Constrained Regression in Satellite Meteorology

Least squares or regression techniques have been used for many problems in satellite meteorology. Because of the large number of variables and the linear dependence among these variables, colinearity causes significant problems in the application of standard regression techniques. In some of the applications there is prior knowledge about the values of the regression parameters. Since there are errors in the predictor variables as well as the predictand variables, the standard assumptions for ordinary least squares are not valid. In this paper the authors examine several techniques that have been developed to ameliorate the effects of colinearity or to make use of prior information. These include ridge regression, shrinkage estimators, rotated regression, and orthogonal regression. In order to illustrate the techniques and their properties, the authors apply them to two simple examples. These techniques are then applied to a real problem in satellite meteorology: that of estimating theoretical computed brightness temperatures from measured brightness temperatures. It is found that the rotated and the shrinkage estimators make good use of the prior information and help solve the colinearity problem. Ordinary least squares, ridge regression, and orthogonal regression give unsatisfactory results. Theoretical results for the various techniques are given in an appendix.