Application to the Detection of Non-linear Coupling

The aim of the detection of non-linear coupling is to highlight hidden frequencies in times series. One of the main interest of this technique is to give a reliable detection of intrinsic frequencies even in noisy and short time series. From the non-linear dynamics point of view, a time series is considered as the output of a dynamical system, it can be the cardiorespiratory signals or, used as an example the Lorenz attractor.

The use embedding techniques permits to reconstruct geometric (like attractor structure) and statistical information of the original system by constructing an equivalent representation of it, using time-delayed coordinates. Given a time series of m values, an an n-dimensional vector is constructed : xi=(xi,xi+t,...,xi+(n-1)t) where t is the time delay. These multivariate vectors in the reconstructed n-dimensional space, called the phase space, are used to trace the orbit of the system. The performing of statistical description of dynamical systems gives importance to the natural probability measures. Partitioning the reconstructed phase space allows to estimate the probability density : (1)

where n(xi) is the number of points in partition element i, which is around xi.
The procedure used to detect the hidden frequencies in the previous time series is basically the following :

• choose a relatively high embedding dimension (high enough to unfold the geometrical structure of the attractor)
• perform the reconstruction for a given range of time lags t (the total should beseveral time the correlation length of the time series)
• calculate the density of points encountered by the trajectory as it evolves
• estimate the probability measure the system encounters as it evolves with : (2)

The partition is done using small spheres (radius from 5% to 20% of the attractor total extent), doing this a new time series can be constructed using the density data in each sphere. Now the information on each region of the reconstructed phase space is consistent with Eq. (1).

• perform a periodogram over the density time series (N representing number of data points in the density time series) : Applying this methodology on a given time series permits to highlight frequencies correlation between different variates like heartrate and respiration rythm in the case of the cardiorespiratory interaction, or the x and z coordinates in the Lorenz system but in this case the major interest is pedagogical.