Relationship and prediction
Linear regression
\[y_{i}=\alpha+\beta _{1}\ x_{i1}+\beta _{2}\ x_{i2}+\cdots +\beta _{p}\ x_{ip}+\varepsilon _{i}\]
\[y_{i}=\alpha+\beta _{1}\ x_{i1}+\beta _{2}\ x_{i2}+\cdots +\beta _{p}\ x_{ip}+\varepsilon _{i}\]
\(\alpha\) = constant
\(x_{1}\) = value of a variable \(1\)
\(x_{2}\) = value of a variable \(2\)
\(\beta\) = regression coefficient
\(\varepsilon\) = error
Prediction
Relationship or effect
Assumptions
Linearity
Dependent variable is assumed to be a linear combination of independent variables
Weak exogeneity
Independent variables are assumed to contain no error
Constant variance
Variance of errors does not depend on the value of the dependent variable
Independence of errors
Residuals are not correlated with independent variables
No perfect multicollinearity
Independent variables are not perfectly multicollinear
Evaluation
\(R^2\)
Coefficient of determination
Variance of dependent that can be explained by independent
0…1
\(\beta\)
Regression coefficient
Captures the relationship between the dependent variable and independent variable
Each \(\beta\) has a \(p\) value
Residuals
\(\varepsilon\)
The error
Deviation of the prediction from the observation
Spatial evaluation
Spatial dimension of residuals
Spatial autocorrelation means a problem
Dependent variable is spatially heterogeneous
The relationship between \(\alpha\), dependent and independent variables varies across space
Spatial heterogeneity
Spatially dependent \(\alpha\)
Spatially dependent \(\beta\)
Spatially dependent \(\alpha\)
Spatial fixed effects
Spatially explicit (categorical) variable accounting for regional differences
Spatially dependent \(\beta\)
Spatial regimes
All \(\alpha\) and \(\beta\) coefficients can vary geographically
Spatially explicit (categorical) variable accounting for regional differences
Spatial dependence
configuration matters
Inlcusion of spatially lagged variables in the model
Lag of independent variables
Lag of dependent variable (violates the OLS assumptions)
Lag of error (violates the OLS assumptions)
Geographically weighted regression
single model across space is too restrictive
GWR
A system of smaller, geographically delimited regressions
Accounts for both spatial heterogeneity and spatial dependence
Spatial kernel function
Kernel type
Kernel size
#### Kernel shape
Evaluation
\(R^2\)
Coefficient of determination
Variance of dependent that can be explained by independent
0…1
Global metric
Local metric per geometry
\(\beta\)
Regression coefficient
Captures the relationship between the dependent variable and independent variable
Estimation of \(\beta\) per geometry
Significance based on \(t\) value and a critical \(t\) value
Residuals
\(\varepsilon\)
The error
Deviation of the prediction from the observation
Multiscale geographically weighted regression
Extension where each variable can have its own kernel bandwidth
Limits
(M)GWR is computationally expensive and does not scale well
Use with caution
import statsmodels
import mgwr
