262
Chapter 14: Patterns in Time: SSA and MSSA
The reconstruction of nonlinear attractors by embedding a single time series
has been demonstrated by Broomhead and King (1986), who compared phase
portraits of the original Lorenz attractor with the reconstructed one. The
result of Takens emphasized the need of analysing the structure of the data
embedded in the delay coordinates. SSA is precisely an analysis of the secondorder moments in this new coordinate system.
14.2.2 Dimension and Noise
The reconstruction problem, in practice, suffers from two major problems:
• One does not have any knowledge of the underlying dimension m of the
system, so that the condition on the embedding dimension is hard to
verify;
• Data are always polluted by noise due to round-off, measurement, estimation errors.
In the absence of noise, and with large sets of data, it is still possible to estimate the required value of m*, by increasing it and cakulating the dimension
ofthe attractor, using the Grassberger-Procaccia algorithm, for instance; this
dimension increases with m* up to a saturation value reached when the attractor is fully resolved in the delay-coordinate space.
Unfortunately, noise and the shortness of records mean that, generally, no
reliable estimate of the proper dimension is possible. In fact, due to these
problems, there is a scale down to which it is impossible to describe the
detailed structure of the attractor (Vautard and Ghil, 1989). With a finite,
noisy data series, only a macroscopic dimension can be estimated. We show
now how SSA may help to reconstruct an approximate attractor: one gives
up the ambition of reconstructing the very fine structure of the attractor
including complex foldings and fractal patterns, but one would like still to
conserve its macroscopic aspect'
14.2.3 The Macroscopic Approximation
A product of the EOF analysis is, as explained in Chapter 13, the distinction
between "signal" and "noise" components. Noisy components are usually reflected by the relatively flat behaviour of the tail of the eigenvalue spectrum.
The truncation of the EOF expansion (13.1) to the significant components
constitutes therefore an approximation of the cloud of data points, by projecting it onto a finite, restricted number of directions, spanned by the significant "guess patterns". In asense, the attractor is approximated by a flat
subspace. This approximation depends of course both on the noise variance
and the sam pie length, and is not characteristic of the physical system.
The EOF expansion in the delay-coordinate space (SSA) has the same
property; the state vector -Vt can be decomposed, as in (13.1) as the sum
Chapter 14: Patterns in Time: SSA and MSSA
The reconstruction of nonlinear attractors by embedding a single time series
has been demonstrated by Broomhead and King (1986), who compared phase
portraits of the original Lorenz attractor with the reconstructed one. The
result of Takens emphasized the need of analysing the structure of the data
embedded in the delay coordinates. SSA is precisely an analysis of the secondorder moments in this new coordinate system.
14.2.2 Dimension and Noise
The reconstruction problem, in practice, suffers from two major problems:
• One does not have any knowledge of the underlying dimension m of the
system, so that the condition on the embedding dimension is hard to
verify;
• Data are always polluted by noise due to round-off, measurement, estimation errors.
In the absence of noise, and with large sets of data, it is still possible to estimate the required value of m*, by increasing it and cakulating the dimension
ofthe attractor, using the Grassberger-Procaccia algorithm, for instance; this
dimension increases with m* up to a saturation value reached when the attractor is fully resolved in the delay-coordinate space.
Unfortunately, noise and the shortness of records mean that, generally, no
reliable estimate of the proper dimension is possible. In fact, due to these
problems, there is a scale down to which it is impossible to describe the
detailed structure of the attractor (Vautard and Ghil, 1989). With a finite,
noisy data series, only a macroscopic dimension can be estimated. We show
now how SSA may help to reconstruct an approximate attractor: one gives
up the ambition of reconstructing the very fine structure of the attractor
including complex foldings and fractal patterns, but one would like still to
conserve its macroscopic aspect'
14.2.3 The Macroscopic Approximation
A product of the EOF analysis is, as explained in Chapter 13, the distinction
between "signal" and "noise" components. Noisy components are usually reflected by the relatively flat behaviour of the tail of the eigenvalue spectrum.
The truncation of the EOF expansion (13.1) to the significant components
constitutes therefore an approximation of the cloud of data points, by projecting it onto a finite, restricted number of directions, spanned by the significant "guess patterns". In asense, the attractor is approximated by a flat
subspace. This approximation depends of course both on the noise variance
and the sam pie length, and is not characteristic of the physical system.
The EOF expansion in the delay-coordinate space (SSA) has the same
property; the state vector -Vt can be decomposed, as in (13.1) as the sum