Singular Systems Analysis (SSA), or time domain
Principal Component Analysis (PCA),
is most appropriately analysed in terms of local, moving-window spectral
analysis. The
behaviour of Empirical Orthogonal Functions (EOF) of this theory are examined,
for
continuously sampled data, in the limits of large and small window length,
and for centre or
end projection. Filters obtained by projecting on to these EOFs are shown
to approximate
local, linear band pass filters, where the EOFs depend
upon the correlation structure (or the
power spectral density) of the signal and the window length. Power in the
spectra is not
generally conserved, and projection to the endpoints of a window may not
converge to the
underlying signal in the absence of noise. The filters are independent
of the phase of the
Fourier transform, and are therefore unable to distinguish dynamically
between a signal and a
surrogate (phase-randomized) transform of it. Iteration of such
local filters using a prediction
error-based stopping criterion can and does lead to improved results,
but the choice of window
length must be made a priori. Hence, we introduce an iterative
local filter with the window
length being determined as part of the filtering procedure. This involves
the determination of
the predictability of the projected time series, and hence allows SSA to
be used in a genuinely
nonlinear way.
