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( ui , vi ) .. 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
0(u ,v )
i
0( u ,v )
i 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 win 0 0 ..
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
# # # ## ### # #
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Density surface of victim-based homicides
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
a Coe fficients
Model 1
(Constant) XNALF
Unstandardized Coefficients B Std. Error 16,064 1,997 4,566 ,292
Standardi zed Coefficien ts Beta ,598
t 8,043 15,657
Sig. ,000 ,000
a. Dependent Variable: TAXA_HOMIC
OLS regression results for TX_homic and X_analf
Linear Regression
15 0,00
10 0,00
50 ,0 0
HOMIC = 16.06 + 4.57 * xna lf TAXA_ 0.3 6 R-Square =
TAXA_HOMIC
0,00 0,00 000 5,00 000 10 ,0 000 0 15 ,0 000 0 20 ,0 000 0
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