Time series analysis consists of a wide variety of techniques that allow to extract useful information from temporal data. The most basic one is the power spectral density (PSD) estimation via Fourier transform (for evenly spaced data) and Lomb-Scargle periodogram (for unevenly spaced). Time series can be characterised by many indicators, starting from the simple mean and standard deviation, to the Hurst exponent (describing the long range memory of the underlying process), Lyapunov exponent (allowing to detect chaotic phenomena), Shannon entropy (measuring the amount of information produced by a process) etc.

If the Hurst exponent, 0<H<1, is greater than 0.5, we say that there is long term memory, or the process is persistent: it remembers what it did in the past, and acts persistently, i.e. if a relatively high value is observed, the probability that the next value will also be high is greater than that it will be low. Similarly, when H<0.5 the process has short term memory (is anti-persistent) and the low and high values alternate rapidly. H is related to the fractal properties of the time series, i.e. to self-similarity or self-affinity.

Figure 1. Example realisations of two types of stochastic processes: fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) with short (H<0.5) and long (H>0.5) term memory.

The Lyapunov exponent (LE) is a key concept of the theory of chaos. It is a measure of the mean exponential divergence of nearby orbits. When LE>0, the system is sensitive to initial conditions (the so-called butterfly effect), and behaves chaotically. Contrary, when LE≤0, the system is regular and acts in a predictable manner. Chaos is a wide-spread phenomenon; in fact it's easier to find a chaotic rather than regular system. Examples can be found in cosmology, financial markets, metheorology, plasma physics, hydrodynamics, celestial mechanics, and many more. A tight correlation between the Hurst and Lyapunov exponents was a surprising finding of mine that yet awaits a formal explanation.

Figure 2. The correlation between the Lyapunov and Hurst exponents for the Chirikov standard map.

Yet another surprising correlation I found is the one between the Abbe value, A (the ratio of the mean square successive difference to the variance), and the proportion of turning points, T (i.e., local extrema, or peaks/valleys, in a time series). Both these values are in turn tightly related to the Hurst exponent. Initially, I intended the A—T correlation to be a fast and simple estimator of the Hurst exponent. It turns out it is very useful by itself, and has already been applied with great success to physiological data (Zunino et al. 2017) and to discriminate between regular, stochastic and chaotic phenomena (Zhao & Morales 2018). This discrimating power, together with the dependence of the location in the A—T plane on the PSD shape of the time series, is a very promising novel tool for astrophysical classification (Tarnopolski et al. 2020). A formal derivation of the A(H) and T(H) relations has provided closed-form expressions and working approximations (Tarnopolski 2019).

Figure 3. Relations between (a) H and A, and (b) H and T (scaled by the expected value). (c-d) The separation in the A—T plane of different types of noise: (c) fBm, fGn, and differentiated fGn (DfGn), (d) stochastic processes with power law PSDs with varying indices.

A—T plane: a new tool for time series analysis" – a lecture given at the Mark Kac Complex Systems Research Centre at the Jagiellonian University, 09.12.2019, and the corresponding slides