# Determining embedding dimension for phase-space reconstruction using a geometrical construction.

@article{Kennel1992DeterminingED, title={Determining embedding dimension for phase-space reconstruction using a geometrical construction.}, author={Kennel and Brown and Abarbanel}, journal={Physical review. A, Atomic, molecular, and optical physics}, year={1992}, volume={45 6}, pages={ 3403-3411 } }

We examine the issue of determining an acceptable minimum embedding dimension by looking at the behavior of near neighbors under changes in the embedding dimension from d\ensuremath{\rightarrow}d+1. When the number of nearest neighbors arising through projection is zero in dimension ${\mathit{d}}_{\mathit{E}}$, the attractor has been unfolded in this dimension. The precise determination of ${\mathit{d}}_{\mathit{E}}$ is clouded by ``noise,'' and we examine the manner in which noise changes the…

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## References

SHOWING 1-4 OF 4 REFERENCES

### Ergodic theory of chaos and strange attractors

- Physics
- 1985

Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the…

### Determining embedding dimension for phase-space reconstruction using a geometrical construction

- Phys Rev A
- 1992