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;

Power (Minkowski distance) : the power of the Minkowski distance (p=1 is manhattan distance, p=2 is euclidean 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

Filtered: default FALSE, the automatic approach is used, if TRUE the filtered data approach is employed, as that described in Fotheringham et al [1]

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, 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.

Videos

Local collinearity diagnostics

Video 1 : Local collinearity diagnostics for basic GWR

Basic Geographically Weighted Regression

Video 2 : Basic GWR

Robust Geographically Weighted Regression

Video 3 : Robust GWR

Generalized Geographically Weighted Regression (GGWR)

Video 4 :Generalized GWR :

Heterocedastic Geographically Weighted Regression (HGWR)

Video 5 : Heterocedastic GWR

Mixed Geographically Weighted Regression (MGWR)

Video 6 : Mixed GWR

Scalable Geographically Weighted Regression (SGWR)

Video 7 : Scalable GWR

References

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

[2] Belsley, D. A., Kuh, E., & Welsch, R. E. (2005). Regression diagnostics: Identifying influential data and sources of collinearity. John Wiley & Sons.

[3] Wheeler, D., & Tiefelsdorf, M. (2005). Multicollinearity and correlation among local regression coefficients in geographically weighted regression. Journal of Geographical Systems, 7(2), 161-187.

[4] Wheeler, D.C., Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environment and Planning A 2007, 39, (10), 2464-2481.

[5] Griffith, D.A., Spatial-filtering-based contributions to a critique of geographically weighted regression (GWR). Environment and Planning A 2008, 40, (11), 2751-2769.

[6] Wheeler, D.; Waller, L., Comparing spatially varying coefficient models: a case study examining violent crime rates and their relationships to alcohol outlets and illegal drug arrests. Journal of Geographical Systems 2009, 11, (1), 1-22.

[7] Wheeler, D.C., Simultaneous coefficient penalization and model selection in geographically weighted regression: the geographically weighted lasso. Environment and Planning A 2009, 41, (3), 722-742.

[8] 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

[9] Oshan, T. M., Li, Z., Kang, W., Wolf, L. J., & Fotheringham, A. S. (2019). mgwr: A Python implementation of multiscale geographically weighted regression for investigating process spatial heterogeneity and scale. ISPRS International Journal of Geo-Information, 8(6), 269.

[10] Zou, H. and Hastie, T. 2005. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 672., pp.301-320.

[11] Friedman J, Hastie T, Tibshirani R 2010. Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software 331:1–22

[12] Dormann, C.F., J. Elith, S. Bacher, et al. 2013. Collinearity: a review of methods to deal with it and a simulation study evaluating their performance. Ecography 36: 27-46.

[13] Comber, A., Brunsdon, C., Charlton, M., Dong, G., Harris, R., Lu, B., … & Harris, P. (2020). The GWR route map: a guide to the informed application of Geographically Weighted Regression. arXiv preprint arXiv:2004.06070.

[14] 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

[15] 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