Geographically-Weighted Models

Description

In GWmodel, we introduce techniques from a particular branch of spatial statistics, termed geographically-weighted (GW) models. GW models suit situations when data are not described well by some global model, but where there are spatial regions where a suitably localised calibration provides a better description. GWmodel includes functions to calibrate: GW summary statistics, GW principal components analysis, GW discriminant analysis and various forms of GW regression; some of which are provided in basic and robust (outlier resistant) forms. In particular, the high-performence computing technologies, including multi-thread and CUDA techniques are started to be adopted for efficient calibrations.

Details

Note

Acknowledgements: We gratefully acknowledge support from National Natural Science Foundation of China (42071368); Science Foundation Ireland under the National Development Plan through the award of a Strategic Research Centre grant 07-SRC-I1168.

Beta versions can always be found at https://github.com/lbb220/GWmodel, which includes all the newly developed functions for GW models.

For latest tutorials on using GWmodel please go to: https://rpubs.com/gwmodel

Author(s)

Binbin Lu, Paul Harris, Martin Charlton, Chris Brunsdon, Tomoki Nakaya, Daisuke Murakami,Isabella Gollini[ctb], Yigong Hu[ctb], Fiona H Evans[ctb]

Maintainer: Binbin Lu <binbinlu@whu.edu.cn>

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, doi: 10.18637/jss.v063.i17

Lu B, Harris P, Charlton M, Brunsdon C (2014) The GWmodel R Package: further topics for exploring Spatial Heterogeneity using Geographically Weighted Models. Geo-spatial Information Science 17(2): 85-101, doi: 10.1080/10095020.2014.917453

Lu, B., Hu, Y., Yang, D., Liu, Y., Ou, G., Harris, P., Brunsdon, C., Comber, A., Dong, G., 2024. Gwmodels: A standalone software to train geographically weighted models. Geo-spatial Information Science, 1-23.

Lu, B., Hu, Y., Murakami, D., Brunsdon, C., Comber, A., Charlton, M., Harris, P., 2022. High-performance solutions of geographically weighted regression in r. Geo-spatial Information Science 25 (4), 536-549.

Voter turnout data in Greater Dublin(SpatialPolygonsDataFrame)

Description

Voter turnout and social characters data in Greater Dublin for the 2002 General election and the 2002 census. Note that this data set was originally thought to relate to 2004, so for continuity we have retained the associated variable names.

Usage

data(DubVoter)

Format

A SpatialPolygonsDataFrame with 322 electoral divisions on the following 11 variables.

DED_ID

a vector of ID

X

a numeric vector of x coordinates

Y

a numeric vector of y coordinates

DiffAdd

percentage of the population in each ED who are one-year migrants (i.e. moved to a different address 1 year ago)

LARent

percentage of the population in each ED who are local authority renters

SC1

percentage of the population in each ED who are social class one (high social class)

Unempl

percentage of the population in each ED who are unemployed

LowEduc

percentage of the population in each ED who are with little formal education

Age18_24

percentage of the population in each ED who are age group 18-24

Age25_44

percentage of the population in each ED who are age group 25-44

Age45_64

percentage of the population in each ED who are age group 45-64

GenEl2004

percentage of population in each ED who voted in 2004 election

Details

Variables are from DubVoter.shp.

References

Kavanagh A (2006) Turnout or turned off? Electoral participation in Dublin in the early 21st Century. Journal of Irish Urban Studies 3(2):1-24

Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25 (10):1717-1736

Examples

data(DubVoter)
ls()
## Not run: 
spplot(Dub.voter,names(Dub.voter)[4:12])

## End(Not run)

House price data set (DataFrame) in England and Wales

Description

A house price data set for England and Wales from 2001 with 9 hedonic (explanatory) variables.

Usage

data(EWHP)

Format

A data frame with 519 observations on the following 12 variables.

Easting

a numeric vector, X coordinate

Northing

a numeric vector, Y coordinate

PurPrice

a numeric vector, the purchase price of the property

BldIntWr

a numeric vector, 1 if the property was built during the world war, 0 otherwise

BldPostW

a numeric vector, 1 if the property was built after the world war, 0 otherwise

Bld60s

a numeric vector, 1 if the property was built between 1960 and 1969, 0 otherwise

Bld70s

a numeric vector, 1 if the property was built between 1970 and 1979, 0 otherwise

Bld80s

a numeric vector, 1 if the property was built between 1980 and 1989, 0 otherwise

TypDetch

a numeric vector, 1 if the property is detached (i.e. it is a stand-alone house), 0 otherwise

TypSemiD

a numeric vector, 1 if the property is semi detached, 0 otherwise

TypFlat

a numeric vector, if the property is a flat (or 'apartment' in the USA), 0 otherwise

FlrArea

a numeric vector, floor area of the property in square metres

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Examples

###
data(EWHP)
head(ewhp)
houses.spdf <- SpatialPointsDataFrame(ewhp[, 1:2], ewhp)
 ####Get the border of England and Wales
data(EWOutline)
plot(ewoutline)
plot(houses.spdf, add = TRUE, pch = 16)

Outline of England and Wales for data EWHP

Description

Outline (SpatialPolygonsDataFrame) of the England and Wales house price data EWHP.

Usage

data(EWOutline)

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Georgia census data set (csv file)

Description

Census data from the county of Georgia, USA

Usage

data(Georgia)

Format

A data frame with 159 observations on the following 13 variables.

AreaKey

An identification number for each county

Latitude

The latitude of the county centroid

Longitud

The longitude of the county centroid

TotPop90

Population of the county in 1990

PctRural

Percentage of the county population defined as rural

PctBach

Percentage of the county population with a bachelors degree

PctEld

Percentage of the county population aged 65 or over

PctFB

Percentage of the county population born outside the US

PctPov

Percentage of the county population living below the poverty line

PctBlack

Percentage of the county population who are black

ID

a numeric vector of IDs

X

a numeric vector of x coordinates

Y

a numeric vector of y coordinates

Details

This data set can also be found in GWR 3 and in spgwr.

References

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

Examples

data(Georgia)
ls()
coords <- cbind(Gedu.df$X, Gedu.df$Y)
educ.spdf <- SpatialPointsDataFrame(coords, Gedu.df)
spplot(educ.spdf, names(educ.spdf)[4:10])

Georgia counties data (SpatialPolygonsDataFrame)

Description

The Georgia census data with boundaries for mapping

Usage

data(GeorgiaCounties)

Details

This data set can also be found in GWR 3 and in spgwr.

Examples

data(GeorgiaCounties)
plot(Gedu.counties)
data(Georgia)
coords <- cbind(Gedu.df$X, Gedu.df$Y)
educ.spdf <- SpatialPointsDataFrame(coords, Gedu.df)
plot(educ.spdf, add=TRUE)

London boroughs data

Description

Outline (SpatialPolygonsDataFrame) of London boroughs for the LondonHP data.

Usage

data(LondonBorough)

Author(s)

Binbin Lu binbinlu@whu.edu.cn

London house price data set (SpatialPointsDataFrame)

Description

A house price data set with 18 hedonic variables for London in 2001.

Usage

data(LondonHP)

Format

A SpatialPointsDataFrame object (proj4string set to "+init=epsg:27700 +datum=OSGB36").

The "data" slot is a data frame with 372 observations on the following 21 variables.

X

a numeric vector, X coordinate

Y

a numeric vector, Y coordinate

PURCHASE

a numeric vector, the purchase price of the property

FLOORSZ

a numeric vector, floor area of the property in square metres

TYPEDETCH

a numeric vector, 1 if the property is detached (i.e. it is a stand-alone house), 0 otherwise

TPSEMIDTCH

a numeric vector, 1 if the property is semi detached, 0 otherwise

TYPETRRD

a numeric vector, 1 if the property is in a terrace of similar houses (commonly referred to as a 'row house' in the USA), 0 otherwise

TYPEBNGLW

a numeric vector, if the property is a bungalow (i.e. it has only one floor), 0 otherwise

TYPEFLAT

a numeric vector, if the property is a flat (or 'apartment' in the USA), 0 otherwise

BLDPWW1

a numeric vector, 1 if the property was built prior to 1914, 0 otherwise

BLDPOSTW

a numeric vector, 1 if the property was built between 1940 and 1959, 0 otherwise

BLD60S

a numeric vector, 1 if the property was built between 1960 and 1969, 0 otherwise

BLD70S

a numeric vector, 1 if the property was built between 1970 and 1979, 0 otherwise

BLD80S

a numeric vector, 1 if the property was built between 1980 and 1989, 0 otherwise

BLD90S

a numeric vector, 1 if the property was built between 1990 and 2000, 0 otherwise

BATH2

a numeric vector, 1 if the property has more than 2 bathrooms, 0 otherwise

GARAGE

a numeric vector,1 if the house has a garage, 0 otherwise

CENTHEAT

a numeric vector, 1 if the house has central heating, 0 otherwise

BEDS2

a numeric vector, 1 if the property has more than 2 bedrooms, 0 otherwise

UNEMPLOY

a numeric vector, the rate of unemployment in the census ward in which the house is located

PROF

a numeric vector, the proportion of the workforce in professional or managerial occupations in the census ward in which the house is located

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Lu, B, Charlton, M, Harris, P, Fotheringham, AS (2014) Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data. International Journal of Geographical Information Science 28(4): 660-681

Examples

data(LondonHP)
data(LondonBorough)
ls()
plot(londonborough)
plot(londonhp, add=TRUE)

Results of the 2004 US presidential election at the county level (SpatialPolygonsDataFrame)

Description

Results of the 2004 US presidential election at the county level, together with five socio-economic (census) variables. This data can be used with GW Discriminant Analysis.

Usage

data(USelect)

Format

A SpatialPolygonsDataFrame with 3111 electoral divisions on the following 6 variables.

winner

Categorical variable with three classes: i) Bush, ii) Kerry and iii) Borderline (supporting ratio for a candidate ranges from 0.45 to 0.55)

unemploy

percentage unemployed

pctcoled

percentage of adults over 25 with 4 or more years of college education

PEROVER65

percentage of persons over the age of 65

pcturban

percentage urban

WHITE

percentage white

References

Robinson, A. C. (2013). Geovisualization of the 2004 Presidential Election. In: NATIONAL INSTITUTES OF HEALTH, P. S. U. (ed.). Penn State.

Foley, P. & Demsar, U. (2012). Using geovisual analytics to compare the performance of geographically weighted discriminant analysis versus its global counterpart, linear discriminant analysis. International Journal of Geographical Information Science, 27, 633-661.

Examples

data(USelect)
ls()

Bandwidth selection for generalised geographically weighted regression (GWR)

Description

A function for automatic bandwidth selection to calibrate a generalised GWR model

Usage

bw.ggwr(formula, data, family ="poisson", approach="CV",
kernel="bisquare",adaptive=FALSE, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Returns the adaptive or fixed distance bandwidth

Note

For a discontinuous kernel function, a bandwidth can be specified either as a fixed (constant) distance or as a fixed (constant) number of local data (i.e. an adaptive distance). For a continuous kernel function, a bandwidth can be specified either as a fixed distance or as a 'fixed quantity that reflects local sample size' (i.e. still an 'adaptive' distance but the actual local sample size will be the sample size as functions are continuous). In practise a fixed bandwidth suits fairly regular sample configurations whilst an adaptive bandwidth suits highly irregular sample configurations. Adaptive bandwidths ensure sufficient (and constant) local information for each local calibration. This note is applicable to all GW models

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Bandwidth selection for GTWR

Description

A function for automatic bandwidth selection to calibrate a GTWR model

Usage

bw.gtwr(formula, data, obs.tv, approach="CV",kernel="bisquare",adaptive=FALSE, 
        p=2, theta=0, longlat=F,lamda=0.05,t.units = "auto",ksi=0, st.dMat,
        verbose=T)

Arguments

Value

Returns the adaptive or fixed distance bandwidth

Note

The function is developed according to the articles by Huang et al. (2010) and Wu et al. (2014).

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Huang, B., Wu, B., & Barry, M. (2010). Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. International Journal of Geographical Information Science, 24, 383-401.

Wu, B., Li, R., & Huang, B. (2014). A geographically and temporally weighted autoregressive model with application to housing prices. International Journal of Geographical Information Science, 28, 1186-1204.

Fotheringham, A. S., Crespo, R., & Yao, J. (2015). Geographical and Temporal Weighted Regression (GTWR). Geographical Analysis, 47, 431-452.

Bandwidth selection for GW Discriminant Analysis

Description

A function for automatic bandwidth selection for GW Discriminant Analysis using a cross-validation approach only

Usage

bw.gwda(formula, data, COV.gw = T, prior.gw = T, mean.gw = T,
                 prior = NULL, wqda = F, kernel = "bisquare", adaptive
                 = FALSE, p = 2, theta = 0, longlat = F,dMat)

Arguments

Value

Returns the adaptive or fixed distance bandwidth.

Note

For a discontinuous kernel function, a bandwidth can be specified either as a fixed (constant) distance or as a fixed (constant) number of local data (i.e. an adaptive distance). For a continuous kernel function, a bandwidth can be specified either as a fixed distance or as a 'fixed quantity that reflects local sample size' (i.e. still an 'adaptive' distance but the actual local sample size will be the sample size as functions are continuous). In practise a fixed bandwidth suits fairly regular sample configurations whilst an adaptive bandwidth suits highly irregular sample configurations. Adaptive bandwidths ensure sufficient (and constant) local information for each local calibration. This note is applicable to all GW models

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Bandwidth selection for Geographically Weighted Principal Components Analysis (GWPCA)

Description

A function for automatic bandwidth selection to calibrate a basic or robust GWPCA via a cross-validation approach only

Usage

bw.gwpca(data,vars,k=2, robust=FALSE, scaling=T, kernel="bisquare",adaptive=FALSE,p=2, 
         theta=0, longlat=F,dMat)

Arguments

Value

Returns the adaptive or fixed distance bandwidth

Note

For a discontinuous kernel function, a bandwidth can be specified either as a fixed (constant) distance or as a fixed (constant) number of local data (i.e. an adaptive distance). For a continuous kernel function, a bandwidth can be specified either as a fixed distance or as a 'fixed quantity that reflects local sample size' (i.e. still an 'adaptive' distance but the actual local sample size will be the sample size as functions are continuous). In practise a fixed bandwidth suits fairly regular sample configurations whilst an adaptive bandwidth suits highly irregular sample configurations. Adaptive bandwidths ensure sufficient (and constant) local information for each local calibration. This note is applicable to all GW models

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Bandwidth selection for basic GWR

Description

A function for automatic bandwidth selection to calibrate a basic GWR model

Usage

bw.gwr(formula, data, approach="CV", kernel="bisquare",
       adaptive=FALSE, p=2, theta=0, longlat=F, dMat,
       parallel.method=F,parallel.arg=NULL)

Arguments

Value

Returns the adaptive or fixed distance bandwidth

Note

For a discontinuous kernel function, a bandwidth can be specified either as a fixed (constant) distance or as a fixed (constant) number of local data (i.e. an adaptive distance). For a continuous kernel function, a bandwidth can be specified either as a fixed distance or as a 'fixed quantity that reflects local sample size' (i.e. still an 'adaptive' distance but the actual local sample size will be the sample size as functions are continuous). In practise a fixed bandwidth suits fairly regular sample configurations whilst an adaptive bandwidth suits highly irregular sample configurations. Adaptive bandwidths ensure sufficient (and constant) local information for each local calibration. This note is applicable to all GW models

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Bandwidth selection for locally compensated ridge GWR (GWR-LCR)

Description

A function for automatic bandwidth selection for gwr.lcr via a cross-validation approach only

Usage

bw.gwr.lcr(formula, data, kernel="bisquare",
        lambda=0,lambda.adjust=FALSE,cn.thresh=NA,
        adaptive=FALSE, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Returns the adaptive or fixed distance bandwidth

Note

For a discontinuous kernel function, a bandwidth can be specified either as a fixed (constant) distance or as a fixed (constant) number of local data (i.e. an adaptive distance). For a continuous kernel function, a bandwidth can be specified either as a fixed distance or as a 'fixed quantity that reflects local sample size' (i.e. still an 'adaptive' distance but the actual local sample size will be the sample size as functions are continuous). In practise a fixed bandwidth suits fairly regular sample configurations whilst an adaptive bandwidth suits highly irregular sample configurations. Adaptive bandwidths ensure sufficient (and constant) local information for each local calibration. This note is applicable to all GW models

Author(s)

Binbin Lu binbinlu@whu.edu.cn

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

Bandwidth selection for GW summary averages

Description

A function for automatic bandwidth selections to calculate GW summary averages, including means and medians, via a cross-validation approach.

Usage

bw.gwss.average(data, summary.locat, vars, kernel = "bisquare", adaptive = FALSE, 
               p = 2, theta = 0, longlat = F, dMat)

Arguments

Value

Returns the adaptive or fixed distance bandwidths (in a two-column matrix) for calculating the averages of each variable.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Generalised GWR models with Poisson and Binomial options

Description

This function implements generalised GWR

Usage

ggwr.basic(formula, data, regression.points, bw, family =
                 "poisson", kernel = "bisquare", adaptive = FALSE, cv =
                 T, tol = 1e-05, maxiter = 20, p = 2, theta = 0,
                 longlat = F, dMat, dMat1)

 ## S3 method for class 'ggwrm'
print(x, ...)
 

Arguments

Value

A list of class “ggwrm”:

Note

Note that this function calibrates a Generalised GWR model via an approximating algorithm, which is different from the back-fitting algorithm used in the GWR4 software by Tomoki Nakaya.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Nakaya, T., A. S. Fotheringham, C. Brunsdon & M. Charlton (2005) Geographically weighted Poisson regression for disease association mapping. Statistics in Medicine, 24, 2695-2717.

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

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

Examples

data(LondonHP)
## Not run: 
DM<-gw.dist(dp.locat=coordinates(londonhp))
bw.f1 <- bw.ggwr(BATH2~FLOORSZ,data=londonhp, dMat=DM)
res.poisson<-ggwr.basic(BATH2~FLOORSZ, bw=bw.f1,data=londonhp, dMat=DM)
bw.f2 <- bw.ggwr(BATH2~FLOORSZ,data=londonhp, dMat=DM,family ="binomial")
res.binomial<-ggwr.basic(BATH2~FLOORSZ, bw=bw.f2,data=londonhp, dMat=DM,
              family ="binomial")

## End(Not run)

Cross-validation score for a specified bandwidth for generalised GWR

Description

This function finds the cross-validation score for a specified bandwidth for generalised GWR. It can be used to construct the bandwidth function across all possible bandwidths and compared to that found automatically.

Usage

ggwr.cv(bw, X, Y,family="poisson", kernel="bisquare",adaptive=F, dp.locat,  
        p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Cross-validation data at each observation location for a generalised GWR model

Description

This function finds the individual cross-validation score at each observation location, for a generalised GWR model, for a specified bandwidth. These data can be mapped to detect unusually high or low cross-validations scores.

Usage

ggwr.cv.contrib(bw, X, Y,family="poisson", kernel="bisquare",adaptive=F, 
                dp.locat, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Geographically and Temporally Weighted Regression

Description

A function for calibrating a Geographically and Temporally Weighted Regression (GTWR) model.

Usage

gtwr(formula, data, regression.points, obs.tv, reg.tv, st.bw, kernel="bisquare",
     adaptive=FALSE, p=2, theta=0, longlat=F,lamda=0.05,t.units = "auto",ksi=0,
     st.dMat)

Arguments

Value

A list of class “gtwrm”:

Note

The function implements GTWR model proposed by Huang et al. (2010) and Wu et al. (2014).

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Huang, B., Wu, B., & Barry, M. (2010). Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. International Journal of Geographical Information Science, 24, 383-401.

Wu, B., Li, R., & Huang, B. (2014). A geographically and temporally weighted autoregressive model with application to housing prices. International Journal of Geographical Information Science, 28, 1186-1204.

Fotheringham, A. S., Crespo, R., & Yao, J. (2015). Geographical and Temporal Weighted Regression (GTWR). Geographical Analysis, 47, 431-452.

Distance matrix calculation

Description

Calculate a distance vector(matrix) between any GW model calibration point(s) and the data points.

Usage

gw.dist(dp.locat, rp.locat, focus=0, p=2, theta=0, longlat=F)

Arguments

Value

Returns a numeric distance matrix or vector; matrix with its rows corresponding to the observations and its columns corresponds to the GW model calibration points.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

See Also

dist in stats

Examples

dp<-cbind(sample(100),sample(100))
rp<-cbind(sample(10),sample(10))
#Euclidean distance metric is used.
dist.v1<-gw.dist(dp.locat=dp, focus=5, p=2, theta=0, longlat=FALSE)
#Manhattan distance metric is used.
#The coordinate system is rotated by an angle 0.5 in radian.
dist.v2<-gw.dist(dp.locat=dp, focus=5, p=1, theta=0.5)
#Great Circle distance metric is used.
dist.v3<-gw.dist(dp.locat=dp, focus=5, longlat=TRUE)
#A generalized Minkowski distance metric is used with p= 0.75 .
#The coordinate system is rotated by an angle 0.8 in radian.
dist.v4<-gw.dist(dp.locat=dp,rp.locat=rp, focus=5, p=0.75,theta=0.8)
################################
#matrix is calculated
#Euclidean distance metric is used.
dist.m1<-gw.dist(dp.locat=dp, p=2, theta=0, longlat=FALSE)
#Manhattan distance metric is used.
#The coordinate system is rotated by an angle 0.5 in radian.
dist.m2<-gw.dist(dp.locat=dp, p=1, theta=0.5)
#Great Circle distance metric is used.
#dist.m3<-gw.dist(dp.locat=dp, longlat=TRUE)
#A generalized Minkowski distance metric is used with p= 0.75 .
#The coordinate system is rotated by an angle 0.8 in radian.
dist.m4<-gw.dist(dp.locat=dp,rp.locat=rp, p=0.75,theta=0.8)

Geographically weighted parallel coordinate plot for investigating multivariate data sets

Description

This function provides a geographically weighted parallel coordinate plot for locally investigating a multivariate data set. It has an option that weights the lines of the plot with increasing levels of transparency, according to their observation's distance from a specified focal/observation point.

Usage

gw.pcplot(data,vars,focus,bw,adaptive = FALSE, ylim=NULL,ylab="",fixtrans=FALSE, 
          p=2, theta=0, longlat=F,dMat,...) 

Arguments

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

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

Weight matrix calculation

Description

Calculate a weight vector(matrix) from a distance vector(matrix).

Usage

gw.weight(vdist,bw,kernel,adaptive=FALSE)

Arguments

Value

Returns a numeric weight matrix or vector; matrix with its rows corresponding to the observations and its columns corresponds to the GW model calibration points.

Note

The gaussian and exponential kernel functions are continuous and valued in the interval (0,1]; while bisquare, tricube and boxcar kernel functions are discontinuous and valued in the interval [0,1]. Notably, the upper limit of the bandwidth is exactly the number of observations when the adaptive kernel is used. In this function, the adaptive bandwidth will be specified as the number of observations even though a larger number is assigned. The function will be the same as a global application function (i.e. all weights are 1) when the adaptive bandwidth is equal to or larger than the number of observations when using the boxcar kernel function.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

GW Discriminant Analysis

Description

This function implements GW discriminant analysis, where location-wise probabilities and their associated entropy are also calculated.

Usage

gwda(formula, data, predict.data,validation = T, COV.gw=T, 
                 mean.gw=T, prior.gw=T, prior=NULL, wqda =F,
                kernel = "bisquare", adaptive = FALSE, bw,
                 p = 2, theta = 0, longlat = F,dMat)
## S3 method for class 'gwda'
print(x, ...)

Arguments

Value

An object of class “gwda”. This includes a SpatialPointsDataFrame (may be gridded) or SpatialPolygonsDataFrame object, SDF, (see package “sp”) or sf object (see package “sf”) with, following the use of new version of gwda, the probabilities for each level, the highest probabiliity and the entropy of the probabilities in its “data” slot.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Brunsdon, C, Fotheringham S, and Charlton, M (2007), Geographically Weighted Discriminant Analysis, Geographical Analysis 39:376-396

Lu B, Harris P, Charlton M, Brunsdon C (2014) The GWmodel R Package: further topics for exploring Spatial Heterogeneity using Geographically Weighted Models. Geo-spatial Information Science 17(2): 85-101

Examples

## Not run: 
 data(USelect)
 dMat <- gw.dist(coordinates(USelect2004))
 bw <- bw.gwda(winner~unemploy+pctcoled+PEROVER65+pcturban+WHITE,data=USelect2004,
 adaptive=TRUE,dMat=dMat)
 ge.gwda <- gwda(winner~unemploy+pctcoled+PEROVER65+pcturban+WHITE,data=USelect2004,
 bw=bw,adaptive=TRUE,dMat=dMat)
 table(USelect2004$winner,ge.gwda$SDF$group.predicted)
 spplot(ge.gwda$SDF, "entropy")
 
## End(Not run)

GWPCA

Description

This function implements basic or robust GWPCA.

Usage

gwpca(data, elocat, vars, k = 2, robust = FALSE, scaling=T, kernel = "bisquare",
                  adaptive = FALSE, bw, p = 2, theta = 0, longlat = F, cv = T, scores=F,
                  dMat)
## S3 method for class 'gwpca'
print(x, ...)

Arguments

Value

A list of class “gwpca”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25:1717-1736

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

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

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

Examples

## Not run: 
if(require("mvoutlier") && require("RColorBrewer"))
{
  data(bsstop)
  Data.1 <- bsstop[, 1:14]
  colnames(Data.1)
  Data.1.scaled <- scale(as.matrix(Data.1[5:14]))  # standardised data...
  rownames(Data.1.scaled) <- Data.1[, 1]
  #compute principal components:
  pca <- princomp(Data.1.scaled, cor = FALSE, scores = TRUE)  
  # use covariance matrix to match the following...
  pca$loadings
  data(bss.background)
  backdrop <- function() 
   plot(bss.background, asp = 1, type = "l", xaxt = "n", yaxt = "n", 
   xlab = "", ylab = "", bty = "n", col = "grey")
  pc1 <- pca$scores[, 1]
  backdrop()
  points(Data.1$XCOO[pc1 > 0], Data.1$YCOO[pc1 > 0], pch = 16, col = "blue")
  points(Data.1$XCOO[pc1 < 0], Data.1$YCOO[pc1 < 0], pch = 16, col = "red")
  
  #Geographically Weighted PCA and mapping the local loadings
  # Coordinates of the sites
  Coords1 <- as.matrix(cbind(Data.1$XCOO,Data.1$YCOO)) 
  d1s <- SpatialPointsDataFrame(Coords1,as.data.frame(Data.1.scaled))
  pca.gw <- gwpca(d1s,vars=colnames(d1s@data),bw=1000000,k=10)
  local.loadings <- pca.gw$loadings[, , 1]  
  
  # Mapping the winning variable with the highest absolute loading
  # note first component only - would need to explore all components..
  
  lead.item <- colnames(local.loadings)[max.col(abs(local.loadings))]
  df1p = SpatialPointsDataFrame(Coords1, data.frame(lead = lead.item))
  backdrop()
  colour <- brewer.pal(8, "Dark2")[match(df1p$lead, unique(df1p$lead))]
  plot(df1p, pch = 18, col = colour, add = TRUE)
  legend("topleft", as.character(unique(df1p$lead)), pch = 18, col = 
      brewer.pal(8, "Dark2"))
  backdrop()
  
  #Glyph plots give a view of all the local loadings together
  glyph.plot(local.loadings, Coords1, add = TRUE)
  
  #it is not immediately clear how to interpret the glyphs fully, 
  #so inter-actively identify the full loading information using:
  check.components(local.loadings, Coords1)
  
  # GWPCA with an optimal bandwidth
  bw.choice <- bw.gwpca(d1s,vars=colnames(d1s@data),k=2) 
  pca.gw.auto  <- gwpca(d1s,vars=colnames(d1s@data),bw=bw.choice,k=2)
  # note first component only - would need to explore all components..
  local.loadings <- pca.gw.auto$loadings[, , 1]  
  
  lead.item <- colnames(local.loadings)[max.col(abs(local.loadings))]
  df1p = SpatialPointsDataFrame(Coords1, data.frame(lead = lead.item))
  backdrop()
  colour <- brewer.pal(8, "Dark2")[match(df1p$lead, unique(df1p$lead))]
  plot(df1p, pch = 18, col = colour, add = TRUE)
  legend("topleft", as.character(unique(df1p$lead)), pch = 18, 
  col = brewer.pal(8, "Dark2"))
  
  # GWPCPLOT for investigating the raw multivariate data
  gw.pcplot(d1s, vars=colnames(d1s@data),focus=359, bw = bw.choice) 
}

## End(Not run)

Interaction tool with the GWPCA glyph map

Description

The function interacts with the multivariate glyph plot of GWPCA loadings.

Usage

gwpca.check.components(ld,loc)

Arguments

Note

The function “check.components” (in the early versions of GWmodel) has been renamed as “gwpca.check.components”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

See Also

gwpca.glyph.plot

Cross-validation score for a specified bandwidth for GWPCA

Description

This function finds the cross-validation score for a specified bandwidth for basic or robust GWPCA. It can be used to construct the bandwidth function across all possible bandwidths and compared to that found automatically.

Usage

gwpca.cv(bw,x,loc,k=2,robust=FALSE,kernel="bisquare",adaptive=FALSE,p=2, 
         theta=0, longlat=F,dMat)  

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Cross-validation data at each observation location for a GWPCA

Description

This function finds the individual cross-validation score at each observation location, for a GWPCA model, for a specified bandwidth. These data can be mapped to detect unusually high or low cross-validations scores.

Usage

gwpca.cv.contrib(x,loc,bw, k=2,robust=FALSE,kernel="bisquare",adaptive=FALSE,
                 p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Multivariate glyph plots of GWPCA loadings

Description

This function provides a multivariate glyph plot of GWPCA loadings at each output location.

Usage

gwpca.glyph.plot(ld,loc, r1=50, add=FALSE,alpha=1,sep.contrasts=FALSE)

Arguments

Note

The function “glyph.plot” (in the early versions of GWmodel) has been renamed as “gwpca.glyph.plot”, while the old name is still kept valid.

References

Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25:1717-1736

Monte Carlo (randomisation) test for significance of GWPCA eigenvalue variability for the first component only - option 1

Description

This function implements a Monte Carlo (randomisation) test for a basic or robust GW PCA with the bandwidth pre-specified and constant. The test evaluates whether the GW eigenvalues vary significantly across space for the first component only.

Usage

gwpca.montecarlo.1(data, bw, vars, k = 2, nsims=99,robust = FALSE, scaling=T, 
                   kernel = "bisquare", adaptive = FALSE,  p = 2, theta = 0, 
                   longlat = F, dMat)
## S3 method for class 'mcsims'
plot(x, sname="SD of local eigenvalues from randomisations", ...)

Arguments

Value

A list of components:

Note

The function “montecarlo.gwpca.1” (in the early versions of GWmodel) has been renamed as “gwpca.montecarlo.1”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25:1717-1736

Examples

## Not run: 
data(DubVoter)
DM<-gw.dist(dp.locat=coordinates(Dub.voter))
gmc.res<-gwpca.montecarlo.1(data=Dub.voter, vars=c("DiffAdd", "LARent",
"SC1", "Unempl", "LowEduc"), bw=20,dMat=DM,adaptive=TRUE)
gmc.res
plot(gmc.res)

## End(Not run)

Monte Carlo (randomisation) test for significance of GWPCA eigenvalue variability for the first component only - option 2

Description

This function implements a Monte Carlo (randomisation) test for a basic or robust GW PCA with the bandwidth automatically re-selected via the cross-validation approach. The test evaluates whether the GW eigenvalues vary significantly across space for the first component only.

Usage

gwpca.montecarlo.2(data, vars, k = 2, nsims=99,robust = FALSE, scaling=T, 
                   kernel = "bisquare", adaptive = FALSE,  p = 2, 
                   theta = 0, longlat = F, dMat)

Arguments

Value

A list of components:

Note

The function “montecarlo.gwpca.2” (in the early versions of GWmodel) has been renamed as “gwpca.montecarlo.2”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. International Journal of Geographical Information Science 25:1717-1736

Examples

## Not run: 
data(DubVoter)
DM<-gw.dist(dp.locat=coordinates(Dub.voter))
gmc.res.autow<-gwpca.montecarlo.2(data=Dub.voter, vars=c("DiffAdd", "LARent",
"SC1", "Unempl", "LowEduc"), dMat=DM,adaptive=TRUE)
gmc.res.autow
plot.mcsims(gmc.res.autow)

## End(Not run)

Basic GWR model

Description

This function implements basic GWR

Usage

gwr.basic(formula, data, regression.points, bw, kernel="bisquare",
adaptive=FALSE, p=2, theta=0, longlat=F,dMat,F123.test=F,cv=F, W.vect=NULL,
parallel.method=FALSE,parallel.arg=NULL)
## S3 method for class 'gwrm'
print(x, ...)

Arguments

Value

A list of class “gwrm”:

Note

Requirements of using CUDA for high-performence computation in GWR functions:

To run GWR-CUDA (i.e. parallel.method is pecified as “cuda”) with gwr.basic , bw.gwr and gwr.model.selection, the following conditions are required:

1. There is at least one NVIDIA GPU supporting CUDA equipped on user's computer.

2. CUDA (>10.2) are installed with the environment variable 'CUDA_HOME' set properly.

3. The package should re-built from source. - For Linux user, 'GWmodelCUDA' will be automatically built if CUDA toolkit could be detected by the complier. - For Windows user, 'GWmodelCUDA.dll' and 'GWmodelCUDA.lib' will be automatically downloaded; however, we would recommend users to build the 'GWmodelCUDA' library manually to avoid some potentially unknown issues, and an 'CMakeLists.txt' file is provided for this procedure.

If any condition above is not satisfied, the GWR-CUDA will not work even though the “parallel.method” is specified as “cuda”.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Brunsdon, C, Fotheringham, S, Charlton, M (1996), Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity. Geographical Analysis 28(4):281-298

Charlton, M, Fotheringham, S, and Brunsdon, C (2007), GWR3.0, http://gwr.nuim.ie/.

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

Leung, Y, Mei, CL, and Zhang, WX (2000), Statistical tests for spatial nonstationarity based on the geographically weighted regression model. Environment and Planning A, 32, 9-32.

Lu, B, Charlton, M, Harris, P, Fotheringham, AS (2014) Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data. International Journal of Geographical Information Science 28(4): 660-681

OpenMP: https://www.openmp.org/

CUDA: https://developer.nvidia.com/cuda-zone

R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.

Examples

data(LondonHP)
DM<-gw.dist(dp.locat=coordinates(londonhp))
##Compare the time consumed with and without a specified distance matrix
## Not run: 
system.time(gwr.res<-gwr.basic(PURCHASE~FLOORSZ, data=londonhp, bw=1000,
            kernel = "gaussian"))
system.time(DM<-gw.dist(dp.locat=coordinates(londonhp)))
system.time(gwr.res<-gwr.basic(PURCHASE~FLOORSZ, data=londonhp, bw=1000,
            kernel = "gaussian", dMat=DM))

## specify an optimum bandwidth by cross-validation appraoch
bw1<-bw.gwr(PURCHASE~FLOORSZ, data=londonhp, kernel = "gaussian",dMat=DM)
gwr.res1<-gwr.basic(PURCHASE~FLOORSZ, data=londonhp, bw=bw1,kernel = "gaussian", 
                   dMat=DM)
gwr.res1 
## End(Not run)
data(LondonBorough)

nsa = list("SpatialPolygonsRescale", layout.north.arrow(), offset = c(561900,200900), 
scale = 500, col=1)
## Not run: 
if(require("RColorBrewer"))
{
  mypalette<-brewer.pal(6,"Spectral")
  x11()
  spplot(gwr.res1$SDF, "FLOORSZ", key.space = "right", cex=1.5, cuts=10,
  ylim=c(155840.8,200933.9), xlim=c(503568.2,561957.5),
  main="GWR estimated coefficients for FLOORSZ with a fixed bandwidth", 
  col.regions=mypalette, sp.layout=list(nsa, londonborough))}

## End(Not run)
## Not run: 
bw2<-bw.gwr(PURCHASE~FLOORSZ,approach="aic",adaptive=TRUE, data=londonhp, 
            kernel = "gaussian", dMat=DM)
gwr.res2<-gwr.basic(PURCHASE~FLOORSZ, data=londonhp, bw=bw2,adaptive=TRUE,
                    kernel = "gaussian", dMat=DM)
gwr.res2
if(require("RColorBrewer"))
{
  x11()
  spplot(gwr.res2$SDF, "FLOORSZ", key.space = "right", cex=1.5, cuts=10,
  ylim=c(155840.8,200933.9), xlim=c(503568.2,561957.5),
  main="GWR estimated coefficients for FLOORSZ with an adaptive bandwidth", 
  col.regions=mypalette, sp.layout=list(nsa,londonborough))}

## End(Not run)
## Not run: 
  ############HP-GWR test code
  simulate.data.generator <- function(data.length) {
  x1 <- rnorm(data.length)
  x2 <- rnorm(data.length)
  x3 <- rnorm(data.length)
  lon <- rnorm(data.length, mean = 533200, sd = 10000)
  lat <- rnorm(data.length, mean = 159400, sd = 10000)
  y <- x1 + 5 * x2 + 2.5 * x3 + rnorm(data.length)
  simulate.data <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3, lon = lon, lat = lat)
  coordinates(simulate.data) <- ~ lon + lat
  names(simulate.data)
  return(simulate.data)
}
simulate.data <- simulate.data.generator(10000)
adaptive = TRUE

## GWR (not parallelized)
bw.CV.s <- bw.gwr(data = simulate.data, formula = y ~ x1 + x2 + x3, approach="CV", 
                  kernel = "gaussian", adaptive = adaptive, parallel.method = FALSE)
model.s <- gwr.model.selection(DeVar = "y", InDeVars = c("x1", "x2", "x3"), data = simulate.data, 
                              bw = bw.CV.s, approach="AIC", kernel = "gaussian", adaptive = T, 
                              parallel.method = FALSE)
system.time(
  betas.s <- gwr.basic(data = simulate.data, formula = y ~ x1 + x2 + x3, bw = bw.CV.s, 
                       kernel = "gaussian", adaptive = TRUE)
)

## GWR-Omp
bw.CV.omp <- bw.gwr(data = simulate.data, formula = y ~ x1 + x2 + x3, approach="CV", 
                    kernel = "gaussian", adaptive = adaptive, parallel.method = "omp")
model.omp <- gwr.model.selection(DeVar = "y", InDeVars = c("x1", "x2", "x3"), data = simulate.data, 
                                bw = bw.CV.omp, approach="AIC", kernel = "gaussian", adaptive = T, 
                                parallel.method = "omp")
system.time(
  betas.omp <- gwr.basic(data = simulate.data, formula = y ~ x1 + x2 + x3, bw = bw.CV.omp, 
                        kernel = "gaussian", adaptive = T, parallel.method = "omp"))

## GWR-CUDA
bw.CV.cuda <- bw.gwr(data = simulate.data, formula = y ~ x1 + x2 + x3, approach="CV", 
                     kernel = "gaussian", adaptive = adaptive, parallel.method = "cuda", 
                     parallel.arg = 6*16)
model.cuda <- gwr.model.selection(DeVar = "y", InDeVars = c("x1", "x2", "x3"), 
                                 data = simulate.data, bw = bw.CV.cuda, approach="AIC", 
                                 kernel = "gaussian", adaptive = T, 
                                 parallel.method = "cuda", parallel.arg = 6*16)
system.time(
  betas.cuda <- gwr.basic(data = simulate.data, formula = y ~ x1 + x2 + x3, bw = bw.CV.cuda, 
                          kernel = "gaussian", adaptive = T, parallel.method = "cuda", 
                          parallel.arg = 6*8))

## End(Not run)

Bootstrap GWR

Description

This function implements bootstrap methods to test for coefficient variability found from GWR under model assumptions for each of four null hypotheses: MLR, ERR, SMA and LAG models. Global test statistic results are found, as well local observation-specific test results that can be mapped.

Usage

gwr.bootstrap(formula, data, kernel = "bisquare", approach = "AIC",
                 R = 99, k.nearneigh = 4, adaptive = FALSE, p = 2,
                 theta = 0, longlat = FALSE, dMat, verbose = FALSE,
                 parallel.method = FALSE, parallel.arg = NULL)

## S3 method for class 'gwrbsm'
print(x, ...)

Arguments

Value

A list of class “gwrbsm”:

Note

This function implements the bootstrap methods introduced in Harris et al. (2017). It provides a global test statistic (the modified one given in Harris et al. 2017) and a complementary localised version that can be mapped. The bootstrap methods test for coefficient variability found from GWR under model assumptions for each of four null hypotheses: i) multiple linear regression model (MLR); ii) simultaneous autoregressive error model (ERR); iii) moving average error model (SMA) and iv) simultaneous autoregressive lag model (LAG).

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Harris, P., Brunsdon, C., Lu, B., Nakaya, T., & Charlton, M. (2017). Introducing bootstrap methods to investigate coefficient non-stationarity in spatial regression models. Spatial Statistics, 21, 241-261.

Examples

## Not run: 
#Example with the Georgia educational attainment data
data(Georgia)
data(GeorgiaCounties)
coords <- cbind(Gedu.df$X, Gedu.df$Y)
Gedu.spdf <- SpatialPointsDataFrame(coords, Gedu.df)
#Make a SpatialPolygonDataFrame
require(RColorBrewer)
gSRDF <- SpatialPolygonsDataFrame(polygons(Gedu.counties), over(Gedu.counties, 
                                  Gedu.spdf),match.ID=T)  
mypalette.1 <- brewer.pal(11,"Spectral")
X11(width=9,height=8)                   
spplot(gSRDF, names(gSRDF)[c(5,7:9)], col.regions=mypalette.1,
cuts=10, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Georgia educational attainment predictor data")))
bsm.res <- gwr.bootstrap(PctBach~PctRural+PctEld+PctFB+PctPov, gSRDF, 
                         R=999, longlat=T)
bsm.res
#local bootstrap tests with respect to: MLR, ERR, SMA and LAG models.
mypalette.local.test <- brewer.pal(10,"Spectral")
X11(width=12,height=16)
spplot(bsm.res$SDF, names(bsm.res$SDF)[14:17], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the MLR model 
                       null hypothesis")))

X11(width=12,height=16)
spplot(bsm.res$SDF, names(bsm.res$SDF)[19:22], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the ERR model 
                       null hypothesis")))
X11(width=12,height=16)
spplot(bsm.res$SDF, names(bsm.res$SDF)[24:27], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the SMA model null
                       hypothesis")))

X11(width=12,height=16)
spplot(bsm.res$SDF, names(bsm.res$SDF)[29:32], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the LAG model null
                       hypothesis")))
################################################################################
#Example with Dublin voter data
data(DubVoter)
X11(width=9,height=8)                   
spplot(Dub.voter, names(Dub.voter)[c(5,7,9,10)], col.regions=mypalette.1,
cuts=10, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Dublin voter turnout predictor data")))
bsm.res1 <- gwr.bootstrap(GenEl2004~LARent+Unempl+Age18_24+Age25_44, Dub.voter
                         , R=999)
bsm.res1

#local bootstrap tests with respect to: MLR, ERR, SMA and LAG models.
X11(width=11,height=8)
spplot(bsm.res1$SDF, names(bsm.res1$SDF)[14:17], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the MLR model null
                        hypothesis")))
X11(width=11,height=8)
spplot(bsm.res1$SDF, names(bsm.res1$SDF)[19:22], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the ERR model null
                        hypothesis")))
X11(width=11,height=8)
spplot(bsm.res1$SDF, names(bsm.res1$SDF)[24:27], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the SMA model 
                            null hypothesis")))
X11(width=11,height=8)
spplot(bsm.res1$SDF, names(bsm.res1$SDF)[29:32], col.regions=mypalette.local.test,
cuts=9, par.settings=list(fontsize=list(text=15)),
main=expression(paste("Local p-values for each coefficient of the LAG model 
                            null hypothesis")))

## End(Not run)

Local collinearity diagnostics for basic GWR

Description

This function provides a series of local collinearity diagnostics for the independent variables of a basic GWR model.

Usage

gwr.collin.diagno(formula, data, bw, kernel="bisquare",
                    adaptive=FALSE, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Wheeler D (2007) Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environment and Planning A 39:2464-2481

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

Cross-validation score for a specified bandwidth for basic GWR

Description

This function finds the cross-validation score for a specified bandwidth for basic GWR. It can be used to construct the bandwidth function across all possible bandwidths and compared to that found automatically.

Usage

gwr.cv(bw, X, Y, kernel="bisquare",adaptive=FALSE, dp.locat, p=2, theta=0, 
       longlat=F,dMat, verbose=T,
       parallel.method=F, parallel.arg=NULL)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Cross-validation data at each observation location for a basic GWR model

Description

This function finds the individual cross-validation score at each observation location, for a basic GWR model, for a specified bandwidth. These data can be mapped to detect unusually high or low cross-validations scores.

Usage

gwr.cv.contrib(bw, X, Y, kernel="bisquare",adaptive=FALSE, dp.locat, p=2, 
               theta=0, longlat=F,dMat,
               parallel.method=F, parallel.arg=NULL)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Heteroskedastic GWR

Description

This function implements a heteroskedastic GWR model

Usage

gwr.hetero(formula, data, regression.points, bw, kernel="bisquare",
                    adaptive=FALSE, tol=0.0001,maxiter=50,verbose=T,
                    p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

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

Harris P, Brunsdon C, Fotheringham AS (2011) Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor. Stochastic Environmental Research and Risk Assessment 25:123-138

GWR with a locally-compensated ridge term

Description

To address possible local collinearity problems in basic GWR, GWR-LCR finds local ridge parameters at affected locations (set by a user-specified threshold for the design matrix condition number).

Usage

gwr.lcr(formula, data, regression.points, bw, kernel="bisquare",
                    lambda=0,lambda.adjust=FALSE,cn.thresh=NA,
                    adaptive=FALSE, p=2, theta=0, longlat=F,cv=T,dMat)
## S3 method for class 'gwrlcr'
print(x, ...)

Arguments

Value

A list of class “rgwr”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Wheeler D (2007) Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environment and Planning A 39:2464-2481

Brunsdon C, Charlton M, Harris P (2012) Living with collinearity in Local Regression Models. GISRUK 2012, Lancaster, UK

Brunsdon C, Charlton M, Harris P (2012) Living with collinearity in Local Regression Models. Spatial Accuracy 2012, Brazil

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

Examples

data(DubVoter)
require(RColorBrewer)

# Function to find the global condition number (CN)
BKW_cn <- function (X) {
  p <- dim(X)[2]
  Xscale <- sweep(X, 2, sqrt(colSums(X^2)), "/")
  Xsvd <- svd(Xscale)$d
  cn <- Xsvd[1] / Xsvd[p]
  cn
}
#
X <- cbind(1,Dub.voter@data[,3:10])
head(X)
CN.global <- BKW_cn(X)
CN.global
## Not run: 
# gwr.lcr function with a global bandwidth to check that the global CN is found
gwr.lcr1 <- gwr.lcr(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64, data=Dub.voter, bw=10000000000)
summary(gwr.lcr1$SDF$Local_CN)

# Find and map the local CNs from a basic GWR fit using the lcr-gwr function 
#(note this is NOT the locally-compensated ridge GWR fit as would need to set 
#lambda.adjust=TRUE and cn.thresh=30, say)

bw.lcr2 <- bw.gwr.lcr(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64, data=Dub.voter, kernel="bisquare", adaptive=TRUE)
gwr.lcr2 <- gwr.lcr(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64, data=Dub.voter, bw=bw.lcr2, kernel="bisquare", adaptive=TRUE)
if(require("RColorBrewer"))
  spplot(gwr.lcr2$SDF,"Local_CN",col.regions=brewer.pal(9,"YlOrRd"),cuts=8,
  main="Local CN")

## End(Not run)

Cross-validation score for a specified bandwidth for GWR-LCR model

Description

This function finds the cross-validation score for a specified bandwidth for GWR-LCR. It can be used to construct the bandwidth function across all possible bandwidths and compared to that found automatically.

Usage

gwr.lcr.cv(bw,X,Y,locs,kernel="bisquare",
                    lambda=0,lambda.adjust=FALSE,cn.thresh=NA,
                    adaptive=FALSE, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Cross-validation data at each observation location for the GWR-LCR model

Description

This function finds the individual cross-validation score at each observation location, for a GWR-LCR model, for a specified bandwidth. These data can be mapped to detect unusually high or low cross-validations scores.

Usage

gwr.lcr.cv.contrib(bw,X,Y,locs,kernel="bisquare",
                    lambda=0,lambda.adjust=FALSE,cn.thresh=NA,
                    adaptive=FALSE, p=2, theta=0, longlat=F,dMat)

Arguments

Value

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Minkovski approach for GWR

Description

This function implements the Minkovski approach to select an 'optimum' distance metric for calibrating a GWR model.

Usage

gwr.mink.approach(formula, data, criterion="AIC", bw, bw.sel.approach = "AIC",adaptive=F, 
              kernel="bisquare", p.vals=seq(from=0.25, to=8, length.out=32), p.inf = T,
                          theta.vals = seq(from=0, to=0.5*pi, length.out=10), verbose=F, 
                          nlower = 10)

Arguments

Value

A list of:

Note

The function “mink.approach” (in the early versions of GWmodel) has been renamed as “gwr.mink.approach”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Lu, B, Charlton, M, Brunsdon, C & Harris, P(2016). The Minkowski approach for choosing the distance metric in Geographically Weighted Regression. International Journal of Geographical Information Science, 30(2): 351-368.

Visualisation of the results from gwr.mink.approach

Description

This function visualises the AICc/CV results from the gwr.mink.approach.

Usage

gwr.mink.matrixview(diag.df, znm=colnames(diag.df)[4], criterion="AIC")

Arguments

Note

The function “mink.matrixview” (in the early versions of GWmodel) has been renamed as “gwr.mink.matrixview”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Lu, B, Charlton, M, Brunsdon, C & Harris, P(2016). The Minkowski approach for choosing the distance metric in Geographically Weighted Regression. International Journal of Geographical Information Science, 30(2): 351-368.

Select the values of p for the Minkowski approach for GWR

Description

These functions implement heuristics to select the values of p from two intervals: (0, 2] in a 'backward' direction and (2, Inf) in a 'forward' direction.

Usage

gwr.mink.pval(formula, data, criterion="AIC", bw, bw.sel.approach = "AIC",
                       adaptive=F, kernel="bisquare", left.interval=0.25,
                       right.interval=0.5,drop.tol=3, theta0=0,verbose=F,nlower = 10)
gwr.mink.pval.forward(formula, data, bw, bw.sel.approach = "AIC",
                       adaptive=F, kernel="bisquare", p.max=Inf,p.min=2,
                       interval=0.5,drop.tol=3, theta0=0,verbose=F,nlower = 10)
gwr.mink.pval.backward(formula, data, bw, bw.sel.approach = "AIC",
                       adaptive=F, kernel="bisquare", p.max=2,p.min=0.1,
                       interval=0.5,drop.tol=3, theta0=0,verbose=F,nlower = 10)
## S3 method for class 'pvlas'
plot(x, ...)
                       

Arguments

Value

A list of:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Lu, B, Charlton, M, Brunsdon, C & Harris, P(2016). The Minkowski approach for choosing the distance metric in Geographically Weighted Regression. International Journal of Geographical Information Science, 30(2): 351-368.

Mixed GWR

Description

This function implements mixed (semiparametric) GWR

Usage

gwr.mixed(formula, data, regression.points, fixed.vars,
                     intercept.fixed=FALSE, bw, diagnostic=T, kernel="bisquare", 
                     adaptive=FALSE, p=2, theta=0, longlat=F,dMat, dMat.rp)

Arguments

Value

A list of class “mgwr”:

Note

For an alternative formulation of mixed GWR, please refer to GWR 4, which provides useful tools for automatic bandwidth selection. This windows-based software also implements generalised mixed GWR.

The mixed GWR in the latest release of GWmodel (2.0-0) has been revised by Dr. Fiona H Evans from Centre for Digital Agriculture, Murdoch and Curtin Universities in terms of its computational efficiency.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

Brunsdon C, Fotheringham AS, Charlton ME (1999) Some notes on parametric signficance tests for geographically weighted regression. Journal of Regional Science 39(3):497-524

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

Mei L-M, Wang N, Zhang W-X (2006) Testing the importance of the explanatory variables in a mixed geographically weighted regression model. Environment and Planning A 38:587-598

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

Nakaya T et al. (2011) GWR4.0, http://gwr.nuim.ie/.

Model selection for GWR with a given set of independent variables

Description

This function selects one GWR model from many alternatives based on the AICc values.

Usage

gwr.model.selection(DeVar=NULL,InDeVars=NULL, data=list(),bw=NULL,approach="CV",
               adaptive=F,kernel="bisquare",dMat=NULL,p=2, theta=0, longlat=F,
               parallel.method=F,parallel.arg=NULL)

Arguments

Value

A list of:

Note

The algorithm for selecting GWR models consists of the following four steps:

Step 1. Start by calibrating all the possible bivariate GWR models by sequentially regressing a single independent variable against the dependent variable;

Step 2. Find the best performing model which produces the minimum AICc value, and permanently include the corresponding independent variable in subsequent models;

Step 3. Sequentially introduce a variable from the remaining group of independent variables to construct new models with the permanently included independent variables, and determine the next permanently included variable from the best fitting model that has the minimum AICc value;

Step 4. Repeat step 3 until all the independent variables are permanently included in the model.

In this procedure, the independent variables are iteratively included into the model in a "forward" direction. Note that there is a clear distinction between the different number of involved variables in a selection step, which can be called model levels.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Lu, B, Charlton, M, Harris, P, Fotheringham, AS (2014) Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data. International Journal of Geographical Information Science 28(4): 660-681

See Also

gwr.model.view, gwr.model.sort

Sort the results of the GWR model selection function gwr.model.selection.

Description

Sort the results from the GWR model selection function gwr.model.selection

Usage

gwr.model.sort(Sorting.list , numVars, ruler.vector)

Arguments

Note

The function sorts the results of model selection within individual levels.

The function “model.sort.gwr” (in the early versions of GWmodel) has been renamed as “gwr.model.sort”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

See Also

gwr.model.selection, gwr.model.view

Visualise the GWR models from gwr.model.selection

Description

This function visualises the GWR models from gwr.model.selection.

Usage

gwr.model.view(DeVar, InDeVars, model.list)

Arguments

Note

The function “model.view.gwr” (in the early versions of GWmodel) has been renamed as “gwr.model.view”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

See Also

gwr.model.selection, gwr.model.sort

Examples

## Not run: 
data(LondonHP)
DM<-gw.dist(dp.locat=coordinates(londonhp))
DeVar<-"PURCHASE"
InDeVars<-c("FLOORSZ","GARAGE1","BLDPWW1","BLDPOSTW")
model.sel<-gwr.model.selection(DeVar,InDeVars, data=londonhp,
kernel = "gaussian", dMat=DM,bw=5000)
model.list<-model.sel[[1]]
gwr.model.view(DeVar, InDeVars, model.list=model.list)

## End(Not run)

Monte Carlo (randomisation) test for significance of GWR parameter variability

Description

This function implements a Monte Carlo (randomisation) test to test for significant (spatial) variability of a GWR model's parameters or coefficients.

Usage

gwr.montecarlo(formula, data = list(),nsims=99, kernel="bisquare",adaptive=F, bw,
                         p=2, theta=0, longlat=F,dMat)

Arguments

Value

Note

The function “montecarlo.gwr” (in the early versions of GWmodel) has been renamed as “gwr.montecarlo”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Brunsdon C, Fotheringham AS, Charlton ME (1998) Geographically weighted regression - modelling spatial non-stationarity. Journal of the Royal Statistical Society, Series D-The Statistician 47(3):431-443

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

Charlton, M, Fotheringham, S, and Brunsdon, C (2007), GWR3.0.

Examples

## Not run: 
data(LondonHP)
DM<-gw.dist(dp.locat=coordinates(londonhp))
bw<-bw.gwr(PURCHASE~FLOORSZ,data=londonhp,dMat=DM, kernel="gaussian")
#See any difference in the next two commands and why?
res.mont1<-gwr.montecarlo(PURCHASE~PROF+FLOORSZ, data = londonhp,dMat=DM,
nsim=99, kernel="gaussian", adaptive=FALSE, bw=3000)
res.mont2<-gwr.montecarlo(PURCHASE~PROF+FLOORSZ, data = londonhp,dMat=DM,
nsim=99, kernel="gaussian", adaptive=FALSE, bw=300000000000)

## End(Not run)

Multiscale GWR

Description

This function implements multiscale GWR to detect variations in regression relationships across different spatial scales. This function can not only find a different bandwidth for each relationship but also (and simultaneously) find a different distance metric for each relationship (if required to do so).

Usage

gwr.multiscale(formula, data, kernel = "bisquare", adaptive = FALSE,
                 criterion = "dCVR", max.iterations = 2000, threshold =
                 1e-05, dMats, var.dMat.indx, p.vals, theta.vals,
                 longlat = FALSE, bws0, bw.seled, approach = "AIC", bws.thresholds, 
                 bws.reOpts = 5, verbose = F,
                 hatmatrix = T, predictor.centered = rep(T,
                 length(bws0) - 1), nlower = 10, parallel.method = F,
                 parallel.arg = NULL, force.armadillo = F)
## S3 method for class 'multiscalegwr'
print(x, ...)

Arguments

Value

A list of class “psdmgwr”:

Note

This function implements multiscale GWR to detect variations in regression relationships across different spatial scales. This function can not only find a different bandwidth for each relationship, but also (and simultaneously), find a different distance metric for each relationship (i.e. Parameter-Specific Distance Metric GWR, i.e. PSDM GWR). Note that multiscale GWR (MGWR) has also been referred to as flexible bandwidth GWR (FBGWR) and conditional GWR (CGWR) in the literature. All are one and the same model, but where PSDM-GWR additionally provides a different distance metric option for each relationship. An MGWR model is calibrated if no “dMats” and “p.vals” are specified; a mixed GWR model will be calibrated if an infinite bandwidth and another regular bandwidth are used for estimating the global and local parameters (again when no “dMats” and “p.vals” are specified). In other words, the gwr.multiscale function is specified with Euclidean distances in both cases. Note that the results from this function for a mixed GWR model and gwr.mixed might be different, as a back-fitting algorithm is used in gwr.multiscale, while an approximating algorithm is applied in gwr.mixed. The gwr.mixed function performs better in computational efficiency, but poorer in prediction accuracy.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Yang, W. (2014). An Extension of Geographically Weighted Regression with Flexible Bandwidths. St Andrews, St Andrews, UK.

Lu, B., Harris, P., Charlton, M., & Brunsdon, C. (2015). Calibrating a Geographically Weighted Regression Model with Parameter-specific Distance Metrics. Procedia Environmental Sciences, 26, 109-114.

Lu, B., Brunsdon, C., Charlton, M., & Harris, P. (2017). Geographically weighted regression with parameter-specific distance metrics. International Journal of Geographical Information Science, 31, 982-998.

Fotheringham, A. S., Yang, W. & Kang, W. (2017). Multiscale Geographically Weighted Regression (MGWR). Annals of the American Association of Geographers, 107, 1247-1265.

Yu, H., A. S. Fotheringham, Z. Li, T. Oshan, W. Kang & L. J. Wolf. 2019. Inference in multiscale geographically weighted regression. Geographical Analysis(In press).

Leong, Y.Y., & Yue, J.C. (2017). A modification to geographically weighted regression. International Journal of Health Geographics, 16 (1), 11.

Lu, B., Yang, W. Ge, Y. & Harris, P. (2018). Improvements to the calibration of a geographically weighted regression with parameter-specific distance metrics and bandwidths. Forthcoming Computers, Environment and Urban Systems.

Wolf, L.J, Oshan, T.M, Fotheringham, A.S. (2018). Single and multiscale models of process spatial heterogeneity. Geographical Analysis, 50(3): 223-246.

Murakami, D., Lu, B., Harris, P., Brunsdon, C., Charlton, M., Nakaya, T., & Griffith, D. (2019) The importance of scale in spatially varying coefficient modelling. Forthcoming Annals of the Association of American Geographers.

Examples

data(LondonHP)
EUDM <- gw.dist(coordinates(londonhp))
#No bandwidth is selected, and bws0 values are used
## Not run: 
###Similar as the basic GWR
res1<-gwr.multiscale(PURCHASE~FLOORSZ+PROF, data=londonhp, criterion="dCVR",kernel="gaussian", 
adaptive=T, bws0=c(100, 100, 100),bw.seled=rep(T, 3), dMats=list(EUDM,EUDM,EUDM))
#FBGWR
res2<-gwr.multiscale(PURCHASE~FLOORSZ+PROF, data=londonhp, criterion="dCVR",kernel="gaussian",
adaptive=T, bws0=c(100, 100, 100), dMats=list(EUDM,EUDM,EUDM))
#Mixed GWR
res3<-gwr.multiscale(PURCHASE~FLOORSZ+PROF, data=londonhp, bws0=c(Inf, 100, 100, Inf),
               bw.seled=rep(T, 3),kernel="gaussian", dMats=list(EUDM,EUDM,EUDM))
#PSDM GWR
res4<- gwr.multiscale(PURCHASE~FLOORSZ+PROF, data=londonhp, kernel="gaussian", p.vals=c(1,2,3))

## End(Not run)

GWR used as a spatial predictor

Description

This function implements basic GWR as a spatial predictor. The GWR prediction function is able to do leave-out-one predictions (when the observation locations are used for prediction) and predictions at a set-aside data set (when unobserved locations are used for prediction).

Usage

gwr.predict(formula, data, predictdata, bw, kernel="bisquare",adaptive=FALSE, p=2,
           theta=0, longlat=F,dMat1, dMat2)
## S3 method for class 'gwrm.pred'
print(x, ...)
           

Arguments

Value

A list of class “gwrm.pred”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Harris P, Fotheringham AS, Crespo R, Charlton M (2010) The use of geographically weighted regression for spatial prediction: an evaluation of models using simulated data sets. Mathematical Geosciences 42:657-680

Harris P, Juggins S (2011) Estimating freshwater critical load exceedance data for Great Britain using space-varying relationship models. Mathematical Geosciences 43: 265-292

Harris P, Brunsdon C, Fotheringham AS (2011) Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor. Stochastic Environmental Research and Risk Assessment 25:123-138

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

Examples

## Not run: 
data(LondonHP)
gwr.pred<-gwr.predict(PURCHASE~FLOORSZ, data=londonhp, bw=2000,kernel = "gaussian")
gwr.pred
#########Global OLS regression results and comparison with gstat functions
if(require("gstat"))
{
  mlr.g <- gstat(id = "xx1", formula = PURCHASE~FLOORSZ,data=londonhp)
  mlr.g1 <- predict(mlr.g, newdata = londonhp, BLUE = TRUE)
  mlr.g1
}
############
ols.pred<-gwr.predict(PURCHASE~FLOORSZ, data=londonhp, bw=100000000000000000000000)
ols.pred$SDF

## End(Not run)

Robust GWR model

Description

This function implements two robust GWR models.

Usage

gwr.robust(formula, data, bw,filtered=FALSE, kernel = "bisquare", adaptive = FALSE, p = 2, 
          theta = 0, longlat = F, dMat, F123.test = F, maxiter=20,cut.filter= 3,cut1=2,
          cut2=3,delta=1.0e-5, parallel.method = FALSE, parallel.arg = NULL)  

Arguments

Value

A list of class “gwrm”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

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

Examples

## Not run: 
data(DubVoter)
bw.a <- bw.gwr(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64,
data=Dub.voter,approach="AICc",kernel="bisquare",adaptive=TRUE)
bw.a
gwr.res <- gwr.basic(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64,
data=Dub.voter,bw=bw.a,kernel="bisquare",adaptive=TRUE,F123.test=TRUE)
print(gwr.res)

# Map of the estimated coefficients for LowEduc
names(gwr.res$SDF)
if(require("RColorBrewer"))
{
  mypalette<-brewer.pal(6,"Spectral")
  X11(width=10,height=12)
  spplot(gwr.res$SDF,"LowEduc",key.space = "right",
  col.regions=mypalette,at=c(-8,-6,-4,-2,0,2,4),
  main="Basic GW regression coefficient estimates for LowEduc")
}
# Robust GW regression and map of the estimated coefficients for LowEduc
rgwr.res <- gwr.robust(GenEl2004~DiffAdd+LARent+SC1+Unempl+LowEduc+Age18_24
+Age25_44+Age45_64, data=Dub.voter,bw=bw.a,kernel="bisquare",
adaptive=TRUE,F123.test=TRUE)
print(rgwr.res)
if(require("RColorBrewer"))
{
  X11(width=10,height=12)
  spplot(rgwr.res$SDF, "LowEduc", key.space = "right",
  col.regions=mypalette,at=c(-8,-6,-4,-2,0,2,4),
  main="Robust GW regression coefficient estimates for LowEduc")
}

## End(Not run)

Scalable GWR

Description

This function implements Scalable GWR for large dataset

Usage

gwr.scalable(formula, data, bw.adapt=100, kernel = "gaussian", polynomial = 4, 
             p = 2, theta = 0, longlat = F, dMat)
## S3 method for class 'scgwrm'
print(x, ...)

Arguments

Value

A list of class “scgwrm”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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.

Examples

## Not run: 
require(spData)
data(boston)
boston <- boston.c
coordinates(boston) <- ~ LON + LAT
res <- gwr.scalable(formula = MEDV ~ CRIM + ZN + INDUS + CHAS + AGE, data = boston, bw.adapt = 100)
res

## End(Not run)

Adjust p-values for multiple hypothesis tests in basic GWR

Description

Given a set of p-values from the pseudo t-tests of basic GWR outputs, this function returns adjusted p-values using: (a) Bonferroni, (b) Benjamini-Hochberg, (c) Benjamini-Yekutieli and (d) Fotheringham-Byrne procedures.

Usage

gwr.t.adjust(gwm.Obj)

Arguments

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

Byrne, G., Charlton, M. and Fotheringham, S., 2009. Multiple dependent hypothesis tests in geographically weighted regression. In: Lees, B. and Laffan, S. eds. 10th International conference on geocomputation. Sydney.

Write the GWR results into files

Description

This function writes the calibration result of function gwr.basic to a text file and shape files

Usage

gwr.write(x,fn="GWRresults")
gwr.write.shp(x,fn="GWRresults")

Arguments

Note

The projection file is missing for the writen shapefiles.

The functions “writeGWR” and “writeGWR.shp” (in the early versions of GWmodel) have been renamed respectively as “gwr.write” and “gwr.write.shp”, while the old names are still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

Geographically weighted summary statistics (GWSS)

Description

This function calculates basic and robust GWSS. This includes geographically weighted means, standard deviations and skew. Robust alternatives include geographically weighted medians, inter-quartile ranges and quantile imbalances. This function also calculates basic geographically weighted covariances together with basic and robust geographically weighted correlations.

Usage

gwss(data, summary.locat,vars,kernel="bisquare",adaptive=FALSE, bw,p=2, 
            theta=0, longlat=F,dMat,quantile=FALSE)
## S3 method for class 'gwss'
print(x, ...)

Arguments

Value

A list of class “lss”:

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

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

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

Examples

## Not run: 
data(EWHP)
data(EWOutline)
head(ewhp)
houses.spdf <- SpatialPointsDataFrame(ewhp[, 1:2], ewhp)
localstats1 <- gwss(houses.spdf, vars = c("PurPrice", "FlrArea"), bw = 50000)
head(data.frame(localstats1$SDF))
localstats1
##A function for mapping data
if(require("RColorBrewer"))
{
   quick.map <- function(spdf,var,legend.title,main.title) 
   {
     x <- spdf@data[,var]
     cut.vals <- pretty(x)
     x.cut <- cut(x,cut.vals)
     cut.levels <- levels(x.cut)
     cut.band <- match(x.cut,cut.levels)
     colors <- brewer.pal(length(cut.levels), "YlOrRd")
     colors <- rev(colors)
     par(mar=c(1,1,1,1))
     plot(ewoutline,col="olivedrab",bg="lightblue1")
     title(main.title)
     plot(spdf,add=TRUE,col=colors[cut.band],pch=16)
     legend("topleft",cut.levels,col=colors,pch=16,bty="n",title=legend.title)
  }
  quick.map(localstats1$SDF, "PurPrice_LM", "1000's Uk Pounds", 
  "Geographically Weighted Mean")
  par(mfrow = c(1, 2))
  quick.map(localstats1$SDF, "PurPrice_LSKe", "Skewness Level", "Local Skewness")
  quick.map(localstats1$SDF, "PurPrice_LSD", "1000's Pounds", "Local Standard Deviation")
  #Exploring Non-Stationarity of Relationships
  quick.map(localstats1$SDF, "Corr_PurPrice.FlrArea", expression(rho), 
  "Geographically Weighted Pearson Correlation")
  #Robust, Quantile Based Local Summary Statistics
  localstats2 <- gwss(houses.spdf, vars = c("PurPrice", "FlrArea"), 
  bw = 50000, quantile = TRUE)
  quick.map(localstats2$SDF, "PurPrice_Median", "1000 UK Pounds", 
  "Geographically Weighted Median House Price")
}

## End(Not run)

Monte Carlo (randomisation) test for gwss

Description

This function implements Monte Carlo (randomisation) tests for the GW summary statistics found in gwss.

Usage

gwss.montecarlo(data, vars, kernel = "bisquare", 
                adaptive = FALSE, bw, p = 2, theta = 0, longlat = F, 
                dMat, quantile=FALSE,nsim=99) 

Arguments

Value

Note

The function “montecarlo.gwss” (in the early versions of GWmodel) has been renamed as “gwss.montecarlo”, while the old name is still kept valid.

Author(s)

Binbin Lu binbinlu@whu.edu.cn

References

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

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

Harris P, Brunsdon C (2010) Exploring spatial variation and spatial relationships in a freshwater acidification critical load data set for Great Britain using geographically weighted summary statistics. Computers & Geosciences 36:54-70

Examples

## Not run: 
data(LondonHP)
DM<-gw.dist(dp.locat=coordinates(londonhp))
test.lss<-gwss.montecarlo(data=londonhp, vars=c("PURCHASE","FLOORSZ"), bw=5000,
          kernel ="gaussian", dMat=DM,nsim=99)
test.lss

## End(Not run)

Spatio-temporal distance matrix calculation

Description

Calculate a distance vector(matrix) between any GW model calibration point(s) and the data points.

Usage

st.dist(dp.locat, rp.locat, obs.tv, reg.tv,focus=0, p=2, 
        theta=0, longlat=F,lamda=0.05,t.units = "auto",
        ksi=0, s.dMat,t.dMat)

Arguments

Value

Returns a numeric spatio-temporal distance matrix or vector; or a matrix with its rows corresponding to the observations and its columns corresponds to the calibration points.

Author(s)

Binbin Lu binbinlu@whu.edu.cn