Geographically Weighted Regression

1 Kernel Functions Used in GWR

Definitions

• \(d_{ij}\) : 位置 \(i\) 到位置 \(j\) 的距离

• \(w_{ij}\) :

• \(b\) : a parameter describing bandwidth

1.1 Fixed Kernel

• Gaussian Kernel

\[w_{ij} = \exp \left[ - \frac{1}{2} \left( \frac{d_{ij}}{b} \right)^2 \right] \]

• Exponential Kernel

\[w_{ij} = \exp \left( - \frac{|d_{ij}|}{b} \right) \]

• Box-car

\[w_{ij} = \begin{cases} 1, & \text{if} \ |d_{ij}| < b \\ 0, & \text{otherwise} \end{cases} \]

• Bi-square Kernel

\[w_{ij} = \begin{cases} \left[ 1 - (d_{ij}/ b)^2 \right]^2, & \text{if} \ |d_{ij}| < b \\ 0, & \text{otherwise} \end{cases} \]

It provides a continuous, near-Gaussian
weighting function up to distance \(b\) from the regression point and then zero
weights any data point beyond \(b\)

• Tri-cube Kernel

\[w_{ij} = \begin{cases} \left[ 1 - (|d_{ij}| / b)^3 \right]^3, & \text{if} \ |d_{ij}| < b \\ 0, & \text{otherwise} \end{cases} \]

1.2 Adaptive Kernel

• Fixed Kernel: a fixed bandwidth \(d_{ij}\) for all local samples

• Adaptive Kernel: a varing bandwidth \(d_{ij}\) with local samples

  • ensure sufficient (and constant) local information for each local calibration of a given GW model

\[w_{ij} = \begin{cases} f(d_{ij}), & \text{if} \ |d_{ij}| < b_{[N]}^{(i)} \\ 0, & \text{otherwise} \end{cases} \]

• where \(b_{[N]}^{(i)}\) is the distance to the \(N\)th nearest neighbours of loction \(i\)