Identification Method - Climate Change: Identification and Projections - page 73

ˆ ˆ ˆ

66%:

θ ± 0.954 σ

, 90%:

θ ± 1.645 σ

, 95%:

θ ± 1.960 σ

i i i i i i

ˆ ˆ

For a pair of parameters, the inequality

T − 1

defines the

( θ − θ ) V ( θ − θ ) < S

θθ

ellipsoid of confidence at:
66%: 90%: 95%:

S = 2.16 S = 4.60 S = 5.99

A hypothesis is rejected or not depending on whether the corresponding
parameters are located outside or within the confidence region. We shall
carry this out for sensitivities S 1 and S 2 .

The degree of adequacy for the complexity of the model can be tested by
observing whether a reduction in order, or the annulment of a given
parameter, can be accepted or rejected. The validity of this detection method
depends on the richness of the measurements available.

6.5. Conditions of application

The above calculation of variance (section 6.3) as well as its
consequences (section 6.4), presume that the noises and disturbances which
affects the input and output data have a minimum amount of statistical
regularity, and that the duration of measurements is sufficient so that
convergence towards normal laws can arise. When we are still far from
convergence, calculations become more complicated, and the size of the
confidence regions can increase in proportions which are difficult to
evaluate.

Yet above all, this statistical regularity not only covers the hypothesis of
stationarity (invariance in time, relatively well-verified by climatic
archives), but also the hypothesis of ergodicity . The latter assumes that the
alinement and calibration of the paleoclimatic reconstructions to modern
data is correct. Indeed, in the event of an error in calibration, the sensitivity
of the output in relation to the input would be irreparably effected by the
initial calibration error, without the possibility of compensating this with an
average over time, whatever the duration of the observation may be. The
estimates and confidence regions are given with this reservation. If
the conditions are not verified (and they are never perfectly so), it may be the
case that, according to data used for identification, a single parameter may
be assigned a number of quasi-certain region of confidence (99%) which are
completely disjointed.