234
Ghapter 13: Spatial Patterns: EOFs and GGA
Also patterns "predicted" by simplified dynamieal theory (for instanee,
linear barotropie equations) are in use (Hannosehöek and Frankignoul,
1985; Hense et al. , 1990).
• A very widely used dass of patterns are orthogonal functions such as
trigonometrie functions or spherieal harmonies. In all "spectral" atmospheric general eirculation models the horizontal fields are expanded
aceording to (13.1) with spherical harmonics as guess patterns. In the
spectral analysis of time series the trigonometrie functions are used to
efficiently represent fields.
• The Empirical Orthogonal Functions (EOFs) and Canonieal Correlation
Patterns (CCPs) are very widely used guess patterns. These choices
will be discussed in so me length in the next two Sections 13.3 and 13.4.
Offsprings ofthese teehniques are Extended EOFs (EEOFs) and Comp/ex
EOFs (CEOFs). In the EEOFs (Weare and Nasstrom, 1982; see also
Chapter 14) the same vector at different times is coneatenated; in the
CEOF (Wallace and Dickinson, 1972; Barnett, 1983; also Section 15.3.4)
the original vector real-valued time series is made complex by adding its
Hilbert transform as imaginary eomponent. The Hilbert transform may
be seen as a kind of "momentum" . Both techniques are successfully
applied in dimate research but we will not go into details in the present
reView.
• The "wavelet" analysis is a teehnique which projects a given time series
on a set of patterns, whieh are eontrolled by a location and a dispersion
parameter. See Meyers et al. (1993) or Farge et al. (1993).
13.2.4 Rotation of Guess Patterns
For EOFs there exists a widely used variant named Rotated EOFs (for instance, Barnston and Livezey, 1987). The name is somewhat misleading as it
indieates that the "rotation" would exploit properties special to the EOFs.
This is not the case. Instead, the general concept of "rotation" is to replace
the patterns pk in (13.1) by "nicer" patterns pA:
K
K
I:akii k = I:afPA
(13.12)
k=l
k=l
The patterns pA are determined such that they maximize a certain (nonlinear) functional of "simplieity" FR and that they span the same spaee as the
original set of vectors {pk}. Constraints like unit length (pA T pk = 1) and,
sometimes, orthogonality (pA T pi = 0) are invoked. Riehman (1986) lists
five vague eriteria for patterns being "simple" and there are many proposals of "simplieity" functionals. If the patterns are not orthogonal the term
oblique is used. The minimization of functionals such as (13.13) is in general
Ghapter 13: Spatial Patterns: EOFs and GGA
Also patterns "predicted" by simplified dynamieal theory (for instanee,
linear barotropie equations) are in use (Hannosehöek and Frankignoul,
1985; Hense et al. , 1990).
• A very widely used dass of patterns are orthogonal functions such as
trigonometrie functions or spherieal harmonies. In all "spectral" atmospheric general eirculation models the horizontal fields are expanded
aceording to (13.1) with spherical harmonics as guess patterns. In the
spectral analysis of time series the trigonometrie functions are used to
efficiently represent fields.
• The Empirical Orthogonal Functions (EOFs) and Canonieal Correlation
Patterns (CCPs) are very widely used guess patterns. These choices
will be discussed in so me length in the next two Sections 13.3 and 13.4.
Offsprings ofthese teehniques are Extended EOFs (EEOFs) and Comp/ex
EOFs (CEOFs). In the EEOFs (Weare and Nasstrom, 1982; see also
Chapter 14) the same vector at different times is coneatenated; in the
CEOF (Wallace and Dickinson, 1972; Barnett, 1983; also Section 15.3.4)
the original vector real-valued time series is made complex by adding its
Hilbert transform as imaginary eomponent. The Hilbert transform may
be seen as a kind of "momentum" . Both techniques are successfully
applied in dimate research but we will not go into details in the present
reView.
• The "wavelet" analysis is a teehnique which projects a given time series
on a set of patterns, whieh are eontrolled by a location and a dispersion
parameter. See Meyers et al. (1993) or Farge et al. (1993).
13.2.4 Rotation of Guess Patterns
For EOFs there exists a widely used variant named Rotated EOFs (for instance, Barnston and Livezey, 1987). The name is somewhat misleading as it
indieates that the "rotation" would exploit properties special to the EOFs.
This is not the case. Instead, the general concept of "rotation" is to replace
the patterns pk in (13.1) by "nicer" patterns pA:
K
K
I:akii k = I:afPA
(13.12)
k=l
k=l
The patterns pA are determined such that they maximize a certain (nonlinear) functional of "simplieity" FR and that they span the same spaee as the
original set of vectors {pk}. Constraints like unit length (pA T pk = 1) and,
sometimes, orthogonality (pA T pi = 0) are invoked. Riehman (1986) lists
five vague eriteria for patterns being "simple" and there are many proposals of "simplieity" functionals. If the patterns are not orthogonal the term
oblique is used. The minimization of functionals such as (13.13) is in general