Elsevier

Remote Sensing of Environment

Volume 88, Issue 3, 15 December 2003, Pages 283-293
Remote Sensing of Environment

Geographical weighting as a further refinement to regression modelling: An example focused on the NDVI–rainfall relationship

https://doi.org/10.1016/j.rse.2003.08.004Get rights and content

Abstract

The regression analyses undertaken commonly in remote sensing are aspatial, ignoring the locational information associated with each sample site at which the variables under study were measured. Typically, basic ordinary least squares regression analysis is used to derive a relationship that is believed to be uniformly applicable across the study area. Although such global analyses may appear satisfactory, often with large coefficients of determination derived, they may provide an inappropriate description of the relationship between the variables under study. In particular, a global regression analysis may miss local detail that can be significant if the relationship is spatially non-stationary. Local statistical approaches, such as geographically weighted regression, include the spatial coordinates of the sample sites in the analysis and may provide a more appropriate basis for the investigation of the relationship between variables. The potential value of geographically weighted regression to the remote sensing community is illustrated with reference to the relationship between the normalised difference vegetation index (NDVI) and rainfall over north Africa and the Middle East over an 8-year period. For each year, spatial non-stationarity was evident, particularly with regard to the slope parameter of the regression model. Moreover, the conventional ordinary least squares regression models, while superficially strong (minimum R2=0.67), were relatively poor local descriptors of the relationship. Relative to this, the geographically weighted approach to regression provided considerably stronger relationships from the same data sets (minimum R2=0.96) as well as highlighting areas of local variation. The implications of the difference in the outputs from the two types of regression analysis are illustrated with reference to the use of the derived NDVI–rainfall relationships in mapping desert extent. For example, with the data relating to 1987 the southern limit of the Sahara was generally estimated to lie at a more southerly position when the relationship derived from OLS rather than geographically weighted regression was used.

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 formy=α+βx+ε.In 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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