Tangle: A Metric for Quantifying Complexity and Erratic Behavior in Short Time Series

Background:Temporal complexity refers to qualities of a time series that are emergent, erratic, or not easily described by linear processes. Quantifying temporal complexity within a system is key to understanding the time based dynamics of said system. However, many current methods of complexity quantification are not widely used due to their technical difficulty, computational intensity, or large number of required data samples. These requirements impede the study of complexity in areas of research such as behavioral science and mental health. A method is presented, tangle, which overcomes these difficulties and allows for complexity quantification in relatively short time series, such as those typically obtained by these research areas. Tangle is a measure of how dissimilar a given process is from simple periodic motion. Methods:Tangle is derived from an iterative scaling and untangling procedure. This procedure relies on the use of a 3-dimensional time delay embedding of a 1-dimensional time series. This embedding is then iteratively scaled and premultiplied by a modified upshift matrix until a convergence criterion is reached. Results:Systems studied are: (A) random noise, (B) a Lorenz attractor, (C) a Rossler attractor, (D) a Hindmarsh-Rose neuron model, and (E) a noisy sine wave. An example application of tangle for studying mental health is given using emotional stability and anxiety time series data obtained from 65 socially anxious participants over a five-week period. Simulation results show tangle is able to distinguish between different complex temporal systems in time series with as few as 50 samples. Conclusions:Tangle shows promise as a reliable quantification of irregular behavior of a time series. Unlike many other complexity quantification metrics, tangle is technically simple to implement and is able to uncover meaningful information about time series derived from behavioral and mental health research studies. Future work should focus on comparing tangle to other temporal complexity metrics and seek to optimize convergence.