Robust Geographically Weighted Regression: A Technique for Quantifying Spatial Relationships Between Freshwater Acidification Critical Loads and Catchment Attributes

Abstract

Geographically weighted regression (GWR) is used to investigate spatial relationships between freshwater acidification critical load data and contextual catchment data across Great Britain. Although this analysis is important in developing a greater understanding of the critical load process, the study also examines the application of the GWR technique itself. In particular, and unlike many previous presentations of GWR, the steps taken in choosing a particular GWR model form are presented in detail. A further important advance here is that the calibration results of the chosen GWR model are scrutinized for robustness to outlying observations. With respect to the critical load process itself, the results of this study largely agree with those of earlier research, where relationships between critical load and catchment data can vary across space. The more sophisticated spatial statistical models used here, however, are shown to be more flexible and informative, allowing a clearer picture of process heterogeneities to be revealed.

La regresión geográfica ponderada (GWR, por su acrónimo en inglés) se utiliza para investigar las relaciones espaciales existentes entre los datos de carga crítica de acidificación de agua dulce y los datos contextuales de desagüe, en Gran Bretaña. Aunque tal tipo de análisis es importante para desarrollar una mejor comprensión del proceso de carga crítica, el estudio examina también la aplicación de la propia técnica GWR. En particular, y a diferencia de muchas presentaciones previas de la GWR, se muestran en detalle los pasos que se siguen para escoger una determinada forma de modelo de GWR. Un importante avance adicional es que aquí la calibración de resultados del modelo GWR escogido es escudriñada por robustez mediante observaciones externas. Respecto al proceso de carga en sí mismo, los resultados del estudio en gran medida concuerdan con los de investigaciones anteriores, donde las relaciones entre la carga crítica y los datos de desagüe pueden variar a través del espacio. Los más sofisticados modelos estadísticos espaciales utilizados aquí, sin embargo, demuestran ser más flexibles e informativos, permitiendo un cuadro más claro del proceso de las heterogeneidades en trance de ser reveladas.

Key Words:

• acidified surface waters
• catchment characteristics
• relationship nonstationarity
• robust
• spatial heterogeneity

关键词:

• 酸化的地表水
• 流域特征
• 非平稳的关系
• 稳定
• 空间的异质性

Palabras clave:

• aguas superficiales acidificadas
• características del desagüe
• relación de estacionariedad
• robustez
• heterogeneidad espacial

Notes

1. Further work on outliers and GWR can be found in Farber and Páez (2007), where the cross-validation approach to bandwidth selection is made robust to outlying observations. Unfortunately, this methodology is not directly transferable to the AIC approach used here but (as the authors note) would be a topic suitable for further research.

2. The approach is similar to that used when replacing a global estimate of the error variance with a local estimate in the spatially heteroscedastic GWR model of Fotheringham, Brunsdon, and Charlton (2002).

3. Base saturation reflects the extent of soil acidification.

4. Regardless of the kernel type used, all first-stage GWR models are calibrated with the small random error addition to the catchment data for an unbiased comparison (although the MLR models still use the original data). Here the GWR models were also rerun a few times to check for any strongly different results with different random error additions (but still within the limits given before).

5. Note that as GWR with an exponential kernel is chosen, it is now unnecessary to have the random error addition for this and further model calibration.

6. This and all other study surfaces are on a rectangular 350E 500N grid.

7. Monte Carlo tests are used to evaluate whether local parameter estimates vary significantly across space. Here the sample data are successively randomized and after each randomization the GWR model is applied and the variance of a given parameter estimate calculated. The actual variance of the same parameter estimate is then included in the ranked distribution of variances. Its position in this ranked distribution relates to whether there is significant (spatial) variation in this parameter estimate.

8. In the spirit of exploration, presenting the GWR surfaces using more than one kernel type would also have been appropriate (e.g., Lloyd and Shuttleworth 2005).