1、Function Fittings by Regressions and Application in Analyzing Urban and Regional Density Patterns(Chapter 6, Project 6)OutlineTheories for Urban & Regional Density PatternsEmpirical Models for Monocentric ModelsStudy on BeijingFunction Fittings for Polycentric ModelsStudy on ShenyangCase Study 6: An

2、alyzing Urban Density Patterns in Chicago RegionTheory I: Urban Economic ModelClarks (1951) observation of urban density patternAlonsos (1967) urban land use theory The monocentric model by Mills (1972) & Muth (1969)Max U = U(h,x), s.t. y = phh+pxx+tr(1/ph) ph/r = -t ph=p0e-trFrom land price to popu

3、lation densityBasic concept tradeoff of commute cost vs. living spaceLand Price (ph)rCBDLand Consumption (h)rCBDrCBDPopulation Density (D)Theory II: Gravity-Based Models for Urban DensityWang and Guldmann (1996)Population at a location is proportional to its potential (location advantage)xi = j (xjd

4、ij-b) for all i=1,2, Higher population density near CBD is due to better accessibility shaped by transportation network (e.g., Chicago)Also see the Garin-Lowry Model in Wang (1998)Road Network(Highway) inChicagoRegion Theories for Regional StructureVon Thnens (1826) rural land use theorySpatial equi

5、librium model by Wang and Guldmann (1997)City produces industrial goodsRural area produces agricultural goodsBoth are sold in the cityWage, population density (labor input) & land price vary in distance from the cityUtility is equalized everywhereWage rateDevelopedDevelopingLand rentAgricultural yie

6、ldPopulation densityEmpirical Models (Monocentric)Linear: D = a-brNegative exponential: D = D0e-brlnD = a-brlogarithmic : D = a-b(lnr) Power (log-linear): D = D0r-blnD = a-b(lnr)All are bivariateExtensions to Monocentric:CBD CraterSuburban Bumps2. Newlings model:lnD = a+br-cr23. Cubic Spline Functio

7、n:D = a+b(x-x0)+c(x-x0)2 +d0(x-x0)3+ di(x-xi)3Zi*1. Tanner-Sherratt model: lnD = a-br2 Linear Regressions (Monocentric)ModelsFunction used in regressionOriginal functionX var(s)Y varRestrictionsLinearDr = a+brSamerDrnoneLogarithmic Dr = a+blnrSamelnrDrr0Power lnDr = A+blnr Dr = arblnrlnDrr0 & Dr0Exp

8、onential lnDr = A+brDr = aebrrlnDrDr0Tanner-SherrattlnDr = A+br2r2lnDrDr0NewlingslnDr = A+b1r+b2r2r , r2lnDrDr0Two Statistical IssuesNon-randomness of sampling (more high-density small tracts)Weighted regression Non-linear regression vs. linearized log-transformationD = D0e-br weights all equal abso

9、lute errors equally lnD = a-br weights equal percentage errors equally Both use non-linear regressions in SAS or SPSSBeijing 1982 Density vs. DistanceRegional growth patternsBeijing 1990 Density vs. DistanceRegressions in Beijing (non-weighted)FunctionForms19821990abR2abR2D=a-br37238-22450.48434444-

10、19440.523D=D0e-br70949-0.2290.67153938-0.1680.649D=a-blnr55448-195900.63949529-164150.634D=D0r-b85385-0.8880.63467071-0.7280.593Regressions in Beijing (weighted)Function Forms19821990ababD=a-br22373-121522984-1217D=D0e-br69907-0.26256772-0.212D=a-blnr42883-1501642011-14368D=D0r-b98108-1.17782066-1.0

11、351982-90 Population Change %General Trend: Suburbanization19821990Density (D)Distance (r)Polycentric Models(a) Substitutable influences from centers only the nearest center matters(b) Complementary influences from centers access to all centers is necessary(c) Between (a) & (b)(d) Access to CBD and

12、the nearest subcenter is valued (central-place theory)(a)(b)(d)(c)CBDCBDCBDCBDEmpirical Models (Polycentric)(a) D = D0e-br within each subarea(b) lnD = a+b1r1+b2r2+(c) D = a1exp( b1r1)+ a2exp( b2r2)+(d) lnD = a+ b1rCBD+b2rsubcPolycentric Assumptions and Corresponding Functions LabelAssumptionModel (

13、exponential as an example)X variablesSample Estimation method(A)Only access to the nearest center is neededDistances ri from the nearest center i (1 variable)Areas in a subregion iLinear regression1(B)Access to all centers is necessary (multiplicative effects)Distances from each center (n variables

14、ri)All areasLinear regression(C)Access to all centers is necessary (additive effects)Distances from each center (n variables ri)All areasNonlinear regression(D)Access to CBD and the nearest center is neededDistances from the major and nearest center (2 variables)All areasLinear regression2Polycentri

15、c Structure in ShenyangWang and Meng (1999), based on 1982 & 1990 Population Census DataPoor fitting power of monocentric modelsR2=0.258, 0.140 for negative exponential in 1982 and 1990 respectively Shenyang 1982 Density ContourShenyang 1990 Density ContourRegression (Shenyang): D = aiexp( biri)Mult

16、i-Centers1982 (R2=0.721)1990 (R2=0.678)aibiaibiA27232*(3.54)-0.8749*(-1.96)13553(1.68)-0.1287(-1.75)B54894*(10.11)-0.4236*(-5.54)51421*(9.66)-0.4069*(-4.85)C35944*(4.54)-0.7593*(-2.92)23374*(3.47)-0.5774(-1.70)D25412*(3.11)-0.7156*(-1.97)30245*(3.31)-1.1622*(-2.15)E42758*(6.95)-0.4225*(-3.74)44006*(

17、6.00)-0.6082*(-3.64)Case Study 6 (Part 1): Monocentric Density Functions in Chicago (Tracts)extracting study area and CBD location population density surface modelingcomputing distances between tract centroids and CBD in ArcGISusing the regression tool in Excel to run simple linear regressionsusing

18、the trendline tool in Excel for generating graphs and function fittings nonlinear and weighted regressions in SAS Case Study 6 (Part 2): Polycentric Density Functions in Chicago (Tracts)Computing distances between tract centroids and their nearest centers in ArcGIS test the polycentric assumptions (A) and (D) Computing distances between tract centroids and each center in ArcGIS test the polycentric assumptions (B) and (C). Fitting polycentric functions in SAS Defining job centers by GIS surface modelingSpatial approach to commuting: Job-housing balance ratio ap