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A pair of analogues are two states in a geophysical system, widely separated in time, that are very close.
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5. Empirical Orthogonal Functions
The purpose of this chapter is to discuss Empirical Orthogonal Functions (EOF), both in
method and application. When dealing with teleconnections in the previous chapter we came very
close to EOF, so it will be a natural extension of that theme. However, EOF opens the way to an
alternative point of view about spacetime relationships, especially correlation across distant times
as in analogues . EOFs have been treated in book size texts, most recently in Jolliffe (2002), a
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principal older reference being Preisendorfer(1988). The subject is extremely interdisciplinary, and
each field has its own nomenclature, habits and notation. Jolliffe’s book is probably the best
attempt to unify various fields. The term EOF appeared first in meteorology in Lorenz(1956).
Zwiers and von Storch(1999) and Wilks(1995) devote lengthy single chapters to the topic.
Here we will only briefly treat EOF or PCA (Principal Component Analysis) as it is called
in most fields. Specifically we discuss how to set up the covariance matrix, how to calculate the
EOF, what are their properties, advantages, disadvantages etc. We will do this in both spacetime
setups already alluded to in Eqs (2.14) and (2.14a). There are no concrete rules as to how one
constructs the covariance matrix. Hence there are in the literature matrices based on correlation,
based on covariance etc. Here we follow the conventions laid out in Chapter 2. The
postprocessing and display conventions of EOFs can also be quite confusing. Examples will be
shown, for both daily and seasonal mean data, for both the Northern and Southern Hemisphere.
EOF may or may not look like teleconnections. Therefore, as a diagnostic tool, EOFs may not
always allow the interpretation some would wish. This has led to many proposed ‘simplifications’
of the EOFs, which hopefully are more like teleconnections.
However, regardless of physical interpretation, since EOFs are maximally efficient in
retaining as much of the data set’s information as possible for as few degrees of freedom as
possible they are ideally suited for empirical modeling.
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5.1 Methods and definitions
5.1.1 Working definition:
Here we cite Jolliffe (2002, p 1). “The central idea of principal component analysis (PCA) is to
reduce the dimensionality of a data set consisting of a large number of interrelated variables, while
retaining as much as possible of the variation present in the data set. This is achieved by
transforming to a new set of variables, the principle components, which are uncorrelated, and
which are ordered so that the first
few
retain most of the variation present in all of the original
variables.” The italics are Jolliffe’s. PCA and EOF analysis is the same.
5.1.2 The Covariance Matrix
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 Spring '11
 V.Ram

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