Section 14.3: Singular Spectrum Analysis
267
wishes to reconstruct astrange attractor, whose spectrum includes periods
of arbitrary length, the larger m* the better, as long as statistical errors do
not dominate the last values of the autocovariance function. To prevent this,
one should not exceed in practice m* = ~.
In many physical and engineering applications, however, one wishes to
concentrate on oscillatory phenomena in a certain frequency band, which may
be associated with the least-unstable periodic orbits embedded in astrange
attractor. Such periodic orbits typically generate oscillations of stronglyvarying amplitude: the system's trajectory approaches and follows them for
a certain time, comparable to or longer than the period in question, only to
wander off into other parts of phase space. When the ratio of m* to the life
time of such an intermittent oscillation, the typical time interval of sustained
high amplitudes speIls, is large, the oscillatory eigenvector pair suffers from
the same Gibbs effect as classical spectral analysis. SpeIls of the oscillation
will be smoothed out. The following arguments should clarify the difficulty
and help in making the correct choice of m* .
If m* is too smaIl, the coarse spectral resolution may mix together several
neighboring peaks in the spectrum of X t . When there is an intermittent oscillation, reflected by a broad spectral peak, on the contrary, large m* values
(high resolution) will split the peak into several components with neighboring frequencies. In fact, it can be shown that, given a peak in the power
spectrum r zU) of X t , with maximal spectral density at /0 and width 21l./,
SSA will isolate correctly the intermittent oscillation if
1
*
1
10 ~ m ~ 21l.1
(14.14)
In other words, the embedding dimension has to be chosen between the period
of the oscillation and the average life time of its speIls. In practice, this
latter quantity cannot be estimated apriori, but SSA is typically successful
at analyzing periods in the range (~. ,m*).
14.3.6 Estimating Time and Space-Time Patterns
The estimation of SSA elements does not really differ from the estimation
of EOFs, since the calculations are the same, except that they are transposed in the delay-coordinate phase space. However, one has to be careful
since generally the SSA eigenvalue spectra are generally flatter than EOFs
eigenvalue spectra. Hence eigenvalues are more "degenerate" and the associated patterns tend to be mixed from one sampIe to another. TypicaIly, for
short sampIes, the order of a "significant" component is statistically unstable,
whereas its spectral properties are not. Also, pairs of oscillatory components
are almost degenerate (see the example of Section 14.3.4). Thus, from one
sampIe to another, one does not in general expect to re cover the right phase
in the same oscillatory T-EOF or ST-EOF.
267
wishes to reconstruct astrange attractor, whose spectrum includes periods
of arbitrary length, the larger m* the better, as long as statistical errors do
not dominate the last values of the autocovariance function. To prevent this,
one should not exceed in practice m* = ~.
In many physical and engineering applications, however, one wishes to
concentrate on oscillatory phenomena in a certain frequency band, which may
be associated with the least-unstable periodic orbits embedded in astrange
attractor. Such periodic orbits typically generate oscillations of stronglyvarying amplitude: the system's trajectory approaches and follows them for
a certain time, comparable to or longer than the period in question, only to
wander off into other parts of phase space. When the ratio of m* to the life
time of such an intermittent oscillation, the typical time interval of sustained
high amplitudes speIls, is large, the oscillatory eigenvector pair suffers from
the same Gibbs effect as classical spectral analysis. SpeIls of the oscillation
will be smoothed out. The following arguments should clarify the difficulty
and help in making the correct choice of m* .
If m* is too smaIl, the coarse spectral resolution may mix together several
neighboring peaks in the spectrum of X t . When there is an intermittent oscillation, reflected by a broad spectral peak, on the contrary, large m* values
(high resolution) will split the peak into several components with neighboring frequencies. In fact, it can be shown that, given a peak in the power
spectrum r zU) of X t , with maximal spectral density at /0 and width 21l./,
SSA will isolate correctly the intermittent oscillation if
1
*
1
10 ~ m ~ 21l.1
(14.14)
In other words, the embedding dimension has to be chosen between the period
of the oscillation and the average life time of its speIls. In practice, this
latter quantity cannot be estimated apriori, but SSA is typically successful
at analyzing periods in the range (~. ,m*).
14.3.6 Estimating Time and Space-Time Patterns
The estimation of SSA elements does not really differ from the estimation
of EOFs, since the calculations are the same, except that they are transposed in the delay-coordinate phase space. However, one has to be careful
since generally the SSA eigenvalue spectra are generally flatter than EOFs
eigenvalue spectra. Hence eigenvalues are more "degenerate" and the associated patterns tend to be mixed from one sampIe to another. TypicaIly, for
short sampIes, the order of a "significant" component is statistically unstable,
whereas its spectral properties are not. Also, pairs of oscillatory components
are almost degenerate (see the example of Section 14.3.4). Thus, from one
sampIe to another, one does not in general expect to re cover the right phase
in the same oscillatory T-EOF or ST-EOF.