244
Ghapter 13: Spatial Patterns: EOFs and GGA
of the kth estimated EOF the equality of eigenvalues and variance of
EOF coefficients is biased (cf. H. von Storch and Hannoschöck, 1986):
- For the largest eigenvalues Ak:
E(~k) > Ak = VAR(ak) > E(VAR(ch))
(13.36)
- for the smallest eigenvalues Ak:
E(~k) < Ak = VAR(ak) < E(VAR(Cik))
(13.37)
The relation (13.36,13.37) means that the large (small) eigenvalues
are systematically over-(under)estimated, and that the variance of the
random variable Cik = (X,t k ) which are expansion coefficients when
projecting X on the random variable "estimated EOFs" is systematically over- or underestimated by the sampie variance VAR(Cik) = ~k
derived from the sampie {i(1) ... i(n)}. Similarly, COV(Cik, Ci;) ICOV{&:, Ci;) = o .
• "Selection Rules"
So-called selection rules have been proposed. One often used is named
"Rule N" (Preisendorfer and Overland, 1982), which is supposed to determine the physically "significant" EOFs. The basic concept is that the full
phase space is the sum of a subset in which all variations are purely noise
and of a subset whose variability is given by dynamical processes. The
signal-subspace is spanned by well-defined EOFs whereas in the noisesubspace no preferred directions exist. For the eigenvalue-spectrum this
assumption implies that the eigenvalues of the EOFs spanning the signalsubspace are unequal and that the eigenvalues in the noise-subspace are
all identical.
The selection rules compare the distributions of sam pie eigenvaluespectra, representative for the situation that all or the m - K smallest
true eigenvalues (K being specified a-priori or determined recursively)
are all alike, with the actually observed sampie eigenvalue spectrum.
All those estimated eigenvalues which are larger than the, say, 95%percentile of the (marginal) distribution of the reference "noise spectra",
are selected as signijicant at the 5%-level.
The problem with this approach is that this selection rule is claimed
to be a statistical test which supposedly is capable of accepting, with
a given risk, the alternative hypothesis that all EOFs with an index
sm aller than some number m-K represent "signals" of the analyzed data
field. The null hypothesis tested would be "all eigenvalues are equal",
and the rejection of this null hypothesis would be the acceptance of
the alternative "not all eigenvalues are equal". The connection between
this alternative and the determination of a "signal subspace" is vague.
Ghapter 13: Spatial Patterns: EOFs and GGA
of the kth estimated EOF the equality of eigenvalues and variance of
EOF coefficients is biased (cf. H. von Storch and Hannoschöck, 1986):
- For the largest eigenvalues Ak:
E(~k) > Ak = VAR(ak) > E(VAR(ch))
(13.36)
- for the smallest eigenvalues Ak:
E(~k) < Ak = VAR(ak) < E(VAR(Cik))
(13.37)
The relation (13.36,13.37) means that the large (small) eigenvalues
are systematically over-(under)estimated, and that the variance of the
random variable Cik = (X,t k ) which are expansion coefficients when
projecting X on the random variable "estimated EOFs" is systematically over- or underestimated by the sampie variance VAR(Cik) = ~k
derived from the sampie {i(1) ... i(n)}. Similarly, COV(Cik, Ci;) ICOV{&:, Ci;) = o .
• "Selection Rules"
So-called selection rules have been proposed. One often used is named
"Rule N" (Preisendorfer and Overland, 1982), which is supposed to determine the physically "significant" EOFs. The basic concept is that the full
phase space is the sum of a subset in which all variations are purely noise
and of a subset whose variability is given by dynamical processes. The
signal-subspace is spanned by well-defined EOFs whereas in the noisesubspace no preferred directions exist. For the eigenvalue-spectrum this
assumption implies that the eigenvalues of the EOFs spanning the signalsubspace are unequal and that the eigenvalues in the noise-subspace are
all identical.
The selection rules compare the distributions of sam pie eigenvaluespectra, representative for the situation that all or the m - K smallest
true eigenvalues (K being specified a-priori or determined recursively)
are all alike, with the actually observed sampie eigenvalue spectrum.
All those estimated eigenvalues which are larger than the, say, 95%percentile of the (marginal) distribution of the reference "noise spectra",
are selected as signijicant at the 5%-level.
The problem with this approach is that this selection rule is claimed
to be a statistical test which supposedly is capable of accepting, with
a given risk, the alternative hypothesis that all EOFs with an index
sm aller than some number m-K represent "signals" of the analyzed data
field. The null hypothesis tested would be "all eigenvalues are equal",
and the rejection of this null hypothesis would be the acceptance of
the alternative "not all eigenvalues are equal". The connection between
this alternative and the determination of a "signal subspace" is vague.