Constructed Analogs and Linear Regression

Abstract

Abstract The constructed analog procedure produces a statistical forecast that is a linear combination of past predictand values. The weights used to form the linear combination depend on the current predictor value and are chosen so that the linear combination of past predictor values approximates the current predictor value. The properties of the constructed analog method have previously been described as being distinct from those of linear regression. However, here the authors show that standard implementations of the constructed analog method give forecasts that are identical to linear regression forecasts. A consequence of this equivalence is that constructed analog forecasts based on many predictors tend to suffer from overfitting just as in linear regression. Differences between linear regression and constructed analog forecasts only result from implementation choices, especially ones related to the preparation and truncation of data. Two particular constructed analog implementations are shown to correspond to principal component regression and ridge regression. The equality of linear regression and constructed analog forecasts is illustrated in a Niño-3.4 prediction example, which also shows that increasing the number of predictors results in low-skill, high-variance forecasts, even at long leads, behavior typical of overfitting. Alternative definitions of the analog weights lead naturally to nonlinear extensions of linear regression such as local linear regression.