Geographical weighting as a further refinement to regression modelling: An example focused on the NDVI–rainfall relationship
Introduction
Regression techniques have been used widely in remote sensing. Frequently, regression has been used to describe the relationship between an environmental variable measured at the Earth's surface (e.g. biomass) and some measure of its associated remotely sensed response (e.g. a vegetation index). Often, the regression analysis is undertaken with the aim of using the model formed to make predictions of the environmental variable at other sites from their remotely sensed response. Although a variety of approaches to regression modelling exist (Curran & Hay, 1986), the remote sensing community has tended to use uncritically conventional ordinary least squared (OLS) regression analysis (Cohen, Maiersperger, Gower, & Turner, 2003). Since OLS regression has important limitations, its use may not always be appropriate and alternatives should be evaluated Cohen et al., 2003, Curran & Hay, 1986.
Cohen et al. (2003) present an improved strategy for regression modelling in remote sensing. Recognizing the need to critically assess the techniques used commonly in research and considering the merits of alternative methods, they illustrate some of the different options to OLS regression that may be of immense value to the remote sensing community. One aspect that is infrequently addressed is that the regression analyses commonly used in remote sensing are global techniques, with a single set of model parameters taken to apply uniformly in space. Such analyses are based implicitly on an assumption that the relationship is spatially stationary. The assumption of spatial stationarity in a relationship may often be untenable, particularly when considering the large area of coverage provided by remote sensing, and a local technique may be more appropriate (Maselli, 2002). The aim of this article is to outline to the remote sensing community a recent refinement to regression modelling, geographical weighting, that has attracted interest within the geographical research community (Fotheringham, Brunsdon, & Charlton, 2002) and which has attractive features for use with remotely sensed data. In particular, geographically weighted regression is a local technique that allows the regression model parameters to vary in space. This paper will briefly introduce the salient features of geographically weighted regression and then illustrate its application in comparison to standard OLS regression with regard to the widely used relationship between the normalised difference vegetation index (NDVI) and rainfall.
Section snippets
Geographically weighted regression
The basic linear regression model that has been used widely in remote sensing may be expressed in the formIn this model, the two variables to be related are y, the dependent variable, and x, the independent variable. Typically in remote sensing studies, y is a remotely sensed variable and x the environmental variable of interest. The remaining parts of the model are its parameters, α which represents the intercept and β which expresses the slope of the relationship between the two
Example: the NDVI–rainfall relationship
Vegetation amount and condition are a function of environmental variables such as rainfall. Consequently, a strong relationship, involving a brief time–lag in the vegetation response to rainfall, would be expected between vegetation indices, such as the NDVI[(infrared reflectance (IR)−red reflectance (R))/(IR+R)] and rainfall Li et al., 2002, Potter & Brooks, 1998, Richard & Poccard, 1998.
Many studies have focused on the relationship between the NDVI and rainfall. These studies have been
Summary and conclusion
Regression analysis is used widely in remote sensing. From the range of regression techniques available, OLS regression is generally used unquestioningly. OLS regression, however, may not always be appropriate and other approaches have been suggested for use by the remote sensing research community Cohen et al., 2003, Curran & Hay, 1986. One further refinement that could be added to the suggestions made by previous authors is the use of geographically weighted regression (Fotheringham et al.,
Acknowledgements
I am grateful for the data sets used that are available in the public domain from major activities funded through the NASA Mission to Planet Earth program. Specifically, the NDVI and rainfall data were extracted from the Climatology Interdisciplinary Data Collection (CIDC), a Goddard Space Flight Center activity sponsored by the NASA Earth Science Enterprise, and the ancillary data on soils and land cover were extracted from the Distributed Active Archive Center (Code 902.2) at Goddard Space
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