Spatial Autocorrelation and Localization of Urban Development

Transcription

1 Chinese Geographical Science 27 17(1) DOI: 1.17/s Spatial Autocorrelation and Localization of Urban Development LIU Jisheng 1, CHEN Yanguang 2 (1. Department of Geography, Northeast Normal University, Changchun 1324, China; 2. Department of Geography, Peking University, Beijing 1871, China) Abstract: A nonlinear analysis of urban evolution is made by using of spatial autocorrelation theory. A first-order nonlinear autoregression model based on Clark s negative exponential model is proposed to show urban population density. The new method and model are applied to Hangzhou City, China, as an example. The average distance of population activities, the auto-correlation coefficient of urban population density, and the auto-regressive function values all show trends of gradual increase from 1964 to 2, but there always is a sharp first-order cutoff in the partial autocorrelations. These results indicate that urban development is a process of localization. The discovery of urban locality is significant to improve the cellular-automata-based urban simulation of modeling spatial complexity. Keywords: urban population density; nonlinear spatial autocorrelation; Clark s law; localization; Hangzhou City 1 Introduction Urban and region are all the complex spatial systems (Wilson, 2). An important task of spatial complexity aims at urban form and structure (Batty, 2). The methods of exploring complexity include mathematics and simulation. Simulation of a complex system may be very helpful practically, but it does not help us conceptually in understanding the rules of behavior at the higher level (Bossomaier and Green, 1998; Hao, 1999). It is significant to combine mathematics and simulation to form a more effective method. One of the principal factors that affect the urban form and structure is human activity. In the 195s, Clark propounded a negative exponential for modeling urban population density (Clark, 1951). From then on, various models on urban population density appeared one after another (Batty and Longley, 1994). However, all these models neglected spatial autocorrelation of urban population distribution so that they cannot pass the test on residuals. In fact, the Clark s model implied the localization of urban growth. The problem of residuals autocorrelation can be solved with spatial autoregression. The research on localization as well as locality can be traced back to P W Anderson (1958), and today the word localization has become an important concept in the sciences of complexity and nonlinearity (Thompson and Virgin, 1988; El Naschie, 2; Bendiksen, 2). The paper is devoted to nonlinear spatial autocorrelation analysis of urban form and structure. A nonlinear autoregression model based on Clark s density-distance relation is presented to model urban population distribution. The results of autocorrelation and autoregression analysis are contributed to the research on urban localization, which will be useful to the simulation of urban evolution and spatial complexity. 2 Spatial Autocorrelation Model 2.1 Spatial-temporal autocorrelation function For a stationary ergodic process {y(t)}, the autocorrelation function (C) is: Cov( τ) C( τ)= (1) σ 2 where τ is time lag, C is autocorrelation function, and Cov(τ) is auto-covariance function, which is defined by: 1 T Cov ( τ) = lim -- τ [ yt ( ) μ ] T T τ [ yt+τ ( )- μ]d t (2) The mean μ is given by: 1 T μ = lim y( t)d t T T (3) The variance σ 2 is given by: Received date: ; accepted date: Foundation item: Under the auspices of the National Natural Science Foundation of China (No ) Corresponding author: LIU Jisheng. liujs362@nenu.edu.cn

2 Spatial Autocorrelation and Localization of Urban Developmentl 35 1 T 2 2 =lim [ ()- ] d T (4) σ yt μ t T where t is the dummy variable of time (e. g., ordinal number), T is the length of sample path. Covariance function is often named as correlation function. It is in fact the autocorrelation function that does not be standardized. For an ergodic transformation, the time mean is equal to the space mean almost everywhere (Walters, 2). Based on the Ergodic Theory, a mathematical model on the temporal process can be used to characterize spatial phenomena (Harvey, 1971). Because of this, we have the spatial autocorrelation function as follows: Cov( s) Cs ()= (5) σ 2 where s is the continuous spatial displacement corresponding to time lag, Cov(s) is the auto-covariance function in the following form: 1 Rs Cov ( s)=lim [ y( r) μ][ y( r+s) μ]dr R R s - (6) - where r is the spatial dummy variable (e.g., displacement or radius), R is the length of space series. The definition of the mean and the variance is similar to those of time series. The spatial data series do not require the stationarity as precondition, but its generalized integral must be convergent. 2.2 Spatial autocorrelation function of urban population density According to the spatial-temporal correlation ideas mentioned above, the covariance function of urban population density can be expressed as: 1 Rr Cov ( r) = lim [ ρ( x) μ] [ ρ( x+ r) μ]dx (7) R R r where ρ(x) is the population density in site x, r is displacement, R is the length of sample path measured by urban maximum radius. Theoretically, R, μ, so the generalized spatial correlation function is: 1 Rr Cov ( r) = lim ρ( x) ρ( x+ r)dx (8) R R r As a special case, we will consider ρ(r) as a onedirectional function on a single line of distance from the city center (x=). If the spatial relation is between the center and any point on the line, then Equation (8) is reduced to the following form: Cov( r) = ρ() ρ( r) (9) where obviously, ρ() denotes population density in urban center, ρ(r) is the population density at distance r from the center of the city. Equation (9) represents the density-distance relationship, which usually is described with negative exponential function known as Clark (1951) s model: ρ( r ) = ρe - rr / (1) where ρ =ρ(), and r is the average traveling distance of urban population. In fact, the parameter r denotes the urban characteristic radius, and its reciprocal, 1/r, is the declining gradient of urban density. Equations (9) and (1) show that the correlation function of urban population is directly proportional to the urban population density, and the generalized integral of covariance, which is based on negative exponential model, is convergent. So the process of spatial autocorrelation of urban population is acceptable. However, from Equation (8) to Equation (9), the general spatial relation of urban population changes into a core-periphery relationship. Despite this, we can draw general conclusions from the special relation. 3 Data Processing 3.1 Study area and data sources Hangzhou City, Zhejiang Province is taken as an example to develop the spatial autocorrelation model. The urban population density data in 1964, 1982, 199 and 2 are listed in Table 1. The census data is based on residential committees of community. We process the data by means of spatial weight and average. The length of sample path is n=26, and the maximum urban radius is 15.3km. The method of processing data is illuminated in detail by Feng (22; 24). (1) Sampling area. The data cover the Hangzhou proper in 2. The spatial scopes of sampling in four years are same in order to make it sure that the parameters from 1964 to 2 are comparable. Because of scale invariance of urban form (Batty and Longley, 1994), we take no account of the borderline between the urban and rural area. (2) Processing method. A series of concentric rings are drawn around urban population center. The ratio of the area of each community to the area between two rings is taken as the weight of computing urban population density (Chen, 2). The weighed arithmetic average can lessen the influence of community scope on the estimated results of population density as much as possible (Fig. 1). (3) Spatial scale. The radius difference between rings is.6km, less than r. The parameter values of r can be estimated with Clark s law.

3 36 LIU Jisheng, CHEN Yanguang Table 1 Average urban population density of Hangzhou Metropolis in gression (AR). The least squares method as a classical analysis is chosen to estimate the AR coefficients. The No. Distance Urban population density (person/km 2 ) procedure of making model is as follows. (km) Step 1: Scatterplots analysis to examine the trend char acters of data series (Feng, 22; 24) Step 2: Correlogram analysis to examine the AC characters of data series Step 3: AR model analysis to determine the AR model formation according to analysis results of AC and scatterplots Step 4: Spatial interpretation to reveal the geographical meaning of the AC analysis and the AR model The population density data of Hangzhou City can be well fitted to Clark s density-distance relation. For example, a least square computation involving the density data in 2 gives the following expression: ( ) 3185e r / ρ r = (11) The determination coefficient R 2 = The parameter indicates that the average distance of urban population activity is estimated as r =3.946km. The results in other years are similar to one another except the model parameters are different According to the density-distance relation of urban population, it is easy to make nonlinear spatial AC analysis based on the linear relationship between lnρ(r) Source: Feng, 22 and lnρ(r+k), where ln means natural logarithms, and k denotes discrete displacement variable. The key step is to determine the maximum displacement of AC and the maximum order of AR in virtue of correlogram analysis. Theoretically, critical values (cf ) can be given as: ±1.96 cf = = ±.384 n Source: Feng, 22 Fig. 1 Sketch map of study area in Hangzhou metropolis 3.2 Steps and results of data processing Based on the mathematic principles and models aforementioned, it is easy to make autocorrelation (AC) analysis of spatial data and compute the autoregressive representation. There are several algorithms for auto-re- according to which we can know whether or not there is significant difference between autocorrelation function (ACF) or partial autocorrelation function (PACF) and zero. For the purpose of intuitive estimate, we can add the positive and negative critical values into the correlogram to form what is called two-standard-error bands. However, because that sample path is not long enough, the critical value can be revised as ±.34 or so. The average spatial distance of urban population activity can be judged through ACF diagram. Taking the data in 2 as an example, the autocorrelation coefficients (ACC) are within the double standard error lines and close to the revised critical values when displacement k=6 or 7, which implies that ACF is not significantly different from (Fig. 2). In the original data, the first value is r=.3km, the step length (distance between

4 rings) is Δr=.6km, thus the distance r p corresponding to k=6 is r= p = 3.9km Fig. 2 Spatial ACF and PACF of population density of Hangzhou in 2 This means that the average distance of population activity of Hangzhou City in 2 is about km. The characteristic distance in other years can be esti- Spatial Autocorrelation and Localization of Urban Developmentl 37 mated in the same fashion. It is shown that the average activity distance is about km from 1964 to 1982, and around km in 199. As the ACF values of all years are not significantly different to at about 6km, the average distance of population activity in Hangzhou is not greater than 6km. Clearly, PACF diagram shows a first-order cutoff. That is, PACC (partial autocorrelation coefficient) is not significantly different from when k>1 (Fig. 2). Thus it can be seen that the biggest order of AR should be p=1. The autoregressive coefficient (ARC) can be estimated with use of least square method. The nonlinear AR model based on the data in 2 is:.9819 ρ( k ) = ρ( k - 1) (12) The goodness of fit is R 2 = The models of the other years are similar to one another except that the parameter values are different in different years:.9396 in 1964: ρ( k) = ρ( k - 1) (13) 2 R = in 1982: ρ( k) = ρ( k - 1) (14) 2 R = in 199: ρ( k ) = ρ( k - 1) (15) 2 R =.9883 The results of computation are presented in Table 2, including ACF, PACF, ACC, ARC, characteristic radius (CR). Spectral exponent β and Hurst exponent H values (Chen and Liu, 26) are added in the table for reference and comparison. Table 2 Parameter values of spatial analyses of Hangzhou s population density Year CR (r ) AAD ACC PACC a b β H Notes: Characteristic radius (CR) is the parameter of Clark s model; average activity distance (AAD) is estimated with ACF; ACC and PACC correspond to the values of ACF and PACF when k=1; ARC is calculated by least square method; a, b are parameters 4 Discussion The ACF histogram displays dual property of auto-regression process of Hangzhou s population. If the time series represents the AR(1) process, then the ACF should display slow one-sided damping; while if the time series represents AR(2) process, then ACF should display an oscillation damping slowly with much richer patterns. The ACF character of Hangzhou s urban population density comes from process between AR(1) and AR(2) (Fig. 2a). On the other hand, the PACF clearly cuts off at a displacement of 2 which corresponds to the AR(1) process (Fig. 2b). According to ACF, the spatial interaction of Hangzhou s population can pass several population units (e.g. community) indirectly. While according to PACF, a population unit only affects the proximate units directly. As is well known, the ACF reflects direct and indirect influences, but PACF only reflects direct

5 38 LIU Jisheng, CHEN Yanguang influence. So an urban population unit only acts directly on its neighboring units, but its indirect action can spread abroad. The action of unit A can transfer to its neighboring unit B, then transfers to apart unit C through unit B, and so on. According to the first-order cutoff of PACF and the Clark model, we build the nonlinear AR model of urban population in the form: ρ( k ) = aρ( k - 1) b (16) where a, b are parameters (a>, b>) in the log-linear model, lna is intercept, b denotes ARC. For the AR(1) process, ARC is equal to ACF(1) and PACF(1) theoretically. In virtue of a recursion relation, we have: β p β ρ( k ) =αρ( k ) =α ρ( k- l) p (17) where l is passing length of urban population unit action, l=1 means that a unit acts on the neighbors, l =2 means that a unit acts on the separate neighbors neighbors. The smaller parameters a and b values are, the weaker the action of location k 1 on location k is, and the shorter the distance of spatial influence becomes. From Equation (12) and the data in Table 1, several inferences can be drawn as follows. 1) The average distance of population activity reflected by ACF agrees with characteristic radius r defined by Clark s model. 2) The parameters of the nonlinear AR model are a>1, b<1, but a, b tend towards 1 gradually with the passion of time. The decrease of a value means that the long distance action of urban population gradually dies down; while the increase of b value means that the long distance action of urban population gradually strengthens. 3) Although ACF(1)= PACF(1)=b in theory, the parameter b corresponds to ACF, while a corresponds to PACF. 4) The trend reflected by the first-order ACC agrees closely with that reflected by Spectral exponent β, the correlation coefficient is ) Along with the city development, especially the urban traffic upgrowth, the spatial interaction of population become stronger, but the direct spatial action distance becomes shorter. An interesting discovery is the first-order cutoff phenomenon of PACF (Fig. 2b), which suggests localization of urban development. The original meaning of localization is that one particle only acts directly on its neighbors, but does not act on neighbors neighbors directly. When each element of a system has a direct action on its neighbors, and its direct action on neighbors neighbors becomes weaker and weaker, the system can be regarded as localizing. The spatial process of Hangzhou City s population activity is not the standard localization but quasi-localization phenomenon, because the characteristic of ACF shows that AR process falls between AR (1) and AR (2). However, the PACF shows that order of AR process is less than 2. The main criteria of urban localization based on Hangzhou s population data are as tnat will be stated next. 1) The urban population density follows the negative exponential distribution (Clark, 1951; Chen, 2; Feng, 22), which implies that the Spectral exponent is approximately equal to 2 (Chen and Liu, 26). Theoretically, if Spectral exponent β 2, then the Hurst exponent H 1/2, thus the second-order ACC approximates to. 2) The proportionality coefficient of the nonlinear AR model a 1 indicates the direct correlation distance of urban population becomes shorter and shorter. 3) The first-order cutoff of PACF means that a population unit acts only on its neighbors. The spectral analysis provides the indirect theoretical basis, while the first-order cutoff of PACF gives direct evidence for the localization of urban development. A possible question is that whether or not the scale and the granularity of spatial sampling will influence the analysis results. In fact, if only r<r, the scale and the granularity will not affect the conclusions. For Hangzhou City, r =3 5km, while r=.6km. In addition, according to Clark (1951), Boston City of USA s characteristic is about r =5.5km in 194, the ACF and PACF and corresponding analysis results agree closely with those based on Hangzhou City. The conclusions drawn from the data of different space and time are same as each other. There is little probability of coincidence. Localization is one of objects of complexity research (Anderson, 1991). Modeling and simulation of spatial complexity are an important task of the urban theory (White and Engelen, 1993b). Almost all of the spatial complexity simulations of urban evolution based cellular automata (CA) and geographic information system (GIS) accepted the assumption of action-at-a-distance (Batty et al., 1997; White and Engelen, 1993a). The discovery of urban locality and localization rules against the assumption. The study of localization suggests a new angle of view to look at urban form as well as spatial interaction. 5 Conclusions Mathematical modeling and computer simulation are two kinds of tools for geographical analysis of urban system.

6 In the past ten years, international urban and geographical researches rely heavily on CA-based simulation. CA is one of the significant methods of geographical simulation. once considered to be a possible paradigm for the 21st century (Batty et al., 1997), but before long the simulation analysis based on CA model comes in a puzzle (Torrens and O'Sullivan, 21). That the original assumption of localization is abandoned can account for the occurrence (Liu and Chen, 22). The main conclusions of the paper can be summarized as follows. 1) The urban population density conforms to the negative exponential distribution, which implies that the Spectral exponent is approximately equal to 2, thus the second-order ACC is close to. This is a theoretical mark of localization. 2) The characteristic of ACF falls between AR(1) and AR(2) process, but PACF corresponds to the AR(1) process. This indicates that population development of realistic city is a process of quasi-localization. 3) The parameter values of the nonlinear AR model in different years show that the locality of city becomes clearer and clearer. The conclusions based on the assumption of circular city and center-periphery relationship can be generalized since the interaction between center and periphery is much stronger than that along other direction. An inference is that it may be a misunderstanding to replace locality of city CA model by action-at-a-distance. It is necessary to resume the localization assumption of CA-based urban simulation. The action-at-a-distance in a CA model can be represented as an AR(1) process. Acknowledgment The authors would like to thank Dr. Feng Jian at the Peking University, for providing the urban population density data of Hangzhou City. References Anderson P W, The absence of diffusion in certain random lattices. Physical Review, 19: Anderson P W, Is complexity physics? Is it science? What is it? Physics Today, 44 (7): Batty M, 2. Less is more, more is different: complexity, morphology, cities, and emergence. Environment and Planning B: Planning and Design, 27: Batty M, Couclelis H, Eichen M, Urban systems as cellular Spatial Autocorrelation and Localization of Urban Developmentl 39 automata. Environment and Planning B: Planning and Design, 24: Batty M, Longley P A, Fractal Cities: A Geometry of Form and Function. London: Academic Press. Bendiksen O O, 2. Localization phenomena in structural dynamics. Chaos, Solitons and Fractals, 11: Bossomaier T, Green D, Patterns in the Sand: Computers, Complexity and Life. Massachusetts: Perseus Books. Chen Yanguang, 2. Derivation and generalization of Clark s model on urban population density using entropy-maximising methods and fractal ideas. Journal of Central China Normal University (Natural Science Edition), 34(4): (in Chinese) Chen Yanguang, Liu Jisheng, 26. Power spectral analyses of spatial auto-correlations of urban density: An application to the Hangzhou Metropolis. Advances in Earth Science, 21(1): 1 9. (in Chinese) Clark C, Urban population densities. Journal of Royal Statistical Society, 114: El Naschie M S, 2. Foreword: A very brief history of localization. Chaos, Solitons and Fractals, 11: Feng Jian, 22. Modeling the spatial distribution of urban population density and its evolution in Hangzhou. Geographical Research, 21(5): (in Chinese) Feng Jian, 24. Reconstruction of Urban Internal Space in China in the Transition Period. Beijing: Science Press. (in Chinese) Hao Bailin, Characterization of complexity and complexity science. Science, 51(3): 3 8. (in Chinese) Harvey D, Explanation in Geography. London: Edward Arnold Ltd. Liu Jisheng, Chen Yanguang, 22. GIS-based cellular automata models and researches on spatial complexity of man-land relationship. Geographical Research, 21(2): (in Chinese) Thompson J M T, Virgin L N, Spatial chaos and localization phenomena in nonlinear elasticity. Physical Letter A, 126: Torrens P M, O'Sullivan D, 21. Cellular automata and urban simulation: where do we go from here? Environment and Planning B: Planning and Design, 28: Walters P, 2. An Introduction to Ergodic Theory. New York: Springer-Verlag. White R, Engelen G, 1993a. Cellular dynamics and GIS: modeling spatial complexity. Geographical Systems, 1: White R, Engelen G, 1993b. Cellular automata and fractal urban form: a cellular modeling approach to the evolution of urban land-use patterns. Environment and Planning A, 25: Wilson A G, 2. Complex Spatial Systems: The Modelling Foundations of Urban and Regional Analysis. Singapore: Pearson Education.