Geographically Weighted Regression

Module Description

This module contain functions for basic and Robust GWR, generalised GWR , Heterocedastic GWR, Mixed GWR, Scalable GWR, and Local collinearity diagnostics for basic GWR

Argument

Argument Basic Robust Generalized Heterocedastic Mixed Scalable
Dependient
Independient
Family x x x x x
Cv x x x x
Kernel
Power
Theta
Longlat
Adaptive x
Distance bandwidth x
Max iter x x x x
Fixed x x x x x
Intercep fixed x x x x x
Diagnostic x x x x x
F123 x x x x x
Filtered x x x x x
bw.adapt x x x x x
Polynomial x x x x x

The same arguments are used in the Local collinearity diagnostics module as in the GWR Basic module, except for CV.

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”.

Cv: if TRUE, cross-validation data will be calculated

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 calculate an adaptive kernel where the bandwidth (bw) corresponds to the number of nearest neighbours (i.e. adaptive distance); default is FALSE, where a fixed kernel is found (bandwidth is a fixed distance)

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.

Max iter: maximum number of iterations for the automatic approach

Fixed: independent variables that appeared in the formula that are to be treated as global

Intercep fixed: logical, if TRUE the intercept will be treated as global

Diagnostic: logical, if TRUE the diagnostics will be calculated

F123:default FALSE, otherwise calculate F-test results (Leung et al. 2000)

Filtered: default FALSE, the automatic approach is used, if TRUE the filtered data approach is employed, as that described in Fotheringham et al. (2002 p.73-80)

bw.adapt: adaptive bandwidth (i.e. number of nearest neighbours) used for geographically weighting

Polynomial: degree of the polyunomial to approximate the kernel function

Value

SDF a SpatialPointsDataFrame (may be gridded) or SpatialPolygonsDataFrame object (see package “sp”) integrated with fit.points,GWR coefficient estimates, y value,predicted values, coefficient standard errors and t-values in its “data” slot.

In the plot tab, the values obtained in the summary can be plotted, customized and downloaded in .pdf or .png format (see video)

Video

Video 1 : Basic GWR
Video 2 : Robust GWR
Video 3 :Generalized GWR :
Video 4 : Heterocedastic GWR
Video 5 : Mixed GWR
Video 6 : Scalable GWR
Video 7 : Local collinearity diagnostics for basic GWR

References

Brunsdon C, Fotheringham AS, Charlton ME (2002) Geographically weighted summary statistics -a framework for localised exploratory data analysis. Computers, Environment and Urban Systems 26:501-524. https://doi.org/10.1016/S0198-9715(01)00009-6

Fotheringham S, Brunsdon, C, and Charlton, M (2002), Geographically Weighted Regression: The Analysis of Spatially Varying Relationships, Chichester: Wiley.

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 (2014) Geographically weighted methods and their use in network re-designs for environmental monitoring. Stochastic Environmental Research and Risk Assessment 28: 1869-1887. https://doi.org/10.1007/s00477-014-0851-1

Nakaya, T., M. Charlton, S. Fotheringham & C. Brunsdon. 2009. How to use SGWRWIN (GWR4.0).Maynooth, Ireland: National Centre for Geocomputation.

Harris P, Fotheringham AS, Juggins S (2010) Robust geographically weighed regression: a technique for quantifying spatial relationships between freshwater acidification critical loads and catchment attributes. Annals of the Association of American Geographers 100(2): 286-306

Mei L-M, He S-Y, Fang K-T (2004) A note on the mixed geographically weighted regression model. Journal of regional science 44(1):143-157

Nakaya T, Fotheringham AS, Brunsdon C, Charlton M (2005) Geographically Weighted Poisson Regression for Disease Association Mapping, Statistics in Medicine 24: 2695-2717

Murakami, D., N. Tsutsumida, T. Yoshida, T. Nakaya & B. Lu (2019) Scalable GWR: A linear-time algorithm for large-scale geographically weighted regression with polynomial kernels. arXiv:1905.00266