Bandwidth Selection

Module Description

This module contain functions for automatic bandwidth selection to calibrate basic GW regresion (bw.gwr) , generalised GWR model (bw.ggwr), GW Principal Components Analysis (bw.gwpca), and GW Discriminant Analysis (bw.gwda)

Argument

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

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

Value

Returns the adaptive or fixed distance bandwidth.

Video

Video 1 : Bandwidth Selection

References

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

Harris P, Clarke A, Juggins S, Brunsdon C, Charlton M (2015) Enhancements to a geographicallyweighted principal components analysis in the context of an application to an environmental data set. Geographical Analysis 47: 146-172. https://doi.org/10.1111/gean.12048