Spatial Data Analysis of Areas: Regression

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Spatial Data Analysis of Areas Regression
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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

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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 ?i0,
• ? 2(?i) ? 2
• error terms are N(0, ?2)

basic model
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Multiple Regression
Basic Model

• Yi is the ith observation of the dependent
  variable
• are parameters
• are observations of the
  ind variables
• are independent and normal

estimated model
ith residual
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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.
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Precision of estimates and fit

• Analysis of variation
• Sum of squares of Y Sum of squares of
  estimate Sum of squares of residuals

• 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.

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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)
• Nonlinearity
• Heteroskedacity (i.e., the variance of the
  residual increases or decreases with the
  independent variable).
• Regression assumes that residuals are constant
  variance and normally distributed.

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Good Residual Plot
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Nonlinearity
0.25
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0.15
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residual
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-0.1
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Heteroskedacity
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residual
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Regression with Spatial Data Understanding
Deforestation in Amazonia
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The forest...
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(No Transcript)
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The rains...
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The rivers...
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Deforestation...
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Fire...
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Fire...
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Amazon Deforestation 2003
Deforestation 2002/2003
Deforestation until 2002
Fonte INPE PRODES Digital, 2004.
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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 5of cases, not depicted here.
sourceGeist Lambin
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1 9 7 3
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1 9 9 1
Courtesy INPE/OBT
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1 9 9 9
Courtesy INPE/OBT
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Deforestation in Amazonia
PRODES (Total 1997) 532.086 km2 PRODES (Total
2001) 607.957 km2
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Modelling Tropical Deforestation
Coarse 100 km x 100 km grid
Fine 25 km x 25 km grid
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Amazônia in 2015?
fonte Aguiar et al., 2004
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Factors Affecting Deforestation
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Coarse resolution candidate models
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Coarse resolution Hot-spots map
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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. 

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Spatial regression models
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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
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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 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

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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)

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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

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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 (Morans I)
  of the regression residuals

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Regression using spatial data

• Try a linear model first
• Adjust the model and calculate residuals
• Are the residuals spatially autocorrelated?
• No, were OK
• Yes, nonspatial model will be biased and we
  should propose a spatial model

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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
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Spatial dependence

• Substantive Spatial Dependence
• lag dependence
• include Wy as explanatory variable in regression
• y ?Wy Xß e
• Dependence as a Nuisance
• error dependence
• non-spherical error variance
• Eee O
• where O incorporates dependence structure

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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
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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
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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
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Spatial regression models

• Incorporate spatial dependency
• Spatial lag model
• Two explanatory terms
• One is the variable at the neighborhood
• Second is the other variables

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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

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A study of the spatially varying relationship
between homicide rates and socio-economic data of
São Paulo using GWR
Frederico Roman Ramos CEDEST/Brasil
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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

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Introducing São Paulo
Some numbers Metropolitan region Population
17,878,703 (ibge,200) 39 municipalities Municipali
ty 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
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Data
4,637 homicide victims residence geoadressed 2001
456 Census Sample Tracts 2000
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Density surface of victim-based homicides
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Victim-based homicide rate (Tx_homic)
Tx_homic count homicide events (2001)
100.000 population (census, 2000)
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LISA Victim-based homicide rate
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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
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LISA Percentage of illiterate house-head
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OLS regression results for TX_homic and X_analf
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OLS regression results for TX_homic and X_analf
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LISA for standardized residuals of the OLS
regression for TX_homic and X_analf
Moran0,2624
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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
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GWR regression results for TX_homic and Xanlf
residuals
Moran -0,0303
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GWR regression results for TX_homic and Xanlf
Local Beta1
Local t-value
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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