Parametric (modified least squares) and non-parametric (Theil-Sen) linear regressions for predicting biophysical parameters in the presence of measurement errors

Remote sensing often involves the estimation of in situ quantities from remote measurements. Linear regression, where there are no non-linear combinations of regressors, is a common approach to this prediction problem in the remote sensing community. A review of recent remote sensing articles using univariate linear regression indicates that in the majority of cases, ordinary least squares (OLS) linear regression has been applied, with approximately half the articles using the in situ observations as regressors and the other half using the inverse regression with remote measurements as regressors. OLS implicitly assume an underlying normal structural data model to arrive at unbiased estimates of the response. OLS regression can be a biased predictor in the presence of measurement errors when the regression problem is based on a functional rather than structural data model. Parametric (Modified Least Squares) and non-parametric (Theil-Sen) consistent predictors are given for linear regression in the presence of measurement errors together with analytical approximations of their prediction confidence intervals. Three case studies involving estimation of leaf area index from nadir reflectance estimates are used to compare these unbiased estimators with OLS linear regression. A comparison to Geometric Mean regression, a standardized version of Reduced Major Axis regression, is also performed. The Theil-Sen approach is suggested as a potential replacement of OLS for linear regression in remote sensing applications. It offers simplicity in computation, analytical estimates of confidence intervals, robustness to outliers, testable assumptions regarding residuals and requires limited a priori information regarding measurement errors. Crown Copyright (c) 2005 Published by Elsevier Inc. All rights reserved.