140
Chapter 8: Statistical Analysis of GCM Output
anticipated from observations, theory, or prior experiments. Since the early
seventies, many response studies have been made with atmospheric GCMs,
and it is in this context that the analysis strategies discussed here were first
developed.
The difliculties are that GCMs, like -reality, usually have a large natural
variability and that the sampie is often limited, so that the ratio of the true
GCM signal to the sampling variability (the signal-to-noise ratio) is generally
smalI. Also, GCM fields, like observations, have large and complex correlation scales, so individual tests performed at grid points need to be interpreted
by taking their multiplicity and their interdependency into account. Finally,
the external forcing may be imperfectly known, resulting in GCM response
uncertainties that need to be distinguished from the effects of model inadequaeies.
If the GCM variables have a known statistical distribution, or approach it
sufliciently, standard parametrie methods can be used. Many GCM variables
are very nearly Gaussian, and hence parametrie tests are often used. However, some variables, like precipitation, are not normally distributed, and
in this case nonparametrie methods (such as permutation techniques) are
needed. In most cases, the multivariate statistic methods are more appropriate than the univariate ones, but univariate tests are easier to implement.
The univariate and multivariate approaches have advantages and limitations,
depending on the problem at hand, and it is often advisable to use both.
In this chapter, the parametrie approach is discussed with emphasis on the
mean, rather than high er-order moments. In Section 8.2, the t-test on the
mean is introduced, and its application to GCM experiments discussed. The
multivariate test is introduced in Section 8.3. Because of the large dimension
of the GCM fields, the sampie size is much sm aller than the dimension, hence
general strategies had to be devised for the use of the standard multivariate
methods: they are discussed in the context of both response studies and
model testing. Permutation procedures are discussed in Chapter 9.
8.2 U nivariate Analysis
8.2.1 The t-Test on the Mean of a Normal Variable
Let {Xl, ... X n } be a sampie of n independent observations on a univariate
random normal variable X with distribution N(J1., (12).1 Then, the null hypothesis Ho that J1. = J1.o can be tested against an alternative hypothesis HA,
say J1. i= J1.o (two-sided test), by considering the test statistic
x-J1.o
t - - -
- s/Fn
where x denotes the sampie mean and
1 i.e., X is Gaussian distributed with a mean IJ. and a standard deviation q.
(8.1)
Chapter 8: Statistical Analysis of GCM Output
anticipated from observations, theory, or prior experiments. Since the early
seventies, many response studies have been made with atmospheric GCMs,
and it is in this context that the analysis strategies discussed here were first
developed.
The difliculties are that GCMs, like -reality, usually have a large natural
variability and that the sampie is often limited, so that the ratio of the true
GCM signal to the sampling variability (the signal-to-noise ratio) is generally
smalI. Also, GCM fields, like observations, have large and complex correlation scales, so individual tests performed at grid points need to be interpreted
by taking their multiplicity and their interdependency into account. Finally,
the external forcing may be imperfectly known, resulting in GCM response
uncertainties that need to be distinguished from the effects of model inadequaeies.
If the GCM variables have a known statistical distribution, or approach it
sufliciently, standard parametrie methods can be used. Many GCM variables
are very nearly Gaussian, and hence parametrie tests are often used. However, some variables, like precipitation, are not normally distributed, and
in this case nonparametrie methods (such as permutation techniques) are
needed. In most cases, the multivariate statistic methods are more appropriate than the univariate ones, but univariate tests are easier to implement.
The univariate and multivariate approaches have advantages and limitations,
depending on the problem at hand, and it is often advisable to use both.
In this chapter, the parametrie approach is discussed with emphasis on the
mean, rather than high er-order moments. In Section 8.2, the t-test on the
mean is introduced, and its application to GCM experiments discussed. The
multivariate test is introduced in Section 8.3. Because of the large dimension
of the GCM fields, the sampie size is much sm aller than the dimension, hence
general strategies had to be devised for the use of the standard multivariate
methods: they are discussed in the context of both response studies and
model testing. Permutation procedures are discussed in Chapter 9.
8.2 U nivariate Analysis
8.2.1 The t-Test on the Mean of a Normal Variable
Let {Xl, ... X n } be a sampie of n independent observations on a univariate
random normal variable X with distribution N(J1., (12).1 Then, the null hypothesis Ho that J1. = J1.o can be tested against an alternative hypothesis HA,
say J1. i= J1.o (two-sided test), by considering the test statistic
x-J1.o
t - - -
- s/Fn
where x denotes the sampie mean and
1 i.e., X is Gaussian distributed with a mean IJ. and a standard deviation q.
(8.1)