Quantitative Methods and Applications in GIS - Chapter 6 pdf

97 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 S TUDIES ON U RBAN D ENSITY F UNCTIONS 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 2795_C006.fm Page 97 Friday, February 3, 2006 12:16 PM © 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: (6.1) where D r 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 β falls within a certain range (0.2 ≤ β ≤ 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 β ) 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 Dae r br = 2795_C006.fm Page 98 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC Function Fittings by Regressions and Application in Analyzing Density Patterns 99 friction coefficient β 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 S TUDIES ON R EGIONAL D ENSITY F UNCTIONS 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 2795_C006.fm Page 99 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC 100 Quantitative Methods and Applications in GIS 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. FIGURE 6.1 Regional growth patterns by the density function approach. (c) (d) Dr r t t + 1 Log-transform lnDr r t t + 1 Backwash (centralization) (a) (b) Dr r t t + 1 t + 1 Log-transform lnDr r t Spread ( decentralization) 2795_C006.fm Page 100 Friday, February 3, 2006 12:16 PM © 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 F OUR S IMPLE B IVARIATE F UNCTIONS In addition to the exponential function (Equation 6.1) introduced earlier, three other simple bivariate functions for the monocentric structure have often been used: D r = a + br (6.2) D r = a + blnr (6.3) D r = ar b (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 lnD r = A + br (6.5) lnD r = 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 = e A . 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 D r = 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. 2795_C006.fm Page 101 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC 102 Quantitative Methods and Applications in GIS 6.2.2 O THER M ONOCENTRIC F UNCTIONS In addition to the four simple bivariate functions discussed above, three other functions are also used widely in the literature. One was proposed by Tanner (1961) and Sherratt (1960) independently, commonly referred to as the Tanner–Sherratt model . The model is written as (6.7) where the density D r declines exponentially with distance squared, r 2 . Newling (1969) incorporated both Clark’s model and the Tanner–Sherratt model and suggested the following model: (6.8) where the constant term b 1 is most likely to be positive and b 2 negative, and other notations remain the same. In Newling’s model , a positive b 1 represents a density crater around the CBD, where population density is comparatively low due to the presence of commercial and other nonresidential land uses. According to Newling’s model, the highest population density does not occur at the city center, but rather at a certain distance away from the city center. The third model is the cubic spline function used by some researchers (e.g., Anderson, 1985; Zheng, 1991) in order to capture the complexity of urban density pattern. The function is written as (6.9) where x is the distance from the city center, D x is the density there, x 0 is the distance of the first density peak from the city center, x i is the distance of the i th knot from the city center (defined by either the second, third, etc., density peak or simply even intervals across the whole region), and Z i * is a dummy variable (= 0, if x is inside the knot; = 1, if x is outside of the knot). The cubic spline function intends to capture more fluctuations of the density pattern (e.g., local peaks in suburban areas), and thus cannot be strictly defined as a monocentric model. However, it is still limited to examining density variations related to distance from the city center regardless of directions, and thus assumes a concentric density pattern. 6.2.3 GIS AND REGRESSION IMPLEMENTATIONS The density function analysis only uses two variables: one is Euclidean distance r from the city center, and the other is the corresponding population density D r . Dae r br = 2 Dae r br br = + 12 2 D a bxx cxx dxx d xx x i =+ − + − + − + − +11 0 1 0 2 10 3 1 ()()() ( iii i k Z) *3 1= ∑ 2795_C006.fm Page 102 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC Function Fittings by Regressions and Application in Analyzing Density Patterns 103 Euclidean distances from the city center can be obtained using the techniques explained in Section 2.1. Identifying the city center requires knowledge of the study area and is often defined as a commonly recognized landmark site by the public. In the absence of a commonly recognized city center, one may use the local government center 2 or the location with the highest level of job concentration to define it, or follow Alperovich (1982) to identify it as the point producing the highest R 2 in density function fittings. Density is simply computed as population divided by area size. Area size is a default item in any ArcGIS polygon coverage and can be added in a shapefile (see step 3 in Section 1.2). Once the two variables are obtained in GIS, the dataset can be exported to an external file for regression analysis. Linear OLS regressions are available in many software packages. For example, one may use the widely available Microsoft Excel. Make sure that the Analysis ToolPak is installed in Excel. Open the distance and density data as an Excel workbook, add two new columns to the workbook (e.g., lnr and lnDr), and compute them as the logarithms of distance and density, respectively. Select Tools from the main menu bar > Data Analysis > Regression to activate the regression dialog window shown in Figure 6.2. By defining the appropriate data ranges for X and Y variables, Equations 6.2, 6.3, 6.5, and 6.6 can be all fitted by OLS linear regressions in Excel. Note that Equations 6.5 and 6.6 are the log-transformations of exponential Equation 6.1 and power Equation 6.4, respectively. Based on the results, Equation 6.1 or 6.4 can be easily recovered by computing the coefficient a = e A and the coefficient b unchanged. Alternatively, one may use the Chart Wizard in Excel to obtain the regression results for all four bivariate functions directly. First, use the Chart Wizard to draw a graph depicting how density varies with distance. Then click the graph and choose Chart from the main menu > Add Trendline to activate the dialog window shown in Figure 6.3. Under the menu Type, all four functions (linear, logarithmic, expo- nential, and power) are available for selection. Under the menu Options, choose “Display equation on chart” and “Display R-squared value on chart” to have regres- sion results shown on the graph. The Add Trendline tool outputs the regression FIGURE 6.2 Excel dialog window for regression. 2795_C006.fm Page 103 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC 104 Quantitative Methods and Applications in GIS results for the four original bivariate functions without log-transformations, but does not report as many statistics as the Regression tool. The regression results reported here are based on linear OLS regressions by using the log-transform Equations 6.5 and 6.6 (though the computation is done internally). This is different from nonlinear regressions, which will be discussed in the next section. Both the Tanner–Sherratt model (Equation 6.7) and Newling’s model (Equation 6.8) can be estimated by linear OLS regression on their log-transformed forms. See Table 6.1. In the Tanner–Sherratt model, the X variable is distance squared (r 2 ), and in Newling’s model, there are two X variables (r and r 2 ). Newling’s model has one more explanatory variable (r 2 ) than Clark’s model (exponential function), and thus always generates a higher R 2 regardless of the significance of the term r 2 . In this sense, these two models are not comparable in terms of fitting power. Table 6.1 summarizes the models. FIGURE 6.3 Excel dialog window for adding trend lines. TABLE 6.1 Linear Regressions for a Monocentric City Models Function Used in Regression Original Function X Variable(s) Y Variable Restrictions Linear D r = a + br Same rD r None Logarithmic D r = a + blnr Same lnr D r r ≠ 0 Power lnD r = A + blnr D r = ar b lnr lnD r r ≠ 0 and D r ≠ 0 Exponential lnD r = A + br D r = ae br r lnD r D r ≠ 0 Tanner–Sherratt lnD r = A + br 2 r 2 lnD r D r ≠ 0 Newling’s lnD r = A + b 1 r + b 2 r 2 r, r 2 lnD r D r ≠ 0 Dae r br = 2 Dae r br br = + 12 2 2795_C006.fm Page 104 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC Function Fittings by Regressions and Application in Analyzing Density Patterns 105 Fitting the cubic spline function (Equation 6.9) is similar to that of other mono- centric functions, with some extra work in preparing the data. First, sort the data by the variable distance in an ascending order. Second, define the constant x 0 and calculate the terms (x – x 0 ), (x – x 0 ) 2 , and (x – x 0 ) 3 . Third, define the constants x i (i.e., x 1 , x 2 , …) and compute the terms . Take one term, , as an example: (1) set the values = 0 for those records with x ≤ x 1 , and (2) compute the values = for those records with x > x 1 . Finally, run a multivariate regression, where the Y variable is density D x and the X variables are (x – x 0 ), (x – x 0 ) 2 , (x – x 0 ) 3 , , , and so on. The cubic spline function contains multiple X variables, and thus its regression R 2 tends to be higher than other models. 6.3 NONLINEAR AND WEIGHTED REGRESSIONS IN FUNCTION FITTINGS In function fittings for the monocentric structure, two statistical issues deserve more discussion. One is the choice between nonlinear regressions directly on the expo- nential and power functions vs. linear regressions on their log-transformations (as discussed in Section 6.2). Generally they yield different results since the two have different dependent variables (D r in nonlinear regressions vs. lnD r in linear regres- sions) and imply different assumptions of error terms (Greene and Barnbrock, 1978). We use the exponential function (Equation 6.1) and its log-transformation (Equation 6.5) to explain the differences. The linear regression on Equation 6.5 assumes multiplicative errors and weights equal percentage errors equally, such as D r = ae br + ε (6.10) The nonlinear regression on the original function (Equation 6.1) assumes that additive errors and weights all equal absolute errors equally, such as D r = ae br + ε (6.11) The ordinary least squares (OLS) linear regression seeks the optimal values of a and b so that residual sum of squares (RSS) is minimized. See Appendix 6B on how the parameters in a bivariate linear function are estimated by the OLS regression. Nonlinear least squares regression has the same objective of minimizing the RSS. For the model in Equation 6.11, it is to minimize where i indexes individual observations. There are several ways to estimate the parameters in nonlinear regression (Griffith and Amrhein, 1997, p. 265), and all methods use iterations to gradually improve guesses. For example, the modified Gauss–Newton method uses linear approximations to estimate how RSS changes with () * xxZ ii − 3 () * xxZ− 1 3 1 ()xx− 1 3 () * xxZ− 1 3 1 () * xx Z− 2 3 2 RSS D ae i br i i =− ∑ () 2 2795_C006.fm Page 105 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC 106 Quantitative Methods and Applications in GIS small shifts from the present set of parameter estimates. Good initial estimates (i.e., those close to the correct parameter values) are critical in finding a successful non- linear fit. The initialization of parameters is often guided by experience and knowledge of similar studies. Which is a better method for estimating density functions, linear or nonlinear regression? The answer depends on the emphasis and objective of a study. The linear regression is based on the log-transformation. By weighting equal percentage errors equally, the errors generated by high-density observations are scaled down (in terms of percentage). However, the differences between the estimated and observed values in those high-density areas tend to be much greater than those in low-density areas (in terms of absolute value). As a result, the total estimated population in the city can be off by a large margin. On the contrary, the nonlinear regression is to minimize the residual sum of squares directly based on densities instead of their logarithms. By weighting all equal absolute errors equally, the regression limits the errors (in terms of absolute value) contributed by high-density samples. As a result, the total estimated population in the city is often closer to the actual value than the one based on linear regression, but the estimated densities in low-density areas may be off by high percentages. Another issue in estimating urban density functions concerns randomness of sample (Frankena, 1978). A common problem for census data (not only in the U.S.) is that high-density observations are many and are clustered in a small area near the city center, whereas low-density ones are fewer and spread in remote areas. In other words, high-density samples may be overrepresented, as they are concentrated within a short distance from the city center, and low-density samples may be underrepre- sented, as they spread across a wide distance range from the city center. A plot of density vs. distance will show many observations in short distances and fewer in long distances. This is referred to as nonrandomness of sample and causes biased (usually upward) estimators. A weighted regression can be used to mitigate the problem. Frankena (1978) suggests weighting observations in proportion to their areas. In the regression, the objective is to minimize the weighted residual sum of squares (RSS). Note that R 2 in a weighted regression can no longer be interpreted as a measure of goodness of fit and is called pseudo-R 2 . See Wang and Zhou (1999) for an example. Some researchers favor samples with uniform area sizes. In case study 6 (Section 6.5.3 in particular), we will also analyze population density func- tions based on survey townships of approximately same area sizes. Estimating the nonlinear regression or weighted regression requires the use of advanced statistical software. For example, in SAS, if the DATA step uses DEN to represent density, DIST to represent distance, and AREA to represent area size, the following SAS statements implement the nonlinear regression for the exponential Equation 6.1: proc MODEL; /* procedure for nonlinear regression */ PARMS a 1000 b -0.1; /*initialize parameters */ DEN = a * exp(b * DIST); /* code the fitting function */ fit DEN; /* define the dependent variable */ 2795_C006.fm Page 106 Friday, February 3, 2006 12:16 PM © 2006 by Taylor & Francis Group, LLC [...]... =1 4 X Variables 2795_C0 06. fm Page 109 Friday, February 3, 20 06 12: 16 PM Label Function Fittings by Regressions and Application in Analyzing Density Patterns © 20 06 by Taylor & Francis Group, LLC TABLE 6. 2 Polycentric Assumptions and Corresponding Functions 109 2795_C0 06. fm Page 110 Friday, February 3, 20 06 12: 16 PM 110 Quantitative Methods and Applications in GIS 6. 4.2 GIS AND REGRESSION IMPLEMENTATIONS... 7.2850*** 0. 160 9*** 0.193 –0.1108*** 1 1 06 7.3 964 *** 0.1529*** 0. 268 –0. 061 5* 2 401 *** 8.3702 –0.0 464 *** 0.110 –0.01 46 3 76 7.71 96* ** –0.0515** 0.114 –0. 068 9** 4 52 *** 5 .61 73 5 6 71 51 7 8 9 0.3 460 0. 260 0.13 86* ** 7.1939*** 7.25 16* ** –0.0535** –0.0010 0.102 0.000 0.1155*** –0.0487 a = 10.98*** 46 58 7.5583*** 7.0 065 *** –0. 069 4* 0.0100 0.132 0.003 0.1027*** –0.0341 R2 = 0.429 100 7.48 16* ** –0. 065 7*** 0.350... by areal weighting interpolator in ArcGIS: Refer to Section 3 .6, Part 2, for detailed instructions on areal © 20 06 by Taylor & Francis Group, LLC 2795_C0 06. fm Page 117 Friday, February 3, 20 06 12: 16 PM Function Fittings by Regressions and Application in Analyzing Density Patterns 117 weighting interpolator In ArcToolbox, use the analysis tool Intersect to overlay cnty6trt and twnshp, and name the output... density, land use intensity (reflected in its price), land productivity, commodity prices, and wage rate may all experience “distance decay effects” (Wang and Guldmann, 1997), and studies testing the spatial patterns may benefit from the © 20 06 by Taylor & Francis Group, LLC 2795_C0 06. fm Page 120 Friday, February 3, 20 06 12: 16 PM 120 Quantitative Methods and Applications in GIS function-fitting approach... the argument discussed in Section 6. 3 that nonlinear regression tends to value high-density areas more than low-density areas in minimizing the residual sum squared (RSS) In other words, the logarithmic transformation in linear © 20 06 by Taylor & Francis Group, LLC 2795_C0 06. fm Page 119 Friday, February 3, 20 06 12: 16 PM Function Fittings by Regressions and Application in Analyzing Density Patterns 119... −0.33 96 r4 with R2 = 0.39 46 (38 .64 ) (–21.14) © 20 06 by Taylor & Francis Group, LLC (–1.55) (–2.37) 2795_C0 06. fm Page 1 16 Friday, February 3, 20 06 12: 16 PM 1 16 Quantitative Methods and Applications in GIS TABLE 6. 4 Regressions Based on Polycentric Assumptions 1 and 2 (1837 Census Tracts) n 2: ln D = a + ∑b r i i i =1 1: ln D = Ai + biri for Center i’s Proximal Area for the Whole Study Area Center index... level (as shown in Figure 6. 7) than at the tract level (as shown in Figure 6. 6) 6. 6 DISCUSSION AND SUMMARY Based on Table 6. 3 and Table 6. 5, among the six bivariate monocentric functions (linear, logarithmic, power, exponential, Tanner–Sherratt, and Newling’s), the exponential function has the best fit overall It generates the highest R2 by linear regressions using both census tract data and survey township... 771.39e−0. 060 7x 400.00 R2 = 0 .69 52 300.00 200.00 100.00 0.00 0 20 40 60 80 100 120 Distance (km) FIGURE 6. 7 Density vs distance exponential trend line (survey townships) higher R2 than the linear counterpart using both the tract and township data The nonlinear regression also generates a higher intercept than the linear regression (10013 > e8.7953 = 66 03.1 for census tracts and 862 .74 > e6 .65 = 771.4... original functions, and recovered to the exponential and power function forms by computing the coefficient a = e A (e.g., for the exponential function, 9157.5 = e9.1223 , see Figure 6. 6) Results from the trend line tool are the same as those obtained by the regression tool © 20 06 by Taylor & Francis Group, LLC 2795_C0 06. fm Page 114 Friday, February 3, 20 06 12: 16 PM 114 Quantitative Methods and Applications. .. area and the major center and the distance between each area and its nearest center The two distances are obtained by using the Near tool in ArcGIS twice See Section 6. 5.2 for details Based on assumption 1, the polycentric model is degraded to monocentric functions (Equation 6. 12) within each center’s proximal area, which can be estimated by the techniques explained in Sections 6. 2 and 6. 3 Equation 6. 13 . 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. e rr =− −− 9 967 63 67 44 84 0 04 56 0 33 96 14 4 2795_C0 06. fm Page 115 Friday, February 3, 20 06 12: 16 PM © 20 06 by Taylor & Francis Group, LLC 1 16 Quantitative Methods and Applications in GIS The. regression dialog window shown in Figure 6. 2. By defining the appropriate data ranges for X and Y variables, Equations 6. 2, 6. 3, 6. 5, and 6. 6 can be all fitted by OLS linear regressions in Excel. Note