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.

Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer

“…The optimal embedding dimension m is selected when the percentage of FNN vanishes. Multiplying this dimension m by the number of components of data-set d, provides the true estimate of dimensionality [59]. The procedure is then repeated decreasing the data dimensionality to d = 2 and then to d = 1, in order to explore the appropriate embedding dimension.…”

Development and Validation of Wearable Systems for Human Postural Sway Analysis

“…Average Mutual Information (AMI) is used to find optimal delay for embedding. Independence is quantified by I(x(t), x(t + τ )) where τ is a constant that shifts the signal by some factor from original x(t) and x(t + τ ) (Wallot & Mønster, 2018). This problem can be evaluated by a nonlinear generalization of the auto-correlation function and is given by equation 4.1 (Wallot & Møn-ster, 2018).…”

“…False Nearest Neighbors (FNN) is used to find the optimal embedding dimension of the signal (Wallot & Mønster, 2018). This is reconstructed by embedding the original one-dimensional time series by taking time delayed signals (Wallot & Mønster, 2018). This is described by the original time series y(t) and follows equation 4.2.…”

“…t and τ are integers that are indexed on the sample data (Wallot & Mønster, 2018). The embedding dimension D can be estimated by examining the distance between points in phase space.…”

“…This requires both finding the dimension to embed the signal in and amount of time to delay the embedding. Two methods are used to find these two methods; Average Mutual Information (AMI) and False Nearest Neighbor(FNN) function (Wallot & Mønster, 2018, Marwan et al, 2007. RQA is used to quantify the duration and number of recurrences in the time series.…”