Uncertainty, Emergence, and Statistics in Dendrochronology - Dendroclimatology - page 92

way for doing so is to use the cumulative eigenvalues product (PVP in French) crite-
rion (Guiot et al. 1982 ) , which is based on the cumulative product of the eigenvalues,
one way of calculating the determinant of a matrix. The actual cutoff occurs where
PVP drops below 1.0, the value of the determinant of a random correlation matrix
of equal rank. In contrast, the EV1 cutoff occurs at the point where the remaining
eigenvectors cannot explain as much variance as those extracted from an equal-rank
random correlation matrix. Both PVP and EV1 are based on appealing asymptotic
arguments that relate to expected values of random correlation matrices. Yet, they
yield quite different results in practice.
In our example, the EV1 cutoff retained 13 of 32 eigenvectors as candidate pre-
dictors, which cumulatively explained 73.4% of the total climate variance. If the
PVP cutoff had been used, it would have retained 25 of the 32 eigenvectors (equal
to 96.5% of the total variance) as candidate predictors. So, is one cutoff better than
the other? Support for EV1 comes from the fact that Monte Carlo estimates of eigen-
value confidence limits based on 'Rule N' (Preisendorfer et al. 1981 ; Preisendorfer
1988 ) always select a cutoff that is consistent with the asymptotic argument under-
lying EV1. On the other hand, there may be some useful climate information in the
deleted eigenvectors below the EV1 cutoff, which is the principal argument for using
PVP (or some other criterion) to retain more candidate eigenvectors (cf. Jolliffe
1973 ) . This argument is appealing because the total information in the original cli-
mate correlation matrix is contained in the complete set of eigenvectors. See Fritts
( 1976 , p. 357) for this mathematical equivalence. However, the increasing orthogo-
nality constraints imposed on the higher-order eigenvectors are likely to distort the
climatic meaning of those modes increasingly away from physical reality, which
may make them more sensitive to chance correlations within the original intercor-
relation matrix of climate variables. This and our 'Rule N' argument above are the
reasons why we prefer the EV1 cutoff.
The choice of EV1 or PVP can also be argued in terms of Type-1 and Type-2
errors. Choosing EV1 is a more conservative choice because it reduces the num-
ber of candidate predictors and the likely inflation of R 2 (Rencher and Pun 1980 ) .
However, the premium paid for protecting against inflated R 2 using EV1 is the
possible loss of additional useful climate information in the higher-order climate
eigenvectors that PVP would retain. Thus, EV1 reduces the chance of Type-1 error
and increases the chance of a Type-2 error in response functions by eliminating
more eigenvectors from the candidate predictor pool. In contrast, PVP increases the
chance of Type-1 error and decreases the chance of Type-2 error by allowing more
potentially spurious candidate eigenvectors to be included in the response function.
Given the selected cutoff used to retain candidate eigenvectors, the criterion for
entering the eigenvectors into the regression-based response function model ulti-
mately determines the final form of the response function. Fritts ( 1976 ) , Guiot et al.
( 1982 ) , and Fekedulegn et al. ( 2002 ) argue for somewhat lenient entry criteria. Fritts
( 1976 ) usedan F -level = 1.0 cutoff for entering eigenvectors, which in our example

has a probability p = 0.30. Guiot et al. ( 1982 ) suggested p = 0.50 for relatively

short datasets ( n 30 years) when the number of variables is comparable to the

number of observations and 0.10-0.20 for longer datasets like that used here ( n

=

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