# Symbolic dynamics techniques for complex systems: Application to share price dynamics

## TL;DR

This paper demonstrates that symbolic dynamics techniques, traditionally used for low-dimensional systems, can effectively analyze high-dimensional complex systems like stock market returns, revealing non-Markovian properties and enabling cross-domain comparisons.

## Contribution

It extends symbolic dynamics methods to high-dimensional complex systems and applies them to financial data, uncovering non-Markovian behavior and spectral properties.

## Key findings

- Nontrivial Renyi entropy spectrum in stock returns
- Dependence of entropy spectrum on time scale and sector
- Transition probabilities depend on entire symbol history

## Abstract

The symbolic dynamics technique is well-known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Our main example is the investigation of share price returns in a coarse-grained way. A nontrivial spectrum of Renyi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a modelling based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of symbol sequences generated by share price returns using 4 symbols can be compared with that of genomic sequences.