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This paper introduces the average predictability time (APT) for characterizing the overall predictability of a system. APT is the integral of a predictability measure over all lead times. The underlying predictability measure is based on the Mahalanobis metric, which is invariant to linear transformation of the prediction variables and hence gives results that are independent of the (arbitrary) basis set used to represent the state. The APT is superior to some integral time scales used to characterize the time scale of a random process because the latter vanishes in situations when it should not, whereas the APT converges to reasonable values. The APT also can be written in terms of the power spectrum, thereby clarifying the connection between predictability and the power spectrum. In essence, predictability is related to the width of spectral peaks, with strong, narrow peaks associated with high predictability and nearly flat spectra associated with low predictability. Closed form expressions for the APT for linear stochastic models are derived. For a given dynamical operator, the stochastic forcing that minimizes APT is one that allows transformation of the original stochastic model into a set of uncoupled, independent stochastic models. Loosely speaking, coupling enhances predictability. A rigorous upper bound on the predictability of linear stochastic models is derived, which clarifies the connection between predictability at short and long lead times, as well as the choice of norm for measuring error growth. Surprisingly, APT can itself be interpreted as the “total variance” of an alternative stochastic model, which means that generalized stability theory and dynamical systems theory can be used to understand APT. The APT can be decomposed into an uncorrelated set of components that maximize predictability time, analogous to the way principle component analysis decomposes variance. Part II of this paper develops a practical method for performing this decomposition and applies it to meteorological data. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for diagnosing predictability on multiple time scales without time averaging. The method finds components that maximize the average predictability time (APT) of a system, where APT is defined as the integral of the average predictability over all lead times. Basing the predictability measure on the Mahalanobis metric leads to a complete, uncorrelated set of components that can be ordered by their contribution to APT, analogous to the way principal components decompose variance. The components and associated APTs are invariant to nonsingular linear transformations, allowing variables with different units and natural variability to be considered in a single state vector without normalization. For prediction models derived from linear regression, maximizing APT is equivalent to maximizing the sum of squared multiple correlations between the component and the time-lagged state vector. The new method is used to diagnose predictability of 1000-hPa zonal velocity on time scales from 6 h to decades. The leading predictable component is dominated by a linear trend and presumably identifies a climate change signal. The next component is strongly correlated with ENSO indices and hence is identified with seasonal-to-interannual predictability. The third component is related to annular modes and presents decadal variability as well as a trend. The next few components have APTs exceeding 10 days. A reconstruction of the tropical zonal wind field based on the leading seven components reveals eastward propagation of anomalies with time scales consistent with the Madden–Julian oscillation. The remaining components have time scales less than a week and hence are identified with weather predictability. The detection of predictability on these time scales without time averaging is possible by virtue of the fact that predictability on different time scales is characterized by different spatial structures, which can be optimally extracted by suitable projections. [ABSTRACT FROM AUTHOR]