The method of empirical eigenfunctions (Karhunen-Loève procedure) is developed within a framework suitable for dealing with large scientific datasets. It is shown that this furnishes an intrinsic representation of any given database which is always, in a well-defined mathematical sense, the optimal description. The methodology is illustrated by a variety of examples, arising out of current research and taken from pattern recognition, turbulent flow, physiology, and oceanographic flow. In each instance examples of the empirical eigenfunctions are presented.
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