Predictable components and singular vectors

This paper shows that if a measure of predictability is invariant to affine transformations and monotonically related to forecast uncertainty, then the component that maximizes this measure for normally distributed variables is independent of the detailed form of the measure. This result explains why different measures of predictability such as anomaly correlation, signal-to-noise ratio, predictive information, and the Mahalanobis error are each maximized by the same components. These components can be determined by applying principal component analysis to a transformed forecast ensemble, a procedure called predictable component analysis (PrCA). The resulting vectors define a complete set of components that can be ordered such that the first maximizes predictability, the second maximizes predictability subject to being uncorrelated of the first, and so on. The transformation in question, called the whitening transformation, can be interpreted as changing the norm in principal component analysis. The resulting norm renders noise variance analysis equivalent to signal variance analysis, whereas these two analyses lead to inconsistent results if other norms are chosen to define variance. Predictable components also can be determined by applying singular value decomposition to a whitened propagator in linear models. The whitening transformation is tantamount to changing the initial and final norms in the singular vector calculation. The norm for measuring forecast uncertainty has not appeared in prior predictability studies. Nevertheless, the norms that emerge from this framework have several attractive properties that make their use compelling. This framework generalizes singular vector methods to models with both stochastic forcing and initial condition error. These and other components of interest to predictability are illustrated with an empirical model for sea surface temperature.