Analysis of observed chaotic data (Abarbanel, H. D.I.)

Comments on "Analysis of observed chaotic data," Abarbanel H.D.I. Institute of Nonlinear Science Springer Study Edition 1995

pg 4 1.1.2 Correlations among data points

The dynamics takes palce in a space of vectors y(t) of larger dimension, and we view it projected down on the axis of observed variables. we can identify a space formally equivalent to the original space of variables using coordinates made out of the observed variables and its time delayed copies. - Phase space reconstruction.

The time delay in the embedding is divided in terms on nonlinear correlation function called average mutual information has its first minimum.

pg 5 1.1.3 number of coordinates

the number of coordinates to use is determined by asking when the projection of the geometrical structure has been completely unfolded. implies that points lying close to one another in the space of the y(t) vectors do so because of the dynamics and not because of the projection. this can be decided on the basis of False Nearest Neighbors Algorithm. [false neighbors are connected with unpredictability as phase space points will be near to each other for non dynamical reasons, namely just because of projection. it may happen that the percentage of false nearest neighbors drops to ZERO is the necessary global dimension to be used to examine the observed data]. pg 7

reconstructed series replace the scalar data measurements with data vectors in Euclidean distance d-dimensional space in which the invariant aspects of the sequence of points s(n) are captured with no loss of information about the properties of the original system. the new space is related to the original space of the s(n) by smooth, differentiable transformations.

what is the time lag Ts to use and what dimension d to use are the central issues of this reconstruction ----
the theorem notes that if the motion lies on a set of dimension Da, which could be fractional, then choosing the integer dimension d of the unfolding space so d > 2Da is sufficient to undo all overlaps and make the orbit unambiguous.

pg25
Time delays
it must be some multiple of the sampling time tau_s, since we only have data at those times, an interpolation scheme to get more data is just as uncertain as estimating the time derivatives of s(t).
if the time delay is too short, the coordinates s(n) and s(n+T) which we wish to use in our reconstructed data vector y(n) will not be independent enough.
finally, since chaotic systems are intrinsically unstable, if T is too large, any connection between the measurements is numerically tantamount to being random with respect to reach other.

Average mutual information -
the actual prescription suggesed is to take the T where the first minimum of the average mutual information I(T) occurs as that valye to use in time delay reconstruction of phase space. Average mutual information is invariant under smooth changes of coordinate system. Thus, the quantity I(T) evaluated in time delay coordinates and in the original, but unknown, coordinates takes on the same values. Also, I(T) will be robust against noise than many other quantities.

choosing the dimension of reconstructed phase space
De is a global dimension and may well be different from the local dimension of the underlying dynamics.
using global false nearest neighbours --
since the attractors for real physical systems is quite compact in phase space, each phase space point will have numerous neighbours as the number of data becomes large enough to populate state space well.

Teh basic geometric idea in the embedding theorem is that we have achieved an acceptable unfolding of the attractor from its values as seen projected on the observation axis when the orbits conposing the attractor are no longer crossing one another in the reconstructed phase space.

filtering out spatial variations on small scales, perhaps even associated with high frequencies as well, ca nlead to a practical system of lower dynamical dimensions.