447
The remainder of this chapter is organized, over all, as follows. Section 2
deals mainly with signal-to-noise ratio (SIN) enhancement and introduces
singular-spectrum analysis (SSA) as an important and flexible tool for this
enhancement. Statistical tests for the reliability of SSA results are also
discussed. In Sec. 3, we present, in succession, three methods of spectral analysis: Blackman-Tukey, maximum entropy and multi-taper. Both
Secs. 2 and 3 use the Southern Oscillation Index (SOl) for the purposes of
illustrating the methods "in action". In Sec. 4, the multivariate extension
of single-channel SSA is introduced and a few additional applications are
mentioned or illustrated. The chapter concludes with a section on open
questions, from the point of view of both the methodology and its applications.
1.2 Time series and nonlinear dynamics
Before proceeding with the technical details, we give in this subsection a
quick perspective on the "nonlinear revolution" in time-series analysis. In
the 1960s and '70s, the scientific community found out that much of the
irregularity in observed time series, which had traditionally been attributed
to the above-mentioned random "pumping" of a linear system by infinitely
many (independent) degrees offreedom (d-o-/), could be generated by the
nonlinear interaction of a few d-o-J (Lorenz 1963, Ruelle and Takens 1971,
Smale 1967). This realization of the possibility of deterministic aperiodicity
or "chaos" (Gleick 1987) created quite a stir and the purpose ofthis review
is to describe briefly some of its implications for time-series analysis (Drazin
and King 1992), with a special view to climatic time series.
A connection between deterministically chaotic time series and the nonlinear, possibly low-order, dynamics generating them was attempted fairly
early in the young history of "chaos theory." The basic idea was to exploit specifically a univariate time series with apparently irregular behavior,
generated by a deterministic or stochastic system, in order to (i) ascertain
whether the underlying system has a finite number of d-o-J, by establishing
an upper bound on this number; (ii) to verify that the observed irregularity
arises from the fractal nature of the deterministic system's invariant set,
as reflected by the fractional, rather than integer, character of (one of) its
dimension(s); and (iii) reconstruct the invariant set or even the equations
governing the dynamics from the data.
This ambitious program (Packard et al. 1980, Roux et al. 1980, Ru-
The remainder of this chapter is organized, over all, as follows. Section 2
deals mainly with signal-to-noise ratio (SIN) enhancement and introduces
singular-spectrum analysis (SSA) as an important and flexible tool for this
enhancement. Statistical tests for the reliability of SSA results are also
discussed. In Sec. 3, we present, in succession, three methods of spectral analysis: Blackman-Tukey, maximum entropy and multi-taper. Both
Secs. 2 and 3 use the Southern Oscillation Index (SOl) for the purposes of
illustrating the methods "in action". In Sec. 4, the multivariate extension
of single-channel SSA is introduced and a few additional applications are
mentioned or illustrated. The chapter concludes with a section on open
questions, from the point of view of both the methodology and its applications.
1.2 Time series and nonlinear dynamics
Before proceeding with the technical details, we give in this subsection a
quick perspective on the "nonlinear revolution" in time-series analysis. In
the 1960s and '70s, the scientific community found out that much of the
irregularity in observed time series, which had traditionally been attributed
to the above-mentioned random "pumping" of a linear system by infinitely
many (independent) degrees offreedom (d-o-/), could be generated by the
nonlinear interaction of a few d-o-J (Lorenz 1963, Ruelle and Takens 1971,
Smale 1967). This realization of the possibility of deterministic aperiodicity
or "chaos" (Gleick 1987) created quite a stir and the purpose ofthis review
is to describe briefly some of its implications for time-series analysis (Drazin
and King 1992), with a special view to climatic time series.
A connection between deterministically chaotic time series and the nonlinear, possibly low-order, dynamics generating them was attempted fairly
early in the young history of "chaos theory." The basic idea was to exploit specifically a univariate time series with apparently irregular behavior,
generated by a deterministic or stochastic system, in order to (i) ascertain
whether the underlying system has a finite number of d-o-J, by establishing
an upper bound on this number; (ii) to verify that the observed irregularity
arises from the fractal nature of the deterministic system's invariant set,
as reflected by the fractional, rather than integer, character of (one of) its
dimension(s); and (iii) reconstruct the invariant set or even the equations
governing the dynamics from the data.
This ambitious program (Packard et al. 1980, Roux et al. 1980, Ru-