PHC6194 SPATIAL EPIDEMIOLOGY

Geographically Weighted Regression

Yi Zheng, MPH

Department of Epidemiology

College of Public Health and Health Professions & College of Medicine

March 27, 2019

Geographically Weighted Regression

 

Lab: GWR

Geographically Weighted Regression

Traditional Regression Models

  • If we apply a traditional regression model to spatial data, we usually assume a stationary process
    -  the same exposure has the same impact on the outcome in all parts of the study region


     
  • The estimated coefficients are constant over space
    -  assume that the values of β are the same everywhere
y_i=\beta_0+\beta_1x_{1i}+...+\beta_nx_{ni}+\varepsilon_i

Measured Associations Might Vary Spatially

  • Sampling variation
     
  • Associations intrinsically different across space
    -  e.g. differences in attitudes, different political or other contextual effects
     
  • Model misspecification

Geographically Weighted Regression

  • To address the issue of spatial non-stationary directly and allow the associations to vary over space



     
  • We can then estimate the values of β by:


     
  • W(i) is a weight matrix specific to location i such that observations nearer to i are given greater weight than observations further away
y_i=\beta_{0(i)}+\beta_{1(i)}x_{1i}+...+\beta_{n(i)}x_{ni}+\varepsilon_i
\beta_{(i)}=(X^TW_{(i)}X)^{-1}X^TW_{(i)}Y

A Typical Spatial Weight Function

Spatial Weight Functions

  • Numerous weight functions can be used
    -  usually use Gaussian or "Gaussian-like" functions to reflect the type of dependency found in most spatial processes
    -  can be either fixed or adaptive

Fixed

Adaptive

W_{ij}=e^{- {{(d_{ij}/h)^2} \over 2}}

h is the bandwidth, as it increases, the gradient of the kernel becomes less steep and more data points are incuded

W_{ij}= \{
[1-(d_{ij}^2/h^2)]^2
0

if j is one of the Nth nearest neighbors of i

otherwise

Calibration

  • Results of GWR appear to be relatively insensitive to the choice of weighting functions
    -  as long as it is a continuous distance-based function
     
  • However, the results will be sensitive to the degree of distance-decay
     
  • Therefore, an optimal value of either h or N has to be obtained
    -  through minimizing a cross-validated score or the AIC

Bandwidth Selection

  • Optimal bandwidth selection is a trade-off between bias and variance
    -  too small a bandwidth leads to large variance in the local estimates
    -  too large a bandwidth leads to large bias in the local estimates

Output from GWR

  • Main output from GWR is a set of location-specific coefficient estimates which can be mapped and analyzed to provide information on spatial non-stationary in associations
     
  • We  can also use GWR to
    -  estimate local standard errors
    -  derive local t statistics
    -  calculate local goodness-of-fit measures
    -  perform tests to assess the significance of the spatial variation in the local parameter estimates
    -  perform tests to determine if the local model performs better than the global one

Lab: GWR

git pull

PHC6194-Spring2019-Lecture10

By Hui Hu

PHC6194-Spring2019-Lecture10

Lecture 10, Spring 2019, PHC6194 Spatial Epidemiology

  • 425
Loading comments...

More from Hui Hu