Introducing the GWmodel R and python packages for modelling spatial heterogeneity

In the very early developments of quantitative geography, statistical techniques were invariably applied at a ‘global’ level, where moments or relationships were assumed constant across the study region (Fotheringham and Brunsdon, 1999). However, the world is not an “average” space but full of variations and as such, statistical techniques need to account for different forms of spatial heterogeneity or non-stationarity (Goodchild, 2004). Consequently, a number of local methods were developed, many of which model non- stationarity relationships via some regression adaptation. Examples include: the expansion method (Casetti, 1972), random coefficient modelling (Swamy et al., 1988), multilevel modelling (Duncan and Jones, 2000) and space varying parameter models (Assunção, 2003). One such localised regression, geographically weighted regression (GWR) (Brunsdon et al., 1996) has become increasingly popular and has been broadly applied in many disciplines outside of its quantitative geography roots. This includes: regional economics, urban and regional analysis, sociology and ecology. There are several toolkits available for applying GWR, such as GWR3.x (Charlton et al., 2007); GWR 4.0 (Nakaya et al., 2009); the GWR toolkit in ArcGIS (ESRI, 2009); the R packages spgwr (Bivand and Yu, 2006) and gwrr (Wheeler, 2011); and STIS (Arbor, 2010). Most focus on the fundamental functions of GWR or some specific issue - for example, gwrr provides tools to diagnose collinearity. As a major extension, we report in this paper the development an integrated framework for handling spatially varying structures, via a wide range of geographically weighted (GW) models, not just GWR. All functions are included in an R package named GWmodel, which is also mirrored with a set of GW modelling tools for ESRI’s ArcGIS written in Python

Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor

In this study, we link and compare the geographically weighted regression (GWR) model with the kriging with an external drift (KED) model of geostatistics. This includes empirical work where models are performance tested with respect to prediction and prediction uncertainty accuracy. In basic forms, GWR and KED (specified with local neighbourhoods) both cater for nonstationary correlations (i.e. the process is heteroskedastic with respect to relationships between the variable of interest and its covariates) and as such, can predict more accurately than models that do not. Furthermore, on specification of an additional heteroskedastic term to the same models (now with respect to a process variance), locallyaccurate measures of prediction uncertainty can result. These heteroskedastic extensions of GWR and KED can be preferred to basic constructions, whose measures of prediction uncertainty are only ever likely to be globallyaccurate. We evaluate both basic and heteroskedastic GWR and KED models using a case study data set, where data relationships are known to vary across space. Here GWR performs well with respect to the more involved KED model and as such, GWR is considered a viable alternative to the more established model in this particular comparison. Our study adds to a growing body of empirical evidence that GWR can be a worthy predictor; complementing its more usual guise as an exploratory technique for investigating relationships in multivariate spatial data sets

Package ‘GWmodel’

In GWmodel, we introduce techniques from a particular branch of spatial statis- tics,termed geographically-weighted (GW) models. GW models suit situa- tions when data are not described well by some global model, but where there are spatial re- gions where a suitably localised calibration provides a better description. GWmodel in- cludes functions to calibrate: GW summary statistics, GW principal components analy- sis, GW discriminant analysis and various forms of GW regression; some of which are pro- vided in basic and robust (outlier resistant) forms

Package ‘GWmodel’

In GWmodel, we introduce techniques from a particular branch of spatial statis- tics,termed geographically-weighted (GW) models. GW models suit situa- tions when data are not described well by some global model, but where there are spatial re- gions where a suitably localised calibration provides a better description. GWmodel in- cludes functions to calibrate: GW summary statistics, GW principal components analy- sis, GW discriminant analysis and various forms of GW regression; some of which are pro- vided in basic and robust (outlier resistant) forms

Novel approaches to investigating spatial variability in channel bank total phosphorus at the catchment scale

Phosphorus (P) is often a limiting nutrient that leads to the eutrophication of aquatic systems. While dissolved P forms are the most bioavailable, the form, mobility, transport and fate of P are directly related to its association with fine-grained riverine sediment. Therefore, to implement successful P catchment management strategies it is important to understand the relative contribution of different sediment sources to P loads across the river continuum. While agricultural topsoil and, to a lesser extent, riverbed sediment are important sources of sediment-associated P, channel banks have been shown to be an important sediment source in some catchments. However, comparatively little is known about the P concentration and corresponding spatial variability in channel bank sediment and the associated implications for catchment management. The present study examines the spatial variability of P associated with channel bank profiles within a series of three nested catchments using both non-spatial and spatial statistical methods, where for the latter, a novel spatial approach was used to estimate the spatial averages and variances of P in channel bank sediment along the stream network. Channel bank P concentrations were compared to factors such as catchment scale, stream order, land use, bank exposure and location along the stream network. Concentrations of TP ranged between 129.6 and 1206.9 mg P kg⁻¹ of which the water extractable P (WEP) content ranged from 0.01 to 0.12%. Stream order was found to influence TP concentrations, while land use and catchment scale provided only a moderate influence. This suggested that focussing channel bank sampling strategies at the largest catchment scale would capture key drivers of TP variability provided stream order is sufficiently represented. Whether the bank was had limited vegetation and was exposed and potentially eroding had a slight influence on TP variability in second-order stream banks in the larger of the two nested catchments. However, the slightly lower TP concentrations measured at these sites indicated that banks that are actually eroding may be contributing less TP than the total channel bank TP values measured across the catchments as a whole. The results of an explicitly spatial analysis demonstrated that local channel bank TP averages and TP variances vary along the stream network. Specifically, the most accurate spatial predictor of TP was local TP means with the use of ‘crow flies’ rather than stream network distances. Local TP variances were used to provide optimal designs for future channel bank TP sampling campaigns, given available resources. Throughout, both standard and outlier-resistant statistical analyses were applied to improve interpretation of the study findings

gwpcorMapper: an interactive mapping tool for exploring geographically weighted correlation and partial correlation in high-dimensional geospatial datasets

Exploratory spatial data analysis (ESDA) plays a key role in research that includes geographic data. In ESDA, analysts often want to be able to visualize observations and local relationships on a map. However, software dedicated to visualizing local spatial relations be-tween multiple variables in high dimensional datasets remains undeveloped. This paper introduces gwpcorMapper, a newly developed software application for mapping geographically weighted correlation and partial correlation in large multivariate datasets. gwpcorMap-per facilitates ESDA by giving researchers the ability to interact with map components that describe local correlative relationships. We built gwpcorMapper using the R Shiny framework. The software inherits its core algorithm from GWpcor, an R library for calculating the geographically weighted correlation and partial correlation statistics. We demonstrate the application of gwpcorMapper by using it to explore census data in order to find meaningful relationships that describe the work-life environment in the 23 special wards of Tokyo, Japan. We show that gwpcorMapper is useful in both variable selection and parameter tuning for geographically weighted statistics. gwpcorMapper highlights that there are strong statistically clear local variations in the relationship between the number of commuters and the total number of hours worked when considering the total population in each district across the 23 special wards of Tokyo. Our application demonstrates that the ESDA process with high-dimensional geospatial data using gwpcorMapper has applications across multiple fields.Comment: 18 pages, 8 figures, 2 table

Investigating spatial error structures in continuous raster data

The objective of this study is to investigate spatial structures of error in the assessment of continuous raster data. The use of conventional diagnostics of error often overlooks the possible spatial variation in error because such diagnostics report only average error or deviation between predicted and reference values. In this respect, this work uses a moving window (kernel) approach to generate geographically weighted (GW) versions of the mean signed deviation, the mean absolute error and the root mean squared error and to quantify their spatial variations. Such approach computes local error diagnostics from data weighted by its distance to the centre of a moving kernel and allows to map spatial surfaces of each type of error. In addition, a GW correlation analysis between predicted and reference values provides an alternative view of local error. These diagnostics are applied to two earth observation case studies. The results reveal important spatial structures of error and unusual clusters of error can be identified through Monte Carlo permutation tests. The first case study demonstrates the use of GW diagnostics to fractional impervious surface area datasets generated by four different models for the Jakarta metropolitan area, Indonesia. The GW diagnostics reveal where the models perform differently and similarly, and found areas of under-prediction in the urban core, with larger errors in peri-urban areas. The second case study uses the GW diagnostics to four remotely sensed aboveground biomass datasets for the Yucatan Peninsula, Mexico. The mapping of GW diagnostics provides a means to compare the accuracy of these four continuous raster datasets locally. The discussion considers the relative nature of diagnostics of error, determining moving window size and issues around the interpretation of different error diagnostic measures. Investigating spatial structures of error hidden in conventional diagnostics of error provides informative descriptions of error in continuous raster data

The GWmodel R package: Further Topics for Exploring Spatial Heterogeneity using Geographically Weighted Models

In this study, we present a collection of local models, termed geographically weighted (GW) models, that can be found within the GWmodel R package. A GW model suits situations when spatial data are poorly described by the global form, and for some regions the localised fit provides a better description. The approach uses a moving window weighting technique, where a collection of local models are estimated at target locations. Commonly, model parameters or outputs are mapped so that the nature of spatial heterogeneity can be explored and assessed. In particular, we present case studies using: (i) GW summary statistics and a GW principal components analysis; (ii) advanced GW regression fits and diagnostics; (iii) associated Monte Carlo significance tests for non-stationarity; (iv) a GW discriminant analysis; and (v) enhanced kernel bandwidth selection procedures. General Election data sets from the Republic of Ireland and US are used for demonstration. This study is designed to complement a companion GWmodel study, which focuses on basic and robust GW models

GWmodel: an R package for exploring spatial heterogeneity using geographically weighted models

Spatial statistics is a growing discipline providing important analytical techniques in a wide range of disciplines in the natural and social sciences. In the R package GWmodel we present 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 localized calibration provides a better description. The approach uses a moving window weighting technique, where localized models are found at target locations. Outputs are mapped to provide a useful exploratory tool into the nature of the data spatial heterogeneity. Currently, GWmodel includes functions for: GW summary statistics, GW principal components analysis, GW regression, and GW discriminant analysis; some of which are provided in basic and robust forms