Abstract
Order, disorder and recurrence are common features observed in complex time series that can be encountered in many fields, like finance, economics, biology and physiology. These phenomena can be modelled by chaotic dynamical systems and one way to undertake a rigorous analysis is via symbolic dynamics, a mathematical-statistical technique that allows the detection of the underlying topological and metrical structures in the time series. Symbolic dynamics is a powerful tool initially developed for the investigation of discrete dynamical systems. The main idea consists in constructing a partition, that is, a finite collection of disjoint subsets whose union is the state space. By identifying each subset with a distinct symbol, we obtain sequences of symbols that correspond to each trajectory of the original system. One of the major problems in defining a “good” symbolic description of the corresponding time series is to obtain a generating partition, that is, the assignment of symbolic sequences to trajectories that is unique, up to a set of measure zero. Unfortunately, this is not a trivial task, and, moreover, for observed time series the notion of a generating partition is no longer well defined in the presence of noise. In this paper we apply symbolic shadowing, a deterministic algorithm using tessellations, in order to estimate a generating partition for a financial time series (PSI20) and consequently to compute its entropy. This algorithm allows producing partitions such that the symbolic sequences uniquely encode all periodic points up to some order. We compare these results with those obtained by considering the Pesin’s identity, that is, the metric entropy is equal to the sum of positive Lyapunov exponents. To obtain the Lyapunov exponents, we reconstruct the state space of the PSI20 data by applying an embedding process and estimate them by using the Wolf et al. algorithm.
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Notes
- 1.
The representatives can be chosen so that
$${\sup }_{x\in X}\left \Vert x - {r}_{{\Phi }_{[-k,k]}\left (x\right)}\right \Vert \rightarrow 0,\ k \rightarrow \infty,$$where Φ [ − k, k] x is the set of all possible subsequences of length 2k + 1 and Φ is localizing, that is
$${\sup }_{z}\mbox{ {\it diam}}\left ({\Phi }_{[-k,k]}^{-1}\left (z\right)\right) \rightarrow 0,\ k \rightarrow \infty.$$Following Eckmann and Ruelle [13], localizing is a sufficient condition for \(\mathcal{P}\) being a generating partition.
- 2.
The algorithm can be stopped when the length of substrings reaches a certain length, or when the discrepancy is sufficiently small.
The discrepancy is defined by
$$\sum_{i=m+1}^{N-n}{\left \Vert {x}_{ i} - {r}_{{s}_{[i-m,i+n]}}\right \Vert }^{2}\big{/}(N - m - n).$$
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Acknowledgements
Financial support from the Fundação Ciência e Tecnologia, Lisbon, is grateful acknowledged by the authors, under the contracts No PTDC/GES/73418/2006 and No PTDC/GES/70529/2006.
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Mendes, D.A., Mendes, V.M., Ferreira, N., Menezes, R. (2010). Symbolic Shadowing and the Computation of Entropy for Observed Time Series. In: Takayasu, M., Watanabe, T., Takayasu, H. (eds) Econophysics Approaches to Large-Scale Business Data and Financial Crisis. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53853-0_12
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