There has been a number of interesting results concerning the modeling
of time series from nonlinear dynamical systems. Since the
formulation of the reconstruction theorem by Takens
Takens:81 it has been clear that a nonlinear model of a
system may be derived directly from a systems time series. The method
of state space reconstruction has already been used for modeling and
analysis of musical sounds [Monro, 1993,Pressing et al. ,
1993].
For other time series prediction tasks the combination of
reconstruction techniques and neural networks has shown good results
[Weigend and Gershenfeld,
1993]. In our work we extend this ideas by
the more demanding task of building models, which are able to
resynthesize the systems time series. Because we are interested in
modeling musical instruments we extended the standard neural network
predictors such that they are able to model instationary dynamics.
In the following, we first give a short review concerning the state
space reconstruction from time series by delay coordinate vectors.
Then we explain the neural networks we used and the modification
necessary to model instationary dynamics. As an example we
investigate the neural models of a saxophone tone. In the end of the
paper we describe further results and applications.