This module contains functions for automatic bandwidth selection to calibrate basic GW regression (bw.gwr), generalised GWR model (bw.ggwr), GW Principal Components Analysis (bw.gwpca), and GW Discriminant Analysis (bw.gwda)
The arguments were taken from Gollini et al.[1]
Argument | bw.gwr | bw.ggwr | bw.gwda | bw.gwpca |
---|---|---|---|---|
Dependient |
✔ | ✔ | ✔ | x |
Independient |
✔ | ✔ | ✔ | x |
Family |
x | ✔ | x | x |
Approach |
✔ | ✔ | x | x |
Kernel |
✔ | ✔ | ✔ | ✔ |
Power |
✔ | ✔ | ✔ | ✔ |
Theta |
✔ | ✔ | ✔ | ✔ |
Longlat |
✔ | ✔ | ✔ | ✔ |
Adaptative |
✔ | ✔ | ✔ | ✔ |
Cov.gw |
x | x | ✔ | x |
Prior.gw |
x | x | ✔ | x |
Mean.gw |
x | x | ✔ | x |
wqda |
x | x | ✔ | x |
Variables |
x | x | x | ✔ |
Robust |
x | x | x | ✔ |
Dependient
: Dependent variable of the regression
model
Independient
: Independent(s) variable(s) of the
regression model.
Family
: a description of the model’s error distribution
and link function, which can be “poisson” or “binomial”.
Approach
: specified by CV for cross-validation approach
or by Akaike Information Criterion corrected (AICc) approach
Kernel
: A set of five commonly used kernel
functions;
Figure 1. Five kernel functions \(w_{ij}\) is the j-th element of the diagonal of the matrix of geographical weights W(\(u_i\),\(v_i\)), and \(d_{ij}\) is the distance between observations i and j, and b is the bandwidth.
Power (Minkowski distance)
: the power of the Minkowski
distance (p=1 is manhattan distance, p=2 is euclidean distance).
Figure 2. Minkowski distance
Theta (Angle in radians)
: an angle in radians to rotate
the coordinate system, default is 0
longlat
: if TRUE, great circle distances will be
calculated
Adaptive
:If TRUE, find an adaptive kernel with a
bandwidth proportional to the number of nearest neighbors (i.e. adaptive
distance); otherwise, find a fixed kernel (bandwidth is a fixed
distance)
Cov.gw
:if TRUE, localised variance-covariance matrix is
used for GW discriminant analysis; otherwise, global variance-covariance
matrix is used
Prior.gw
: if TRUE, localised prior probability is used
for GW discriminant analysis; otherwise, fixed prior probability is
used
Mean.gw
: if true, localised mean is used for GW
discriminant analysis; otherwise, global mean is used
wqda
: if TRUE, a weighted quadratic discriminant
analysis will be applied; otherwise a weighted linear discriminant
analysis will be applied
Variables
: a vector of variable names to be
evaluated
Robust
: if TRUE, robust GWPCA will be applied; otherwise
basic GWPCA will be applied
Returns the adaptive or fixed distance bandwidth.
A critical issue is deciding between two types of spatial kernels: fixed kernels and adaptive kernels. A fixed kernel, by definition, uses a fixed bandwidth to define a region around all regression points. The distance to a given point determines the extent of the kernel, which is identical at any point in space. An adaptive kernel defines a region around points by using varying bandwidth. The number of nearest neighbors from a given regression point determines the kernel’s size. Where the data is sparse, the kernels have larger bandwidths.
Continuous kernels and kernels with compact support are two different types of kernel functions. The uniform, Gaussian, and exponential kernel functions are used to weight all data in the study region. Kernels with compact support are used to give nonzero weight to observations within a specific distance and zero weight to observations beyond that distance.
[1] Gollini, I., Lu, B., Charlton, M., Brunsdon, C., & Harris, P. (2015). GWmodel: An R Package for Exploring Spatial Heterogeneity Using Geographically Weighted Models. Journal of Statistical Software, 63(17), 1–50. https://doi.org/10.18637/jss.v063.i17