Quantitative Methods and Applications in GIS - Chapter 6

6 Function Fittings by Regressions and Application in Analyzing Urban and Regional Density Patterns Urban and regional studies begin with analyzing the spatial structure, particularly population density patterns. As population serves as both supply (labor) and demand (consumers) in an economic system, the distribution of population represents that of economic activities. Analysis of changing population distribution patterns is a starting point for examining economic development patterns in a city or region. Urban and regional density patterns mirror each other: the central business district (CBD) is the center of a city, whereas the whole city itself is the center of a region, and densities decline with distances both from the CBD in a city and from the central city in a region. While the theoretical foundations for declining urban and regional density patterns are different (see Section 6.1), the methods for empirical studies are similar and closely related. This chapter discusses how we can find a function capturing the density patterns best, and what we can learn about urban and regional growth patterns from this approach. The methodological focus is on function fittings by regressions and related statistical issues. Section 6.1 explains how density functions are used to examine urban and regional structures. Section 6.2 presents various functions for a monocentric structure. Section 6.3 discusses some statistical concerns on monocentric function fittings and introduces nonlinear regression and weighted regression. Section 6.4 examines various assumptions for a polycentric structure and corresponding function forms. Section 6.5 uses a case study in the Chicago region to illustrate the techniques (monocentric vs. polycentric models, linear vs. nonlinear and weighted regressions). The chapter is concluded in Section 6.6 with discussion and a brief summary. 6.1 THE DENSITY FUNCTION APPROACH TO URBAN AND REGIONAL STRUCTURES 6.1.1 STUDIES ON URBAN DENSITY FUNCTIONS Since the classic study by Clark (1951), there has been great interest in empirical studies of urban population density functions. This cannot be solely explained by 97 © 2006 by Taylor & Francis Group, LLC 98 Quantitative Methods and Applications in GIS the easy availability of data. Many are attracted to the research topic because of its power of revealing urban structure and its solid foundation in economic theory.1 McDonald (1989, p. 361) considers the population density pattern as “a critical economic and social feature of an urban area.” Among all functions, the exponential function or Clark’s model is the one used most widely: Dr = aebr (6.1) where Dr is the density at distance r from the city center (i.e., CBD), a is a constant (the CBD intercept), and b is also a constant for the density gradient. Since the density gradient b is often a negative value, the function is also referred to as the negative exponential function. Empirical studies show that it is a good fit for most cities in both developed and developing countries (Mills and Tan, 1980). The economic model by Mills (1972) and Muth (1969), often referred to as the Mills–Muth model, is developed to explain the empirical pattern of urban densities as a negative exponential function. The model assumes a monocentric structure: a city has only one center, where all employment is concentrated. Intuitively, as everyone commutes to the city center for work, a household farther away from the CBD spends more on commuting and is compensated by living in a larger-lot house (also cheaper in terms of price per area unit). The resulting population density exhibits a declining pattern with distance from the city center. Appendix 6A shows how the negative exponential urban density function is derived in the economic model. From the deriving process, parameter b in Equation 6.1 is the unit cost of transportation. Therefore, declining transportation costs over time, as a result of improvements in transportation technologies and road networks, lead to a flatter density gradient. This clearly explains that urban sprawl and suburbanization are mainly attributable to transportation improvements. However, economic models are “simplification and abstractions that may prove too limiting and confining when it comes to understanding and modifying complex realities” (Casetti, 1993, p. 527). The main criticisms lie in its assumptions of the monocentric city and unit price elasticity for housing, neither of which is supported by empirical studies. Wang and Guldmann (1996) developed a gravity-based model to explain the urban density pattern (also see Appendix 6A). The basic assumption of the gravity-based model is that population at a particular location is proportional to its accessibility to all other locations in a city, measured as a gravity potential. Simulated density patterns from the model conform to the negative exponential func-tion when the distance friction coefficient b falls within a certain range (0.2 £ b £ 1.0 in the simulated example). The gravity-based model does not make the restrictive assumptions as in the economic model, and thus implies wide applicability. It also explains two important empirical findings: (1) flattening density gradient over times (corresponding to smaller b) and (2) flatter gradients in larger cities. The economic model explains the first finding well, but not the second (McDonald, 1989, p. 380). Both the economic model and the gravity-based model explain the change of density gradient over time through transportation improvements. Note that both the distance © 2006 by Taylor & Francis Group, LLC Function Fittings by Regressions and Application in Analyzing Density Patterns 99 friction coefficient b in the gravity model and the unit cost of transportation in the economic model decline over time. Earlier empirical studies of urban density patterns are based on the monocentric model, i.e., how population density varies with distance from the city center. It emphasizes the impact of the primary center (CBD) on citywide population distri-bution. Since the 1970s, more and more researchers recognize the changing urban form from monocentricity to polycentricity (Ladd and Wheaton, 1991; Berry and Kim, 1993). In addition to the major center in the CBD, most large cities have secondary centers or subcenters, and thus are better characterized as polycentric cities. In a polycentric city, assumptions of whether residents need to access all centers or some of the centers lead to various function forms. Section 6.4 will examine the polycentric models in detail. 6.1.2 STUDIES ON REGIONAL DENSITY FUNCTIONS The study of regional density patterns is a natural extension to that of urban density patterns as the study area is expanded to include rural areas. The urban population density patterns, particularly the negative exponential function, are empirically observed first, and then explained by theoretical models (either the economic model or the gravity-based model). Even considering the Alonso’s (1964) urban land use model as the precedent of the Mills–Muth urban economic model, the theoretical explanation lags behind the empirical finding on urban density patterns. In contrast, following the rural land use theory by von Thünen (1966, English version), economic models for the regional density pattern by Beckmann (1971) and Webber (1973) were developed before the work of empirical models for regional population density functions by Parr (1985), Parr et al. (1988), and Parr and O’Neill (1989). The city center used in the urban density models remains as the center in regional density models. The declining regional density pattern has a different explanation. In essence, rural residents farther away from a city pay higher transportation costs for the shipment of agricultural products to the urban market and for gaining access to industrial goods and urban services in the city, and are compensated by occupying cheaper, and hence more, land. See Wang and Guldmann (1997) for a recent model. Similarly, empirical studies of regional density patterns can be based on a monocentric or a polycentric structure. Obviously, as the territory for a region is much larger than a city, it is less likely for physical environments (e.g., topography, weather, and land use suitability) to be uniform across a region than a city. Therefore, population density patterns in a region tend to exhibit less regularity than in a city. An ideal study area for empirical studies of regional density functions would be an area with uniform physical environments, like the “isolated state” in the von Thünen model (Wang, 2001a, p. 233). Analyzing the function change over time has important implications for both urban and regional structures. For urban areas, we can examine the trend of urban polarization vs. suburbanization. The former represents an increasing percentage of population in the urban core relative to its suburbia, and the latter refers to a reverse trend, with an increasing portion in the suburbia. For regions, we can identify the process of centralization vs. decentralization. Similarly, the former © 2006 by Taylor & Francis Group, LLC 100 Quantitative Methods and Applications in GIS Spread (decentralization) Dr lnDr Log-transform t + 1 t + 1 t t r r (a) (b) Backwash (centralization) Dr lnDr Log-transform t t t + 1 t + 1 r r (c) (d) FIGURE 6.1 Regional growth patterns by the density function approach. refers to the migration trend from peripheral rural to central urban areas, and the latter is the reverse. Both can be synthesized into a framework of core vs. periphery. According to Gaile (1980), economic development in the core (city) impacts the surrounding (suburban and rural) region through a complex set of dynamic spatial processes (i.e., intraregional flows of capital, goods and services, information and technology, and residents). If the processes result in an increase in activity (e.g., population) in the periphery, the impact is spread. If the activity in the periphery declines while the core expands, the effect is backwash. Such concepts help us understand core–hinterland interdependencies and various relationships between them (Barkley et al., 1996). If the exponential function is a good fit for regional density patterns, the changes can be illustrated as in Figure 6.1, where t + 1 represents a more recent time than t. In a monocentric model, we can see the relative importance of the city center; in a polycentric model, we can understand the strengthening or weakening of various centers. © 2006 by Taylor & Francis Group, LLC Function Fittings by Regressions and Application in Analyzing Density Patterns 101 In the reminder of this chapter, the discussion focuses on urban density patterns. However, similar techniques can be applied to studies of regional density patterns. 6.2 FUNCTION FITTINGS FOR MONOCENTRIC MODELS 6.2.1 FOUR SIMPLE BIVARIATE FUNCTIONS In addition to the exponential function (Equation 6.1) introduced earlier, three other simple bivariate functions for the monocentric structure have often been used: Dr = a + br (6.2) Dr = a + blnr (6.3) Dr = arb (6.4) Equation 6.2 is a linear function, Equation 6.3 is a logarithmic function, and Equation 6.4 is a power function. Parameter b in all the above four functions is expected to be negative, indicating declining densities with distances from the city center. Equation 6.2 and 6.3 can be easily estimated by ordinary least squares (OLS) linear regressions. Equations 6.1 and 6.4 can be transformed to linear functions by taking the logarithms on both sides, such as lnDr = A + br (6.5) lnDr = A + blnr (6.6) Equation 6.5 is the log-transform of exponential Equation 6.1, and Equation 6.6 is the log-transform of power Equation 6.4. The intercept A in both Equations 6.5 and 6.6 is just the log-transform of constant a (i.e., A = lna) in Equations 6.1 and 6.4. The value of a can be easily recovered by taking the reverse of logarithm, i.e., a = eA. Equations 6.5 and 6.6 can also be estimated by linear OLS regressions. In regressions for Equations 6.3 and 6.6 containing the term lnr, samples should not include observations where r = 0 (exactly the city center), to avoid taking logarithms of zero. Similarly, in Equations 6.5 and 6.6 containing the term lnDr, samples should not include those where Dr = 0 (with zero population). Take the log-transform of exponential function in Equation 6.5 for an example. The two parameters, intercept A and gradient b, characterize the density pattern in a city. A lower value of A indicates declining densities around the central city; a lower value of b (in terms of absolute value) represents a flatter density pattern. Many cities have experienced lower intercept A and flatter gradient b over time, representing a common trend of urban sprawl and suburbanization. The changing pattern is similar to Figure 6.1a, which also depicts decentralization in the context of regional growth patterns. © 2006 by Taylor & Francis Group, LLC ... - tailieumienphi.vn