Theory of Optimal Weighting of Data to Detect Climatic Change

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Abstract
A search for climatic change predicted by climate models can easily yield unconvincing results because of “climatic noise,” the inherent, unpredictable variability of time-averaged atmospheric data. We describe a weighted average of data that maximizes the probability of detecting predicted climatic change. To obtain the optimal weights, an estimate of the covariance matrix of the data from a prior data set is needed. This introduces additional sampling error into the method. We show how to take this into account. A form of the weighted average is found whose probability distribution is independent of the true (but unknown) covariance statistics of the data and of the climate model prediction. A table of critical values for statistical testing of the weighted average is given, based on Monte Carlo calculations. The results an exact when the prior data set consists of temporary uncorrelated samples. Abstract A search for climatic change predicted by climate models can easily yield unconvincing results because of “climatic noise,” the inherent, unpredictable variability of time-averaged atmospheric data. We describe a weighted average of data that maximizes the probability of detecting predicted climatic change. To obtain the optimal weights, an estimate of the covariance matrix of the data from a prior data set is needed. This introduces additional sampling error into the method. We show how to take this into account. A form of the weighted average is found whose probability distribution is independent of the true (but unknown) covariance statistics of the data and of the climate model prediction. A table of critical values for statistical testing of the weighted average is given, based on Monte Carlo calculations. The results an exact when the prior data set consists of temporary uncorrelated samples.

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