Geographically Weighted Regression (GWR) is a regression technique that extends the traditional regression framework by allowing the estimation of local rather than global parameters. In other words, GWR runs a regression for each location, instead of a sole regression for the entire study area. GWR is a useful regression model to work with non-stationary data. The term stationarity refers to relationships in which the influences of the independent variables remain constant over the dependent variable throughout time and space. On the other hand, local or non-stationary models (e.g. GWR) account for different responses in different parts of the study region that the independent variables produce over the dependent variable. In GWR, observations are weighted in accordance with their proximity to point I (determined by the kernel size). This ensures that the weighting of an observation is no longer constant in the calibration, but instead varies with I. As a result, observations closer to I have a stronger influence on the estimation of the parameters for location i. Basically, GWR uses a kernel (also called window or bandwidth) that moves over the study area and seeks to fit the best results for each subarea. The Geographic Weighted Regression tool in ArcGIS offers the opportunity to work with an adjusted kernel (set by the user) that changes its size as it moves throughout the area under analysis. The kernel size defines the rate at which the influence of the coefficients decreases as the distance increases.
Figure 1. Regression framework with independent variables and the dependent variable, crime.
Every single location within the study area has its own set of coefficients; this allows the model to produce an individual r2 value for each location. It is recommended to map the coefficients and r2 values to observe how the relationship between the dependent and independent/s variables fluctuates throughout the area under study. This procedure also allows the user to observe how the predictive capabilities of the model vary across space (Figure 2). GWR also provides an overall R2 output value that can be compared to R2 values obtained from different regression models, such as Ordinary Least Squares (OLS). The GWR analysis on ArcGIS also provides a t-score output. The researcher can map the t-scores for each location to detect confidence levels (0.01 and 0.05). The t-score map would show where the dependence is statistically significant for each independent variable.
Figure 2. Local R2 output.
Other outputs produced by the GWR tool on ArcGIS are the residuals and the standard deviations of those residuals. The residuals represent the amount of variability that remains unexplained by the model (i.e. variability on the dependent variable not explained by the independent variables). The standard deviations of the residuals also show the locations where the model over and unpredicted the value of the dependent variable (Figure 3).
Figure 3. GWR Residuals.
The table below shows how Brinkmann et al. (2011) Improved the explanatory power of their regression model (R2 values) when using GWR instead of OLS. The authors studied rangeland productivity and anthropogenic degradation in the Arabian Peninsula.
Kupfer and Farris (2007) also used OLS and GWR to predict patterns of montane ponderosa pine (Pinus ponderosa) basal area in Saguaro National Park, AZ, USA on the basis of variables related to topography (elevation, slope steepness, aspect) and fire history (fire frequency, time since fire). See Figure 4 below. The authors concluded that GWR provides a better way to understand relationships affecting ponderosa pine distribution in the study area than OLS.
Prediction map, map of prediction variance,regression residuals, summary statistics of model performance
Brinkmann, K., U. Dickhoefer, E. Schlecht, and A. Buerkert. 2011. Quantification of aboveground rangeland productivity and anthropogenic degradation on the Arabian Peninsula using Landsat imagery and field inventory data. In Remote Sensing of Environment. 115:2 (465-475).
Kazmierczak, T. 2010. Rangeland to cropland conversion in the central Great Plains. Northern Illinois University Press.
Kupfer, J. and C. Farris. 2010. Incorporating spatial non-stationarity of regression coefficients into predictive vegetation models. In Landscape Ecology. 22:6 (837-852). DOI: 10.1007/s10980-006-9058-2
Sharma, V., S. Irmak, I. Kabenge. 2011. Application of GIS and Geographically Weighted Regression to Evaluate the Spatial Non-Stationarity Relationships between Precipitation Vs. Irrigated and Rainfed Maize and Soybean Yields. In American Society of Agricultural and Biological Engineers. 54:3 (953-972).
ESRI. 2010. Arc GIS Resource Center. ESRI.
Fotheringham, A., M. Charlton, and C. Brundson. 1998. Geographically weighted regression: a natural evolution of the expansion method for spatial data analysis. In Environment and Planning. 30: 1905-1927.
Gonzalez, L. 2012. An Analysis of the Sociolinguistic Situation of Galicia, Spain. Las Cruces, NM: New Mexico State University Press.
McGrew, J. and C. Monroe. 2009. An Introduction to Statistical Problem Solving in Geography. 2nd edn. Long Grove, IL: Waveland Press, Inc.
Mitchell, A. 2009. The ESRI Guide to GIS Analysis. Volume 2: Spatial Measurements and Statistics. Redlands, CA: Esri Press.
GWR analysis can be performed in several GIS applications like ESRI’s ArcGIS Spatial Analyst
Applying GWR in ArcGIS http://www.esri.com/news/arcuser/0309/re_gwr.html
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