142
Cbapter 8: StatisticaJ Analysis of GCM Output
some GCM experiments aI:e performed in "perpetual conditions", providing
instead long time se ries that can be more effectively dealt with by taking into
ac count their finite correlation time when estimating the standard errors of
time averages.
The variance of the sampie me an x of a statistically stationary process
x(l) .. . x(n) is given by
2
1 n-1
lil.
O"j( = - I: (1- -)-yx(t)
n
n
i=-(n-1)
(8.5)
where IX (i) is the lagged covariance of Xi. Thus, if xis approximately normal
(which should always be the case for large sampies, because of the central
limit theorem) and if a reliable estimator si of (8.5) could be found, the null
hypothesis /lx = /lo could be tested since
x - /lo
t = - -
Sj(
(8.6)
would be again distributed as at variable if Ho is true, with an approximate
equivalent number of degrees of freedom v given by
S2
v = -t -1,
sj(
(8.7)
where s; is the sampie variance. The main problem is that there is no
unbiased estimator of IX (i) when the true mean is not known, and that
the use of traditional estimators in (8.5) results in unacceptable biases for
small sampies (Anderson, 1971). When sampies are sufficiently large, (8.6) is
asymptotically distributed as a N(O, 1) variable if Ho is true, and the usual
i-test can be used. Guidelines for the application of the latter and alternative
tests for the small sampie case are given in Zwiers and H. von Storch (1994).
Alternatively, one can use spectral analysis to evaluate (8.5), since one has
asymptotically for large n
(8.8)
where r X (0) is the spectral density near zero frequency. Atmospheric spectra
are approximately white at low frequencies, hence r x(O) can be estimated if
the time series are long enough, providing a direct estimate of the denominator in (8.6), or its generalization to the two-sample case. Many degrees of
freedom are needed to get a stable estimate of r x(O), which can be obtained
by averaging spectral estimates at low frequencies. However, the averaging
may then include frequencies for which the spectrum is not entirely Hat, resulting in an underestimation or an overestimation (depending on the sampie
size and the shape of the spectrum near zero frequency) of the percentage of
rejections of the null hypothesis (Jones, 1976; Zwiers and Thiebaux, 1987).
Cbapter 8: StatisticaJ Analysis of GCM Output
some GCM experiments aI:e performed in "perpetual conditions", providing
instead long time se ries that can be more effectively dealt with by taking into
ac count their finite correlation time when estimating the standard errors of
time averages.
The variance of the sampie me an x of a statistically stationary process
x(l) .. . x(n) is given by
2
1 n-1
lil.
O"j( = - I: (1- -)-yx(t)
n
n
i=-(n-1)
(8.5)
where IX (i) is the lagged covariance of Xi. Thus, if xis approximately normal
(which should always be the case for large sampies, because of the central
limit theorem) and if a reliable estimator si of (8.5) could be found, the null
hypothesis /lx = /lo could be tested since
x - /lo
t = - -
Sj(
(8.6)
would be again distributed as at variable if Ho is true, with an approximate
equivalent number of degrees of freedom v given by
S2
v = -t -1,
sj(
(8.7)
where s; is the sampie variance. The main problem is that there is no
unbiased estimator of IX (i) when the true mean is not known, and that
the use of traditional estimators in (8.5) results in unacceptable biases for
small sampies (Anderson, 1971). When sampies are sufficiently large, (8.6) is
asymptotically distributed as a N(O, 1) variable if Ho is true, and the usual
i-test can be used. Guidelines for the application of the latter and alternative
tests for the small sampie case are given in Zwiers and H. von Storch (1994).
Alternatively, one can use spectral analysis to evaluate (8.5), since one has
asymptotically for large n
(8.8)
where r X (0) is the spectral density near zero frequency. Atmospheric spectra
are approximately white at low frequencies, hence r x(O) can be estimated if
the time series are long enough, providing a direct estimate of the denominator in (8.6), or its generalization to the two-sample case. Many degrees of
freedom are needed to get a stable estimate of r x(O), which can be obtained
by averaging spectral estimates at low frequencies. However, the averaging
may then include frequencies for which the spectrum is not entirely Hat, resulting in an underestimation or an overestimation (depending on the sampie
size and the shape of the spectrum near zero frequency) of the percentage of
rejections of the null hypothesis (Jones, 1976; Zwiers and Thiebaux, 1987).