460
Sec. 1); here as there, the difficulty is due to the shortness and noisiness
of climatic time series, but the estimation of "peaks" is intuitively more
robust and easy to interpret than that of the continuous background.
The computation of the power spectrum of a random process is an
ill-posed inverse problem (Jenkins and Watts 1968, Thomson 1982). For
example, a straightforward calculation of the discrete Fourier transform of
a random time series (which has a continuous spectral density) will provide
a spectral estimate whose variance is equal to the estimate itself (Jenkins
and Watts 1968, Box and Jenkins 1970). In the remainder of this section,
we outline three techniques to reduce this variance that are commonly
used for the spectral analysis of climatic time series and point out their
respective properties, advantages and failings.
Each one of the three techniques outlined in Secs. 3.1-3.3 below -
Blackman-Tukey, maximum entropy, and multi-taper - can provide error
bars for the estimates it produces. Still, these bars are based on certain
assumptions about the process generating the time series that is being
analyzed. These assumptions are rarely, if ever, met in practice by the
physical processes one wishes to examine. Therefore, we highly recommend
to apply several independent techniques to any time series before drawing
any conclusion about its spectrum.
3.1 Blackman-Tukey spectral estimate
A generic problem of time series analysis is the finiteness of the time interval on which the series is known. When the spectrum is estimated by a
discrete Fourier transform, this corresponds to convoluting the true spectrum with the spectrum of a box-car function, which induces power leakage,
due to the lobes of the Fourier transform of the box car. This systematic
distortion of the spectrum adds to the problem of the variance of the spectral estimate. The Blackman-Tukey (1958) method gives an estimate of
the power spectrum of a given time series X(t) which reduces the estimate's variance and attenuates these leakage effects (Chatfield 1984). The
starting point of this method is the so-called Bochner-Khinchin-Wiener
identity, which states that the power spectrum Px is equal to the Fourier
transform of the auto-correlation function
based on a discretized version of this identity, by using the first M + 1
Sec. 1); here as there, the difficulty is due to the shortness and noisiness
of climatic time series, but the estimation of "peaks" is intuitively more
robust and easy to interpret than that of the continuous background.
The computation of the power spectrum of a random process is an
ill-posed inverse problem (Jenkins and Watts 1968, Thomson 1982). For
example, a straightforward calculation of the discrete Fourier transform of
a random time series (which has a continuous spectral density) will provide
a spectral estimate whose variance is equal to the estimate itself (Jenkins
and Watts 1968, Box and Jenkins 1970). In the remainder of this section,
we outline three techniques to reduce this variance that are commonly
used for the spectral analysis of climatic time series and point out their
respective properties, advantages and failings.
Each one of the three techniques outlined in Secs. 3.1-3.3 below -
Blackman-Tukey, maximum entropy, and multi-taper - can provide error
bars for the estimates it produces. Still, these bars are based on certain
assumptions about the process generating the time series that is being
analyzed. These assumptions are rarely, if ever, met in practice by the
physical processes one wishes to examine. Therefore, we highly recommend
to apply several independent techniques to any time series before drawing
any conclusion about its spectrum.
3.1 Blackman-Tukey spectral estimate
A generic problem of time series analysis is the finiteness of the time interval on which the series is known. When the spectrum is estimated by a
discrete Fourier transform, this corresponds to convoluting the true spectrum with the spectrum of a box-car function, which induces power leakage,
due to the lobes of the Fourier transform of the box car. This systematic
distortion of the spectrum adds to the problem of the variance of the spectral estimate. The Blackman-Tukey (1958) method gives an estimate of
the power spectrum of a given time series X(t) which reduces the estimate's variance and attenuates these leakage effects (Chatfield 1984). The
starting point of this method is the so-called Bochner-Khinchin-Wiener
identity, which states that the power spectrum Px is equal to the Fourier
transform of the auto-correlation function
based on a discretized version of this identity, by using the first M + 1