Geoscience Reference
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66). Fekedulegn et al. ( 2002 ) suggested using p
0.15 for entering eigenvectors.
Here, we argue for the use of the minimum Akaike information criterion (Akaike
1974 ) , with a correction for small sample bias (Hurvich and Tsay 1989 ) . This is a
totally objective way of determining the order of the model, one that is based on
sound and well-tested information theoretic principles. For this reason, we prefer
the minimum AIC cutoff. Interestingly, the original uncorrected AIC (Akaike 1974 )
also allows variables to enter into a model if p < 0.15 (Jones 1985 ) ,thesameasthe
variable entry criterion of Fekedulegn et al. ( 2002 ) . So the use of the AIC should
not change the response function results nearly as much as the chosen eigenvector
cutoff.
Table 4.1 provides more detail on the four steps used in calculating our hem-
lock response function. Eigenvectors #3, #9, #10, and #6 were added to the model
in that order, with a cumulative explained variance of 34.2% and a minimum AIC
of
=
16.63. Perhaps the most interesting result here is what is not included in the
model; i.e., climate eigenvectors #1 and #2, which together explain 19.2% of the
total variance in the correlation matrix of temperature and precipitation at Mohonk
Lake. Even though those eigenvectors are the two most important modes of covari-
ance among the monthly temperature and precipitation variables, they correlate
extremely poorly with hemlock growth ( r
0.01 and 0.06, respectively). This
result illustrates another aspect of response function analysis. The most important
modes of monthly climate variability defined by the eigenvectors need not relate to
the needs of tree growth. Thus, it is always dangerous to impose a priori expecta-
tions on how trees respond to climate. We should allow the climate response of a
tree-ring series to objectively emerge from our analysis.
The bottom two rows of Table 4.1 provide additional steps in the response func-
tion model that would have occurred if the PVP criterion were used as the cutoff
for candidate eigenvectors instead of EV1. As was mentioned earlier, PVP retains
25 of the 32 eigenvectors (96.5% of the total variance) as candidate predictors. In
this case, eigenvectors #15 and #18 would have also been added to the model, with
a minimum AIC now at
=−
19.37. This result suggests that PVP is better than EV1
because the final model with six eigenvectors has a smaller AIC. Is this true? In this
Table 4.1 The four regression steps used in calculating the hemlock response function
PART R 2
R 2
STEP
EIG
CORR
T-STAT
PROB
AIC
10.09
1
3 (7.2)
0.441
3.931
0.0003
0.195
0.195
2
9 (4.6)
0.235
1.932
0.0549
0.055
0.250
12.57
3
10 (4.1)
0.232
1.907
0.0580
0.054
0.303
15.20
4
6 (5.2)
0.197
1.605
0.1093
0.039
0.342
16.63
5
15 (2.8)
0.197
1.605
0.1094
0.039
0.381
18.21
6
18 (2.1)
0.183
1.490
0.1371
0.034
0.413
19.37
STEP = response function step; EIG = the eigenvector number entered and its (percent variance);
CORR = simple correlation of the eigenvector with tree rings; T-STAT = Student's t statistic for
Corr; PROB = Probability of t-stat; PART R 2
= partial R 2 or fractional variance contributed by
each step; R 2
= cumulative fractional variance of the model; AIC = Akaike information criterion.
 
 
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