Regression II

Abstract

Regression serves in this chapter to relate two climate variables, X(i) and Y(i). This is a standard tool for formulating a quantitative "climate theory" based on equations. Owing to the complexity of the climate system, such a theory can never be derived alone from the pure laws of physics-it requires to establish empirical relations between observed climate processes. Since not only Y(i) but also X(i) are observed with error, the relation has to be formulated as an errors-in-variables model, and the estimation has to be carried out using adaptions of the OLS technique. This chapter focuses on the linear model and studies three estimation techniques (denoted as OLSBC, WLSXY and Wald-Bartlett procedure). It presents a novel bivariate resampling approach (pairwise-MBBres), which enhances the coverage performance of bootstrap CIs for the estimated regression parameters. Monte Carlo simulations allow to assess the role of various aspects of the estimation. First, prior knowledge about the size of the measurement errors is indispensable to yield a consistent estimation. If this knowledge is not exact, which is typical for a situation in the climatological practice, it contributes to the estimation error of the slope (RMSE and CI length). This contribution persists even when the data size goes to infinity; the RMSE does then not approach zero. Second, autocorrelation has to be taken into account to prevent estimation errors unrealistically small and CIs too narrow. This chapter studies two extensions of high relevance for climatological applications: linear prediction and lagged regression. Regression as a method to estimate the trend in the climate equation (Eq. 1.2) is presented in Chap. 4. PU - SPRINGER PI - DORDRECHT PA - PO BOX 17, 3300 AA DORDRECHT, NETHERLANDS