Section 5.6: Quantifying Climate Signals
89
tree growth in one year is often influenced by climate variability over one or
more previous years (seasons). Prior "growing-season" climate can predispose the tree's response to current climate in many ways, such as through
its effect on stored resources, leaf development (and hence photosynthetic
potential), frost hardiness, drought resistanee etc. Attempts to model such
effects, often manifest as statistical persistence in tree-ring time series, ean
involve the use of multiple tree-ring predictors so that climate in one year
is estimated as a function of tree growth in some combination of preceding,
current and following years. Where the dependent data series displays significant persistence, this lagged-predictor approach has the potential to reduee
the regression degrees of freedom severely.
In addition to lagged predictors, reeonstruetions of single area-average
climate series might use a number of chronology predictors from different
chronology sites. The typieally restricted length of many observational climate series (probably no more than 100 years and often considerably less)
then exacerbates the problem of low degrees of freedom.
To reduee the effective number of predictors in such regressions, principal
components analysis, PCA (see Section 13.3) is often used on the predictor
data set. Only the amplitudes 1 of the "significant" predictor components are
then offered as candidate predictors. As is frequently the case in PC regression (e.g. Preisendorfer et al., 1981) various criteria have been used to judge
"significance" (e.g. see Cook et al., 1994; see also the discussion in Section
13.3.3). High intercorrelation among the original predictors ensures that a
notable reduction in the number of remaining "candidate" PC predictors and
a corresponding improvement in the regression degrees of freedom is achieved.
However, in many applications this is often followed by further elimination
of candidate predictors, with only those PC amplitudes correlating with the
predictand better than some predetermined significanee threshold being retained. Because this further reduction in predictors is clearly based on an a
posteriori decision, however, it would not be appropriate to assurne a further
corresponding improvement in the regression degrees offreedom (Barnett and
Hasseiman, 1979).
The most complex reconstruetion scenario is that in which a network or
grid of climate data series is direetly expressed as a function of tree-ring variability over a number of sites (Figure 5.3). The final transfer function matrix
(comprising a set of individual equations, one for each predictand point expressed as a function ofvariability at each chronology site) is derived through
the initial orthogonalization of the dependent networkj selection of "signifieant" predictand amplitudes; calibration of only the significant predictand
components - either individually (i.e., Orthogonal Spatial Regressionj Briffa
et al., 1983, 1986) or as canonical variates (Fritts etal., 1971) using only the
signifieant predictor PCs as candidate predictors; and ultimate backtransforlln the terminology of Chapter 13 the predictors are the EOFs pi and the amplitudes
are the EOF coefficlents Qi.
89
tree growth in one year is often influenced by climate variability over one or
more previous years (seasons). Prior "growing-season" climate can predispose the tree's response to current climate in many ways, such as through
its effect on stored resources, leaf development (and hence photosynthetic
potential), frost hardiness, drought resistanee etc. Attempts to model such
effects, often manifest as statistical persistence in tree-ring time series, ean
involve the use of multiple tree-ring predictors so that climate in one year
is estimated as a function of tree growth in some combination of preceding,
current and following years. Where the dependent data series displays significant persistence, this lagged-predictor approach has the potential to reduee
the regression degrees of freedom severely.
In addition to lagged predictors, reeonstruetions of single area-average
climate series might use a number of chronology predictors from different
chronology sites. The typieally restricted length of many observational climate series (probably no more than 100 years and often considerably less)
then exacerbates the problem of low degrees of freedom.
To reduee the effective number of predictors in such regressions, principal
components analysis, PCA (see Section 13.3) is often used on the predictor
data set. Only the amplitudes 1 of the "significant" predictor components are
then offered as candidate predictors. As is frequently the case in PC regression (e.g. Preisendorfer et al., 1981) various criteria have been used to judge
"significance" (e.g. see Cook et al., 1994; see also the discussion in Section
13.3.3). High intercorrelation among the original predictors ensures that a
notable reduction in the number of remaining "candidate" PC predictors and
a corresponding improvement in the regression degrees of freedom is achieved.
However, in many applications this is often followed by further elimination
of candidate predictors, with only those PC amplitudes correlating with the
predictand better than some predetermined significanee threshold being retained. Because this further reduction in predictors is clearly based on an a
posteriori decision, however, it would not be appropriate to assurne a further
corresponding improvement in the regression degrees offreedom (Barnett and
Hasseiman, 1979).
The most complex reconstruetion scenario is that in which a network or
grid of climate data series is direetly expressed as a function of tree-ring variability over a number of sites (Figure 5.3). The final transfer function matrix
(comprising a set of individual equations, one for each predictand point expressed as a function ofvariability at each chronology site) is derived through
the initial orthogonalization of the dependent networkj selection of "signifieant" predictand amplitudes; calibration of only the significant predictand
components - either individually (i.e., Orthogonal Spatial Regressionj Briffa
et al., 1983, 1986) or as canonical variates (Fritts etal., 1971) using only the
signifieant predictor PCs as candidate predictors; and ultimate backtransforlln the terminology of Chapter 13 the predictors are the EOFs pi and the amplitudes
are the EOF coefficlents Qi.