Phase space reconstruction

The phase space reconstruction of a time series is accomplished by the method of delays. An embedding dimension de is chosen, plus a time delay t°, and then the phase space is constructed using as variables y(t), y(t + t°),... 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

The phase space reconstruction approach, making use only of the hormone plasma profiles, was utilized in order to assess the dimensionality and thus expose the chaotic nature of the underlying dynamics of various hormones. In... 

In that spirit, Widman et al. [576] adapted a prescription for an overall index of nonlinear coherence that has been found powerful for anticipating epileptic seizures from implanted electrode recordings. This index based on phase space reconstruction and correlation sums was called d, and it contains many ingredients familiar from the Grassberger-Procaccia algorithm for the correlation dimension [577]. 

The scalar data, Xj(t), and the values X/(t+i) and X/ H-lt), i.e., die values of Xi separated by a time-delay, x, between successive measurements (this procedure is called a pseudo-phase-space reconstruction). 

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