Am. Albano et al., KOLMOGOROV-SMIRNOV TEST DISTINGUISHES ATTRACTORS WITH SIMILAR DIMENSIONS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 52(1), 1995, pp. 196-206
Recent advances in nonlinear dynamics have led to more informative cha
racterizations of complex signals making it possible to probe correlat
ions in data to which traditional linear statistical and spectral anal
yses were not sensitive. Many of these new tools require detailed know
ledge of small scale structures of the attractor; knowledge that can b
e acquired only from relatively large amounts of precise data that are
not contaminated by noise-not the kind of data one usually obtains fr
om experiments. There is a need for tools that can take advantage of '
'coarse-grained'' information, but which nevertheless remain sensitive
to higher-order correlations in the data. We propose that the correla
tion integral, now much used as an intermediate step in the calculatio
n of dimensions and entropies, can be used as such a tool and that the
Kolmogorov-Smirnov test is a convenient and reliable way of comparing
correlation integrals quantitatively. This procedure makes it possibl
e to distinguish between attractors with similar dimensions. For examp
le, it can unambiguously distinguish (p < 10(-8)) the Lorenz, Rossler,
and Mackey-Glass (delay = 17) attractors whose correlation dimensions
are within 1% of each other. We also show that the Kolmogorov-Smirnov
test is a convenient way of comparing a data set with its surrogates.