This paper provides a concise description of the philosophy, mathematics, and algorithms for estimating,detecting, and attributing climate changes. The estimation follows the spectral method by using empirical orthogonal functions, also called the method of reduced space optimal averaging. The detection follows the linear regression method, which can be found in most textbooks about multivariate statistical techniques.The detection algorithms are described by using the space-time approach to avoid the non-stationarity problem. The paper includes (1) the optimal averaging method for minimizing the uncertainties of the global change estimate, (2) the weighted least square detection of both single and multiple signals, (3)numerical examples, and (4) the limitations of the linear optimal averaging and detection methods.
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