Sign patterns symbolization and its use in improved dependence test for complex network inference.

Inferring the dependence structure of complex networks from the observation of the non-linear dynamics of its components is among the common, yet far from resolved challenges faced when studying real-world complex systems. While a range of methods using the ordinal patterns framework has been proposed to particularly tackle the problem of dependence inference in the presence of non-linearity, they come with important restrictions in the scope of their application. Hereby, we introduce the sign patterns as an extension of the ordinal patterns, arising from a more flexible symbolization which is able to encode longer sequences with lower number of symbols. After transforming time series into sequences of sign patterns, we derive improved estimates for statistical quantities by considering necessary constraints on the probabilities of occurrence of combinations of symbols in a symbolic process with prohibited transitions. We utilize these to design an asymptotic chi-squared test to evaluate dependence between two time series and then apply it to the construction of climate networks, illustrating that the developed method can capture both linear and non-linear dependences, while avoiding bias present in the naive application of the often used Pearson correlation coefficient or mutual information.

Assessing serial dependence in ordinal patterns processes using chi-squared tests with application to EEG data analysis.

We extend Elsinger's work on chi-squared tests for independence using ordinal patterns and investigate the general class of m-dependent ordinal patterns processes, to which belong ordinal patterns processes derived from random walk, white noise, and moving average processes. We describe chi-squared asymptotically distributed statistics for such processes that take into account necessary constraints on ordinal patterns probabilities and propose a test for m-dependence, with which we are able to quantify the range of serial dependence in a process. We apply the test to epilepsy electroencephalography time series data and observe shorter m-dependence associated with seizures, suggesting that the range of serial dependence decreases during those events.

Sigma-Pi Structure with Bernoulli Random Variables: Power-Law Bounds for Probability Distributions and Growth Models with Interdependent Entities.

The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.

Segmentation of time series in up- and down-trends using the epsilon-tau procedure with application to USD/JPY foreign exchange market data.

We propose the epsilon-tau procedure to determine up- and down-trends in a time series, working as a tool for its segmentation. The method denomination reflects the use of a tolerance level ε for the series values and a patience level τ in the time axis to delimit the trends. We first illustrate the procedure in discrete random walks, deriving the exact probability distributions of trend lengths and trend amplitudes, and then apply it to segment and analyze the trends of U.S. dollar (USD)/Japanese yen (JPY) market time series from 2015 to 2018. Besides studying the statistics of trend lengths and amplitudes, we investigate the internal structure of the trends by grouping trends with similar shapes and selecting clusters of shapes that rarely occur in the randomized data. Particularly, we identify a set of down-trends presenting similar sharp appreciation of the yen that are associated with exceptional events such as the Brexit Referendum in 2016.

Measuring Statistical Asymmetries of Stochastic Processes: Study of the Autoregressive Process.

We use the definition of statistical symmetry as the invariance of a probability distribution under a given transformation and apply the concept to the underlying probability distribution of stochastic processes. To measure the degree of statistical asymmetry, we take the Kullback-Leibler divergence of a given probability distribution with respect to the corresponding transformed one and study it for the Gaussian autoregressive process using transformations on the temporal correlations' structure. We then illustrate the employment of this notion as a time series analysis tool by measuring local statistical asymmetries of foreign exchange market price data for three transformations that capture distinct autocorrelation behaviors of the series-independence, non-negative correlations and Markovianity-obtaining a characterization of price movements in terms of each statistical symmetry.

Detection of statistical asymmetries in non-stationary sign time series: Analysis of foreign exchange data.

We extend the concept of statistical symmetry as the invariance of a probability distribution under transformation to analyze binary sign time series data of price difference from the foreign exchange market. We model segments of the sign time series as Markov sequences and apply a local hypothesis test to evaluate the symmetries of independence and time reversion in different periods of the market. For the test, we derive the probability of a binary Markov process to generate a given set of number of symbol pairs. Using such analysis, we could not only segment the time series according the different behaviors but also characterize the segments in terms of statistical symmetries. As a particular result, we find that the foreign exchange market is essentially time reversible but this symmetry is broken when there is a strong external influence.