Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Acknowledgements
- I Basic topics
- II Advanced topics
- 9 Advanced embedding methods
- 10 Chaotic data and noise
- 11 More about invariant quantities
- 12 Modelling and forecasting
- 13 Non-stationary signals
- 14 Coupling and synchronisation of nonlinear systems
- 15 Chaos control
- A Using the TISEAN programs
- B Description of the experimental data sets
- References
- Index
9 - Advanced embedding methods
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Acknowledgements
- I Basic topics
- II Advanced topics
- 9 Advanced embedding methods
- 10 Chaotic data and noise
- 11 More about invariant quantities
- 12 Modelling and forecasting
- 13 Non-stationary signals
- 14 Coupling and synchronisation of nonlinear systems
- 15 Chaos control
- A Using the TISEAN programs
- B Description of the experimental data sets
- References
- Index
Summary
The reconstruction of a vector space which is equivalent to the original state space of a system from a scalar time series is the basis of almost all of the methods in this book. Obviously, such a reconstruction is required for all methods exploiting dynamical (such as determinism) or metric (such as dimensions) state space properties of the data. In the first part of the book we introduced the time delay embedding as the way to find such a space. Because of the outstanding importance of the state space reconstruction we want to devote the first section of this chapter to a deeper mathematical understanding of this aspect. In the following sections we want to discuss modifications known as filtered embeddings, the problem of unevenly sampled data, and the possibility of reconstructing state space equivalents from multichannel data.
Embedding theorems
A scalar measurement is a projection of the unobserved internal variables of a system onto an interval on the real axis. Apart from this reduction in dimensionality the projection process may be nonlinear and may mix different internal variables, giving rise to additional distortion of the output. It is obvious that even with a precise knowledge of the measurement process it may be impossible to reconstruct the state space of the original system from the data. Fortunately, a reconstruction of the original phase space is not really necessary for data analysis and sometimes not even desirable, namely, when the attractor dimension is much smaller than the dimension of this space.
- Type
- Chapter
- Information
- Nonlinear Time Series Analysis , pp. 143 - 173Publisher: Cambridge University PressPrint publication year: 2003
- 1
- Cited by