the problem will be solved immediately. Thus our question can be turned
into the process of finding conditional extremum because that the value of
entropy depends on the density of spatial distribution of urban population. A
Lagrangian function is constructed as (2.8)L(ρi)=lnP!−∑ilnρi!+λ1(P−∑iρi)+λ2(Pt−ρ0−∑i2πiρi), where λ is the Lagrangian multiplier
(LM). Theoretically ρi and P are both large
enough in terms of our postulates. According to the well-known Stirling’s formula N!=(2π)1/2NN+1/2e−N,
we have an approximate relation, ∂lnN!/∂N=lnN, where N is considerably big. So, differentiating (2.8)
partially with respect to ρi yields (2.9)∂L(ρi)∂ρi=−lnρi−λ1−2πλ2i. Considering the condition of extremum ∂L(ρi)/∂ρi=0,
we have (2.10)ρi=e−λ1e−2πλ2i.In theory, we
can improve the resolution of digital map unlimitedly, and thus, the
cell/pixels become infinitesimal. That is to say, for simplicity, the discrete
distance variable represented byi can be replaced with a continuous one
represented by r for the time
being, that is, i→r,ρi→ρ(r).
Inserted with (2.10), the discrete (2.7) in which r is used as a substitute for i can be rewritten as an integration
expression (2.11)∫0R2πrρ(r)dr=e−λ1∫0R2πre−2πλ2rdr=Pt, where R is the radius of urban field and it can be
defined by R=F/2, here F is the Feret’s diameter [3], (see Kaye [19]). Equation (2.11) is the continuous
expression replacing (2.7). In keeping with the first postulate, R is large enough. Using
integration by decomposition and taking l’Hospital’s rule into account, we
can find the solution of (2.11) such as (see Appendix B) (2.12)e−λ1=2πλ22Pt. Substituting (2.12) into (2.10) yields (2.13)ρ(r)=2πλ22Pte−2πλ2r. If r=0
as given, then (2.13) collapses to (2.14)ρ(0)=2πλ22Pt=ρ0. The characteristic radius of the city, r0, can be defined by (2.15)ρ0=e−λ1=Pt2πr02. Inserting (2.15) into (2.14) gives (2.16)2πλ2=b=1r0. Substituting (2.14) and (2.16)
into (2.13) immediately yields Clark’s
law, that is, (2.2). Further,
inserting (2.2)
into (2.6), we
can derive r0=P/ρ0. The maximum of entropy can be proved to be Hmax=elnr0, where e is the base of the natural system of
logarithms, having a numerical value of 2.7183 approximately.Entropy maximization suggests the most probable
distribution on some conditions. The negative exponential distribution of urban
population density is not inevitable, but it is the most probable state for at least the
monocentric city. This kind of distribution suggests a special fractal profile,
which can be brought to light by Fourier analysis and scaling wave-spectrum
relation.


## 2.2. Wave-Spectrum Function of Urban Density

The negative exponential model of urban density is in essence
a special spatial correlation function. A power-law relation between wave
number and spectrum density can be derived from the exponential function.
Considering the interaction between the cells along the radial in Figure1, we can construct a density-density correlation
function as follows: (2.17)C(r)=∫−∞∞ρ(x)ρ(x+r)dx=2ρ02∫0∞e−2x/r0−r/r0dx, where ρ(x) denotes the population density of
cell X at distance x from the city
center, ρ(x+r) refers to the population
density of another cell at distance r from X. Given x=0, it follows that one cell becomes the center of the
city, and the spatial correlation function collapses to an exponential function (2.18)C(r)=ρ(0)ρ(r)=ρ02e−r/r0. If the data are so normalized that ρ0=1, we
have C(r)=ρ(r), and thus
(2.18) is equivalent to (2.2). In this case,
Clark’s law is just a special density-density
correlation function, which indicates spatial correlating action between the
city center and the location at distance r from the center. The distance parameter, r0,
is relative to the spatial correlation length. A larger value of the
characteristic radius (r0)
suggests a longer correlation distance.Note that the autocorrelation
function and the energy spectrum can be converted to each other through Fourier’s
cosine transform:(2.19)S(k)=∫−∞∞C(r)e−j2πkrdr=2∫0∞C(r)cos2πkrdr, where j=−1 is the unit of complex number, k denotes
wave number, that is, the reciprocal of the wavelength, S(k)
represents corresponding energy spectral density.
The concept of energy
spectrum comes from engineering mathematics. The product of Fourier
transform of a function and its conjugate bears an analogy with the mathematical
form of energy in physics [20]. In
the light of the symmetry of correlation function, the Fourier transform of (2.18) can be given in the form (2.20)F(k)=ρ02∫−∞∞e−r/r0e−j2πkrdr=2r0ρ021+j2πkr0. As 2r0ρ02 is large enough, we have (2.21)[12r0ρ02]2→0. Thus the energy spectral density can be gained according to
energy integral such as [21] (2.22)S(k)=|F(k)|2=(2r0ρ02)21+r02(2πk)2≈1(πk)2/(ρ04)∝k−2. In practice, the length of sample path, L, is generally limited, therefore the wave spectrum density W(k)=|G(k)|2/L is always employed to substitute for the energy spectrum density S(k).
Then (2.22) can be rewritten as [22] (2.23)W(k)∝k−2. Equation (2.23) is an approximate expression
based on ideal conditions, and it can be generalized to the following scaling
relation: (2.24)W(k)∝k−β, where β is called “spectral
exponent” which usually ranges from 0 to 3. When β value is near 1, (2.24)
indicates what is called 1/β noise (see, e.g., [23, 24]). In fact, the spectral
exponent is associated with a fractal dimension of urban population profiles.For a time series or spatial
series, if the relation between spectral density and frequency or wave number
follows the scaling law defined by (2.24), a
fractal structure can be revealed. It has been demonstrated that, for dE=1 dimension variables, the connection between β and D is given by [25–27] (2.25)D=dE+3−β2=5−β2=2−H, where dE refers to the dimension of Euclidean space. Accordingly, β=5−2D, where D is the fractal dimension of urban population profiles (dE<D<dE+1), and H denotes
the Hurst exponent (0≤H≤1). Further, the autocorrelation coefficients
of the rate of changes can be derived from the fractional Brownian motion as in [25] (2.26)CΔ(r)=〈−ρ(r−1)ρ(r+1)〉〈ρ(r)2〉=22H−1−1. This is a special density-density
correlation function, which can be understood by means of the knowledge of time
series analysis. Many methods of analyzing times series, including
autocorrelation analysis, autoregression analysis, and spectral analysis, can
be employed to deal with spatial series [28]. If D=1.5
or β=2, then we have H=1/2, and thus CΔ=0. In this
case, the ith cells act directly on
and only on the (i±1)th cells, and do not act on the (i±2)th cells or more. If D<1.5 or β>2,
then we have H>1/2, and
thus CΔ>0. In
this case, for u>1, the ith cell can act directly on the (i±u)th
cells positively. If D>1.5 or β<2, then we have H<1/2, and thus CΔ<0. In
this case, the ith cell will act
directly on the (i±u)th
cells negatively.For the negative exponential function of urban population
density, the expected result of spectral exponent isβ≈2, thus the fractal dimension is D≈3/2=1.5, and the
Hurst exponent is H=2−D≈0.5, which gives the
autocorrelation coefficient CΔ≈0. That suggests
a spatial locality of city systems. In physics, the principle of locality coming from Einstein
[29] is that distant objects cannot have
direct influence on each another. In other words, an object is influenced
directly only by its immediate surroundings. The fact of spatial
autocorrelation coefficient CΔ→0 implies that
a population cell tends to interact only on the immediate cells.


## 3. Empirical Analysis

### 3.1. Study Area and Data Resource

The city of Hangzhou  is taken as an example to verify the wave-spectrum relation of urban population
density and related theory. Hangzhou is the capital of Zhejiang province, China. The urban density data in
1964, 1982, 1990, and 2000 come from census, which is processed by Feng [14]. The census tract data are based on jiedao, which bears an analogy with the UK enumeration districts [30], or the US
subdistricts [13], while the system of jiedao has an analogy with the urban zonal
system in Western literature (see [3, page 325]). In the demographic sense, a jiedao is a census
tract. The data are processed by means of spatial weighed average. The length
of sample path is 26, and the maximum urban radius is 15.3 km. The method of processing data is illuminated
in detail by Feng [14]. Some necessary
explanations on sampling and data processing are made as follows.(1) Sampling
area. The data cover the metropolitan area (MA) of Hangzhou, which is greater than the urbanized
area (UA). The spatial scopes of sampling in four years are same in order to
make it sure that the parameters from 1964 through 2000 are comparable. Because
of scaling invariance of urban form [3], we take no account of the borderline between the urban
and rural areas.(2) Calculation method. A series of concentric
rings are drawn around city center in proportional spacing (see Figure2). The ratio of
each partial zone to the whole area between two rings is taken as the weight of
computing urban population density. A region between two adjacent rings
can be named a circular belt, which will be numbered as p=0,1,…,n, where n is the number of circular belt. The zones
can be numbered as q=1,2,…,m, where m is the number of zones. Let Spq be the common area of the pth
circular belt and the qth zone, that
is, (3.1)Spq=Bp⋂Zq, where Bp represents the pth belt, Zq denotes the qth zone, both of them are measured with area;
therefore, Sij is the area of the intersection of Bp and Zq. Defining a weighted coefficient wpq as (3.2)wpq=SpqSp=Spq∑qSpq=Spqπ(rp+12−rp2), we have (3.3)ρ¯p=∑q=1mwpqρq=1Sp∑q=1mSpqρq, in which ρq is the population density of the qth zone (jiedao), which can be known from the census datum. Thus (3.4)Sp=∑q=1mSpq=π(rp+12−rp2), where rp refers to the radius of the pth ring.
It is evident that ρ¯p denotes the average density of the pth circular belt. The weighted arithmetic average can lessen the influence
of zone’s scope on the estimated results of population density as much as
possible.Figure 2
Study area and zonal system in the Hangzhou metropolis (from Feng [14]).(3) Spatial
scale. The radius difference between rings is 0.6 km, less thanr0,
that is, the average distance of urban population activity. The parameter
values of r0 can  be
estimated with Clark’s model, namely, (2.2).


### 3.2. Data Processing Method and
Results

The population density of Hangzhou
city will be analyzed from three
angles of view: spectral analysis, correlogram analysis, and information
entropy analysis. Accordingly, we will compute wave-spectral density,
autocorrelation function, and information entropy. The procedure of wave
spectrum analysis based on fast Fourier transform (FFT) can be summed up as
five steps.Step 1 (sample path extension).
The
symmetrical rule of the FFT’s recursive algorithm requires the length of time
series to be an integer power of 2, that is,L=2z(z=1,2,3,…). However, there are 26 data points in our
spatial sample path (n=26≈24.7).
A process called “zero-padding” can be used to bring the number up to the next
power of 2. In this case, the best way is to add 6 zeros at the end of the data
series to bring the number to 32 (i.e., L=25).Step 2 (FFT of spatial series).
Performing the FFT on the extended population density data of Hangzhou city yields a
complex data seriesF(k). The processing method is so accessible that MS Excel can give the
results conveniently.Step 3 (spectrum density calculation).
The formula is such asW(k)=|F(k)|2/L=|F(k)|2/32. It is not difficult for us to compute
the spectral density based on the FFT results (see Table 1). The spectral density is just the product
of FFT result and its conjugate divided by the extended sample path length (L=32).Table 1
Wave number and spectral density of Hangzhou
urban population density: 1964–2000.
Wave number(k)Spectral densityW(k)1964198219902000047084702968701576896946908614947031960.031252667351753761241655186291226751089490.06251630380912311016942836303732947666510.09375801987371098726111246171491025741540.125340916484678720552487145318783180.15625117132241742326024322424247567030.1875105271751496882521343643298455170.21875125922591823262922844713244579830.25110685781701071415856897117151620.2812510257519159645751243443862379970.312510488070165506551451961789076070.3437593959331480208214191199107277560.37574312401235594511425873104251790.4062560061559418308838911794202980.437548406287937045606073064937480.4687556676418597673632905547890050.56772368977635172451944698112Step 4 (making wave-spectra plots).
As soon as the population density is transformed into spectral density, a plot
reflecting the relation between wave number and spectral density can be given
easily (see Figure3). Let the circular belts be
numbered as p=0,1,2,…,L−1, where L=25=32.
Thus the wave number will be defined by k=p/L.Wave-spectra plots of Hangzhou urban population density distribution: 1964–2000.
(a)
1964(b)
1982(c)
1990(d)
2000Step 5 (modeling the wave-spectra relations).
If the wave-spectra plots displayed in Figure3 show some attenuation trend, we can fit the data of Table 1 to (2.24). A least square computation will give the spectral
exponent β, from which we can
estimate the fractal dimension D and Hurst
exponent H by means of (2.25).All the
population density data of Hangzhou
city in four years satisfy the power law in the form
stipulated by (2.24) to a great extent. A
least squares calculation utilizing the data of Table 1 gives four spectral exponent β values. The fractal dimension D, Hurst exponent H, and the autocorrelation
coefficient CΔ can be evaluated
consequently (see Table 2). From 1964 to 2000, the spectral exponent values become
closer and closer to β=2, the fractal
dimension values become closer and closer to D=3/2, and the Hurst exponent values become closer and closer to H=1/2.
All of these suggest a phenomenon of localization of urban population evolution:
a population cell is inclined towards acting directly on the immediate cells,
and not on the alternate cells, that is, nonimmediate
cells. What is more, the wave-spectrum relations and spectral exponent values indicating
1/f noise [23] remind us of the
self-organized criticality (SOC) of urban evolution [31–33].Table 2
Estimated values of model parameters and related statistics of Hangzhou urban density. The characteristic radius(r0) values are estimated by
means of least squares computation based on (2.2), and using r0 values, we can compute the
maximum entropy with the formula Hmax=elnr0.
The unit of entropy is “nat.”
YearCharacteristic radius(r0)Spectral
exponent(β)Goodness of fit(R2)Fractal
dimension(D)Hurst exponent(H)Autocorrelation
coefficient(CΔ)Maximum entropy(Hmax)19643.5641.4890.9251.7560.245−0.2983.45519823.6711.4350.9201.7830.218−0.3243.53519903.6281.6640.9661.6680.332−0.2083.50320003.9461.7980.9491.6010.399−0.1303.731It should be made clear that the fractal dimension used
here is different from those employed to characterize two-dimension urban form
such asbox dimension and radial dimension ([34], White and Engelen [35]). Generally speaking, we need three kinds of fractal
dimensions at least to characterize the city form with fractal structure. The
first is the box dimensionDb, which can be estimated by
the box-counting method [36]; the second is radial
dimensionDf, which is
defined by the area-radius scaling [3, 5]; and the
third is the dimension of fractal lines [26, 27], the author of this paper calls
it profile dimensionDs when it is applied to
urban morphology. The third type of dimension can be estimated easily through
the wave-spectrum relation (see Appendix C).Spectral analysis and correlation analysis represent
different sides of the same coin in theory, while empirically correlation
analysis and spectral analysis supplement each other. Therefore, a correlogram analysis of Hangzhou
urban density should be made to consolidate
the results of wave-spectra analysis. A spatial autocorrelation function can be
based on the relationship betweenρ(r) and ρ(r+l),
where l refers to displacement
analogous to time lag in time series analysis. Part of the autocorrelation
function (ACF) and partial autocorrelation function (PACF) values is listed in
Table 3, and the results in 2000 are shown in Figure 4. The ACF attenuates gradually when the
displacement becomes long and displays some damped oscillation. What interests
us is the PACF, which cuts off at a displacement of 1.
That is, partial autocorrelation coefficients (PACCs) are not
significantly different from 0 when the displacement l>1 (see
Table 3 and Figure 4(b)). The cutoff of PACF at
a displacement of 1 suggests a possible locality in spatial activities of urban
population: a population cell acts directly on and only on the proximate
population cells, not on alternate cells. Evidently, the correlogram analysis
lends further support to the conclusion drawn from the wave-spectrum analysis.Table 3
Autocorrelation function (ACF) and partial autocorrelation function (PACF) values of Hangzhou’s population
density: 1964–2000. As the sample path is not too
long, only the first five values are really significant (261/2≈5). In time or spatial series analysis, we can judge the nature
of series by standard error or by Box-Ljung statistic including Q-statistic and
corresponding significance. Generally, it is easier and more visual to use the two standard-error bands shown in the histograms.
Distance
(r)Lag
(l)1964198219902000ACFPACFACFPACFACFPACFACFPACF0.910.8820.8820.8780.8780.8920.8920.9030.9031.520.757−0.0930.753−0.0750.773−0.1100.796−0.1052.130.626−0.0990.622−0.0990.656−0.0580.683−0.0842.740.496−0.0730.486−0.0990.532−0.1070.571−0.0663.350.365−0.0900.359−0.0570.410−0.0650.462−0.0523.960.253−0.0140.246−0.0280.292−0.0720.339−0.1524.570.142−0.0840.142−0.0500.177−0.0710.216−0.091Histograms of spatial ACF and PACF of Hangzhou’s population density (2000). The two lines in the histograms are called “two standard-error bands,” according to which we can know whether or not there is significant difference between ACF or PACF values and zero.
(a)
Spatial ACF histogram(b)
Spatial PACF histogramThere exists a
mathematical relation between fractal dimension and information entropy. In a
sense, Hausdorff dimension can be proved to be equivalent to Shannon’s
information entropy [37]. It is hard
to evaluate the population distribution entropy by using (2.4), we can only estimate the maximum entropy by
using the formula Hmax=elnr0,
which is based on one-dimensional continuous measure (see Table 2). However, it is easy to calculate
the one-dimensional discrete information entropy of population profile along the radial (see Figure 1). Defining a probability such as
(3.5)Pi=ρi∑i=0nρi,where variables ρ and n fulfill the same
roles as in (2.3) or (2.4), then we have an
information entropy
(3.6)He=−∑i=0nPilnPi,in which He refers to the Shannon’s entropy. The results of spatial entropy for Hangzhou’s population distribution in four
years are as follows: He=2.459 nat in 1964, He=2.484 nat
in 1982, He=2.549 nat in
1990, and He=2.725 nat in
2000. The maximum information entropy based on discrete measure is Hm=ln(26)=3.258 nat. The
redundancy Z measuring the ratio of
actual entropy to the maximum entropy and subtracting this ratio from 1 can be
applied to spatial entropy statistics [38]. Using the formula Z=1−H/Hm, the redundancy is
computed as follows: Z1964=0.245, Z1982=0.237, Z1990=0.218, Z2000=0.164. The information
entropy values become larger and larger, and the redundancy values approach the
minimum value 0. This trend gives further weight to the viewpoint that the
dynamics of urban population evolvement in Hangzhou
is actually a process of entropy
maximization.As stated above,
information entropy maximization implies negative exponential distribution of
urban population density, and the exponential distribution denotes spectral
exponentβ=2 and thus fractal
dimension D=1.5. On the other hand, the PACF based on
the exponential function shows a cutoff at a displacement of 1. All of these
suggest a localization tendency of urban population distribution in Hangzhou. The reasoning
process from entropy maximization to localization of spatial distributions of
urban population is illustrated as Figure 5. In physics, localization is a phenomenon according to which
the stationary quantum states of electrons in an extended system are localized
due to disorder [39]. As for cities, localization can be defined as follows: a
system of nonlocality changes gradually to that of locality.Figure 5
Entropy maximization suggesting localization of urban population distribution.


### 4. Questions and Discussion

As indicated above, urban population density can be modeled
by different functions under different conditions. The diversity or variability
of urban models suggests asymmetry or symmetry breaking of geographical
systems, which thus suggests spatial complexity of city systems and
complication of urban evolution. Besides the negative exponential function, the
inverse power function is also very important in modeling urban form. The
relations between the exponential function and the power function were
expounded by Batty and Kim [40]. Two questions will be discussed and answered here. The
first is the difference between the negative exponential distribution and the
inverse power-law distribution where the spatial dynamics is concerned, and the
second is the locality and localization of urban population evolution.The negative exponential function known as Clark’s model and the inverse power function known as
Smeed’s model are two types of special spatial correlation functions. Exponential
correlation function implies simple structure, while power-law correlation
function suggests complex dynamics. If and only if a system falls into the
self-organized critical state, the spatial correlation will follow a power law.
Otherwise it follows an exponential law [22, 23]. In urban studies,
the exponential function is always used to characterize urban population
density, while the power function can be used to model urban land use density. Research
into the relation between exponential law and power law is instructive for us
to explore deeply the spatial dynamics of urban morphology.The negative
exponential distribution indicates locality in theory, while the power law
implies action at a distance. This viewpoint can be verified by the correlogram
analysis based on simple simulation computation. Both the ACF and PACF can
provide a summary of a time or spatial series’ dynamics [41]. The ACF based on the standard exponential
distribution displays a gradual one-sided damping (see Figure 6(a)), while the PACF of the exponential distribution cuts off at a
displacement of 1—the partial
autocorrelations drop abruptly to 0 beyond displacement 1 (see Figure 6(b)). The PACF seems to suggest a property of
locality associated with the exponential distribution of urban density. As for Hangzhou
city, the PACF is
consistent with the result based on the standard exponential distribution, but
the ACF differs in the damping way just because that
the exponential distribution in the real world is not often very standard. In other
words, the urban population dynamics of Hangzhou
from 1964 to 2000 is only gradual localization without proper locality.Histograms of spatial ACF and PACF based on the exponential distribution.
(a)
ACF histogram(b)
PACF histogramIt is revealing
to compare the correlogram of the exponential distribution with that of the
power-law distribution. The ACF and PACF based on the standard power-law function
differ from those based on the standard exponential function in an important
way. The ACF of the power-law distribution displays a slow one-sided damping (see
Figure7(a)), while the corresponding PACF
displays rapid one-sided damping without cutoff (see Figure 7(b)).
In short, both the ACF and PACF of the power-law distribution are trailing, and
this phenomenon reminds us of the action at a distance of spatial activities.Histograms of spatial ACF and PACF based on the power-law distribution.
(a)
ACF histogram(b)
PACF histogramThe differences
of ACF and PACF between the exponential distribution and the power-law
distribution are obvious and interesting. The ACF of the power-law distribution
decays more slowly than that of the exponential distribution. In particular,
the PACF of the power-law distribution is trailing, while the PACF of the
exponential distribution cuts it off at the displacement of 1. The former
suggests an action at a distance, while the latter reminds us of locality of
spatial interaction (see Figure8). The similarities
and differences between the correlograms of the exponential distribution and
that of the power-law distribution are tabulated as follows (see Table 4). The correlograms of population density
distributions of Hangzhou
are more similar to those of the exponential distribution than those of the
power-law distribution.Table 4
Autocorrelation function (ACF) and partial autocorrelation function (PACF) values.
DistributionFunctionCorrelogramSuggestionExponential distributionACFTailing: gradual one-sided dampingPACFCutoff at a displacement of 1LocalityPower-law distributionACFTailing: slow one-sided dampingPACFTailing: rapid one-sided dampingAction at a distanceUrban density of Hangzhou cityACFDamped oscillationPACFCutoff at a displacement of 1LocalitySketch maps of locality and action at a distance of urban dynamics. In (a) indicative of locality,Cell1 only acts on Cell2, not on Cell3, while in (b) indicating action at a distance, Cell1 not only acts on Cell2, but also on Cell3, Cell4,
and so on.
(a)
Locality(b)
Action at a distanceAs indicated
above, the exponential distribution has a characteristic length,r0, which indicates simple
geometrical patterns, while the power-law distribution has no characteristic
length, which indicates complex patterns associated with fractal form and
structure. Revealing the relationship between locality and action at a distance
of urban evolution is very important for modeling spatial complexity by using
cellular automata (CA). The original CA model possesses locality. In urban
simulation, the CA’s locality is gradually replaced by action at a distance [35, 42, 43]. For urban-land dynamics, the CA model with action at a distance
is suitable, but for urban population dynamics, the things may be more
complicated because that urban population models are not one and only.The scaling wave-spectrum
relation and fractal properties of urban density suggest a dual character of
urban evolution. On the one hand, the growth of cities look like particle
motion, which can be simulated by means of CA technique, including
diffusion-limited aggregation (DLA) and dielectric breakdown model (DBM), and
so forth [3]. On the other hand, the statistical average of urban population
distribution reminds us of the wave motion, or a ripple spreading from the
center to the periphery. A city seems to be a set of dynamic particles indicating
chaos or disorder distributed on the ripple indicative of order. In fact,
intuitively, the spatial complexity displayed by city seems to express a
struggle between order and chaos. An urban model of ripple-particle duality
should be proposed to address temporal-spatial evolution of cities. As space is
limited, the related questions will be made clear in the future work.


### 5. Conclusions

The study of this paper may be of revelation for modeling
spatial complexity and simulating the urban growth and form. Geographers used
to rely heavily on the rules associated withaction at a distance, but neglect the rules based on the locality of urban population activity. However,
urban spatial dynamics seems to be the unity of opposites of locality and action at a distance. The keys of
comprehending this paper rest with three aspects. (1) Density is a zero-dimension
measure, but urban density function is defined in one-dimension space, from which
we can learn the information of two-dimension space. (2) Urban density models
are in essence spatial correlation function, which can be converted into energy spectrum by Fourier transform and
vice versa. Energy spectral
density divided by sample path length is the wave spectral density. (3) If the
relation between wave-spectrum
density and wave number shows scaling invariance, fractal dimension can be
estimated indirectly through the spectral exponent. The main points
of the paper can be summarized as follows.Firstly, one of the
important physical mechanisms of urban growth and population diffusion is
information entropy maximization indicating spatial optimization. From the
viewpoint of statistical average, urban population density distributions of monocentric
cities always satisfy the negative exponential function, which can be derived
by using entropy-maximizing
methods. Entropy maximization actually implies minimum
cost when benefit is certain, or maximum benefit when cost is determinate. In other
words, entropy maximizing in human systems suggests a process of optimization.
Urban population density tends to evolve into an optimum distribution through
self-organization.Secondly, the negative
exponential distribution implies locality or localization of urban population
activities. Entropy maximization interprets the negative exponential distribution,
and the scaling wave-spectrum relation coming from the negative exponential
function predicts a locality of urban population activities in theory. In terms
of the empirical evidences, the wave spectral analysis shows a localization
process of urban population evolution, while the spatial autocorrelation
analysis associated with wave spectral analysis demonstrates a locality of
spatial interaction of population cells.Thirdly, urban
evolution seems to possess a dual nature of locality and action at a distance.
The concept of locality should be as important as the idea of action at a
distance for urban modeling and simulation. Locality is to urban population
what action at a distance is to urban land use. The former relates to the
negative exponential distribution, while the latter to the inverse power-law
distribution. The power law indicates fractal structure, and the exponential
law can be connected with fractals by Fourier transform. A conjecture is that if
a city evolves into a self-organized critical state, the negative exponential
distribution may change to the inverse power distribution.


## 3.1. Study Area and Data Resource

The city of Hangzhou  is taken as an example to verify the wave-spectrum relation of urban population
density and related theory. Hangzhou is the capital of Zhejiang province, China. The urban density data in
1964, 1982, 1990, and 2000 come from census, which is processed by Feng [14]. The census tract data are based on jiedao, which bears an analogy with the UK enumeration districts [30], or the US
subdistricts [13], while the system of jiedao has an analogy with the urban zonal
system in Western literature (see [3, page 325]). In the demographic sense, a jiedao is a census
tract. The data are processed by means of spatial weighed average. The length
of sample path is 26, and the maximum urban radius is 15.3 km. The method of processing data is illuminated
in detail by Feng [14]. Some necessary
explanations on sampling and data processing are made as follows.(1) Sampling
area. The data cover the metropolitan area (MA) of Hangzhou, which is greater than the urbanized
area (UA). The spatial scopes of sampling in four years are same in order to
make it sure that the parameters from 1964 through 2000 are comparable. Because
of scaling invariance of urban form [3], we take no account of the borderline between the urban
and rural areas.(2) Calculation method. A series of concentric
rings are drawn around city center in proportional spacing (see Figure2). The ratio of
each partial zone to the whole area between two rings is taken as the weight of
computing urban population density. A region between two adjacent rings
can be named a circular belt, which will be numbered as p=0,1,…,n, where n is the number of circular belt. The zones
can be numbered as q=1,2,…,m, where m is the number of zones. Let Spq be the common area of the pth
circular belt and the qth zone, that
is, (3.1)Spq=Bp⋂Zq, where Bp represents the pth belt, Zq denotes the qth zone, both of them are measured with area;
therefore, Sij is the area of the intersection of Bp and Zq. Defining a weighted coefficient wpq as (3.2)wpq=SpqSp=Spq∑qSpq=Spqπ(rp+12−rp2), we have (3.3)ρ¯p=∑q=1mwpqρq=1Sp∑q=1mSpqρq, in which ρq is the population density of the qth zone (jiedao), which can be known from the census datum. Thus (3.4)Sp=∑q=1mSpq=π(rp+12−rp2), where rp refers to the radius of the pth ring.
It is evident that ρ¯p denotes the average density of the pth circular belt. The weighted arithmetic average can lessen the influence
of zone’s scope on the estimated results of population density as much as
possible.Figure 2
Study area and zonal system in the Hangzhou metropolis (from Feng [14]).(3) Spatial
scale. The radius difference between rings is 0.6 km, less thanr0,
that is, the average distance of urban population activity. The parameter
values of r0 can  be
estimated with Clark’s model, namely, (2.2).


## 3.2. Data Processing Method and
Results

The population density of Hangzhou
city will be analyzed from three
angles of view: spectral analysis, correlogram analysis, and information
entropy analysis. Accordingly, we will compute wave-spectral density,
autocorrelation function, and information entropy. The procedure of wave
spectrum analysis based on fast Fourier transform (FFT) can be summed up as
five steps.Step 1 (sample path extension).
The
symmetrical rule of the FFT’s recursive algorithm requires the length of time
series to be an integer power of 2, that is,L=2z(z=1,2,3,…). However, there are 26 data points in our
spatial sample path (n=26≈24.7).
A process called “zero-padding” can be used to bring the number up to the next
power of 2. In this case, the best way is to add 6 zeros at the end of the data
series to bring the number to 32 (i.e., L=25).Step 2 (FFT of spatial series).
Performing the FFT on the extended population density data of Hangzhou city yields a
complex data seriesF(k). The processing method is so accessible that MS Excel can give the
results conveniently.Step 3 (spectrum density calculation).
The formula is such asW(k)=|F(k)|2/L=|F(k)|2/32. It is not difficult for us to compute
the spectral density based on the FFT results (see Table 1). The spectral density is just the product
of FFT result and its conjugate divided by the extended sample path length (L=32).Table 1
Wave number and spectral density of Hangzhou
urban population density: 1964–2000.
Wave number(k)Spectral densityW(k)1964198219902000047084702968701576896946908614947031960.031252667351753761241655186291226751089490.06251630380912311016942836303732947666510.09375801987371098726111246171491025741540.125340916484678720552487145318783180.15625117132241742326024322424247567030.1875105271751496882521343643298455170.21875125922591823262922844713244579830.25110685781701071415856897117151620.2812510257519159645751243443862379970.312510488070165506551451961789076070.3437593959331480208214191199107277560.37574312401235594511425873104251790.4062560061559418308838911794202980.437548406287937045606073064937480.4687556676418597673632905547890050.56772368977635172451944698112Step 4 (making wave-spectra plots).
As soon as the population density is transformed into spectral density, a plot
reflecting the relation between wave number and spectral density can be given
easily (see Figure3). Let the circular belts be
numbered as p=0,1,2,…,L−1, where L=25=32.
Thus the wave number will be defined by k=p/L.Wave-spectra plots of Hangzhou urban population density distribution: 1964–2000.
(a)
1964(b)
1982(c)
1990(d)
2000Step 5 (modeling the wave-spectra relations).
If the wave-spectra plots displayed in Figure3 show some attenuation trend, we can fit the data of Table 1 to (2.24). A least square computation will give the spectral
exponent β, from which we can
estimate the fractal dimension D and Hurst
exponent H by means of (2.25).All the
population density data of Hangzhou
city in four years satisfy the power law in the form
stipulated by (2.24) to a great extent. A
least squares calculation utilizing the data of Table 1 gives four spectral exponent β values. The fractal dimension D, Hurst exponent H, and the autocorrelation
coefficient CΔ can be evaluated
consequently (see Table 2). From 1964 to 2000, the spectral exponent values become
closer and closer to β=2, the fractal
dimension values become closer and closer to D=3/2, and the Hurst exponent values become closer and closer to H=1/2.
All of these suggest a phenomenon of localization of urban population evolution:
a population cell is inclined towards acting directly on the immediate cells,
and not on the alternate cells, that is, nonimmediate
cells. What is more, the wave-spectrum relations and spectral exponent values indicating
1/f noise [23] remind us of the
self-organized criticality (SOC) of urban evolution [31–33].Table 2
Estimated values of model parameters and related statistics of Hangzhou urban density. The characteristic radius(r0) values are estimated by
means of least squares computation based on (2.2), and using r0 values, we can compute the
maximum entropy with the formula Hmax=elnr0.
The unit of entropy is “nat.”
YearCharacteristic radius(r0)Spectral
exponent(β)Goodness of fit(R2)Fractal
dimension(D)Hurst exponent(H)Autocorrelation
coefficient(CΔ)Maximum entropy(Hmax)19643.5641.4890.9251.7560.245−0.2983.45519823.6711.4350.9201.7830.218−0.3243.53519903.6281.6640.9661.6680.332−0.2083.50320003.9461.7980.9491.6010.399−0.1303.731It should be made clear that the fractal dimension used
here is different from those employed to characterize two-dimension urban form
such asbox dimension and radial dimension ([34], White and Engelen [35]). Generally speaking, we need three kinds of fractal
dimensions at least to characterize the city form with fractal structure. The
first is the box dimensionDb, which can be estimated by
the box-counting method [36]; the second is radial
dimensionDf, which is
defined by the area-radius scaling [3, 5]; and the
third is the dimension of fractal lines [26, 27], the author of this paper calls
it profile dimensionDs when it is applied to
urban morphology. The third type of dimension can be estimated easily through
the wave-spectrum relation (see Appendix C).Spectral analysis and correlation analysis represent
different sides of the same coin in theory, while empirically correlation
analysis and spectral analysis supplement each other. Therefore, a correlogram analysis of Hangzhou
urban density should be made to consolidate
the results of wave-spectra analysis. A spatial autocorrelation function can be
based on the relationship betweenρ(r) and ρ(r+l),
where l refers to displacement
analogous to time lag in time series analysis. Part of the autocorrelation
function (ACF) and partial autocorrelation function (PACF) values is listed in
Table 3, and the results in 2000 are shown in Figure 4. The ACF attenuates gradually when the
displacement becomes long and displays some damped oscillation. What interests
us is the PACF, which cuts off at a displacement of 1.
That is, partial autocorrelation coefficients (PACCs) are not
significantly different from 0 when the displacement l>1 (see
Table 3 and Figure 4(b)). The cutoff of PACF at
a displacement of 1 suggests a possible locality in spatial activities of urban
population: a population cell acts directly on and only on the proximate
population cells, not on alternate cells. Evidently, the correlogram analysis
lends further support to the conclusion drawn from the wave-spectrum analysis.Table 3
Autocorrelation function (ACF) and partial autocorrelation function (PACF) values of Hangzhou’s population
density: 1964–2000. As the sample path is not too
long, only the first five values are really significant (261/2≈5). In time or spatial series analysis, we can judge the nature
of series by standard error or by Box-Ljung statistic including Q-statistic and
corresponding significance. Generally, it is easier and more visual to use the two standard-error bands shown in the histograms.
Distance
(r)Lag
(l)1964198219902000ACFPACFACFPACFACFPACFACFPACF0.910.8820.8820.8780.8780.8920.8920.9030.9031.520.757−0.0930.753−0.0750.773−0.1100.796−0.1052.130.626−0.0990.622−0.0990.656−0.0580.683−0.0842.740.496−0.0730.486−0.0990.532−0.1070.571−0.0663.350.365−0.0900.359−0.0570.410−0.0650.462−0.0523.960.253−0.0140.246−0.0280.292−0.0720.339−0.1524.570.142−0.0840.142−0.0500.177−0.0710.216−0.091Histograms of spatial ACF and PACF of Hangzhou’s population density (2000). The two lines in the histograms are called “two standard-error bands,” according to which we can know whether or not there is significant difference between ACF or PACF values and zero.
(a)
Spatial ACF histogram(b)
Spatial PACF histogramThere exists a
mathematical relation between fractal dimension and information entropy. In a
sense, Hausdorff dimension can be proved to be equivalent to Shannon’s
information entropy [37]. It is hard
to evaluate the population distribution entropy by using (2.4), we can only estimate the maximum entropy by
using the formula Hmax=elnr0,
which is based on one-dimensional continuous measure (see Table 2). However, it is easy to calculate
the one-dimensional discrete information entropy of population profile along the radial (see Figure 1). Defining a probability such as
(3.5)Pi=ρi∑i=0nρi,where variables ρ and n fulfill the same
roles as in (2.3) or (2.4), then we have an
information entropy
(3.6)He=−∑i=0nPilnPi,in which He refers to the Shannon’s entropy. The results of spatial entropy for Hangzhou’s population distribution in four
years are as follows: He=2.459 nat in 1964, He=2.484 nat
in 1982, He=2.549 nat in
1990, and He=2.725 nat in
2000. The maximum information entropy based on discrete measure is Hm=ln(26)=3.258 nat. The
redundancy Z measuring the ratio of
actual entropy to the maximum entropy and subtracting this ratio from 1 can be
applied to spatial entropy statistics [38]. Using the formula Z=1−H/Hm, the redundancy is
computed as follows: Z1964=0.245, Z1982=0.237, Z1990=0.218, Z2000=0.164. The information
entropy values become larger and larger, and the redundancy values approach the
minimum value 0. This trend gives further weight to the viewpoint that the
dynamics of urban population evolvement in Hangzhou
is actually a process of entropy
maximization.As stated above,
information entropy maximization implies negative exponential distribution of
urban population density, and the exponential distribution denotes spectral
exponentβ=2 and thus fractal
dimension D=1.5. On the other hand, the PACF based on
the exponential function shows a cutoff at a displacement of 1. All of these
suggest a localization tendency of urban population distribution in Hangzhou. The reasoning
process from entropy maximization to localization of spatial distributions of
urban population is illustrated as Figure 5. In physics, localization is a phenomenon according to which
the stationary quantum states of electrons in an extended system are localized
due to disorder [39]. As for cities, localization can be defined as follows: a
system of nonlocality changes gradually to that of locality.Figure 5
Entropy maximization suggesting localization of urban population distribution.


## 4. Questions and Discussion

As indicated above, urban population density can be modeled
by different functions under different conditions. The diversity or variability
of urban models suggests asymmetry or symmetry breaking of geographical
systems, which thus suggests spatial complexity of city systems and
complication of urban evolution. Besides the negative exponential function, the
inverse power function is also very important in modeling urban form. The
relations between the exponential function and the power function were
expounded by Batty and Kim [40]. Two questions will be discussed and answered here. The
first is the difference between the negative exponential distribution and the
inverse power-law distribution where the spatial dynamics is concerned, and the
second is the locality and localization of urban population evolution.The negative exponential function known as Clark’s model and the inverse power function known as
Smeed’s model are two types of special spatial correlation functions. Exponential
correlation function implies simple structure, while power-law correlation
function suggests complex dynamics. If and only if a system falls into the
self-organized critical state, the spatial correlation will follow a power law.
Otherwise it follows an exponential law [22, 23]. In urban studies,
the exponential function is always used to characterize urban population
density, while the power function can be used to model urban land use density. Research
into the relation between exponential law and power law is instructive for us
to explore deeply the spatial dynamics of urban morphology.The negative
exponential distribution indicates locality in theory, while the power law
implies action at a distance. This viewpoint can be verified by the correlogram
analysis based on simple simulation computation. Both the ACF and PACF can
provide a summary of a time or spatial series’ dynamics [41]. The ACF based on the standard exponential
distribution displays a gradual one-sided damping (see Figure 6(a)), while the PACF of the exponential distribution cuts off at a
displacement of 1—the partial
autocorrelations drop abruptly to 0 beyond displacement 1 (see Figure 6(b)). The PACF seems to suggest a property of
locality associated with the exponential distribution of urban density. As for Hangzhou
city, the PACF is
consistent with the result based on the standard exponential distribution, but
the ACF differs in the damping way just because that
the exponential distribution in the real world is not often very standard. In other
words, the urban population dynamics of Hangzhou
from 1964 to 2000 is only gradual localization without proper locality.Histograms of spatial ACF and PACF based on the exponential distribution.
(a)
ACF histogram(b)
PACF histogramIt is revealing
to compare the correlogram of the exponential distribution with that of the
power-law distribution. The ACF and PACF based on the standard power-law function
differ from those based on the standard exponential function in an important
way. The ACF of the power-law distribution displays a slow one-sided damping (see
Figure7(a)), while the corresponding PACF
displays rapid one-sided damping without cutoff (see Figure 7(b)).
In short, both the ACF and PACF of the power-law distribution are trailing, and
this phenomenon reminds us of the action at a distance of spatial activities.Histograms of spatial ACF and PACF based on the power-law distribution.
(a)
ACF histogram(b)
PACF histogramThe differences
of ACF and PACF between the exponential distribution and the power-law
distribution are obvious and interesting. The ACF of the power-law distribution
decays more slowly than that of the exponential distribution. In particular,
the PACF of the power-law distribution is trailing, while the PACF of the
exponential distribution cuts it off at the displacement of 1. The former
suggests an action at a distance, while the latter reminds us of locality of
spatial interaction (see Figure8). The similarities
and differences between the correlograms of the exponential distribution and
that of the power-law distribution are tabulated as follows (see Table 4). The correlograms of population density
distributions of Hangzhou
are more similar to those of the exponential distribution than those of the
power-law distribution.Table 4
Autocorrelation function (ACF) and partial autocorrelation function (PACF) values.
DistributionFunctionCorrelogramSuggestionExponential distributionACFTailing: gradual one-sided dampingPACFCutoff at a displacement of 1LocalityPower-law distributionACFTailing: slow one-sided dampingPACFTailing: rapid one-sided dampingAction at a distanceUrban density of Hangzhou cityACFDamped oscillationPACFCutoff at a displacement of 1LocalitySketch maps of locality and action at a distance of urban dynamics. In (a) indicative of locality,Cell1 only acts on Cell2, not on Cell3, while in (b) indicating action at a distance, Cell1 not only acts on Cell2, but also on Cell3, Cell4,
and so on.
(a)
Locality(b)
Action at a distanceAs indicated
above, the exponential distribution has a characteristic length,r0, which indicates simple
geometrical patterns, while the power-law distribution has no characteristic
length, which indicates complex patterns associated with fractal form and
structure. Revealing the relationship between locality and action at a distance
of urban evolution is very important for modeling spatial complexity by using
cellular automata (CA). The original CA model possesses locality. In urban
simulation, the CA’s locality is gradually replaced by action at a distance [35, 42, 43]. For urban-land dynamics, the CA model with action at a distance
is suitable, but for urban population dynamics, the things may be more
complicated because that urban population models are not one and only.The scaling wave-spectrum
relation and fractal properties of urban density suggest a dual character of
urban evolution. On the one hand, the growth of cities look like particle
motion, which can be simulated by means of CA technique, including
diffusion-limited aggregation (DLA) and dielectric breakdown model (DBM), and
so forth [3]. On the other hand, the statistical average of urban population
distribution reminds us of the wave motion, or a ripple spreading from the
center to the periphery. A city seems to be a set of dynamic particles indicating
chaos or disorder distributed on the ripple indicative of order. In fact,
intuitively, the spatial complexity displayed by city seems to express a
struggle between order and chaos. An urban model of ripple-particle duality
should be proposed to address temporal-spatial evolution of cities. As space is
limited, the related questions will be made clear in the future work.


## 5. Conclusions

The study of this paper may be of revelation for modeling
spatial complexity and simulating the urban growth and form. Geographers used
to rely heavily on the rules associated withaction at a distance, but neglect the rules based on the locality of urban population activity. However,
urban spatial dynamics seems to be the unity of opposites of locality and action at a distance. The keys of
comprehending this paper rest with three aspects. (1) Density is a zero-dimension
measure, but urban density function is defined in one-dimension space, from which
we can learn the information of two-dimension space. (2) Urban density models
are in essence spatial correlation function, which can be converted into energy spectrum by Fourier transform and
vice versa. Energy spectral
density divided by sample path length is the wave spectral density. (3) If the
relation between wave-spectrum
density and wave number shows scaling invariance, fractal dimension can be
estimated indirectly through the spectral exponent. The main points
of the paper can be summarized as follows.Firstly, one of the
important physical mechanisms of urban growth and population diffusion is
information entropy maximization indicating spatial optimization. From the
viewpoint of statistical average, urban population density distributions of monocentric
cities always satisfy the negative exponential function, which can be derived
by using entropy-maximizing
methods. Entropy maximization actually implies minimum
cost when benefit is certain, or maximum benefit when cost is determinate. In other
words, entropy maximizing in human systems suggests a process of optimization.
Urban population density tends to evolve into an optimum distribution through
self-organization.Secondly, the negative
exponential distribution implies locality or localization of urban population
activities. Entropy maximization interprets the negative exponential distribution,
and the scaling wave-spectrum relation coming from the negative exponential
function predicts a locality of urban population activities in theory. In terms
of the empirical evidences, the wave spectral analysis shows a localization
process of urban population evolution, while the spatial autocorrelation
analysis associated with wave spectral analysis demonstrates a locality of
spatial interaction of population cells.Thirdly, urban
evolution seems to possess a dual nature of locality and action at a distance.
The concept of locality should be as important as the idea of action at a
distance for urban modeling and simulation. Locality is to urban population
what action at a distance is to urban land use. The former relates to the
negative exponential distribution, while the latter to the inverse power-law
distribution. The power law indicates fractal structure, and the exponential
law can be connected with fractals by Fourier transform. A conjecture is that if
a city evolves into a self-organized critical state, the negative exponential
distribution may change to the inverse power distribution.



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*Source: 728420-2008-10-16.xml*