1

An Allometric Algorithm for Fractal-based Cobb-Douglas

Function of Geographical Systems

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Abstracts:The generalized Cobb-Douglasproduction functionhas been derived from a general

input-output relationbased onfractality assumptions. It proved to be a usefulself-affine model for

geographical analysis. However, the ordinary least square calculation is always an

ineffectualmethod forthe Cobb-Douglas modeling because ofthe multicollinerityin the logarithmic

linear regression. In this paper, a novel approach is proposed tobuild the geographical

Cobb-Douglas models.Combiningthe concept of allometric scaling withthe linear regression

technique, we obtain a simple algorithm that can be employed toestimate the parameters of the

Cobb-Douglas function. As a case, the algorithm and models are applied to the public

transportation of China’s cities, and the results validate the allometric algorithm. A conclusion can

be drawn that the allometric analysis is an effective way of modeling geographical systems with

the general Cobb-Douglas function. This study is significant for integrating the notions

ofallometry, fractals, and scaling into a new framework to form a quantitative methodology of

spatial analys is.

Keywords :allometric growth; Cobb-Douglas function; fractal; fractal dimension; geographical

system; scaling analysis; Urban transportation of China

1 Introduction

The well-knownproduction function is proposed by Cobb and Douglas (1928)and consolidated

by many economists. Today, the Cobb-Douglas function proved to be a tool of geographical

2

analys isbecause geographical systems are always associated with economic systems(Chen, 2008;

Chen and Lin, 2009).In fact, many geographical theoriessuch ascentral place theorycame from

economics. By geographical assumptions, the generalized Cobb-Douglas production model can be

derived in a simple way by using the ideasfrom fractals and scaling. Thus fractal concept and the

production function were combined to form a new model.This model may be significant to

geographical information analys is in the future. Fractals are everywhere (Barnsley, 1988; Batty,

2008; Mandelbrot, 1983), and fractal dimension is related to information entropy (Ryabko, 1986).

Fractal geometry can be employed to reveal spatial information ofgeographical systems, especially,

cities as systems and systems of cities (Batty, 2005;Batty and Longley, 1988;Batty and Longley,

1994; Benguiguiet al, 2000; Chen, 2012; Chen and Wang, 2013; Frankhauser, 1990;Frankhauser,

1994; Frankhauser, 1998; Feng and Chen, 2010; Liu and Chen, 1995; Liu and Chen, 2000; Liu and

Chen, 2003; Luo and Chen, 2002; Thomas et al, 2007; Thomas et al, 2008; White and Engelen,

1993; White and Engelen, 1994). What is more, the general Cobb-Douglas function can be

generalized to modeling natural systems, e.g., ecological systems (Chen, 2011).

A good mathematical model is always based on an effective algorithm. However, the ordinary

least square (OLS) method cannot be used to estimate the parameter values of the fractal-based

Cobb-Douglas modelin many cases because of logarithmic multicollinearity. In other words, there

are allometric scaling relationships between any two independent variables, which are employed

to make models. On the other hand, the allometricscaling is one ofthebasic laws in geography.

Itcan be utilized to make spatial analyses for many geographical systems (Batty et al, 2008; Batty

and Longley, 1994; Chen, 2008; Chen, 2010; Lee, 1989; Liu and Chen, 2004; Liu and Chen, 2005;

Lo and Welch, 1977;Longley et al, 1991 Narol and Bertalanffy, 1956; Tobler, 1969).The key to

solving the problem of the Cobb-Douglas modeling rests with the allometry. In this paper, the

allometric analysis is adopted to develop a new approach to evaluating the parameters of the

general production function. The other parts are organized as follows. In Section 2, the allometric

scaling is demonstrated to be way of estimating the parameters of the Cobb-Douglas model. In

Section 3, the model based on theallometric algorithm is applied to the public transportation of

Chinese cities, and the empirical results validatethe theoretical derivation. Finally, in Section 4, the

related questions are discussed, and the paper is concluded with several pieces of comments.

3

2Mathematical models

2.1 New form of the production function

In geographyas well aseconomics, themathematical models similar to the Cobb-Douglas

production functions are widely employed to describe the relationship of one output to many

inputs.Suppose that there are melementsXi in a geographical system,and each elementaffects the

output Yto someextent (i=1,2,…,m). An assumption can be made as follows:the geographical

system is a fractal system in a broad sense.That is, the relationships between the inputs represented

byelements Xi and the output Y satisfy some self-similar or self-affine conditions (Chen,

2008).Becauseall the elements contribute to the output of thegeographical system, the output as

response to the elements can be generallyexpressedas follows

),,,( 21 mXXXkfY ,(1)

inwhich k is aproportionality coefficient. Taking the complete differential of equation (1) yields

i

m

i i

m

i

i

i

XX

YX

X

fkY ddd

11

.(2)

This result is easy to understand by the knowledge of complete differential in higher

mathematics.Dividing equation (2) on both sides withYgives

m

i i

ii

m

i i

ii

i X

X

X

X

Y

X

X

Y

Y

Y

11

d]

d)[(

d . (3)

The precondition of equation (3) is that the system depicted by equation (1) is of fractal structure.

If and only if the system is of self-similar or self-affine structure, the parameter σiwill be a

constant. According to our assumption aforementioned, the fractal parameterσican be defined by

iii

i

i

iX

Y

XX

YY

Y

X

X

Y

ln

ln

/

/

.(4)

This parameter is in fact an elastic coefficient. Apparently,equation (4)reflectsa partial scaling

relation, just indicating that the outputY depends to scale on the element Xiregardless of other

elements.Equation (3)can be equivalently expressedas

m

i

ii XY1

lndlnd . (5)

4

Then taking integral on both sides of this equation (5)yields

m

i

i

m

i

iiiXXY

11

lndlndlnd .(6)

Apparently, from equation (6) it follows

CXYm

i

ii

1

lnln

,(7)

whereC refers to the integral constant. A simple mathematical transform of equation (7)yields

m

i

iiXKY

1

(8)

which is just the generalized Cobb-Douglas function indicative of self-affine patterns (Chen, 2008;

Chen and Lin, 2009).In the function,the constant K=expCrefers to the output coefficient. If m=2 as

given, then equation (8) can be reduced to the common Cobb-Douglas function

21

21

XKXY ,(9)

whereX1 and X2 represent capital and labor respectively.In a city system, X1 and X2 can represent

population size and land-use area.

The traditional production function is a special case of the generalized production function. It is

clear that equation (8)denotes a complete relation, while equation (4)represents a partial relation.

In this case, ifequation (8)is true, we cannotintegrateequation (4)over Xi.Otherwise a

contradictionwill take place if we take integral of equation (4). Now supposethat only Xi

influencesY with other elements are invariable.Then the equation dlnY=σidlnXican be derived from

equation (4), and integrating this relation gives the following power function

i

ii XCY

.(10)

This is a generalizedallometricfunction(Bettencourt, 2013; Bettencourt et al, 2007; Chen, 2008;

Chen and Zhou, 2003). There is a logical contradiction between equation (8)andequation (10).

Comparing equation (10) with equation (8)shows why equation (4)cannot be taken

integralwhenequation (10)holds. Evidently, we can get a multivariate linear regression equationby

taking logarithmson both sides ofequation (8);in the same way, a univariatelinearregression

equationcan be gotten by taking logs of equation (10). According to the knowledge of

statisticsrelated, if a response variable depends on many explanatory variables, but only one of

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whichis considered, then the estimationresults of the parameters of theregressionequation will

greatly deviates from the true values.In short,equation (10)will be trueif and only if others

elements Xj (j≠i) are all fixed.

A general allometric scaling relation can be derived from the generalized Cobb-Douglas

function. Taking two elements, say, Xi and Xj, into consideration in terms of equation (10), we

have

ij

ji XX /

.(11)

This is the well-known allometric relation of any two elements (Bertalanffy, 1968; Chen, 2010).

Suppose the dimension of Xiis Di, and the dimension of Xjis Dj. According to the principle of

dimensional homogeneity, we have

ji DD

ji XX/

.(12)

The principle of dimens ional homogeneity is also called the“principle of dimension consistency”

(Lee, 1989), which comes from the proportion axiom in mathematics.Comparing equation

(12)with equation (11) yields a fractal parameter relation such as (Chen, 2008)

jjii DD ,(13)

which suggests that the ratio of two fractal dimensions equals the ratio of two elastic coefficients,

but a fractal dimension is be reciprocally proportional to its corresponding elastic

coefficient.Substituting equation (11)or equation (12)intoequation (8) repeatedlyyields

id

ii XcY ,(14)

whereci is a proportionality coefficient, and diis generalized allometric scaling exponent. Clearly,

the scaling exponent di is relative to fractal dimension Diand Dj. Equations (14)shares the same

formasequation (10),but they have different physical meaning.Equation (10)is true if and only

ifthe final output depends only on one element, whereas equation (14) is true when the final output

depends on many elements, betweenwhichthere exist allometricscaling relations.However, if

theallometricrelation appearsbetween any pair of urban elements,the parameters of the log-linear

expression of equation (8)cannot be estimatedby using the OLS method. Otherwise, we will

inevitablycome across the problem of multicollinearity.Thus we need a new algorithm for the

Cobb-Douglas modeling of geographical systems.

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2.2Allometric algorithm

The fractal-based Cobb-Douglas equation can be formally solved by the OLS algorithm, but the

results may be unacceptable. Taking natural logarithm on both sides of equation (8) gives

mm XXXKY lnlnlnlnln 2211 , (15)

which is clearly a multivariable linear equation. It seems as if the technique of multivariate linear

regression can be employed to estimate the values of model parameters. However, the independent

variables are always related to one another in practice so that the parameters cannot be properly

evaluated because of multicollinearity. A discovery is that the allometric analys is can be used to

solve the generalized Cobb-Douglas equations. Let i=1, 2, …,m.Then multiplying equation (14)by

equation (14) again and againyields

mdm

mdmdmm

mXXXcccY//

2

/

12121 .(16)

This suggests a partial scaling relation. In fact, multiplyingXiby a scale factor λ gives

)()( ii XYXY i ,(17)

which suggest a self-affine fractal structure (Chen and Lin, 2009). Comparing equation (16) with

equation (8)shows

m

d ii .(18)

An allometric algorithm can be presented for solving the generalized production model.In fact,

equations (14) and (16) indicate a new approach to estimating model parameters, and equation (18)

suggests a simple method of evaluating the partial scaling exponents. In short, by means of the

allometric scaling relations between each independent variable and the dependent variable, we

canreplace the multiplelog-linear regressionwith a set of simplelog-linear regressions toestimate

the model parameters and then build the Cobb-Douglas model.

2.3Fractal meaning of model parameters

It is necessary to demonstrate that the parameters of the Cobb-Douglas model are associated

with fractal dimens ion. The Hausdorff dimension has been proved to be equivalent to the

Shannon’s information entropy (Ryabko, 1986). Revealing the dimension meaning of the model

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parameters is helpful forspatial analys is of geographical systems. The key to understanding the

property of fractional dimension of the Cobb-Douglas model parameters lies in the geometric

measure relation, which is also termed fractal measure relation(Takayasu, 1990). Suppose thatDY

denotesthe dimension of Y.Comparing equation (14) with the geometric measure relation based on

the principle of dimensional consistency

iDD

iXY/Y ,(19)

we have

i

i

i mD

Dd Y .(20)

This suggests thatthe parameterdihas fractal dimension property in a broad sense. In fact,the

generalized allometric scaling exponent is a ratio of the dimens ion of the output variable to that of

an input variable.It is obvious that equation (12)can be derived from equation (19).Thus

ageneralized fractal dimension equation can be obtained as follows

jjiijjiiY DmDmDdDdD . (21)

which gives the mathematical relationships between different fractal dimension, different elastic

coefficients, and different allometri scaling exponents.

The fractal parameter equations are revealing for us to link the generalized Cobb-Douglas

function with allometric analys is. Although the scaling exponent in equation (16)may not be

equivalent to the corresponding parameters in equation (8), the ratio of these parameters equals

each other.For example, considering two elements Xi and Xj, we have

j

i

j

i

d

d

.(22)

which suggests that the ratio of two allometric scaling exponents equals the ratio of two elastic

coefficients, but anallometric scaling exponent is be directly proportional to its corresponding

elastic coefficient.This relation is important for fractal study of geographical systems. In fact,

when we make an allometric analysis of different components in a system,what we really

concerned is theratio of two fractal dimensionsinstead ofeach fractal dimensionitself(Chen, 2010;

Chen and Lin, 2009). Therefore, we canexplorethe ratio of dimensions by substitutingequation

(14)forequation (8).That is, a fractal study can be made from allometricanalyas is to the

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Cobb-Douglas modelling.

The main points of the mathematical process can be summarized as follows (Chen and Lin,

2009). First, the Cobb-Douglas function is a fractal model indicating self-affinity. Second, the

model parameters are relative to the fractal dimension in a broad sense. Third, the relationships

between one element and other elements follow the law of allometric growth. Last but not the least,

if the input-output relations of a geographical system can be characterized with the Cobb-Douglas

function, we can investigate the system by usingonly a few variables, which can simplify the

analytical process to a great extent.

3Empirical analysis

3.1 Materials and methods

A case study can be made by applying the general Cobb-Douglas model and allometric

algorithm to thepublic transportation of Chinese cities.The study area includes the whole mainland

of China (Figure 1).Urban transportation is one of important aspects of new science of cities

(Batty, 2013). The observational data come from China Statistical Yearbook ofNational Bureau of

Statistics of the People’s Republic of China. Three variables are adopted in this statistics. First, the

year-end numberof public vehicles in operation(unit), which is abbreviated to “vehic le number”;

Second, the total length of roads in operation(km), which is represented by“road length” for short;

Third, the volume of passengers transportedby publictraffic vehicles(10000person-times), which is

abbreviated to “passenger volume”. The three variables fall into two types: two input (independent)

variables and one output (dependent) variable (Table 1). The public transportation can be divided

into two groups: the major group includes bus and trolley bus, and the miner group includes

subways, light rail, and streetcar. Based on these variables, partial basic statistical datasets on the

public transportation of China’s cities are tabulated by regionas below (Table 2).

Table 1The input variables and output variable for quanti tative analysisof

thepublictransportation of China’s cities (2006-2012)

Type Variable Abbreviation Measurement unit Symbol

9

Input

(independent

variables)

Year-end numberof public

vehicles in operation

Vehicle number unit x1

Total length of roads in

operation

Road length km x2

Output

(dependent

variable)

Volume of passengers

transportedby publictraffic

vehicles

Passenger

volume

10000person-times y

Figure 1 A sketch map of 31 regionsincluding the 27 provinces, autonomous regions, and 4

munipalities directly under the Central Government of China

Usinglog-linear regression analys is and allometric analysis, we can build two kinds of

Cobb-Douglas model for the urban transportation of China. One is for the whole transportation,

including the major group and minor group (global analys is); the other is for the major group,

including bus and trolley bus (local analysis). Because of the incomplete of statistic dataset, we

don’t make models forthe minor group includingsubway, light rail, and streetcar. First of all, a

structural analysis is made; then, a simple dynamical analys is is implemented. All these analyses

are based on the cross-sectional datasets of thepublic transportation of China’s cities.

Table 2Basic datasets onthepublic transportation in China’s cities by region (2012)

Islands in South

China Sea

National capital

Provincial capital

Great port city

Legend

Yellow River

Yangtse River

Xinjiang

Xizang (Tibet)

Neimenggu (Inner Mongolia)

Heilongjiang

Jilin

LiaoningBeijing

Tianjin

Shanghai

Chongqing

Sichuan

Yunnan

Guangxi Guangdong

Hainan

Taiwan

Fujian

Zhejiang

Jiangsu

Shandong

Hebei

Henan

Shanxi

ShaanxiNingxia

Gansu

Qinghai

Hubei

Hunan Jiangxi

Anhui

Guizhou

10

Region Global datasets(public transportation) Local datasets(bus and trolley bus)

Vehicle

number

Road

length

Passenger

volume

Vehicle

number

Road

length

Passenger

volume

Beijing 25831 19989 761578 22146 19547 515416

Tianjin 9031 12870.7 129951 8405 12732 118721

Hebei 16493 18812 203954 16493 18812 203954

Shanxi 7851 13369 124838 7851 13369 124838

Inner Mongolia 5586 10650 96349 5586 10650 96349

Liaoning 20968 21520.8 428367 20500 21384 401457

Jilin 10912 11254.5 170561 10532 11200 165336

Heilongjiang 14364 15087 223956 14364 15087 223956

Shanghai 19825 23658.2 507933 16695 23190 280360

Jiangsu 30956 49903.2 470233 30380 49793 427578

Zhejiang 23060 40606 311024 22892 40558 310463

Anhui 11992 10535 212719 11992 10535 212719

Fujian 11823 15627 224703 11823 15627 224703

Jiangxi 7852 11648 127961 7852 11648 127961

Shandong 32869 44682 398268 32869 44682 398268

Henan 18137 18337 263718 18137 18337 263718

Hubei 16982 17354.1 338901 16670 17298 330613

Hunan 13148 14132 272165 13148 14132 272165

Guangdong 53089 87797 1003098 50729 87384 739359

Guangxi 7430 9323 142505 7430 9323 142505

Hainan 2614 5600 43306 2614 5600 43306

Chongqing 8540 8959 201331 7982 8828 176968

Sichuan 19628 19179.5 357333 19388 19140 347025

Guizhou 5031 5305 132200 5031 5305 132200

Yunnan 8187 16329 148409 8187 16329 148409

Tibet 396 834 7139 396 834 7139

Shaanxi 10948 9206.9 254599 10840 9187 248687

Gansu 5214 4907 102846 5214 4907 102846

Qinghai 2067 1937 39135 2067 1937 39135

Ningxia 3042 4813 37441 3042 4813 37441

Xin jiang 8155 7568 151397 8155 7568 151397

Source: National Bureau of Statistics of the People’s Republic of China (2013). China Statistical Yearbook.

Beijing: China Statistics Press

3.2Results

For comparison, let’s examine the wrong approach to modeling the urban transportation in the

first place. If the multivariablelog-linearregression is employed to estimate the model parameters,

the results will be deviant. For the global modeling (public transportation), the mathematical

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expression is as follows

3663.0

2

3306.1

19052.26ˆ xxy , (23)

where x1 refers to vehicle number, x2 to road length, and y-hat to passenger volume. The hat

symbol “^” indicates “predicted value”. For thelocal modeling (bus and trolley bus), the model is

3690.0

2

2959.1

17876.36ˆ xxy , (24)

in which the notation is the same as in equation (23). The main statistics for these models are

displayed in Table 3. Obviously, based on the statistics, the levels of confidence of the two

modelsare both greater than 99%. However, the scaling exponents are abnormal because the

elasticity of x2 vs y is minus. If the models are true, it will suggest that the length of roads makes a

negative contribution towards the traffic volume of passengers. This case cannot take place in the

real world. The deviantparameter values mean that the multivariablelinear regression is improper

and it is necessary to adopt a valid approach to evaluating model parameters.

Table 3Mainstatistics based on the multiple linear regressionsfor two types of models on the

publictrans portation ofChina’s cities (2012)

Item Global modeling (public

transportation)

Local modeling (bus and

trolley bus)

Type Object Statistic P values (sig.) Statistic P values (sig.)

Multiple R2 model 0.9606 -- 0.9703 --

Standard errors model 0.2009 -- 0.1642 --

F statistics model 341.5719 2.1518*10-20

457.8210 4.1012*10-22

t statistics

Constant 8.6391 2.1902*10-9 11.5756 3.4669*10

-12

x1 10.5482 2.9246*10-11

12.3132 8.0917*10-13

x2 -2.7743 0.0097 -3.3900 0.0021

Note: The standard error s is for the linear regression prediction; P value or sig. implies the corresponding

significance of F and t statistics, i.e., a kind of probability for null hypothesis.

In contrast, if we employ the allometric analys isbased on the simple linear regressionto estimate

model parameters, the result will be different and convincing. The relationship between vehicle

number as well as road length and passenger volume follows the law of allometric scaling (Figure

2, Figure 3). This is a kind of cross-sectional allometric growth (Chen, 2010). Using equations

(14)to make log-linear regression analyses, we can estimate the parameter values of the

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Cobb-Douglas function easily. According to the mathematical process from equations (14) to

equations (16), two allometric models can be integrated into a Cobb-Douglas model. The first step

is to calculate the allometric scaling exponents, and the second step, evaluate the parameters of the

general production function. For the global modeling (public transportation), the allometric

modelsare as below:

9972.0

13409.18ˆ xy , (25)

9604.0

28673.20ˆ xy . (26)

The values of the goodness of fit are R2=0.9498 and R

2=0.8042, respectively (Figure 2).

Multiplying equation (25) by equation (26)yields the global modelling result:

4802.0

2

4986.0

15633.19ˆ xxy . (27)

For thelocal modeling (bus and trolley bus), the allometric models are as follows

9558.0

18559.25ˆ xy . (28)

9088.0

21484.32ˆ xy . (29)

The determination coefficient values are R2=0.9581 and R

2=0.8097, respectively (Figure 3).

Multiplying equation (28) by equation (29)gives the local modelling result in the following form

4544.0

2

4779.0

18310.28ˆ xxy . (30)

So far, we have finished the process of mathematical modelling from theallometric scaling

relations to the Cobb-Douglas function. The two models, equations (27) and (30), are fair and

reasonable, and thus acceptable. Inthe models, a proportionality coefficient of the Cobb-Douglas

model is the geometrical average of allometric proportionality coefficients, and each cross elastic

coefficient is the corresponding allometric scaling exponent divided by the number of independent

variables (Table 4).

Before using the parameter values to explain the system of China’s public urban transportation,

we clarify the relationships between the parameters and the geographical system. Generally

speaking, it is measurement that makes a link between mathematical models and empirical

phenomena (Taylor, 1983). In theory, each system can be made a measurement, and thus we have

one or more measure values (e.g. length, area, size, density); each measure corresponds to a

dimension, and thus we have one or more dimension values (e.g. 1, 2, 3, 1.262, 1.585). If a system

13

has a characteristic scale, the dimension will be an integer (0, 1, 2, 3) and give no useful

information, and we can use the measure values to make an analysis; on the contrary, if a system

has no characteristic scale, the measure values will be uncertain, and we can use the dimension

values to make an analysis (Chen, 2008). In the latter case, the dimension will be a fractional

value instead of an integral number. In this case study, the spatial measures that we employ are

vehicle number, road length, and passenger volume, as indicated above. Because of human

geographical systems bear no characteristic scale, we had better utilize fractal dimens ions or

scaling exponents based on these measures to make analyses. The relationships between the

scaling exponents, elastic coefficient, and fractal dimension are shown by equations (13), (20),

(21), and (22), which have been explained above.The best approach to understanding the

mathematical models and parameters is the mathematical process. According to these equations,

we can understand the fractal parameters; and using these fractal parameters, especially the scaling

exponents, we can research into a geographical system.

a. Vehicles and passengers b.Roads and passengers

Figure2Theallometric scaling relations between vehicles /roads and passengers: global modeling

of the urban public trans portation of China (2012)

y = 18.3409x10.9972

R² = 0.9498

0

200000

400000

600000

800000

1000000

1200000

0 20000 40000 60000

Pas

senger

volu

me

y

Vehicle number x1

y = 20.8673x20.9604

R² = 0.8042

0

200000

400000

600000

800000

1000000

1200000

1400000

0 20000 40000 60000 80000 100000

Pas

seng

er v

olu

me

y

Road length x2

14

a. Vehicles and passengers b.Roads and passengers

Figure3Theallometric scaling relations between vehicles /roads and passengers: local modelingof

the urban public transportation of China (2012)

In terms of the modeling processes and the scaling exponent values, we can reveal the

geographical features of public transportation of China’s cities. First, the system of public urban

transportation follows the allometric scaling law. Thus the system can be described with

generalized fractal dimension. As far as the case in 2012is concerned, according to the allometric

scaling exponent values (Figures 2 and 3), the fractal dimension of passenger volume is greater

than those of vehicle number and road length, and the fractal dimension of vehic le number

islessthan that of road length. The less fractal dimension indicates a larger developing space. This

suggests that the volume of passengers depends more on the number of vehicles than on the length

of roads, and increasing a unit of vehicles or roads can lead to increasing lessthan one unit of

passengers. Second, the allometric scaling and the Cobb-Douglas relation are all evolutive

patterns rather than inherent patterns. From 2010 on, both the vehicle-passenger relation and

the road-passenger relation follow the allometric scaling law, thus the datasets can be fitted to the

Cobb-Douglas function. However, before 2009, the relationships between the road length and

passenger volume did not conform to the law of allometricgrowth, and the Cobb-Douglas model

cannot be built for early public urban transportation (Table 5). What is more, the goodness of fit

ofthe allometric regression modeling has been escalating in a f luctuant way (Figure 4).From the

above analyses, twoconclusions can be drawn as follows. First, the structure of thepublic

y = 25.8559x10.9558

R² = 0.9581

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

0 20000 40000 60000

Pas

senger

volu

me

y

Vehicle number x1

y = 32.1484x20.9088

R² = 0.8097

0

200000

400000

600000

800000

1000000

1200000

0 20000 40000 60000 80000 100000

Pas

senger

volu

me

y

Road length x2

15

transportation of China’s cities has been improvedyear after year; second, it is more efficient

tooptimize vehicles than to increase loads of Chinese urban transportation.

Table 4Parameters and mainstatistics based on allometric scaling for two types of models

onpublictrans portation ofChina’s cities (2012)

Parameter and statistic Global modeling (public

transportation)

Local modeling (bus and

trolley bus)

x1v.s.y x2v.s.y x1v.s.y x2v.s.y

Proportionality coefficientc 18.3409 20.8673 25.8559 32.1484

Scaling exponentd 0.9972 0.9604 0.9558 0.9088

Determination coefficient R2 0.9498 0.8042 0.9581 0.8097

Cross elastic coefficientσ 0.4986 0.4802 0.4779 0.4544

Table 5Parameters and statistics of allometricmodels on public trans portation ofChina’s cities

(2006-2012)

Model parameter 2006 2007 2008 2009 2010 2011 2012

Vehicle

and

passenger

c 3.8349 11.3699 42.1560 12.1662 6.1807 20.6520 18.3409

d 1.1445 1.0344 0.9083 1.0393 1.1115 0.9841 0.9972

R2 0.8393 0.9137 0.7235 0.8471 0.9240 0.9524 0.9498

Road

and

passenger

c 1112.9227 4890.0248 7285.5613 3182.1909 13.5510 21.4487 20.8673

d 0.5763 0.4150 0.3908 0.4609 1.0025 0.9573 0.9604

R2 0.3358 0.3151 0.2682 0.2322 0.8084 0.8064 0.8042

Note: Before 2006, the datasets are not incomplete, that is, there is no observational data for the total length of

roads in operation.

Figure4A histogram of the goodness of fit for the allometric scaling relations between

vehicle/road and passenger (2006-2012)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2006 2007 2008 2009 2010 2011 2012

Goodnes

s of fi

t

Year

Vehicle/passenger Road/passenger

16

4Discussion and conclusions

The Cobb-Douglas production function comes from economical problems, but it can be applied

to many kinds of systems such as city systems and ecosystems (Chen, 2008; Chen, 2011; Chen

and Lin, 2009). In other words, the Cobb-Douglas function is a universal model, which can be

used to describe the input-output relations of natural and social systemsin various fields. In this

sense, the abovementioned results of derivation can be generalized to other fields, and the

precondition is that the systemswhich will be studied haveself-affine fractality. Compared with the

previous works (Chen and Lin, 2009), the novelty of this study is thata simple allometric

algorithm is advanced to estimate the parameters of the Cobb-Douglas models for geographical

analys is. The case of public transportation of cities shows the effect of this algorithm. The

limitation of this study lies ina lack of dynamical analysis based on time series despite the

estimated values of the allometric scaling exponent based on the temporal dimension.

From the theoretical derivation and empirical evidence, the main conclusions of this studycan

be reached as follows. First, the allometric scaling is associated with the Cobb-Douglas

function. The cross elastic coefficients of the generalized production function are correlated with

the allometric scaling exponents. The allometric analysis provides an efficient approach to

estimating the parameters of the Cobb-Douglas model. Second,bothfractal dimension and

scaling exponent are important indexes for the geographical analysis using the

Cobb-Douglas function. An allometric scaling exponent is a ratio of generalized fractal

parameters. If the fractal dimension cannot be directly evaluated for a geographical system, the

scaling exponents can be used as substitutes. Third, fractal geometry, allometric scaling, and

the production function can be integrated into a new framework . Based on this framework,

we can develop aneffective method for spatial analysis of geographical systems. It is hard to

explain every question in a few lines of words.Due to limited space, the related problems remain

to be solved in future studies.

17

Acknowledgment

*****************************************************************************

***************.

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