Time-delayed embedding

The fundamental logic of empirical dynamic modelling are based on Taken's theorem, and the core idea of Taken's theorem is to create these things called time-delayed embeddings (or time-delayed reconstructions). Essentially, we take a time series and break it into many overlapping short trajectories of a fixed length. This page steps through a basic example of this process, highlighting some relevant hyperparameters which can be set.

Imagine that we observe a time series \(a\).

Choose the number of observations

In tabular form, the data looks like:

So each one of \(a_i\) is an observation of the \(a\) time series.

To create a time-delayed embedding based on any of these time series, we first need to choose the size of the embedding \(E\).

Choose a value for \(E\)

The data may be too finely sampled in time. So we select a \(\tau\) which means we only look at every \(\tau\)th observation for each time series.

Choose a value for \(\tau\)

The time-delayed embedding of the \(a\) time series with

is the manifold:

The manifold is a collection of these time-delayed embedding vectors. For short, we just refer to each vector as a point on the manifold. While the manifold notation above is the most accurate (a set of vectors) we will henceforward use the more convenient matrix notation:

Note that the manifold has \(E\) columns, and the number of rows depends on the number of observations in the \(a\) time series.

Why does this look backwards?

You may wonder why we have each point in the manifold going backwards in time when reading left-to-right. This is simply an unfortunate convention in the EDM literature which we adhere to.