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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
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)
basic model
iippiii XXXY 22110
• Yi is the ith observation of the dependent variable
• are parameters
• are observations of the ind variables
• are independent and normal
p ,......,0
ipi XX ,........,1
i ),0( 2
iii
ippii
YY
XbXbXbbY
ˆ
ˆ22110
Multiple Regression
BasicModel
estimated model
ith residual
Sometimes we need to transform the data
Predicted versus Observed Plots: (a) model with variables not transformed): R2 = 0.61; (b) Model 7: R2 = 0.85.
Scatter plots: (a) Y versus PORC3_NR (percentage of large farms in number ); (b) log10 Y versus log 10 (PORC3_NR).
Precision of estimates and fitAnalysis of variationSum of squares of Y = Sum of squares of estimate + Sum of
squares of residuals
2ii
2i
2
i)Y(Y)YY()Y(Y ˆˆ
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:Correlation (e.g., the higher the independent variable,
the higher the residual)NonlinearityHeteroskedacity (i.e., the variance of the residual
increases or decreases with the independent variable). Regression assumes that residuals are constant
variance and normally distributed.
Amazon Deforestation 2003Amazon Deforestation 2003
Fonte: INPE PRODES Digital, 2004.Fonte: INPE PRODES Digital, 2004.
Deforestation 2002/2003Deforestation 2002/2003
Deforestation until 2002Deforestation until 2002
What Drives Tropical Deforestation?
Underlying Factorsdriving proximate causes
Causative interlinkages atproximate/underlying levels
Internal drivers
*If less than 5%of cases,not depicted here.
source:Geist &Lambin
5% 10% 50%
% of the cases
Modelling Tropical Deforestation
Fine: 25 km x 25 km grid
Coarse: 100 km x 100 km grid
•Análise de tendências•Modelos econômicos
Factors Affecting Deforestation
Category VariablesDemographic Population Density
Proportion of urban populationProportion of migrant population (before 1991, from 1991 to 1996)
Technology Number of tractors per number of farmsPercentage of farms with technical assistance
Agrarian strutucture Percentage of small, medium and large properties in terms of areaPercentage of small, medium and large properties in terms of number
Infra-structure Distance to paved and non-paved roadsDistance to urban centersDistance to ports
Economy Distance to wood extraction polesDistance to mining activities in operation (*)Connection index to national markets
Political Percentage cover of protected areas (National Forests, Reserves, Presence of INCRA settlementsNumber of families settled (*)
Environmental Soils (classes of fertility, texture, slope)Climatic (avarage precipitation, temperature*, relative umidity*)
Coarse resolution: candidate models
MODEL 7: R² = .86Variables Description stb p-level
PORC3_ARPercentage of large farms, in terms of area 0,27 0,00
LOG_DENS Population density (log 10) 0,38 0,00
PRECIPIT Avarege precipitation -0,32 0,00
LOG_NR1Percentage of small farms, in terms of number (log 10) 0,29 0,00
DIST_EST Distance to roads -0,10 0,00
LOG2_FER Percentage of medium fertility soil (log 10) -0,06 0,01
PORC1_UC Percantage of Indigenous land -0,06 0,01
MODEL 4: R² = .83Variables Description stb p-level
CONEX_ME Connectivity to national markets index 0,26 0,00
LOG_DENS Population density (log 10) 0,41 0,00
LOG_NR1Percentage of small farms, in terms of number (log 10) 0,38 0,00
PORC1_ARPercentage of small farms, in terms of area -0,37 0,00
LOG_MIG2Percentage of migrant population from 91 to 96 (log 10) 0,12 0,00
LOG2_FER Percentage of medium fertility soil (log 10) -0,06 0,01
Coarse resolution: Hot-spots map
Terra do Meio, Pará State
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
Specifying the Structure of Spatial dependence which locations/observations interact
Testing for the Presence of Spatial Dependence what type of dependence, what is the alternative
Estimating Models with Spatial Dependence spatial lag, spatial error, higher order
Spatial Prediction interpolation, missing values
source: Luc Anselin
Nonspatial regression
Objective Predict the behaviour of a response variable, given a
set of known factors (explanatory variables).
Multivariate nonspatial modelsyk = 0 + 1x1k +… + ixik + i yk = estimate of response variable for object k i = regression coefficient for factor i xi = explanatory variable i for region k k = random error
Adjustment quality
R2 = 1 –(yi – yi)
i = 1
n
(yi – yi)i = 1
n 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, i
2 ) (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
Adjust the model and calculate residuals
Are the residuals spatially autocorrelated? No, we’re OK Yes, nonspatial model will be biased and we should propose
a spatial model
itii xy
iii yyr ˆ
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
Two explanatory terms One is the variable at the neighborhood Second is the other variables
iti
jjiji xywy
Spatial regimes
Extension of the non-spatial regression model
Considers “clusters” of areas
Groups each “cluster” in a different explanatory variableyi = 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 RamosCEDEST/Brasil
0 ( , ) ( , )i i i k i i ik iky u v u v x
0( , ) 0( , ) 0( , ) 0( , )
0( , ) 0( , ) 0( , ) 0( , )
0( , ) 0( , ) 0( , ) 0( , )
..
..
.. .. .. .. ..
..
i i i i i i i i
i i i i i i i i
i i i i i i i i
u v u v u v u v
u v u v u v u v
u v u v u v u v
1( ) ( ( ) ) ( )T Ti i iX W X X W Y
1
2
0 .. 0
0 .. 0( )
.. .. .. ..
0 0 0
i
i
in
w
wi
w
W
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
Geographically Weighted Regression
Introducing São Paulo
30 Km
70 Km
Some numbers:
Metropolitan region:
Population: 17,878,703 (ibge,200)
39 municipalities
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 victimsresidence geoadressed2001
456 Census Sample Tracts2000
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Density surface of victim-based homicides
Kernel Density Function
Bandwidth = 3 Km
Critical areas Critical areas
Critical areas
Victim-based homicide rate (Tx_homic)
0
10
20
30
40
50
60
70
Tx_homic
Tx_homic = count homicide events (2001) *100.000 population (census, 2000)
Percentage of illiterate house-head (Xanlf)
0
10
20
30
40
50
60
DefinitionHouse-head is the person responsible for the house. Generally, but not necessarily, who has the highest income of the house
OLS regression results for TX_homic and X_analf
Model Summaryb
,598a ,357 ,356 22,5033Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), XNALFa.
Dependent Variable: TAXA_HOMICb.
ANOVAb
124145,0 1 124144,979 245,153 ,000a
223321,9 441 506,399
347466,9 442
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), XNALFa.
Dependent Variable: TAXA_HOMICb.
Coefficientsa
16,064 1,997 8,043 ,000
4,566 ,292 ,598 15,657 ,000
(Constant)
XNALF
Model1
B Std. Error
UnstandardizedCoefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: TAXA_HOMICa.
OLS regression results for TX_homic and X_analf
Linear Regression
0,00000 5,00000 10,00000 15,00000 20,00000
xnalf
0,00
50,00
100,00
150,00
TAX
A_H
OM
IC
TAXA_HOMIC = 16.06 + 4.57 * xnalfR-Square = 0.36
5 0 5 10 15 Kilometers
Area_ po.shp< -3 Std. Dev.-3 - -2 Std. Dev.-2 - -1 Std. Dev.-1 - 0 Std. Dev.Mean0 - 1 Std. Dev.1 - 2 Std. Dev.2 - 3 Std. Dev.> 3 Std. Dev.
View1
Moran=0,2624
LISA for standardized residuals of the OLS regression for TX_homic and X_analf
*********************************************************** 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
5 0 5 10 Kilometers
Area_ po.shp-6.396 - -1.855-1.855 - 00 - 3.5323.532 - 5.8435.843 - 15.765
GWR regression results for TX_homic and Xanlf
Local Beta1 Local t-value
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