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# ho - Information-Theoretic Strategies for Quantifying...

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Information-Theoretic Strategies for Quantifying Variability and Model-Reality Comparison in the Climate System 1 , 2 , 3 J. W. Larson 1 Mathematics and Computer Science Division, Argonne National Laboratory 9700 S. Cass Avenue, Argonne, IL 60439, USA E-Mail: [email protected] 2 Computation Institute, University of Chicago Chicago, IL, USA 3 Department of Computer Science, The Australian National University Canberra ACT 0200, Australia Keywords: Information Theory; Statistics; Climate Data Analysis Model-reality comparison can be viewed in a communications context. In this analogy, the observed “real” data are a sent message, and the model output are the received message. The model plays the role of a noisy channel over which the message is transmitted (Figure 1). Information theory offers a way to assess literally the “information content” of any system, and offers a means for objective quantification of model- observational data fidelity. The Shannon entropy (SE) H ( X ) is the measure of the amount of uncertainty, variability, or “surprise” present in a system variable X , while the Mutual Information (MI) I ( X, Y ) measures the amount of shared information or redundancy between two variables X and Y . Information theory’s roots lie in the analysis of communication of data across a noisy channel (Figure 1), and offer a scheme for quantifying how well a message X coming from a transmitter arrives as Y at the receiver. A more general information- theoretic measure of message degradation is the Kullback-Leibler Divergence (KLD), which quan- tifies insufficiency of agreement in the probatility desnity functions associated with X and Y . The ratio of MI to SE yields the amount of information shared by two datasets versus the information content of one alone. Alas, the aforementioned information- theoretic techniques work best for discrete rather than continuous systems. This is because evaluation of the Shannon Entropy (SE) for continuous systems–the differential entropy–does not constitute the continuum limit of the SE. Relative quantities such as the MI and KLD are always valid in the continuum case, and are the continuum limit of their discrete counterparts, but are just that– relative . This begs the question: Is there some way I can benchmark it against some continuum surrogate for the SE? Thus, one faces a choice when using information theory for model validation and intercomparison: (1) adopt coarse- graining strategies that are physically relevant, always aware that computed SE results re specific to a given discretisation; or (2) treat the data as continuous and use the MI combined with some benchmark quantity. In this paper, I adopt strategy (1), and restrict scope to a variable that has well-agreed-upon discretisations— total cloud cover, which by observational convention is frequently coarse-grained by oktas, tenths, or percent.

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