Biomedical Engineering Reference
In-Depth Information
if apparently pessimistic, calculation of (74), that attractor reconstruction with
an
embedding
dimension of
k
needs 42
k
data-points!) In the early days of the
application of embedding methods to experimental data, these limitations were
not well appreciated, leading to many calculations of low-dimensional determi-
nistic chaos in EEG and EKG series, economic time series, etc., which did not
stand up to further scrutiny. This in turn brought some discredit on the methods
themselves, which was not really fair. More positively, it also led to the devel-
opment of ideas such as
surrogate-data methods
. Suppose you have found
what seems like a good embedding, and it appears that your series was produced
by an underlying deterministic attractor of dimension
D
. One way to test this
hypothesis would be to see what kind of results your embedding method would
give if applied to similar but
non
-deterministic data. Concretely, you find a sto-
chastic model with similar statistical properties (e.g., an ARMA model with the
same power spectrum), and simulate many time series from this model. You
apply your embedding method to each of these
surrogate data
series, getting
the approximate distribution of apparent "attractor" dimensions when there
really is no attractor. If the dimension measured from the original data is not
significantly different from what one would expect under this null hypothesis,
the evidence for an attractor (at least from this source) is weak. To apply surro-
gate data tests well, one must be very careful in constructing the null model, as it
is easy to use over-simple null models, biasing the test towards apparent deter-
minism.
A few further cautions on embedding methods are in order. While
in princi-
ple
any lag U is suitable, in practice both very long and very short lags lead to
pathologies. A common practice is to set the lag to the autocorrelation time (see
above), or the first minimum of the mutual information function (see ยง7 below),
the notion being that this most nearly achieves a genuinely "new" measurement
(75). There is some evidence that the mutual information method works better
(76). Again, while in principle almost any smooth observation function will do,
given enough data, in practice some make it much easier to reconstruct the dy-
namics; several
indices of observability
try to quantify this (77). Finally, it
strictly applies only to deterministic observations of deterministic systems. Em-
bedding approaches are reasonably robust to a degree of noise in the observa-
tions. They do not cope at all well, however, to noise in the dynamics itself. To
anthropomorphize a little, when confronted by apparent non-determinism, they
respond by adding more dimensions, and so distinguishing apparently similar
cases. Thus, when confronted with data that really are stochastic, they will infer
an infinite number of dimensions, which is correct in a way, but definitely not
helpful. These remarks should not be taken to belittle the very real power of
nonlinear dynamics methods. Applied skillfully, they are powerful tools for un-
derstanding the behavior of complex systems, especially for probing aspects of
their structure which are not directly accessible.
