Oscillations, Waves, and Interactions - GWDG

attractor in the
unknown
state space M
flow φ t
x
measurement
h: M IR
observable s=h(x)
Complex dynamics of nonlinear systems 421
h
s
F
measured time series
reconstruction
of the attractor in IR
d
Figure 11. Delay reconstruction of states from scalar time series.
t
y
delay coordinates
y(t) = (s(t), s(t - τ), ... , s(t -( d-1)
τ))
d reconstruction dimension
τ delay time
about the reconstruction of states, in particular in the presence of noise, may be
found in Refs. [46,47].
If the data are measured with a high sampling rate Broomhead-King-coordinates [48,
49] may be advantageous. With this method a very high-dimensional reconstruction
is used and then a new coordinate system is introduced where the origin is shifted
to the center of mass of the reconstructed states and the axes are given by the
(dominant) principal components of the distribution of points (states). This new
coordinate system is based on a Karhunen-Loève transformation 6 that may be computed
by a singular-value decomposition. A discussion of the advantages (e. g., noise
reduction) and disadvantages of this “post-processing” of the reconstructed states
may, for example, be found in Ref. [50].
For time series that consist of a sequence of sharp spikes (e. g. from firing neurons)
delay embedding may lead to very inhomogeneous sets of points in the reconstructed
state space that are difficult to analyse. As an alternative one may use in this
case the time intervals between the spikes as components for (re-)constructed state
vectors [51–53].
4.2 Forecasting and Modelling
After successful state space reconstruction one can approximate the dynamics in reconstruction
space to forecast or control the underlying dynamical process. Very
simple but efficient algorithms for nonlinear prediction are nearest neighbours methods
(also called local models). Let’s assume that we want to forecast the future
evolution of a given (reconstructed) state for some time horizon T . Available (i. e.,
6 Also called Proper Orthogonal Decomposition (POD) or Principle Component Analysis
(PCA).