Geographically Weighted Principal Components Analysis

Module Description

This module implements basic or robust Geographically Weighted Principal Components Analysis (GWPCA)

Argument

Variables: a vector of variable names to be evaluated

k: the number of retained components; k must be less than the number of variable

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.

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.

Robust: if TRUE, robust GWPCA will be applied; otherwise basic GWPCA will be applied.

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).

longlat : if TRUE, great circle distances will be calculated.

Distance bandwidth: bandwidth used in the weighting function. It has two options, automatic which is calculated in the Bandwidth selection module and manual in which the user enter the value.

Power: the power of the Minkowski distance,default is 2, i.e. the 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

Value

Returns a list of class “gwpca” as Gwmodel Package:

Video

Video 1 : Geographically Weighted Principal Components Analysis

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, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25:1717-1736. https://doi.org/10.1080/13658816.2011.554838

Harris P, Brunsdon C, Charlton M, Juggins S, Clarke A (2014) Multivariate spatial outlier detection using robust geographically weighted methods. Mathematical Geosciences 46(1) 1-31. https://doi.org/10.1007/s11004-013-9491-0

Harris P, Clarke A, Juggins S, Brunsdon C, Charlton M (2015) Enhancements to a geographically weighted 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