This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

(Thanks to Jim Crutchfield for contributing to this answer)

This is the application of dynamical systems techniques to a data series,

usually obtained by "measuring" the value of a single observable as a function

of time. The major tool in a dynamicist's toolkit is "delay coordinate

embedding" which creates a phase space portrait from a single data series. It

seems remarkable at first, but one can reconstruct a picture equivalent

(topologically) to the full Lorenz attractor (see [2.12])in three-dimensional

space by measuring only one of its coordinates, say x(t), and plotting the

delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.

It is important to emphasize that the idea of using derivatives or delay

coordinates in time series modeling is nothing new. It goes back at least to

the work of Yule, who in 1927 used an autoregressive (AR) model to make a

predictive model for the sunspot cycle. AR models are nothing more than delay

coordinates used with a linear model. Delays, derivatives, principal

components, and a variety of other methods of reconstruction have been widely

used in time series analysis since the early 50's, and are described in

several hundred books. The new aspects raised by dynamical systems theory are

(i) the implied geometric view of temporal behavior and (ii) the existence of

"geometric invariants", such as dimension and Lyapunov exponents. The central

question was not whether delay coordinates are useful for time series

analysis, but rather whether reconstruction methods preserve the geometry and

the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer &

Shaw)

G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.

N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry

from a time series", Phys. Rev. Lett. 45, no. 9 (1980) 712.

F. Takens, "Detecting strange attractors in fluid turbulence", in: Dynamical

Systems and Turbulence, eds. D. Rand and L.-S. Young

(Springer, Berlin, 1981)

Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T.

"The analysis of observed chaotic data in physical systems",

Rev. Modern Physics 65 (1993) 1331-1392.

D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,

Springer-Verlag http://www.cnd.mcgill.ca/books_understanding.html

E. Peters (1994) Fractal Market Analysis : Applying Chaos Theory to

Investment and Economics, Wiley

http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471585246.html

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