Comparing climate time series – Part 3: Discriminant analysis

Abstract

Abstract. In parts I and II of this paper series, rigorous tests for equality of stochastic processes were proposed. These tests provide objective criteria for deciding whether two processes differ, but they provide no information about the nature of those differences. This paper develops a systematic and optimal approach to diagnosing differences between multivariate stochastic processes. Like the tests, the diagnostics are framed in terms of vector autoregressive (VAR) models, which can be viewed as a dynamical system forced by random noise. The tests depend on two statistics, one that measures dissimilarity in dynamical operators and another that measures dissimilarity in noise covariances. Under suitable assumptions, these statistics are independent and can be tested separately for significance. If a term is significant, then the linear combination of variables that maximizes that term is obtained. The resulting indices contain all relevant information about differences between data sets. These techniques are applied to diagnose how the variability of annual-mean North Atlantic sea surface temperature differs between climate models and observations. For most models, differences in both noise processes and dynamics are important. Over 40 % of the differences in noise statistics can be explained by one or two discriminant components, though these components can be model dependent. Maximizing dissimilarity in dynamical operators identifies situations in which some climate models predict large-scale anomalies with the wrong sign.