Spatial Data Analysis of Areas Regression

Spatial Data Analysis of Areas: Regression Introduction  Basic Idea  Dependent variable (Y) determined by independent variables X1,X2 (e.g., Y = mX + b).  Uses of regression:    Description Control Prediction Simple Linear Regression Yi=0+1Xi +i basic model Yi value of dependent variable on trial i 0, 1 (unknown parameters) Xi value of independent variable on trial i i ith error term (unexplained variation), where E [i]=0,  2(i)=  2 error terms are N(0, 2) Multiple Regression Yi   0  1 X i1   2 X i 2       p X ip   i Basic Model • Yi is the ith observation of the dependent variable •  0 ,......,  p are parameters • X i1,........, X ip are observations of the ind variables •  i are independent and normal (0,  2 ) ˆ Y  b0  b1 X i1  b2 X i 2      b p X ip ˆ  i  Yi  Yi ith residual estimated model Sometimes we need to transform the data Scatter plots: (a) Y versus PORC3_NR (percentage of large farms in number ); (b) log10 Y versus log 10 (PORC3_NR). Predicted versus Observed Plots: (a) model with variables not transformed): R2 = 0.61; (b) Model 7: R2 = 0.85. Precision of estimates and fit Analysis of variation Sum of squares of Y = residuals 2 Sum of squares of estimate + Sum of squares of ˆ  Y)2  (Y  Y )2 ˆ (Yi  Y)  (Yi   i i  Dividing both sides by TSS (sum of squares of Y): 1 = ESS/TSS + RSS/TSS where ESS/TSS = r2 (coefficient of determination)   r2 gives the proportion of total variation “explained” by the sample regression equation. The closer is r2 to 1.00, the better the fit. Analysis of Residuals   It is a good idea to plot the residuals against the independent variables to see if they show a trend. Possible behaviors: (e.g., the higher the independent variable, the higher the residual) Nonlinearity Heteroskedacity (i.e., the variance of the residual increases or decreases with the independent variable). Correlation  Regression assumes that residuals are constant variance and normally distributed. Good Residual Plot 6 4 2 Y 0 -2 0 20 40 60 -4 -6 X Nonlinearity 0.25 0.2 0.15 0.1 residual 0.05 0 -0.05 0 -0.1 -0.15 20 40 60 X Heteroskedacity 1 0.5 residual 0 -0.5 -1 0 20 X 40 60 Regression with Spatial Data: Understanding Deforestation in Amazonia The forest... The rains... The rivers... Deforestation... Fire... Fire... Amazon Deforestation 2003 Deforestation 2002/2003 Deforestation until 2002 Fonte: INPE PRODES Digital, 2004. What Drives Tropical Deforestation? % of the cases  5% 10% 50% Underlying Factors driving proximate causes Causative interlinkages at proximate/underlying levels Internal drivers *If less than 5%of cases, not depicted here. source:Geist &Lambin 1973 Courtesy: INPE/OBT 1991 Courtesy: INPE/OBT 1999 Deforestation in Amazonia PRODES (Total 1997) = 532.086 km2 PRODES (Total 2001) = 607.957 km2 Modelling Tropical Deforestation •Análise de tendências •Modelos econômicos Coarse: 100 km x 100 km grid Fine: 25 km x 25 km grid Amazônia in 2015? fonte: Aguiar et al., 2004 Factors Affecting Deforestation Category Demographic Variables Population Density Proportion of urban population Proportion of migrant population (before 1991, from 1991 to 1996) Number of tractors per number of farms Percentage of farms with technical assistance Technology Agrarian strutucture Percentage of small, medium and large properties in terms of area Percentage of small, medium and large properties in terms of number Infra-structure Distance to paved and non-paved roads Distance to urban centers Distance to ports Economy Distance to wood extraction poles Distance to mining activities in operation (*) Connection index to national markets Percentage cover of protected areas (National Forests, Reserves, Political Presence of INCRA settlements Number of families settled (*) Environmental Soils (classes of fertility, texture, slope) Climatic (avarage precipitation, temperature*, relative umidity*) Coarse resolution: candidate models MODEL 7: Variables PORC3_AR LOG_DENS PRECIPIT LOG_NR1 DIST_EST LOG2_FER PORC1_UC R² = .86 Description Percentage of large farms, in terms of area Population density (log 10) Avarege precipitation Percentage of small farms, in terms of number (log 10) Distance to roads Percentage of medium fertility soil (log 10) Percantage of Indigenous land stb 0,27 0,38 -0,32 0,29 -0,10 -0,06 -0,06 p-level 0,00 0,00 0,00 0,00 0,00 0,01 MODEL 4: Variables 0,01 CONEX_ME LOG_DENS LOG_NR1 PORC1_AR LOG_MIG2 LOG2_FER R² = .83 Description Connectivity to national markets index Population density (log 10) Percentage of small farms, in terms of number (log 10) Percentage of small farms, in terms of area Percentage of migrant population from 91 to 96 (log 10) Percentage of medium fertility soil (log 10) stb 0,26 0,41 0,38 -0,37 0,12 -0,06 p-level 0,00 0,00 0,00 0,00 0,00 0,01 Coarse resolution: Hot-spots map Terra do Meio South of Amazonas State Hot-spots map for Model 7: (lighter cells have regression residual < -0.4) Modelling Deforestation in Amazonia  High coefficients of multiple determination were obtained on all models built (R2 from 0.80 to 0.86). The main factors identified were:   Population density;  Connection to national markets;  Climatic conditions;  Indicators related to land distribution between large and small farmers.  The main current agricultural frontier areas, in Pará and Amazonas States, where intense deforestation processes are taking place now were correctly identified as hot-spots of change. Spatial regression models Spatial regression     Specifying the Structure of Spatial dependence  which locations/observations interact what type of dependence, what is the alternative spatial lag, spatial error, higher order interpolation, missing values Testing for the Presence of Spatial Dependence  Estimating Models with Spatial Dependence  Spatial Prediction  source: Luc Anselin Nonspatial regression  Objective  Predict the behaviour of a response variable, given a set of known factors (explanatory variables).  Multivariate nonspatial models yk = 0 + 1x1k +… + ixik + i     yk i xi k = estimate of response variable for object k = regression coefficient for factor i = explanatory variable i for region k = random error  Adjustment quality R2 = 1  S (y – y ) – S (y – y ) n i=1 n i=1 i i i i 2 2 Nonspatial regression: hypotheses  Y = X +  (model)      Explanatory variables are linearly independent Y - vector of samples of response variable (n x 1) X – matrix of explanatory variables (n x k)  - coefficient vector (k x 1)  - error vector (n x 1)   E(i ) = 0 ( expected value) i ~ N( 0, i2 ) (normal distribution) Generalized linear models  g(Y) = X + U    Response is some function of the explanatory variables g(.) is a link function Ex: logarithm function U = error vector     (U) = 0 (expected value) (UUT ) = C (covariance matrix) if C= 2 I, the error is homoskedastic Spatial regression  Spatial effects    What happens if the original data is spatially autocorrelated? The results will be influenced, showing statistical associated where there is none  How can we evaluate the spatial effects?  Measure the spatial autocorrelation (Moran’s I) of the regression residuals Regression using spatial data Try a linear model first  yi  x    i t i  Adjust the model and calculate residuals ˆ ri  yi  yi  Are the residuals spatially autocorrelated?   No, we’re OK Yes, nonspatial model will be biased and we should propose a spatial model Spatial dependence  Estimating the Form/Extent of Spatial Interaction   substantive spatial dependence spatial lag models  Correcting for the Effect of Spatial Spill-overs   spatial dependence as a nuisance spatial error models source: Luc Anselin Spatial dependence  Substantive Spatial Dependence    lag dependence include Wy as explanatory variable in regression y = ρWy + Xβ + ε  Dependence as a Nuisance     error dependence non-spherical error variance E[εε’] = Ω where Ω incorporates dependence structure Interpretation of spatial lag  True Contagion    related to economic-behavioral process only meaningful if areal units appropriate (ecological fallacy) interesting economic interpretation (substantive)  Apparent Contagion  scale problem, spatial filtering source: Luc Anselin Interpretation of Spatial Error  Spill-Over in “Ignored” Variables    poor match process with unit of observation or level of aggregation apparent contagion: regional structural change economic interpretation less interesting nuisance parameter  Common in Empirical Practice source: Luc Anselin Cost of ignoring spatial dependence  Ignoring Spatial Lag   omitted variable problem OLS estimates biased and inconsistent  Ignoring Spatial Error    efficiency problem OLS still unbiased, but inefficient OLS standard errors and t-tests biased source: Luc Anselin Spatial regression models   Incorporate spatial dependency Spatial lag model   t yi     wij y j   xi    i    j   Two explanatory terms   One is the variable at the neighborhood Second is the other variables Spatial regimes  Extension of the non-spatial regression model Considers “clusters” of areas Groups each “cluster” in a different explanatory variable yi = 0 + 1x1 +… + ixi + i Gets different parameters for each “cluster”    A study of the spatially varying relationship between homicide rates and socio-economic data of São Paulo using GWR Frederico Roman Ramos CEDEST/Brasil Geographically Weighted Regression Extensão of traditional regression model where the parameters are estimaded locally (ui,vi) are the geographical coordinates of point i. The betas vary in space (each location has a different coeficient) We estimate an ordinary regression for each point where the neighbours have more weight yi  0 (ui , vi )  k k (ui , vi ) xik  i   0(ui ,vi )  0( u ,v )  i i  ..    0(ui ,vi )    0(u ,v )  0(u ,v ) i i i i  0(u ,v ) ..  0(u ,v )   0(u ,v ) ..  0(u ,v )   i i i i i i .. .. i .. i  0(u ,v ) i  0(u ,v ) i   ..  0(ui ,vi )   .. i i   (i)  ( X TW (i) X )1 X TW (i)Y  wi1 0 W (i)    ..  0  0 wi 2 .. 0 0 .. 0   .. ..   0 win   .. Introducing São Paulo Some numbers: Metropolitan region: Population: 17,878,703 (ibge,200) 39 municipalities 70 Km 30 Km Municipality of São Paulo: Population: 10,434,252 HDI_M: 0.841 (pnud, 2000) 96 districts IEX: 74 out of 96 districts were classified as socially excluded (cedest,2002) 4,637 homicide victims in 2001 Data # # # ### # # # # # # ## #### # # ## # # # # # # # # ## # # # # # # ## ## ## # # # # # # ## # # # ## # # # ## # # ## # # # # # # # # ## 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# # # # # #### # # ## # # # # ## # # # # # # # # # # # # # # # #### # # # # # # # # ## # # ## # ### # # ## ## ## ## # ## # # # # ## ## # # # # # # # ## # # # # ### # # # # # # # # # # # # ## # # # ## # # # # # # # ### # ## # # # # # ## # # ## # # ## ## # # # # # ## # ### # # # # # # # # # # # # ## # ## ## # # ## # ## # # # # # # # # ## ## # # # # # # # ## # # # ### # ## # # # ## # ## # # # # # # # ## ## # # # # ## # ## # # ## # ## ## # # # # # ### # ## # ## ## # # # #### # ## ### # # # ## # ## # # ## # # # # # # # # # # # # # # # # ## # # ## # # ### ### ## # ### # ## # # ## # ### # # # # # # ## # # # # # # ## # # # ## # # # # ## # # # # # ### # # ## # # # # # ## # # # ## # # ## # ## # # # # # # 4,637 homicide victims residence geoadressed 2001 456 Census Sample Tracts 2000 ## # # # # ## ### # # # # # # Density surface of victim-based homicides Critical areas # # # ### # ## # # # # ## # # # # # # # ## # # # Critical areas # ## # # # ## # # # ## # # ## # # # # # ## # # # # # # # ## ## # # # # # # # ### # # ## 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# # ## # ###### # # ## # # # # # ## ## ### # # ### # #### # # # # # # # ### # ## # # # ## # ## # # ##### # # # # # # # # # ## # # ## # # # # # # # ### # ### # # ## # # # # # # ## # ## ## # # ## # ## # # ## ### # # # # # # # # ## ## # # # ### # # # ## # # # ## # # # # ## # # # # # # # # ## ## # # # ## # ### ## # # ## # # # # # # ## # # # # # # ## # Kernel Density Function Bandwidth = 3 Km Critical areas # # # # # ## # # # # # Victim-based homicide rate (Tx_homic) Tx_homic = count homicide events (2001) *100.000 population (census, 2000) 70 60 50 40 30 20 10 0 0, 00 16 ,1 0 32 ,2 0 48 ,3 0 64 ,4 0 80 ,5 0 96 ,6 1 11 2, 7 12 1 8, 8 14 1 4, 9 16 1 1, 01 Tx_homic LISA Victim-based homicide rate Percentage of illiterate house-head (Xanlf) Definition House-head is the person responsible for the house. Generally, but not necessarily, who has the highest income of the house 60 50 40 30 20 10 0 0, 04 1, 89 3, 73 5, 57 7, 41 9, 25 11 ,0 9 12 ,9 3 14 ,7 7 16 ,6 1 18 ,4 6 LISA Percentage of illiterate house-head OLS regression results for TX_homic and X_analf b Model Summary Model 1 R R Square a ,598 ,357 Adjusted R Square ,356 Std. Error of the Estimate 22,5033 a. Predictors: (Constant), XNALF b. Dependent Variable: TAXA_HOMIC ANOVAb Sum of Squares 124145,0 223321,9 347466,9 Model 1 df 1 441 442 Regression Residual Total Mean Square 124144,979 506,399 F 245,153 Sig. ,000 a a. Predictors: (Constant), XNALF b. Dependent Variable: TAXA_HOMIC Coefficientsa Standardi zed Coefficien ts Beta ,598 Model 1 (Constant) XNALF Unstandardized Coefficients B Std. Error 16,064 1,997 4,566 ,292 t 8,043 15,657 Sig. ,000 ,000 a. Dependent Variable: TAXA_HOMIC OLS regression results for TX_homic and X_analf   Linear Regression 150,00        100,00 50,00              XA TA _HOMIC = 16. 06 +     R-Square = 0.36                                                                                                                                                                                                                                                                                                                                        TAXA_HOMIC  4. 57 * xn al f     0,00 0,00000 5,00000 10,00000 15,00000 20,00000 xnalf LISA for standardized residuals of the OLS regression for TX_homic and X_analf View1 Moran=0,2624 Area_po.shp < -3 Std. Dev. -3 - -2 Std. Dev. -2 - -1 Std. Dev. -1 - 0 Std. Dev. Mean 0 - 1 Std. Dev. 1 - 2 Std. Dev. 2 - 3 Std. Dev. > 3 Std. Dev. 5 0 5 10 15 Kilometers GWR regression results for TX_homic and Xanlf ********************************************************** * GWR ESTIMATION * ********************************************************** Fitting Geographically Weighted Regression Model... Number of observations............ 456 Number of independent variables... 2 (Intercept is variable 1) Bandwidth (in data units)......... 0.0246524516 Number of locations to fit model.. 456 Diagnostic information... Residual sum of squares........ 111179.875 Effective number of parameters.. 83.1309998 Sigma.......................... 17.2677182 Akaike Information Criterion... 4007.32139 Coefficient of Determination... 0.699720224 GWR regression results for TX_homic and Xanlf residuals Moran= -0,0303 GWR regression results for TX_homic and Xanlf Local Beta1 Local t-value Area_po.shp -6.396 - -1.855 -1.855 - 0 0 - 3.532 3.532 - 5.843 5.843 - 15.765 5 0 5 10 Kilometers CONCLUSIONS -There are significant differences in the relationship between violence rates and social territorial data over the intra-urban area of São Paulo -This results reinforces our hypotheses that we should avoid using general concepts -The GWR technique is a useful instrument in social territorial analysis