Calculation of Average Mutual Information (AMI) and False-Nearest Neighbors (FNN) for the Estimation of Embedding Parameters of Multidimensional Time Series in Matlab

Abstract

Using the method or time-delayed embedding, a signal can be embedded into higher-dimensional space in order to study its dynamics. This requires knowledge of two parameters: The delay parameter τ, and the embedding dimension parameter D. Two standard methods to estimate these parameters in one-dimensional time series involve the inspection of the Average Mutual Information (AMI) function and the False Nearest Neighbor (FNN) function. In some contexts, however, such as phase-space reconstruction for Multidimensional Recurrence Quantification Analysis (MdRQA), the empirical time series that need to be embedded already possess a dimensionality higher than one. In the current article, we present extensions of the AMI and FNN functions for higher dimensional time series and their application to data from the Lorenz system coded in Matlab.

Keywords: Multidimensional Recurrence Quantification Analysis; Multidimensional Time series; average mutual information; code:Matlab; false-nearest neighbors; time-delayed embedding.

Figure 1

The original three-dimensional Lorenz system (A), a time series corresponding to the dynamics of the x-axis of the Lorenz system (B), two surrogate series of the x-axis data by shifting the time series for a number of lags equal to τ (C) and 2τ (D). Note the loss of data points in creating the surrogate data in (C,D) evident through missing data points at the end of the time series. When the time series in (B–D) are plotted against each other, the resulting phase-space (E) approximates the original phase-space of the Lorenz system (A).

Figure 2

(A) Shows the graphical output of the mdDelay function for the three-dimensional time series from the Lorenz system. Since the function was called with the option to show the AMI function (Equation 1) for each dimension in the data, there are three curves. The default threshold value (1/e) is shown as the horizontal line in the plot. (B) Shows the graphical output of the mdFnn function for the three-dimensional time series from the Lorenz system. The function was called with the parameters maxEmb = 10, tau = 15 using all three variables x, y and z with 10⁴ number of data points each. The function shows an immediate drop-off of the percentage of false-nearest neighbors to 0, indicating that no additional embedding is necessary for the three-dimensional time series from the Lorenz system.