Delay Coordinate Embedding

This page is under construction.

In this section, we consider the application of delay coordinate embedding to time-series prediction.

The state of a deterministic dynamical system is the information required to determine the entire future evolution of the system.

The state space is the set of all possible states.

A time series is a sequence of numbers or vectors corresponding to observable quantities that are deterministically related to the states of the dynamical system.

Delay coordinate embedding approaches to prediction (attempt to) identify the state s of the system at some time t, search the past history of observations for similar states, and, by studying the evolution of the observable outputs following the similar states, infer information about the future of the system.

If s_t is the state of the system at time t, then let h(s_t) be the observation of the state at time t.

Simple Prediction

We assume that the dynamical system is deterministic and that for an appropriate integer m the vector of observations called the delay coordinate vector and defined by

	d_t=(h(state_t),...,h(s_{t-(m-1)tau}))

where 1/tau is the sampling rate, serves as a proxy for the invisible states.

Let A denote a compact, finite-dimensional set of states of the system corresponding to the attractor of the dynamical system.

Under fairly reasonable conditions on the dynamics (see page 21 of [Gershenfeld and Weigend, 1993]) the correspondence D(s_t)=d_t is one-to-one which means that the behavior of the attractor is accounted for in the behavior of the delay coordinate embedding defined by D.

Assume that we know an appropriate m to define the length of our delay coordinate vector.

Suppose that our sequence of observations ends at time t and that we would like to predict h(s_{t + delta}.

The delay coordinate vector d_t acts as a surrogate for s_t.

We use d_t to find ``similar'' delay vectors earlier in the time series.

In the best of worlds, we find an exact match d_{t'} and we use d_{t' + delta} (suppose that t' + delta <= t and that delta = K tau for some constant K) as our prediction for d_{t + delta} from which we can read off h(s_{t' + delta} as our preduction for h(s_{t + delta}.

In practice, of course, prediction is much more complicated. See [Sauer, 1994] in [Gershenfeld and Weigend, 1993]) and the discussion in DelayCoordinateEmbedding.m for more details.

Exercises

The notebook DelayCoordinateEmbedding.m provides an introduction to delay-coordinate embedding techniques for analyzing time series generated by dynamical systems.

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