Examples of Additionally Constrained Multiple Linear Regression

Abstract

An additionally constrained multiple linear regression technique is outlined for use with the prediction or specification of fields of geophysical variables. The additional constraints are the requirement that the predictions or specifications have spatial interrelationships which are similar to the dominant empirical orthogonal functions of the dependent variables. Since the traditional multiple linear regression estimates by definition minimize the mean square errors for the data used to develop the model, the utility of the additional constraint can be evaluated only when the regression models are tested on new data. A so-called jackknife technique is used in this regard. This additionally constrained multiple linear regression technique is tested on two examples of the specification of monthly values using independent data for the same month. The first is the specification of North Pacific sector 700-mb geopotential heights using the time coefficients of the first two empirical orthogonal functions of Pacific sea surface temperature. The second is the specification of monthly precipitation totals at 36 stations in the western United States using 700-mb heights. Use of the additional constraint leads to average increases in observed-predicted correlations of between 5 and 30% when used with new data. These improvements are quite evenly distributed over nearly all of the points of the fields of the dependent variable.