Computers, Environment and Urban Systems 45 (2014) 89–100

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Computers, Environment and Urban Systems

journal homepage: www.elsevier .com/locate /compenvurbsys

Analyzing urbanization data using rural–urban interaction modeland logistic growth model

0198-9715/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compenvurbsys.2014.01.002

⇑ Address: No. 1, Changda Rd., Gueiren District, Tainan City 71101, Taiwan. Tel.:+886 6 2785 123x2316; fax: +886 6 2785 902.

E-mail address: sch@mail.cjcu.edu.tw

Shun-Chieh Hsieh ⇑Department of Land Management and Development, Chang Jung Christian University, Tainan City 71101, Taiwan

a r t i c l e i n f o

Article history:Received 22 June 2013Received in revised form 10 January 2014Accepted 12 January 2014Available online 1 February 2014

Keywords:Urbanization curveUrbanization dynamicsUrban populationSelf-organized criticality

a b s t r a c t

The level of urbanization is a valuable indicator for projections of some global trends. However, urbani-zation levels may be based on unreliable data. This study proposes a simple method for identifying prob-lems in the time series of urban and rural populations of a country. The time series were fitted to a rural–urban interaction population model, and improper model coefficients indicated that the time series werequestionable. The upper limit of the urbanization level was calculated to determine whether the trend ofthe urbanization level follows the logistic growth model. An analysis of the frequency–spectrum relation-ship was performed to determine whether the urbanization process is a self-organized criticality and toconsolidate the low possibility for chaos in the urbanization model. Empirical analyses were conductedusing data from the United States, China, and India to verify data reliability and to determine the dynam-ical mechanism of urbanization. This is critical for demographers, geographers, other scientists, andpolicymakers.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

There are multiple indices of the urbanization of a country. Theconcentration index, which is related to the distribution andconcentration of urban populations, is the size of the cities relativeto the total population (Casis & Davis, 1946). Ledent (1980)proposed an alternative measure of urban concentration, anagglomeration index, which is based on three factors: populationdensity, population of a large city center, and travel time to thelarge city center. The degree or level of urbanization is the percent-age of urban population in its total population at any fixed date(Davis & Hertz, 1951). The rate or speed of urbanization refers tothe change in the degree of urbanization during a period of time(Durand & Pelaez, 1965). Chen, Ye, and Zhou (2013) differentiatedthe urbanization curve to derive the speed of urbanization curve.To define aforementioned indices of urbanization, in addition toconsidering the urban proportion of the population, Arriaga(1970) also considered the size of the city where an urbanpopulation lives. The tempo of urbanization is defined as the netdifference between the rate of growth in the urban populationand that in the rural population (United Nations, 1974). The scaleof urbanization is defined as RXY, where X is the proportion ofthe urban population in units greater than a certain size and Y isthe proportion of the total population in the same units (Gibbs,

1966). The level of urbanization is a common demographic defini-tion of urbanization because it is easy to calculate and interpret,and because of the high availability of data.

In this study, urbanization differs from exogenous urbangrowth. Urbanization is an increase in the proportion of a country’spopulation that resides in urban areas, in which the city size is notconsidered, whereas exogenous urban growth is an increase in thenumber of people who live in urban areas. For example, if theurban population and total population of a country are 4,000,000and 8,000,000, respectively, then the urban population and totalpopulation will be 8,000,000 and 16,000,000, respectively, fiftyyears later. Accordingly, the level of urbanization does not change,whereas urban growth increases by 4,000,000. The country isexpected to reach a high urbanization level and low urban growthat the terminal stage of urbanization. Recently, much research hasbeen conducted on urban size dynamics. Schaffar and Dimou(2012) studied the dynamics of Chinese and Indian urbanhierarchies from 1981 to 2004, and examined the urban growthpatterns of the rank-size relationship for cities in these countries.To eliminate problems of urban definitions, Mulligan (2006) pro-jected the urban population above high thresholds and exploredthe influence of city-specific initial conditions and national-levelfactors on population growth. However, the proportion of urbandwellers living in large cities exhibits a substantially low correla-tion with the level of urbanization (Bloom, Canning, & Fink,2008), which is investigated in this study. Urbanization has abeginning and an end. By contrast, urban growth is limitless(Northam, 1975). In current study, no cross-country analysis was

USA China

India

Fig. 1. Percentage of urban population and agglomerations by size class in 1960. Source: United Nations, 2012.

India

USA China

Fig. 2. Percentage of urban population and agglomerations by size class in 2011. Source: United Nations, 2012.

90 S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100

conducted using a list of cities ranked according to size for eachcountry, as shown in Figs. 1 and 2, nor were the aggregate urban-ization statuses of the regions and the world determined. Stoto(1979) indicated that the date that the forecast is made is the prin-cipal factor determining error. Keyfitz (1981) argued that compar-ing individual forecasters is essentially futile. This studyinvestigated the time series reliability of the urbanization level;however, the urbanization level was not forecasted for the futureand forecasting methods were not compared.

The curves of the change in the level of urbanization over timeare called ‘‘urbanization curves’’ (Knox, 1994; Northam, 1975). Therelationship between the urbanization level and various topics,namely socioeconomic development (Annez & Buckley, 2009;Black & Henderson, 1999; Bloom et al., 2008; Chenery & Syrquin,1975; Fay & Opal, 2000; Henderson, 2003; Jones & Kone, 1996;Ledent, 1982; Njoh, 2003; Polèse, 2005; Woods, 2003), theenvironment and resources (Alig, 2010; Shen, Peng, Zhang, & Wu,2012; Zhou et al., 2004), and energy consumption and emissions(Cole & Neumayer, 2004; Krey et al., 2012; Poumanyvong &Kaneko, 2010; York, 2007), has been explored extensively. There-fore, the level of urbanization has been used as an indicator for pro-jecting various global trends, such as energy use, poverty, andenvironment and resource use. (Energy Information Administra-tion, 2012; World Bank, 2011; World Resources Institute, 2003).

Currently, the United Nations (UN) is the only institution thatproduces projections of urban and rural population growth on aglobal scale. The World Urbanization Prospects (WUP) data set

published biannually by the United Nations Population Divisionis the most comprehensive source of estimates and projections ofthe urban and rural populations of every country, region, and con-tinent in the world. The published statistics follows the nationalcensus definition of urban population, which differs considerablyamong nations (geographical variations) and varies over timewithin a single country (historical variations). National definitionsare generally based on demographic, administrative, economic,sociocultural, and geographic criteria (Frey & Zimmer, 2001). TheUN (1974) detailed discussions on the problems of urban defini-tions. After discussing numerous definitional problems and thelack of reliable and current census data, Cohen (2004) concludedthat nearly any statistic on an urban population is merely anapproximation of reality. Bocquier (2005) indicated that the UNprojections were systematically biased, and the problem primarilyoriginated in the linear regression model used in the projectionmethod. Montgomery (2008) also indicated that the urbanizationlevels were significantly overestimated in the UN projections. Thisproblem arising from the UN projections raises obvious concernsregarding data reliability and makes cross-country comparisonsproblematic. Because the WUP data set is widely used and refer-enced, methods for identifying definition and measurement prob-lems in the time series of urban and rural populations are required.

Time-series analyses of empirical population data have indi-cated that chaos is rare in natural populations (Ellner & Turchin,1995; Upadhyay & Rai, 1997). Holland (1995) believed that theinteractions that form a city are typically stable. Furthermore, by

S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100 91

demonstrating the dynamics of urbanization based on the logisticgrowth model (LGM), which has been used to forecast the urbani-zation level of every country in the world (United Nations, 2002,2012) to create no possibility for chaos, Chen (2009a) inferred thatthe probability of urban chaos is considerably low in the realworld. Reliable statistics for nations’ urban population andurbanization levels depend on reliable data. Therefore, the primaryobjective of this study was to develop a simple method foridentifying problems in the time series of urban and rural popula-tions based on the aforementioned argument. The followingsection details a series of systematic approaches to developing arural–urban interaction model, the calculation of the upper limitof the urbanization level, and the development of a frequency–spectrum relationship (FSR). A schematic framework was appliedto the WUP data set and census data of the United States, China,and India, and a detailed discussion is subsequently provided. Fi-nally, the paper concludes with a brief summary of this study. A listof acronyms used in this paper is provided in Table 1.

2. Methods

This section details a series of systematic approaches to analyz-ing urbanization dynamics. The analysis commenced by fitting theurban and rural population data to a two-dimensional map todetermine a nation’s urbanization process. The parameter valuesin the model were then verified to lie within the bounds of areasonable scale. To research the rural–urban interaction model,logical and empirical analyses were conducted. The upper limitof the urbanization level was estimated based on the LGM to becompared with the results derived from the rural–urban interac-tion model. Finally, fast Fourier transform (FFT) was used to deter-mine an approximate power-law relationship between frequencyand spectral density to obtain a spectral exponent.

2.1. Rural–urban interaction model

The spatial interaction between the rural and urban populationresults in urbanization dynamics and can be characterized usingtwo nonlinear differential equations that are constructed basedon observations and statistical data. The rural–urban interactionmodel can be expressed as (Chen, 2009a)

drðtÞdt ¼ arðtÞ þ buðtÞ � /rðtÞuðtÞ

duðtÞdt ¼ cuðtÞ þ drðtÞ þurðtÞuðtÞ

(; ð1Þ

where r(t) and u(t) denote the rural and urban population at time t,respectively; and a, b, c, d, /, and u are parameters. The rural–urbaninteraction reduces the rate of rural population growth and raisesthe rate of urban growth. Therefore, it produces faster growth inthe urban population than in the rural population (i.e., urbaniza-tion). If / and u are constants, then the aforementioned model isanalogous to the Lotka-Volterra model (LVM; Dendrinos & Mullally,1985; Volterra, 1938), and the urbanization curve is a J-shapedcurve. An analogy can be drawn between the rural–urban interac-tion in urban systems and the predator–prey interaction in ecosys-

Table 1Acronyms used in the text.

Acronym Full description

FFT Fast Fourier transformFSR Frequency–spectrum relationshipLGM Logistic growth modelLVM Lotka-Volterra modelSOC Self-organized criticalityUNM United Nations modelWUP World Urbanization Prospects

tems (Chen, 2009b). The sizes of the urban and rural populationsaffect each other. If / = /�/[r(t) + u(t)] and u = u�/[r(t) + u(t)], thenthe rural–urban interaction model corresponds with the UN model(UNM; Karmeshu, 1988; Ledent, 1980; United Nations, 1980), andthe urbanization curve is an attenuated S-shaped curve. The formerhits its carrying capacity and continues causing a populationincrease, whereas the latter reaches its carrying capacity and stabi-lizes. Discretizing Eq. (1) yields a two-dimensional map, such as

DrðtÞDt ¼ arðtÞ þ buðtÞ � /rðtÞuðtÞ

DuðtÞDt ¼ cuðtÞ þ drðtÞ þurðtÞuðtÞ

(: ð2Þ

For simplicity, the notation of parameters is not changeddespite the error caused by the continuous-discrete conversion.For dimensional uniformization, the rural and urban populationsare divided by the initial value of the rural population. Thus, we ob-tain r(0) = 1. The model exhibits no periodic oscillation or chaoticbehaviors when the parameter values used in the model lie withinthe following reasonable ranges (Chen, 2009a): 0 < a; /; u < 1and 0 6 b; c; d < 1.

The level of urbanization can be expressed as

LðtÞ ¼ uðtÞuðtÞ þ rðtÞ c: ð3Þ

where c indicates the upper limit of urbanization (usually set at100%). Taking the derivative of Eq. (3) yields

dLðtÞdt¼ ðu� � aÞLðtÞ 1� LðtÞ

c

� �; ð4Þ

for UNM with b = c = d = 0 and /� = u� (closed system). Eq. (4) isanalogous to the logistic equation first created by Verhulst (1838).In a closed system such as the entire world, the decrease in ruralpopulation is equal to the increase in urban population caused bythe rural–urban interaction. The difference between parametersu� and a dominates the behavioral features of the closed urbaniza-tion dynamics. Thus, the rural region and rural–urban interactiondetermine the progress of urbanization.

2.2. Logistic growth model

Because the urbanization level of a nation exhibits clear upperand lower limits and its growth is not of uniform speed, theincrease in the urbanization level over time exhibits an S-shapedcurve, as demonstrated by Northam (1975). The curve can beformulated as a sigmoid function. The LGM is the most commonrepresentation of the urbanization process because it can be esti-mated in a straightforward manner by using ordinary least-squaresregression (Keyfitz & Caswell, 2005), and because ‘‘most othermodels of S-shaped curves are much more complicated to esti-mate’’ (Mulligan, 2013). Although the LGM has often been criti-cized for being applied to population forecasts (Keyfitz & Caswell,2005), it has been proved useful in summarizing historical changesin population size and for short-term projections (Berry, 1973;Keyfitz, 1980; Leach, 1981; Marchetti, Meyer, & Ausubel, 1996;Mulligan, 2013; Rogers, 1995a). For several years, the UN assumedthat the urban–rural growth difference of countries follows a logis-tic path and estimated it based on the experience of numerouscountries (United Nations, 2002, 2012). The standard three-param-eter LGM can be expressed as

LðtÞ ¼ c1þ ae�bt

; ð5Þ

where a is the location parameter (it shifts the model in time butdoes not affect the model’s shape; Oliver, 1966), b is the relativegrowth rate at substantially low urbanization levels, and c refersto the upper limit of the urbanization level. By letting t = 0, we

92 S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100

obtain a = c/L0 � 1, where L0 represents the initial value of L(t). Alower (high) value for a indicates that the growth process beginsearlier (later) and a low (high) value for b indicates that the growthprocess of reaching the upper limit of the urbanization level occursslowly (rapidly). Each country follows its own urban transition,which leads to various urban saturation levels. Based on cross-sec-tional time-series data from 11 countries, Rao, Kanneshu, and Jain(1989) determined that the value of b is below 0.05 in practice. Inother words, the value of b is generally low. This finding supportsthe assumption that urbanization dynamics based on the LGM cre-ate no possibility for chaos. Therefore, the urbanization level is notsensitive to initial conditions and the upper limit of the urbaniza-tion level is asymptotically reached independently of the initial va-lue of L(t).

The upper limit of the urbanization level can be estimated usingnatural logarithms on both sides of Eq. (5) to obtain the followinglinear regression equation:

lnc

LðtÞ � 1� �

¼ lna� b t; ð6Þ

where the intercept indicates the estimated starting time of theurbanization process and the slope indicates the rate of change dur-ing the process. When the process is half complete (L(t) = c/2) andthe logistic curve reaches its inflection point (where the growth rateis maximal), the left side of Eq. (6) can be set to zero. This indicatesthat the acceleration stage changes to the deceleration stage at timet = ln a/b. Therefore, the urbanization process without counterur-banization can be divided into four stages: the initial stage, acceler-ation stage, deceleration stage, and terminal stage (Chen, 2012). TheJ-shaped curve of urbanization at the initial and acceleration stagesis an exponential growth curve, which indicates that the urbaniza-tion process is not sustainable. When a J-shaped curve reaches thedeceleration stage, it converts to an S-shaped curve with inflectionnear the upper limit of the urbanization level.

The derivative of Eq. (5) is

dLðtÞdt¼ bLðtÞ 1� LðtÞ

c

� �: ð7Þ

This equation is analogous to Eq. (4) and indicates how the instan-taneous change dL(t)/dt in the urbanization level is related to theintrinsic growth bL(t), which in turn is constrained by the continu-ally diminishing factor 1 � L(t)/c. The right side of Eq. (7) equalszero when L(t) = c, which must be the point at which growth stopson the urbanization curve. Thus, the urbanization level grows expo-nentially under the constraints of an upper limit, producing a typi-cal S-shaped curve. Discretizing Eq. (7) yields a one-dimensionalmap in the form

Ltþ1 ¼ ð1þ bÞLt �bc

L2t : ð8Þ

Let xt = bLt/[(1 + b)c], then Eq. (8) can be normalized and we obtain

xtþ1 ¼ ð1þ bÞxtð1� xtÞ: ð9Þ

The quadratic map approaches a fixed state b/(1 + b) when 0 < b < 2(May, 1976).

2.3. Frequency–spectrum relationship

Self-organized criticality (SOC) refers to the tendency of largecomplex systems with numerous degrees of freedom naturally todrive systems to a critical state in which minor events can causechain reactions of various sizes, and complements the concept ofchaos, in which simple systems with few degrees of freedom candisplay complex behavior (Bak & Chen, 1991; Bak, Tang, & Wiesen-feld, 1987). Complex systems tend to naturally evolve toward a

narrow regime near the boundary between chaos and order, calledthe ‘‘edge of chaos’’ (Packard, 1988). Kauffman (1993) indicatedthat the rate of evolution of evolving systems is maximized nearthe edge of chaos. By conducting a parameter analysis of urbaniza-tion dynamics, Chen (2009a) concluded that the spatial complexityof a self-organized urban system occurs on the edge of chaos ratherthan in a chaotic state. The urbanization process can be regarded asa phase transition from a rural to an urban settlement (Andersson,Rasmussen, & White, 2002) and an SOC (Allen, 1997; Portugali,2000). This phase transition may explain why an observed urban-ization process often displays no characteristic time or lengthscale. SOC is observed in several simple cellular automation modelsand its chain reaction is a fractal process (Batty & Xie, 1994, 1999;Portugali, 2000). An analysis of a power-spectrum relationship canbe conducted using the urbanization data of a country. The powerspectra of such urbanization processes obey a power-law relation-ship as follows

Pðf Þ / f�g; ð10Þ

where f refers to the frequency and g is the spectral exponent. Thespectral exponent is associated with the profile dimension Ds

according to the following formula (Peitgen & Saupe, 1988):

g ¼ 5� 2Ds ¼ 2H þ 1; ð11Þ

where H is the Hurst exponent (Feder, 1988). The FSR is a typicalmathematical indication of the SOC of urban systems (Chen & Zhou,2008). The 1/f fluctuation and fractal growth are regarded as the‘‘fingerprint’’ and the ‘‘signature’’ of SOC in time and space, respec-tively (Bak, 1996). SOC is characterized by uncontrolled fractalgrowth independent of scale (Batty, 2005; Batty & Xie, 1999).According to Bak (1996), if only 0 < g < 2, then Eq. (10) can beconsidered to indicate a 1/f fluctuation in practice.

3. Empirical analysis

The United States, China, and India are the top three most pop-ulated countries accounting for 75% of the urban population of theworld in 1950, 2000 and 2030 (United Nations, 2002). The reliabil-ity of the rural and urban population data from these countries’population censuses and the WUP 2011 data set (United Nations,2012) were verified in this study.

3.1. United States

Table 2 shows the rural and urban populations of the UnitedStates reported in the population censuses and the WUP 2011 dataset. The definition of U.S. cities was changed in 1950 and adoptedin 1970. We used only the census data from 1790 to 1960. Let r(t),u(t), r(t)u(t) and r(t)u(t)/[r(t) + u(t)] be independent variables andentry statistics, and Dr(t)/Dt, and Du(t)/Dt be dependent variablesand exit statistics. A multivariate stepwise regression analysisbased on least squares computation yielded the following models

DrðtÞDt ¼ 0:02584rðtÞ � 0:03615 rðtÞuðtÞ

rðtÞþuðtÞDuðtÞDt ¼ 0:05044 rðtÞuðtÞ

rðtÞþuðtÞ

8<: : ð12Þ

This model is a UNM in which all types of statistic can pass thetests at a .01 significance level. Stepwise regression involves multi-ple regressions and the weakest correlated variable is removed ineach regression. The regression yields a UNM or an LVM, whichis a combination of independent variables that most accuratelyexplains the dependent variables. The estimates depend on timeseries, which must be a sufficient length, and change if the dataset conforming to the same urban definition is divided into twoperiods (e.g., 1790–1870 vs. 1880–1960 in the United States), thus

Table 2The United States’ rural and urban populations and urbanization levels. Source: http://www.census.gov/population.

Census data WUP data (unit: thousands)

t Dt r(t) u(t) L(t) t Dt r(t) u(t) L(t)

1790 10 3727559 201655 .0513 1950 5 56570.772 101242.268 .64151800 10 4986112 322371 .0607 1955 5 56200.599 114951.692 .67161810 10 6714422 525459 .0726 1960 5 55906.195 130420.02 .70001820 9.8125 8945198 693255 .0719 1965 5 56088.792 143363.716 .71881830 10 11733455 1127247 .0877 1970 5 55293.233 154170.632 .73601840 10 15218298 1845055 .1081 1975 5 57729.547 161378.811 .73651850 10 19617380 3574496 .1541 1980 5 60356.639 169468.365 .73741860 10 25226803 6216518 .1977 1985 5 61498.878 179620.874 .74491870 10 28656010 9902361 .2568 1990 5 62574.509 190764.588 .75301880 10 36059474 14129735 .2815 1995 5 60589.249 205734.468 .77251890 10 40873501 22106265 .3510 2000 5 59073.418 223422.892 .79091900 9.7917 45997336 30214832 .3965 2005 5 57192.943 239627.353 .80731910 9.7917 50164495 42064001 .4561 2010 5 55424.913 254959.035 .82141920 10.25 51768255 54253282 .5117 2015 5 53932.505 269952.635 .83351930 10 54042025 69160599 .5614 2020 5 52691.099 284410.734 .84371940 10 57459231 74705338 .5652 2025 5 51656.798 298101.47 .85231950 10 61197604 90128194 .5956 2030 5 50539.187 311140.702 .8603

54478981 96846817 .6400 2035 5 49273.034 323615.877 .86791960 10 66259582 113063593 .6305 2040 5 47892.552 335567.923 .8751

54045425 125268750 .6986 2045 5 46425.864 347027.900 .88201970 10 53565309 149646617 .7364 2050 5 44916.864 358183.662 .88861980 10 59494813 167050992 .73741990 10 61656386 187053487 .75212000 10 59061367 222360539 .7901

Note: Census data based on new urban definition are indicated in Italic.

1

S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100 93

indicating a bias. The growth rate of the rural population dependson rural population size and the rural–urban interaction, whereasthat of the urban population is proportional only to the rural–urban interaction but not directly associated with rural and urbanpopulation sizes. Therefore, the United States’ urbanization processwas primarily migration-lead, and population migration betweenrural and urban sectors depended only on rural–urban interaction.Consequently, the original UNM was simplified to the form shownin Eq. (12). According to Eq. (12), the rural population cannot spon-taneously flow into the urban sector, and vice versa. The phase por-trait of the United States’ urbanization, as shown in Fig. 3,indicated that the urban population increased slowly as the ruralpopulation increased until the urbanization level was 71.66%,and then increased rapidly as the rural population decreased. Inother words, rural-to-urban migration was initially the primarycontributor to urbanization, but urban natural increase subse-quently became the chief cause of urbanization. By using a similarapproach, we also obtained a UNM based on the WUP data from1950 to 2050. However, the model with unreasonable parameterswas abnormal, as shown in Table 5.

0

50

100

150

0 5 10 15

r (t)

u(t

)

L = 71.66%

Fig. 3. Phase portrait of the United States’ rural–urban interaction model.

The fit of the LGM based on the census data from 1790 to 1960and the original urban definition yielded the upper limit of theurbanization level c = 0.7073. The acceleration stage changed tothe deceleration stage in 1891. When we used the census databased on the urban definition from 1950 to 2000, the fit of theLGM yielded the upper limit of the urbanization level c = 0.7945.The urbanization trend was better fit when using the 1950 urbandefinition than when using the original definition. When we usedthe WUP data from 1950 to 2050, the fit of the LGM yielded anabnormal upper limit of the urbanization level, as shown in Table5. The changing trend of the United States’ urbanization level, asshown Fig. 4, indicated that the UNM can clearly describe the Uni-ted States’ rural–urban interaction process of the recent 200 years.

By using FFT and least squares computation, we obtained theFSR based on the census data as

Pðf Þ ¼ 0:0009f�1:8348 ðR2 ¼ 0:9547Þ: ð13Þ

0

0.2

0.4

0.6

0.8

1750 1800 1850 1900 1950 2000 2050 2100

Year

Urb

aniz

atio

n le

vel

Census 1790-1960

Census 1970-2000

WUP 2011

UNM-Census 1790-1960

LGM-Census 1790-1960

LGM-Census 1950-2000

Fig. 4. The changing trend of the United States’ urbanization levels based on theUNM and the LGM.

Table 3China’s rural and urban populations and urbanization levels. Source: http://www.stats.gov.cn/tjsj/ndsj.

Census data(unit: ten thousands) WUP data (unit: thousands)

t Dt r(t) u(t) L(t) t Dt r(t) u(t) L(t)

1949 1 48404 5763 .1064 1950 5 485765.396 65006.037 .11801950 1 49031 6165 .1117 1955 5 524062.767 84296.918 .13861951 1 49668 6632 .1178 1960 5 551613.736 106656.358 .16201952 1 50320 7162 .1246 1965 5 581825.231 128465.068 .18091953 1 50970 7826 .1331 1970 5 672878.467 141744.374 .17401954 1 52016 8250 .1369 1975 5 755823.828 159217.126 .17401955 1 53180 8285 .1348 1980 5 792851.113 190319.525 .19361956 1 53643 9185 .1462 1985 5 814900.298 241678.921 .22871957 1 54703 9950 .1539 1990 5 842378.617 302816.612 .26441958 1 55270 10724 .1625 1995 5 838120.410 375866.200 .30961959 1 54834 12373 .1841 2000 5 813792.013 455324.724 .35881960 1 53131 13076 .1975 2005 5 751576.031 556017.458 .42521961 1 53155 12704 .1929 2010 5 681049.007 660286.145 .49231962 1 55633 11662 .1733 2015 5 608163.067 761579.451 .55601963 1 57523 11649 .1684 2020 5 541428.390 846363.122 .60991964 1 57548 12951 .1837 2025 5 483452.501 911803.946 .65351965 1 59496 13042 .1798 2030 5 435427.009 957649.059 .68741966 1 61229 13313 .1786 2035 5 397142.173 984445.798 .71251967 1 62820 13548 .1774 2040 5 362325.039 998581.458 .73381968 1 64696 13838 .1762 2045 5 327678.511 1004089.577 .75401969 1 66554 14117 .1750 2050 5 293992.031 1001611.732 .77311970 1 68568 14424 .17381971 1 70518 14711 .17261972 1 72244 14933 .17131973 1 73867 15344 .17201974 1 75268 15591 .17161975 1 76394 16026 .17341976 1 77373 16344 .17441977 1 78306 16668 .17551978 1 79009 17250 .17921979 1 79048 18494 .18961980 1 79566 19139 .19391981 1 79897 20175 .20161982 1 80175 21479 .21131983 1 80738 22270 .21621984 1 80344 24013 .23011985 1 80754 25097 .23711986 1 81146 26361 .24521987 1 81625 27675 .25321988 1 82370 28656 .25811989 1 83164 29540 .26211990 1 83898 30435 .26621991 1 84574 31249 .26981992 1 83566 33605 .28681993 1 83400 35117 .29631994 1 83212 36638 .30571995 1 83041 38080 .31441996 1 82943 39446 .32231997 1 82483 41143 .33281998 1 82043 42718 .34241999 1 81472 44314 .35232000 1 80786 45957 .36262001 1 79563 48064 .37662002 1 78241 50212 .39092003 1 76851 52376 .40532004 1 75705 54283 .41762005 1 74544 56212 .42992006 1 73160 58288 .44342007 1 71496 60633 .45892008 1 70399 62403 .46992009 1 68938 64512 .48342010 1 67113 66978 .4995

94 S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100

Another FSR based on the WUP data was obtained as follows:

Pðf Þ ¼ 0:0002f�1:4745 ðR2 ¼ 0:9872Þ: ð14Þ

Eqs. (13) and (14) indicate that the mathematical criteria of SOC fitwell with the United States’ urbanization for both the census dataand WUP data. The profile dimension of the United States’ urbani-zation level was 1.5826, based on the census data, and 1.7628,based on WUP data.

3.2. China

Table 3 shows the rural and urban populations of China re-ported in the population census and the WUP data. China is expe-riencing a process of rapid urbanization. The urbanization levelincreased from 26.62% to 49.95% between 1990 and 2010. A mul-tivariate stepwise regression analysis based on the census datafrom 1949 to 2010 yielded the following model:

S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100 95

DrðtÞDt ¼ 0:02515rðtÞ � 0:03656rðtÞuðtÞ

DuðtÞDt ¼ 0:02446rðtÞuðtÞ

(: ð15Þ

This model is an LVM in which all types of statistic can passthe tests at a .01 significance level. Shen (2005) also argued thatthe urban–rural difference in the population growth rate was notstable in the period of 1982–2000. Therefore, to achieve a preciseestimate of the urbanization level by using the UNM is impossible.The phase portrait of China’s urbanization, as shown in Fig. 5,indicates that the urban population increased slowly as the ruralpopulation increased until the urbanization level was 26.11%, whichis considerably lower than that of the United States, and thenincreased rapidly as the rural population decreased. In other words,migration ceased to dominate the urban increase at a point at whichthe urban population was still much lower than the rural popula-tion. By using a similar approach, we also obtained an LVM basedon the WUP data from 1950 to 2050. However, the model withunreasonable parameters was abnormal, as shown in Table 5.

From 1961 to 1977, the antiurbanization period, rural to urbanmigration was tightly restricted in China and urbanization was in arapidly declining stage (Chan & Zhang, 1999). In late 1978,economic reforms were initiated in China and the urbanizationlevel ascended. From 1978 to 1999, rural–urban migration

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2

r (t)

u(t

)

L = 26.11%

Fig. 5. Phase portrait of China’s rural–urban interaction model.

0

0.2

0.4

0.6

0.8

1

1940 1960 1980 2000 2020 2040 2060

Year

Urb

aniz

atio

n le

vel

Census 1949-2010

WUP 2011

LVM-Census

LGM-WUP

Fig. 6. The changing trend of China’s urbanization levels based on the LVM and theLGM.

predominantly contributed to urban population growth in China(Zhang & Song, 2003). Since the onset of the economic reform erain 1979, China’s urbanization has developed rapidly and acceler-ated after 1995, as shown in Fig. 6. Generally, China’s urbanizationprocess can be divided into three parts: the random process, peri-odic process, and trend process (Chen, 2007). The estimated regres-sion relationship between the urbanization level and years from1978 to 2010 based on census data follows exponential growth.Furthermore, the definition of urban population has changed sub-stantially over time (Chan & Hu, 2003; Shen, 2005; Zhang & Zhao,1998). Therefore, the fit of the LGM based on census data from1949 to 2010 yielded unreasonable parameters, as shown in Table5. It is difficult for the world’s most populous country to exceed itsurbanization level of 80% (Chen & Luo, 2006). When we used theWUP data from 1950 to 2050, the fit of the LGM also yielded anunreasonable upper limit of the urbanization level, as shown in Ta-ble 5. The changing trend of China’s urbanization level, as shown inFig. 6, indicates that China’s rural–urban interaction process in therecent 60 years cannot be adequately described using the LVM.Chen (2007) also observed that the common model cannot de-scribe China’s urbanization process because of autocorrelationand random disturbance. China’s urbanization process is a compo-sition of the first-order autoregressive model and the high-order(even infinite-order) moving-average model.

By using FFT and least squares computation, we obtained theFSR based on the census data as

Pðf Þ ¼ 2:5462f�1:6149 ðR2 ¼ 0:9824Þ: ð16Þ

Another FSR based on the WUP data was obtained as follows:

Pðf Þ ¼ 0:0012f�1:6786 ðR2 ¼ 0:9742Þ: ð17Þ

Eqs. (16) and (17) indicate that the mathematical criteria of SOC fitwell with China’s urbanization for both the census data and theWUP data. The profile dimension of China’s urbanization levelwas 1.6926, based on the census data, and 1.6607, based on theWUP data.

3.3. India

Table 4 shows the rural and urban populations of India reportedin the population census and WUP data. India exhibits low urban-ization levels. A multivariate stepwise regression analysis based onthe census data from 1901 to 2011 yielded the following model:

DrðtÞDt ¼ �0:01852rðtÞ þ 0:25567uðtÞ � 0:05053rðtÞuðtÞ

DuðtÞDt ¼ 0:03333rðtÞuðtÞ

(: ð18Þ

This model is an LVM and it is abnormal because a few parametervalues in the model lay outside reasonable ranges, as shown inTable 5. By using a similar approach, we also obtained an LVM basedon the WUP data from 1950 to 2050. However, the model withunreasonable parameters was also abnormal, as shown in Table 5.

During the four decades from 1961 to 2001, natural increase(the difference of births and deaths) accounted for greater than50% of urban population growth in India (Kundu, 2011). India’surbanization process is not primarily migration lead, but is aproduct of demographic explosion caused by natural increase.Because India’s urbanization curve is a J-shaped curve rather thanan S-shaped curve, as shown in Fig. 7, the fit of the LGM based onthe census data from 1901 to 2011 yielded unreasonable parame-ters, as shown in Table 5. When we used the WUP data from 1950to 2050, the fit of the LGM also yielded an unreasonable upper limitof the urbanization level, as shown in Table 5.

By using FFT and least squares computation, the FSR based onthe census data was obtained as

Table 4India’s rural and urban populations and urbanization levels. Source: http://censusindia.gov.in/Census_Data_2011/.

Census data WUP data (unit: thousands)

t Dt r(t) u(t) L(t) t Dt r(t) u(t) L(t)

1901 10 212544454 25851873 .1084 1950 5 308483.929 63372.571 .17041911 10 226151757 25941633 .1029 1955 5 334924.536 71449.484 .17581921 10 223235046 28086167 .1118 1960 5 367571.756 80272.403 .17921931 10 245521249 33455989 .1199 1965 5 403151.116 93249.265 .18791941 10 274507283 44153297 .1386 1970 5 444426.573 109447.317 .19761951 10 298644156 62443934 .1729 1975 5 489393.294 132703.388 .21331961 10 360298168 78936603 .1797 1980 5 538360.177 161698.412 .23101971 10 439045675 109113977 .1991 1985 5 593481.426 191009.416 .24351981 10 523866550 159462547 .2334 1990 5 650555.737 223229.712 .25551991 10 628836076 217551812 .2570 1995 5 707862.144 256624.011 .26612001 10 741660293 285354954 .2778 2000 5 762313.429 291584.678 .27672011 10 833087662 377105760 .3116 2005 5 806755.007 333287.856 .2923

2010 5 845838.901 378775.426 .30932015 5 879711.939 428508.756 .32762020 5 903865.56 483043.512 .34832025 5 916766.695 542190.888 .37162030 5 917669.536 605812.799 .39772035 5 906218.493 673583.693 .42642040 5 884362.011 742667.375 .45652045 5 854129.722 810389.351 .48692050 5 816624.748 875382.883 .5174

Table 5The parameter values in the rural–urban interaction model and the LGM.

Country Data Model a b c d / or /� u or u� a b c

US Census UNM 0.02584 0 0 0 0.03615 0.05044 21.2505 0.0303 0.7073WUP UNM 0 �0.00067 0 0 0 0.05962 0.5740 0.0126 1.0371

China Census LVM 0.02515 0 0 0 0.03656 0.02446 �0.5166 �0.0092 0.0655WUP LVM 0 0.01079 0 0 0.00960 0.00645 10.7014 0.0394 0.9674

India Census LVM �0.01852 0.25567 0.03333 0 0.05053 0 92.0234 0.0112 8.7289WUP LVM 0.02502 0 0 0.00307 0 0.00813 �0.9926 0 0.0014

Note: Abnormal values are indicated in bold.

96 S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100

Pðf Þ ¼ 0:0003f�1:7779 ðR2 ¼ 0:8176Þ: ð19Þ

Another FSR based on the WUP data was obtained as

Pðf Þ ¼ 0:0002f�1:6021 ðR2 ¼ 0:9729Þ: ð20Þ

Eqs. (19) and (20) indicate that the mathematical criteria of SOC fitbetter with China’s urbanization for the WUP data because the timeseries of the census data was not sufficiently long. The profiledimension of India’s urbanization level was 1.6111, based on thecensus data, and 1.6990, based on the WUP data.

4. Discussion

In general, the mathematical criteria of SOC fit well with theurbanization data. This study demonstrated that the values of theparameters in the urbanization model could be changed widelywithout affecting the emergence of SOC. The FSR also consolidatedthe rare possibility for chaos that is characterized by a power spec-trum 1/f0 in the urbanization model.

The phase plots indicate that the lower the value of the ruralpopulation is, the sooner natural increase exceeds migration. Inthe United States, when the country was mostly urbanized and lit-tle rural population was left to migrate to cities, natural increasebegan to exceed migration during urbanization, which occurssimilarly in most developed countries. However, in China, naturalincrease began to exceed migration even when the country wasstill primarily rural, which occurs in most developing countries(Bocquier, 2005).

Based on the census and the WUP data set, the urban popula-tion and urbanization level in the United States, China, and Indiaare shown in Figs. 8 and 9. The coefficients of variation of theroot-mean-square error were 2.41% (United States) and 3.05%(China) for urban population, and 0.14% (United States) and4.30% (China) for urbanization level. A comparison of the projectedurbanization levels in the United States, China, and India is shownin Table 6. Clearly, the explanatory powers of the UNM and LVMdecreased when the estimation period was extended from 40 to90 years. The projection model for the urbanization level in devel-oping countries is enhanced by including exogenous variables(Mulligan, 2013) and uncertainty assessment (Alkema, Gerland,et al., 2011). The upper limits of the urbanization level estimatedby Alkema, Gerland, et al. (2011) are more reasonable than thoseestimated in this study, which were 84.4% and 64.4% for Chinaand India, respectively.

Although the LGM generates a curve that tends toward an expo-nential form at low values, its maximal slope, or point of inflection(where the growth rate is maximal), is always at half of the value ofthe upper limit. This is unsatisfactory, because the factors thatdetermine the urbanization at which each country grows fastestare complex; therefore, it is unlikely that all urbanization occursfastest when countries are at half of their urban saturation level.Mulligan (2006) asserted that the logistic model should be usedwith caution when examining urbanization growth in less devel-oped countries. Birch (1999) proposed a generalized LGM thatcould potentially generate its point of inflection at any point.However, the low asymptote is not equal to zero, and obtaining

L (t ) = 1E-10e0.0108t

R 2 = 0.9776

L (t ) = 4E-11e0.0113t

R 2 = 0.9952

0

0.1

0.2

0.3

0.4

0.5

0.6

1900 1950 2000 2050

Year

Urb

aniz

atio

n le

vel

Census 1901-2011

WUP 2011

Expon. (Census 1901-2011)

Expon. (WUP 2011)

Fig. 7. The exponential growth trend of India’s urbanization level.

50000000

150000000

250000000

350000000

450000000

550000000

650000000

750000000

1950 1960 1970 1980 1990 2000 2010 2020

Year

Urb

an p

opul

atio

n

WUP-China

Census-China

WUP-US

Census-US

WUP-India

Census-India

Fig. 8. Urban populations based on the census and WUP data for the United States,China, and India.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1950 1960 1970 1980 1990 2000 2010 2020

Year

Urb

aniz

atio

n le

vel

WUP-China

Census-China

WUP-US

Census-US

WUP-India

Census-India

Fig. 9. Urbanization levels based on the census and WUP data for the United States,China, and India.

S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100 97

separate estimates for all the parameters in this sigmoid equationis difficult. In addition, Meade and Islam (1995) observed that thesimple LGM often outperformed generalized LGMs, which have thedisadvantage of losing clear physical interpretations for theirparameters.

O’Neill, Balk, Brickman, and Ezra (2001) systematically com-pared the approaches and results of various projections. Some sce-narios and probabilistic projection models can account foruncertainty in projected trends of fertility, mortality, and migra-tion but require additional detailed historical information (Alkema,Gerland, et al., 2011; Goldstein, 2004; Lee, 1998; Lutz, Sanderson, &Scherbov, 1998; Rogers, 1995b). However, no generally acceptedapproach to characterizing this uncertainty is available. Further-more, detailed data on fertility, mortality, and migration may notbe available, adequate, or reliable in some countries.

Because the UN projections are based on the extrapolation ofhistorical trends, Bocquier (2005) proved that this can be attrib-uted mainly to an inappropriate projection model that systemati-cally biases the urban estimates upward, and also to the qualityof the available data. In countries such as China, the problemmay be caused by the data used in the projection than by the mod-el. The understanding of the level of urbanization and its scale indeveloping countries is challenged by differences in the definitionof the term urban and, consequently, the lack of reliable data. Thedefinition of urban plays a crucial role.

The types of urbanization in countries differ across time andspace. Urbanization occurs in cities of various sizes and types.Some projections have indicated that urbanization is concentratedin the large cities of developing countries (Henderson, 2002). Otherprojections have indicated that most urbanization is expected tooccur in small- and medium-sized cities of one million or fewerpeople (Montgomery, 2008). Urbanization can occur through rur-al–urban migration, natural population increase, and annexation(Cohen, 2004). Urbanization during and after the Industrial Revolu-tion can be attributed solely to rural–urban migration; however,today, in many parts of the world, urbanization is fueled by bothrural–urban migration and high urban fertility (Mulligan, 2013).Urbanization in East Asia and the Pacific has been caused mostlyby industrialization and job opportunities in urban areas, whereasseveral countries in Africa have experienced urbanization with lit-tle or no economic growth (Cohen, 2004; Fay & Opal, 2000; Weeks,1994). Urbanization caused by migration may be a result of urbanpull, which was the chief cause of urbanization in Europe and theUSA, whereas migration is attributed to rural push in Africa andother developing countries, resulting from agricultural stress,political instability, and natural disasters (Dutt & Parai, 1994).Almost all projections have indicated that urbanization is occur-ring rapidly and at large scales in developing countries, which un-dergo rapid demographic changes (Angel et al., 2005). Therefore,the projection model for urbanization levels may not be enhancedby including exogenous variables, such as socioeconomic and tech-nological factors (Kelley & Williamson, 1984). The urbanizationthat occurs in developing countries is primarily the outcome ofdemographic transition. Economic and other considerations aresecondary influences (Dyson, 2011).

Including exogenous variables in the projection model forurbanization levels would complicate the calculation and interpre-tation. The proposed projection model is endogenous and basedonly on available data. The objective of this study was not to offera projection model with explanatory power, using several exoge-nous socioeconomic variables that could explain the urbanizationlevel and its trend, but rather to identify problems in the projec-tions using only known quantities without knowing the character-istics of the initial population (size, age structure, and vital rates).

5. Conclusion

This study analyzed the trends in urbanization levels in theUnited States, China, and India. Based on the census and WUP data

Table 6The comparison of projected urbanization levels in the United States, China, and India.

Country Urbanization level in 2030 (%) Urbanization level in 2050 (%)

WUP 2011 Alkema, Gerland, et al. (2011) and Alkema,Raftery, et al. (2011) Bocquier (2005) This study WUP 2011 Mulligan (2013) This study

USA 86.0 – 75.2 81.3 88.9 90.6 84.9China 68.7 51.1 39.8 76.0 77.3 77.3 94.7India 39.8 35.6 30.9 – 51.7 51.7 –

98 S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100

sets, urbanization dynamics were examined by using the rural–ur-ban interaction model and the LGM. Insight into the patterns andprocesses of the urbanization trends in the United States, China,and India were obtained by systematically examining of the censusand WUP data. However, the cases in which counterurbanizationpatterns exist, such as in Denmark, France, the Netherlands, Swe-den, and Switzerland, (Champion, 2001; Kontuly, 1998) were notconsidered in this study.

Based on the WUP data for the United States, China, and India,the derived rural–urban interaction model and the upper limit ofthe urbanization level were all abnormal or unreasonable. Theavailable WUP data set was not satisfactory which may be causedby the various definitions of urban areas, the lack of reliable andcurrent demographic data, or the projection method. For projectingthe urban population, the urban–rural growth difference methodwas adopted by the UN. Disregarding certain country-specific as-pects, this method is based on the assumption that the level ofurbanization follows a logistic growth pattern (United Nations,1980). The actual urban–rural growth difference for a countrymay not be as uniform as the UN method assumes. The mannerin which the UN projection manages the issue of uncertainty isunsatisfactory. Alkema, Gerland, et al. (2011) determined thatChina‘s growth differential was projected to decrease linearly untilit reached the global norm, whereas India‘s growth differential wasprojected to increase linearly until it reached the global norm. Theoverestimation of the urban population based on the UNM wasparticularly more pronounced for developing countries (Bocquier,2005). Therefore, the WUP data for China and India were not reli-able. Most projections were based on the WUP data with five-yearintervals, and are, thus, partly based on interpolations; therefore,using the original estimation from the census data is preferred.

The LGM performs well for most animal populations becausethe niches that encase their populations are of constant size(Marchetti et al., 1996). Thus, the LGM is more suitable for closedurbanization dynamics in which the rural region and rural–urbaninteraction determine the progress of urbanization. The growthof human population exhibits the elasticity of the human niche;thus, the LGM may exhibit fleeting limits. Therefore, a fixed carry-ing capacity should not be considered in long-term projections ofurbanization levels (Cohen, 1998). However, an advantage of theaggregate methods, such as the LGM, is that longer time seriesare available for the total population, compared with the lengthof series for variables such as age-specific fertility, mortality, andmigration. Additionally, if ecological and economic factors limit agrowth trend more directly on the total population than on thevital rates, then projecting total population directly may be sensi-ble (O’Neill et al., 2001). In other words, aggregate methods aremore suitable in such a situation.

The management of uncertainty in population forecasting hasrecently received increased interest among researchers (Keilman,Pham, & Hetland, 2002; Lutz & Goldstein, 2004; Raftery, Li,Ševcíková, Gerland, & Heilig, 2012; Wilson & Rees, 2005). Forexample, Alkema, Raftery, et al. (2011) addressed uncertainties inthe factors that cause drops in the fertility level. Lutz andSamir (2010) advocated probabilistic projections to addressuncertainty and indicated that a universally accepted approach

to quantitatively describing the uncertainty of population projec-tions has not yet been developed. New techniques for the probabi-listic projection of urbanization levels must be developed.

The urbanization process in rapidly urbanizing countries can bedescribed using the J-shaped curve rather than the S-shaped curve(Cadwallader, 1996; Haggett, 2001; Mulligan, 2013). Additionalstudies on the S-shaped curve have been conducted. Another re-search challenge that was also considered by Chen (2012) was toexplore the general principle of the J-shaped curve of urbanizationand its underlying rationale. This is critical for demographers,geographers, other scientists, and policymakers because an insightinto the urbanization process is the basis of social, economic, cul-tural, and environmental planning and policy making.

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