Mackey-Glass Predictability

MGpredictability is a Julia library for time series analysis

To learn how yo use MGpredictability you can see Getting started and Contents for further references.

Getting started

To install MGpredictability you can do:

using Pkg; Pkg.add("https://github.com/jTarigo/MGpredictability")

The library is composed of

The Mackey-Glass System

The Mackey-Glass (MG) system was proposed in 1977 to model the process of crations of blood cells in the bone marrow and its realse to the blood stream Mackey, M. & Glass L. (1977). It can be modeled as the following DDE:

\[\frac{dx}{dt} = \alpha \frac{x_{\Gamma}}{1 + x^{n}_{\Gamma}} - x\]

where $\alpha$, $\Gamma$ and $n$ are parameters and $x_{\Gamma} = x(t - \Gamma)$.

DDEs can be solved by many methods used for solving ODEs like the Runge-Kutta methods. However, for the MG system there is an exact discrete implementation as described in Amil, P., Cabeza, C., & Marti, A. C. (2015).

This implementation leaves the equation as a map of $N$ variables as:

\[x_{j}(t + 1) = \begin{cases} x_{j + 1} \quad \quad \quad \quad \qquad \quad \quad \quad \quad \quad if \quad j < N \\ \beta x_j + (1 - \beta) \alpha \frac{x_{j - N +1}}{1 + x^{n}_{j - N +1}} \quad \quad if \quad j = N \end{cases}\]

where $N$ is the number of points in a time interval $\Gamma$ and $\beta = \exp(\Gamma/N)$. This map has a jacobian of the form:

\[J = \begin{bmatrix} 0 & 1 & 0 & \dots & 0\\ 0 & 0 & 1 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \dots & 1\\ \alpha (1 - \beta) \frac{1 - (n - 1) x^n_1}{(1 + x^n_1)^2} & \dots & \dots & \dots & \beta \end{bmatrix}\]