Abstract
Stochastic time series are ubiquitous in nature. In particular, random walks with time-varying statistical properties are found in many scientific disciplines. Here we present a superstatistical approach to analyse and model such heterogeneous random walks. The time-dependent statistical parameters can be extracted from measured random walk trajectories with a Bayesian method of sequential inference. The distributions and correlations of these parameters reveal subtle features of the random process that are not captured by conventional measures, such as the mean-squared displacement or the step width distribution. We apply our new approach to migration trajectories of tumour cells in two and three dimensions, and demonstrate the superior ability of the superstatistical method to discriminate cell migration strategies in different environments. Finally, we show how the resulting insights can be used to design simple and meaningful models of the underlying random processes.
Highlights
Stochastic time series are ubiquitous in nature
The most frequently used statistical measures for random walks, in particular the step width distribution (SWD), the mean-squared displacement (MSD) and the velocity autocorrelation function, are implicitly assuming that the stochastic process can be globally described by a few characteristic parameters, such as a constant variance and a constant correlation time
We demonstrate in this paper that the application of these conventional methods to heterogeneous random walks generates ‘anomalous’ results, such as non-Gaussian SWDs or power-law MSDs with fractional exponents[5,6,7]
Summary
Stochastic time series are ubiquitous in nature. In particular, random walks with time-varying statistical properties are found in many scientific disciplines. Stochastic time series, here used synonymously with random walks, play an important role in earth- and life sciences, technology, medicine and economics Most of these disciplines deal with complex systems in which multiple hierarchical processes are interacting at different timescales. We demonstrate in this paper that the application of these conventional methods to heterogeneous random walks generates ‘anomalous’ results, such as non-Gaussian SWDs or power-law MSDs with fractional exponents[5,6,7] These anomalies emerge inevitably from the temporal averaging over changing local statistics during the evaluation period (Supplementary Note 1), and do not provide meaningful insights into the random walk apart from its heterogeneous nature. Heterogeneous time series of arbitrary complexity can be described (Supplementary Note 2)
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