Characteristic scales of reconstruction distortions
Alexei Potapov
Keldysh Institute of Applied Mathematics, Moscow
Quality of attractor reconstruction and performance of
time series processing algorithms strongly depends on the
choice of reconstruction parameters -- delay 
and embedding
dimension m. Improper choice causes reconstruction
distortions and for 
and 
reconstruction
becomes useless. 
can be determined by many existing
methods, though other rather simple can be proposed.
On small scales linear distortions can be corrected by reconstruction with relevance weighing (Farmer & Sidorowich) using
though on larger scales 
nonlinear distortions can not be corrected.
It is shown for the problem of estimating of the correlation dimension
for the model data, that weighting sometimes can improve the results,
but it require the precise knowledge of 
, too large
values lead to false dimension estimates. In the paper
two algorithms are proposed: for determining 
and
a time scale 
related with 
from a time series. They
are based upon calculating two values: one resembles
correlation integral and the other is related with averaging
of w over different reconstructions fixing the scale 
.
The results for both model and experimental data are presented. The knowledge of characteristic scales may be important e.g. for calculation of Lyapunov exponents from a time series: it states the upper limit for the neighbourhood size for which local approximation of "equations of motion" is done.