457
above this theoretical noise spectrum, it is generally considered as "significant." This approach can be quite deceptive because a single realization of
a noise process can have a spectrum that differs greatly from the theoretical one, especially if the number of data points is small; it is the (suitably
weighted) average of such sample spectra over many realizations that will
tend to the theoretical spectrum of the ideal noise process. Indeed, the
Fourier transform of a single realization of a red-noise process can yield
arbitrarily high peaks at arbitrarily low frequencies; such peaks could be
attributed, quite erroneously, to periodic components. Therefore, more
laborious tests have to be used, to determine error bars that are better
adapted to each data set. Allen (1992) devised such tests comparing the
statistics of simulated (or surrogate) red-noise time series with those of
a given climatic time series. As mentioned in the previous paragraph, a
number of Monte-Carlo-based tests against white-noise hypotheses have
been published in the SSA literature. Still, Allen and Smith (1994, 1995)
prefer to refer to their particular approach to such tests, against red noise,
as "Monte Carlo SSA (MC-SSA)."
Red noise is a first-order auto-regressive, or AR(l), process Ut whose
value at a time t depends on the value at time t - 1 only,
Ut = 'Y( Ut-l - uo) + aZt + uo;
(10)
here Zt is a white-noise, or AR(O), process for which each value is independent of the previous one, while Uo is the initial value and a and 'Yare
deterministic coefficients.
The first step in MC-SSA is to determine the red-noise coefficients a
and ' Y from the time series X(t) using a maximum-likelihood criterion.
Heuristic formulae with low bias are given by Allen (1992) and Allen and
Smith (1996). Based on these coefficients, an ensemble of surrogate rednoise data can be simulated and, for each realization, a covariance matrix
CR is computed. These covariance matrices are then projected onto the
eigenvector basis ED of the original data by using Eq. (7) for their SVD,
(11)
Since (11) is not the SVD of that realization, the matrix AR is not necessarily diagonal, like in Eq. (7), but it measures the resemblance of a given
surrogate set with the data set. This resemblance can be quantified by
computing the statistics of the diagonal elements of AR. The statistical
distribution of these elements, determined from the ensemble of Monte
above this theoretical noise spectrum, it is generally considered as "significant." This approach can be quite deceptive because a single realization of
a noise process can have a spectrum that differs greatly from the theoretical one, especially if the number of data points is small; it is the (suitably
weighted) average of such sample spectra over many realizations that will
tend to the theoretical spectrum of the ideal noise process. Indeed, the
Fourier transform of a single realization of a red-noise process can yield
arbitrarily high peaks at arbitrarily low frequencies; such peaks could be
attributed, quite erroneously, to periodic components. Therefore, more
laborious tests have to be used, to determine error bars that are better
adapted to each data set. Allen (1992) devised such tests comparing the
statistics of simulated (or surrogate) red-noise time series with those of
a given climatic time series. As mentioned in the previous paragraph, a
number of Monte-Carlo-based tests against white-noise hypotheses have
been published in the SSA literature. Still, Allen and Smith (1994, 1995)
prefer to refer to their particular approach to such tests, against red noise,
as "Monte Carlo SSA (MC-SSA)."
Red noise is a first-order auto-regressive, or AR(l), process Ut whose
value at a time t depends on the value at time t - 1 only,
Ut = 'Y( Ut-l - uo) + aZt + uo;
(10)
here Zt is a white-noise, or AR(O), process for which each value is independent of the previous one, while Uo is the initial value and a and 'Yare
deterministic coefficients.
The first step in MC-SSA is to determine the red-noise coefficients a
and ' Y from the time series X(t) using a maximum-likelihood criterion.
Heuristic formulae with low bias are given by Allen (1992) and Allen and
Smith (1996). Based on these coefficients, an ensemble of surrogate rednoise data can be simulated and, for each realization, a covariance matrix
CR is computed. These covariance matrices are then projected onto the
eigenvector basis ED of the original data by using Eq. (7) for their SVD,
(11)
Since (11) is not the SVD of that realization, the matrix AR is not necessarily diagonal, like in Eq. (7), but it measures the resemblance of a given
surrogate set with the data set. This resemblance can be quantified by
computing the statistics of the diagonal elements of AR. The statistical
distribution of these elements, determined from the ensemble of Monte