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This paper aims to introduce the concept of symbolic correlation integral SC that is extensively used in many scientific fields. The new correlation integral SC avoids the noisy parameter 𝜀 of the classical correlation integral, defined by Grassberger and Procaccia (1983) and extensively used for constructing correlation-integral-based statistics, as in the BDS test. Once the free parameter 𝜀 disappears, it is possible to construct a nonparametric powerful test for independence that can also be used as a diagnostic tool for model selection. The symbolic correlation integral is also extended to deal with multivariate models, and a test for causality is proposed as an example of the theoretical power of the new concept. With extensive Monte Carlo simulations, the paper shows the good size and power performance of symbolic correlation-integral-based tests. [ABSTRACT FROM AUTHOR]

López et al. (Reg Sci Urban Econ 40(2-3):106-115, ) introduce a nonparametric test of spatial dependence, called SG( m). The test is claimed to be consistent and asymptotically Chi-square distributed. Elsinger (Reg Sci Urban Econ 43(5):838-840, ) raises doubts about the two properties. Using a particular counterexample, he shows that the asymptotic distribution of the SG( m) test may be far from the Chi-square family; the property of consistency is also questioned. In this note, the authors want to clarify the properties of the SG( m) test. We argue that the cause of the conflict is in the specification of the symbolization map. The discrepancies can be solved by adjusting some of the definitions made in the original paper. Moreover, we introduce a permutational bootstrapped version of the SG( m) test, which is powerful and robust to the underlying statistical assumptions. This bootstrapped version may be very useful in an applied context. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to propose a newly developed non-parametric test for linear and nonlinear causality based on permutation entropy and to show its usefulness in analyzing the potential causal relationship between trading volume and security prices. Most of the empirical applications and tests for causality rely on using Granger causality based test for linear models. Although these tests have high power in uncovering linear causal relations, their power against nonlinear causal relations can be low. Our test is designed to deal with the detection of linear and non-linear causality. We also compare our permutation entropy based test with other Granger causality tests. Monte Carlo simulations show excellent performance (in terms of size and power) of the new test for detecting linear and non-linear causality under different scenarios. Our conclusions point that there is a bidirectional causal relation from volume to price returns not only in the mean but also in the variance. [Copyright &y& Elsevier]

This article addresses the question of improving the detection of nonlinear dependence by means of recently developed nonparametric tests. To this end a generalized version of BDS test and a new test based on symbolic dynamics are used on realizations from a well-known artificial market for which the dynamic equation governing the market is known. Comparisons with other tests for detecting nonlinearity are also provided. We show that the test based on symbolic dynamics outperforms other tests with the advantage that it depends only on one free parameter, namely the embedding dimension. This does not hold for other tests for nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: We propose a novel test to determine, given a time series, if the dynamics are generated by a deterministic (including low dimensional chaos), rather than a stochastic, process. In addition, we introduce a new nonparametric bootstrap test for independence which is consistent against a broad class of alternatives. The conditions under which the tests can be applied are very weak. The advantages of the presented methods are simplicity, invariance with respect to monotonic transformations and the applicability of the tests regardless of the discrete or continuous nature of the data generating process. We conduct several simulation studies to evaluate the performance of our tests on well-known dynamic processes. Finally, our tests are applied to several sets of financial returns that have been recently studied. [ABSTRACT FROM AUTHOR]

In this article we introduce a test for independence between two processes { X t} and { Y t}. To this end we rely on symbolic dynamics and permutation entropy as a measure of dependence. As a result, a nonparametric (model-free) test for either linear or nonlinear processes is presented. The test is consistent for a broad range of dependent alternatives. Empirical simulations indicate and highlight the general utility of the test for time-series analysts. [ABSTRACT FROM AUTHOR]

Background: The etiology of complex diseases is due to the combination of genetic and environmental factors, usually many of them, and each with a small effect. The identification of these small-effect contributing factors is still a demanding task. Clearly, there is a need for more powerful tests of genetic association, and especially for the identification of rare effects Results: We introduce a new genetic association test based on symbolic dynamics and symbolic entropy. Using a freely available software, we have applied this entropy test, and a conventional test, to simulated and real datasets, to illustrate the method and estimate type I error and power. We have also compared this new entropy test to the Fisher exact test for assessment of association with low-frequency SNPs. The entropy test is generally more powerful than the conventional test, and can be significantly more powerful when the genotypic test is applied to low allele-frequency markers. We have also shown that both the Fisher and Entropy methods are optimal to test for association with lowfrequency SNPs (MAF around 1-5%), and both are conservative for very rare SNPs (MAF<1%) Conclusions: We have developed a new, simple, consistent and powerful test to detect genetic association of biallelic/ SNP markers in case-control data, by using symbolic dynamics and symbolic entropy as a measure of gene dependence. We also provide a standard asymptotic distribution of this test statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. The entropy test is generally as good or even more powerful than the conventional and Fisher tests. Furthermore, the entropy test is more computationally efficient than the Fisher's Exact test, especially for large number of markers. Therefore, this entropy-based test has the advantage of being optimal for most SNPs, regardless of their allele frequency (Minor Allele Frequency (MAF) between 1-50%). This property is quite beneficial, since many researchers tend to discard low allele-frequency SNPs from their analysis. Now they can apply the same statistical test of association to all SNPs in a single analysis., which can be especially helpful to detect rare effects. [ABSTRACT FROM AUTHOR]

This Special Issue (SI) brings together contributions from researchers working in symbolic analysis, complex dynamics, and information theory, from both theoretical and applied perspectives. References 1 Bandt C. Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic. C. Bandt [[1]] puts forward an entropy-based concentration coefficient C that avoids the limitations of other classical concentration measures such as Gini's index. [Extracted from the article]

A novel general method for constructing nonparametric hypotheses tests based on the field of symbolic analysis is introduced in this paper. Several existing tests based on symbolic entropy that have been used for testing central hypotheses in several branches of science (particularly in economics and statistics) are particular cases of this general approach. This family of symbolic tests uses few assumptions, which increases the general applicability of any symbolic-based test. Additionally, as a theoretical application of this method, we construct and put forward four new statistics to test for the null hypothesis of spatiotemporal independence. There are very few tests in the specialized literature in this regard. The new tests were evaluated with the mean of several Monte Carlo experiments. The results highlight the outstanding performance of the proposed test. [ABSTRACT FROM AUTHOR]

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay τ ∗ and embedding dimension p for phase space reconstruction. The value of τ ∗ can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that τ ∗ should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay τ w = (p − 1) τ ∗ , which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally τ w and τ ∗ . In this paper, we suggest a simple method for estimating τ ∗ and τ w based on symbolic analysis and symbolic entropy. As in the C-C method, τ ∗ is estimated as the first local optimal time delay and τ w as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes [ABSTRACT FROM AUTHOR]