Section 8.3: Multivariate Analysis
147
ipated GCM response is represented by an apriori sequence of guessed patterns derived from former knowledge or simpler dynamical models. The
guesses are characterized by only a few parameters, and they must be ordered apriori in a sequence reflecting their anticipated contribution to the
total response. Suppose that nx independent realizations of a GCM field X
in control conditions are available, and ny independent realizations of the
same field in anomaly conditions (denoted by Y). It is then assumed that
the GCM response, described by the mean differences between anomaly and
control runs, i - y, can be represented by a linear combination of J{ guess
vectors p"; k = 1,2, ... , J{(J{ ~ m):
K
i-y= LQ"p"+Ti
(8.15)
"=1
where the Q,,' are scalar parameters estimated by minimizing the residual error
11 Ti 11, and where the cut-off J{ must be determined (see also Section 13.2.1).
The null hypo thesis that there is no atmospheric response ( Ho : j1 X = j1y)
is that the true parameters Q" k = 1 ... J{ are all equal to zero, Ci = O. In
the limited sam pIe case, the null hypothesis should be tested by using the
statistic (8.13) or (8.14) (Hannoschöck and Frankignoul, 1985); the X 2 test of
Hasselmann (1979) and H. von Storch and Kruse (1985) is only applicable for
very large sampIes. The selection criteria for J{ in (8.15) may be sequential
or not, with slightly different significance levels, as discussed in Barnett et al.
(1981). One often chooses the highest value of J{ for which the null hypothesis
is rejected. Alternatively, all the J{ guess vectors may be prescribed apriori,
thereby defining a reduced base of fixed dimension for the analysis.
As reviewed below, three kinds of guess patterns have been considered:
empirical guesses, theoretical predictions and patterns based on observations
(see also Section 13.2.3). In each case the main limitation of the approach is
that it requires apriori insights on the expected response. The multivariate
test of significance can tell us whether the GCM response is consistent with
our assumptions, but it does not indicate whether there is any significant
response at all.
Note that Hasselmann (1979, 1993) has also suggested using a pattern
recognition method to optimize the statistical significance of the response,
but the results may become biased. Thus, much caution is required to apply the optimization procedure to the small sampIe case (Hannoschöck and
Frankignoul, 1985; H. von Storch and Kruse, 1985).
a) Empirical Guesses
In their study of the GISS GCM Model I response to the North Pacific SST
anomaly above, Hannoschöck and Frankignoul (1985) have assumed that the
response is largest at the largest scales and used as guess vectors an ordered
sequence of spherical harmonics Y/ of decreasing spatial scales, where j is the
zonal wavenumber and i the total wavenumber. A more truncated expansion
147
ipated GCM response is represented by an apriori sequence of guessed patterns derived from former knowledge or simpler dynamical models. The
guesses are characterized by only a few parameters, and they must be ordered apriori in a sequence reflecting their anticipated contribution to the
total response. Suppose that nx independent realizations of a GCM field X
in control conditions are available, and ny independent realizations of the
same field in anomaly conditions (denoted by Y). It is then assumed that
the GCM response, described by the mean differences between anomaly and
control runs, i - y, can be represented by a linear combination of J{ guess
vectors p"; k = 1,2, ... , J{(J{ ~ m):
K
i-y= LQ"p"+Ti
(8.15)
"=1
where the Q,,' are scalar parameters estimated by minimizing the residual error
11 Ti 11, and where the cut-off J{ must be determined (see also Section 13.2.1).
The null hypo thesis that there is no atmospheric response ( Ho : j1 X = j1y)
is that the true parameters Q" k = 1 ... J{ are all equal to zero, Ci = O. In
the limited sam pIe case, the null hypothesis should be tested by using the
statistic (8.13) or (8.14) (Hannoschöck and Frankignoul, 1985); the X 2 test of
Hasselmann (1979) and H. von Storch and Kruse (1985) is only applicable for
very large sampIes. The selection criteria for J{ in (8.15) may be sequential
or not, with slightly different significance levels, as discussed in Barnett et al.
(1981). One often chooses the highest value of J{ for which the null hypothesis
is rejected. Alternatively, all the J{ guess vectors may be prescribed apriori,
thereby defining a reduced base of fixed dimension for the analysis.
As reviewed below, three kinds of guess patterns have been considered:
empirical guesses, theoretical predictions and patterns based on observations
(see also Section 13.2.3). In each case the main limitation of the approach is
that it requires apriori insights on the expected response. The multivariate
test of significance can tell us whether the GCM response is consistent with
our assumptions, but it does not indicate whether there is any significant
response at all.
Note that Hasselmann (1979, 1993) has also suggested using a pattern
recognition method to optimize the statistical significance of the response,
but the results may become biased. Thus, much caution is required to apply the optimization procedure to the small sampIe case (Hannoschöck and
Frankignoul, 1985; H. von Storch and Kruse, 1985).
a) Empirical Guesses
In their study of the GISS GCM Model I response to the North Pacific SST
anomaly above, Hannoschöck and Frankignoul (1985) have assumed that the
response is largest at the largest scales and used as guess vectors an ordered
sequence of spherical harmonics Y/ of decreasing spatial scales, where j is the
zonal wavenumber and i the total wavenumber. A more truncated expansion