Roberto Dale

Algebraic representations of time series are symbolic representations whose symbols belong to a finite group. Precisely, the framework of the present paper is the analysis of coupled time series in algebraic representations and, more generally, group-valued time series. The prototype of an algebraic representation is an ordinal representation, whose symbols are permutations, also called ordinal patterns in the context of time series analysis. In fact, permutations, endowed with function composition, build a group called a symmetric group. A simple way to harness the algebraic structure of the alphabet in such cases is the concept of transcript from one group element to another. Since transcripts involve two group elements, they are very suitable for studying couplings between time series in the same algebraic representation. In this paper, we outline several existing entropic and algebraic transcript-based tools for analyzing coupled time series and systems. In addition to entropy, the entropic tools include divergence, statistical complexity, and mutual information. The algebraic tools comprise order classes and, most recently, the Cayley and Kendall distances. We use the detection of generalized synchronization in a well-studied coupled system to compare the performances of some of those tools. To this end, we also provide an alternative tool called the similarity distance between time series, which is a mean Kendall distance. We found that the novel similarity distance outperforms the other tools tested.

Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley's theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation-based distances compared to the conventional generator-based distances in finite groups. The reason why we are interested in distances on groups is that finite groups appear in symbolic representations of time series, most notably in the so-called ordinal representations, whose symbols are precisely permutations, usually called ordinal patterns in that context. The natural extension from groups to group-valued time series is also discussed, as well as how such metric tools can be applied in time series analysis. Both theory and applications are illustrated with examples and numerical simulations.

Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley’s theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation-based distances compared to the conventional generator-based distances in finite groups. The reason why we are interested in distances on groups is that finite groups appear in symbolic representations of time series, most notably in the so-called ordinal representations, whose symbols are precisely permutations, usually called ordinal patterns in that context. The natural extension from groups to group-valued time series is also discussed, as well as how such metric tools can be applied in time series analysis. Both theory and applications are illustrated with examples and numerical simulations.

In a recent paper [R. Dale. R. Lapiedra, and J. A. Morales-Lladosa, Phys. Rev. D 107 (2023) 023506] a cosmic-like Clauser-Horne-Shimony-Holt (CHSH) inequality was proved for the temperature fluctuations δT of the perturbed Cosmic Microwave Background (CMB), assuming local realism. This inequality can be tested from observational data. In fact, no violation for it has been found from the CMB data provided by the COBE satellite in different sky directions. This is a result which, being negative, is not conclusive in relation to the possible violation of the CHSH inequality in a cosmological context. So, the preceding analysis needs to be extended by using more precise current CMB data, like those provided by WMAP and Planck satellites. This is a task of considerable importance but, technically, much more involved in data treatments. In the present, assuming again local realism behind measurements and observational data, this extension is accomplished. The result is that a sound violation of the CHSH inequality is found, which would mean the failure of the assumed local realism in accordance with the quantum origin of the primordial temperature fluctuations in the framework of standard inflation and the orthodox interpretation of the Quantum Mechanics.

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder \textquotedblleft follows\textquotedblright\ the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the \textquotedblleft noiseless scenario\textquotedblright\ just described when noise is present in the driver, a more realistic situation that we call the \textquotedblleft noisy scenario\textquotedblright . Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for \textquotedblleft generic\textquotedblright\ driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real world data demonstrate the capability of our method.

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder “follows” the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the “noiseless scenario” just described when noise is present in the driver, a more realistic situation that we call the “noisy scenario.” Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for “generic” driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real-world data demonstrate the capability of our method.

The objective of this paper is the construction of new indicators that can be useful to operate in the cryptocurrency market. These indicators are based on public data obtained from the blockchain network, specifically from the nodes that make up Bitcoin mining. Therefore, our analysis is unique to that network. The results obtained with numerical simulations of algorithmic trading and prediction via statistical models and Machine Learning demonstrate the importance of variables such as the hash rate, the difficulty of mining or the cost per transaction when it comes to trade Bitcoin assets or predict the direction of price. Variables obtained from the blockchain network will be called here blockchain metrics. The corresponding indicators (inspired by the "Hash Ribbon") perform well in locating buy signals. From our results, we conclude that such blockchain indicators allow obtaining information with a statistical advantage in the highly volatile cryptocurrency market.

This is a review of group entropy and its application to permutation complexity. Specifically we revisit a new approach to the notion of complexity in time serie analysis, based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviours.

The Cosmic microwave background (CMB) anisotropies predicted by two cosmological models are compared, one of them is the standard model of general relativity with cold dark matter and cosmological constant, whereas the second model is based on a consistent vector-tensor theory of gravitation explaining solar system and cosmological observations. It is proved that the resulting differences -- between the anisotropies of both models -- are due to the so-called late integrated Sachs Wolfe effect and, consequently, cross correlations between maps of CMB temperatures and tracers of the dark matter distribution could be used in future to select one of the above models. The role of reionization is analysed in detail.