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Testing independence between two or more random variables or random vectors receives a lot of attention in the literature. A new test statistic is proposed based on the consistency of the sample distance multivariance to test mutual independence. The bootstrap method is used to obtain the critical value of the test statistic. The simulation results show that our method performs well in the low-dimensional and large-sample cases and gives reasonable results in some cases where other tests do not work. Three real examples are analyzed to show the usefulness of the new test statistic. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to diagnose the perception, knowledge, awareness and position of the bioeconomy in university education and research. Design/methodology/approach: This study is based on a questionnaire survey conducted at universities in Poland, the Czech Republic, Spain and Portugal (n = 464). The questionnaire consisted of open-ended, dichotomous and Likert-type questions. Variable frequency distribution methods and the non-parametric chi-square test were used to test the independence of the characteristics. The Cramer's V contingency coefficient was used to determine the degree of dependence between the variables. Findings: The researched academic community is dominated by a traditional approach to the bioeconomy, which relates to agriculture and ecological aspects. Respondents believe in the positive environmental impacts of the bioeconomy, while less often being aware of its importance from a socio-economic perspective. Insufficient teaching and research in the field of the bioeconomy can be widely observed. The presumed link between the existence of a bioeconomy strategy at national level and awareness of the bioeconomy was not confirmed. Research limitations/implications: The limited sample, the narrow geographical scope of the study does not allow for a comprehensive analysis of the topic. Another limitation is the lack of representativeness of the results in relation to all university representatives in the countries studied and the uneven composition of the samples. Originality/value: This study fills the knowledge gap about the status of the bioeconomy in European academic communities by analysing its perception among both teachers and students of social, natural and applied sciences. [ABSTRACT FROM AUTHOR]

Wildlife populations can be unmarked, meaning individuals lack distinguishing features for individual identification. Populations may also exhibit non‐independent movements, meaning individuals move together. For populations of either unmarked or non‐independent individuals, models based on spatial capture–recapture (SCR) approaches can be used to estimate abundance, density, and other parameters critical for monitoring, management, and conservation. However, when individuals are both unmarked and non‐independent, few model options are available. One approach has been to apply unmarked models and not address the non‐independence despite unquantified impacts on bias, precision, and the ability to make robust ecological inferences. We conducted a simulation study to quantify the impact of non‐independence on the performance of spatial count (SC) and spatial partial identity models (SPIM) – two SCR‐based unmarked modeling approaches – and used the performance of fully marked and independent SCR as a reference. We varied the levels of non‐independence (aggregation and cohesion), detection probability, and the number of partial identity covariates used to resolve identities in SPIM estimation. We expected abundance estimates to be increasingly biased and precise as aggregation and cohesion increased. Results showed that models indeed became less robust to increasing non‐independence, but importantly suggested that only SPIM could be reliably applied under low levels of cohesion when sufficient partial identity covariates are available. SC yielded consistently biased estimates with poor precision. SCR was consistently robust across combinations of aggregation and cohesion, as expected. We therefore advise against the use of SC models for estimating population parameters when individuals are known to be non‐independent, caution that SPIM may be used under narrow ecological conditions, and encourage continued investigations into sampling design and methods development for estimating populations of unmarked and non‐independent individuals. [ABSTRACT FROM AUTHOR]

In this article, we investigate meandric systems having one shallow side: the arch configuration on that side has depth at most two. This class of meandric systems was introduced and extensively examined by I. P. Goulden, A. Nica, and D. Puder [Int. Math. Res. Not. IMRN 2020 (2020), 983-1034]. Shallow arch configurations are in bijection with the set of interval partitions. We study meandric systems by using moment-cumulant transforms for noncrossing and interval partitions, corresponding to the notions of free and Boolean independence, respectively, in non-commutative probability. We obtain formulas for the generating series of different classes of meandric systems with one shallow side by explicitly enumerating the simpler, irreducible objects. In addition, we propose random matrix models for the corresponding meandric polynomials, which can be described in the language of quantum information theory, in particular that of quantum channels. [ABSTRACT FROM AUTHOR]

The local independence assumption is crucial for the consistent estimation of item parameters in item response theory models. This article explores a pairwise likelihood estimation approach for the two-parameter logistic (2PL) model that treats the local dependence structure as a nuisance in the optimization function. Hence, item parameters can be consistently estimated without explicit modeling assumptions of the dependence structure. Two simulation studies demonstrate that the proposed pairwise likelihood estimation approach allows nearly unbiased and consistent item parameter estimation. Our proposed method performs similarly to the marginal maximum likelihood and pairwise likelihood estimation approaches, which also estimate the parameters for the local dependence structure. [ABSTRACT FROM AUTHOR]

The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture introducing jointly to several fields of applications, some of which are probably not familiar (at least in the form we present them) to the readers. Lastly, we take the opportunity to present various results obtained recently that use group and bialgebra techniques to handle notions such as cumulants or Wick polynomials in the various noncommutative probability theories. [ABSTRACT FROM AUTHOR]

Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic Calabi–Yau fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U (1) gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in P 2 over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the independence of the intersection pairing of the vertical part of the middle cohomology for the resolutions we study. In each model, we find that vertical flux backgrounds can produce nonzero multiplicities for a spanning set of anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints on allowed matter representations beyond those implied by 4D anomaly cancellation. [ABSTRACT FROM AUTHOR]

A multivariate folded normal distribution is a distribution of the absolute value of a Gaussian random vector. In this paper, we provide the marginal and conditional distributions of the multivariate folded normal distribution, and also prove that independence and non-correlation are equivalent for it. In addition, we provide a numerical approach using the R language to fit a multivariate folded normal distribution. The accuracy of the estimated mean and variance parameters is then examined. Finally, a real data application to body mass index data are presented. [ABSTRACT FROM AUTHOR]

Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering. Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the Hilbert–Schmidt independence criterion (hsic) and its d-variate version (d-hsic), but also allows us to introduce a test for conditional independence by defining a novel statistic for the conditional permutation test (cpt) based on the Hilbert–Schmidt conditional independence criterion (hscic), with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socioeconomic data. [ABSTRACT FROM AUTHOR]

We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed N = 1 superfields for the case of N = 4 super-Yang–Mills theory. Under "scale independence" of the effective action of dressed mean superfields, we mean its "finiteness in the off-shell limit of removing all the regularizations". This off-shell limit is scale independent because no scale remains inside these kernels after removing the regularizations. We use two types of regularization: regularization by dimensional reduction and regularization by higher derivatives in its supersymmetric form. Based on the Slavnov–Taylor identity, we show that dressed fields of matter and of vector multiplets can be introduced to express the effective action in terms of them. Kernels of the effective action expressed in terms of such dressed effective fields do not depend on the ultraviolet scale. In the case of dimensional reduction, by using the developed technique, we show how the problem of inconsistency of the dimensional reduction can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on the infrared scale disappears off shell in both the regularizations. [ABSTRACT FROM AUTHOR]

In this article we prove a generalization of the Ejsmont characterization (Ejsmont in Stat Probab Lett 114:1–5, 2016) of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. The test uses an integral of the squared modulus of the difference between the product of empirical characteristic functions and some constant. Special attention is given to the case of testing for univariate normality in which we derive the test statistic explicitly in terms of Bessel function and explore asymptotic properties. The simulation study also includes the cases of testing for bivariate and trivariate normality and independence, as well as multivariate normality. We show the quality performance of our test in comparison to some popular powerful competitors. The practical application of the proposed normality and independence test is discussed and illustrated using a real dataset. [ABSTRACT FROM AUTHOR]

Conducting biodiversity surveys using a fully randomised design can be difficult due to budgetary constraints (e.g., the cost of labour), site accessibility, and other constraints. To this end, ecologists usually select representative line transects or quadrats from a studied area to collect individuals of a given species and use this information to estimate the levels of biodiversity over an entire region. However, commonly used biodiversity estimators such as Rao's quadratic diversity index (and especially the Gini-Simpson index) were developed based on the assumption of independent sampling of individuals. Therefore, their performance can be compromised or even misleading when applied to species abundance datasets that are collected from non-independent sampling. In this study, we utilise a Markov chain model and derive an associated parameter estimator to account for non-independence in sequential sampling. Empirical tests on two forest plots in tropical (Barro Colorado, Island of Panama) and subtropical (Heishiding Nature Reserve of Guangdong, China) regions and the continental-scale spatial distribution of Acacia species in Australia showed that our estimators performed reasonably well. The estimated parameter measuring the degree of non-independence of subsequent sampling showed that a non-independent effect is very likely to occur when using line transects to sample organisms in subtropical regions at both local and regional spatial scales. In summary, based on a first-order Markov sampling model and using Rao's quadratic diversity index as an example, our study provides an improvement in diversity estimation while simultaneously accounting for the non-independence of sampling in field biodiversity surveys. Our study presents one possible solution for addressing the non-independent sampling of individuals in biodiversity surveys. [ABSTRACT FROM AUTHOR]

The commonly used way to test for independence of two random variables is by means of Kendall's tau. Given a sample from a biviariate distribution, permutation test using its empirical value under independence is known to be exact. However, in the problem for testing the null hypothesis that the samples are uncorrelated, the permutation test can have asymptotic null rejection probability that is far from the nominal level. To deal with this issue, we advocate using appropriately studentized statistic to yield permutation test. We will show that the studentized permutation test is asymptotically consistent in the general setting when two paired variables are uncorrelated but dependent. Simulation studies demonstrate this desired property across a range of distributional assumptions and sample sizes. [ABSTRACT FROM AUTHOR]

We develop a simple non-parametric test for testing independence of two random variables against the alternative that they are positive quadrant dependent. The asymptotic distribution of the proposed test statistic is distribution free under the null hypothesis of independence. A Monte Carlo simulation study is carried out to evaluate the finite sample performance of the proposed test. We illustrate the proposed method with a real data set. [ABSTRACT FROM AUTHOR]

A subclass of lattice conditional independence models is introduced. The new class of models is called totally ordered independence models. The class is based on an assumption that the index set which orders the random variables is a chain. It is shown that there is a jump in the chain if and only if there is a conditional independence relation. Some comparisons between the lattice conditional independence models and totally ordered independence models are presented. [ABSTRACT FROM AUTHOR]

A commonly used test for the independence of two sets of variables from a normal multivariate population is the Wilks Lambda test or the multivariate likelihood ratio test (LRT). However, this test's performance is highly influenced by outliers and non-normality of the data and thus, robust test statistics should be used, allied to Monte-Carlo methods. In this paper, we proposed and evaluated three new likelihood ratio test for the independence of two sets of multivariate variables (LRTR, T and TR), replacing the traditional covariance matrix estimator with the robust comedian estimators. The results showed that the proposed test T and TR control type I error rates also for non-normal distributions outperforming the ordinary test. [ABSTRACT FROM AUTHOR]

The induced odd cycle packing numberiocp(G) $\text{iocp}(G)$ of a graph G $G$ is the maximum integer k $k$ such that G $G$ contains an induced subgraph consisting of k $k$ pairwise vertex‐disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded Vapnik–Chervonenkis (VC) dimension, and linear independence number admit a randomized efficient polynomial‐time approximation scheme for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers k≥0 $k\ge 0$ and t≥1 $t\ge 1$ and any n $n$‐vertex graph G $G$ of induced odd cycle packing number at most k $k$ returns in time Ok,t(nk+4) ${O}_{k,t}({n}^{k+4})$ an independent set of G $G$ whose size is at least α(G)−n∕t $\alpha (G)-n\unicode{x02215}t$ with high probability. In addition, we present χ $\chi $‐boundedness results for graphs with bounded odd cycle packing number, and use them to design a quasipolynomial‐time approximation scheme for the independence number only assuming bounded induced odd cycle packing number. [ABSTRACT FROM AUTHOR]

This study explores the hypothesis related to a punctuated equilibrium process for conceptual change in science learning, in conjunction with the effects of four cognitive variables: logical thinking and field-dependence/fieldindependence, divergent and convergent thinking. The participants were fifth and sixth-grade elementary school pupils involved in different tasks, who were asked to describe and interpret chemical phenomena. Latent Class Analysis (LCA) was applied to children's responses, and three clusters or latent classes (LC1, LC2, and LC3) were identified, corresponding to hierarchical levels of conceptual understanding. The ensued LCs align with the theoretical conjecture about a stepwise conceptual change process that might go through various stages or mental models. These levels or stages are conceptualized as attractors, and changes between them were modeled as cusp catastrophes using the four cognitive variables as controls. The analysis showed that logical thinking acted as asymmetry factor, while field-dependence/field-independence, divergent and convergent thinking acted as bifurcation variables. This analytic approach presents a methodology for investigating conceptual change as a punctuated equilibrium process that adds to the nonlinear dynamical research with important implications for theories of conceptual change in science education and psychology as well. Discussion on the new perspective embracing the metatheoretical framework of complex adaptive systems (CDS) is provided. [ABSTRACT FROM AUTHOR]

Partial correlation is a common tool in studying conditional dependence for Gaussian distributed data. However, partial correlation being zero may not be equivalent to conditional independence under non‐Gaussian distributions. In this paper, we propose a statistical inference procedure for partial correlations under the high‐dimensional nonparanormal (NPN) model where the observed data are normally distributed after certain monotone transformations. The NPN partial correlation is the partial correlation of the normal transformed data under the NPN model, which is a more general measure of conditional dependence. We estimate the NPN partial correlations by regularized nodewise regression based on the empirical ranks of the original data. A multiple testing procedure is proposed to identify the nonzero NPN partial correlations. The proposed method can be carried out by a simple coordinate descent algorithm for lasso optimization. It is easy‐to‐implement and computationally more efficient compared to the existing methods for estimating NPN graphical models. Theoretical results are developed to show the asymptotic normality of the proposed estimator and to justify the proposed multiple testing procedure. Numerical simulations and a case study on brain imaging data demonstrate the utility of the proposed procedure and evaluate its performance compared to the existing methods. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. [ABSTRACT FROM AUTHOR]

The ingenious approach of Chatterjee (2021) to estimate a measure of dependence first proposed by Dette et al. (2013) based on simple rank statistics has quickly caught attention. This measure of dependence has the appealing property of being between 0 and 1, and being 0 or 1 if and only if the corresponding pair of random variables is independent or one is a measurable function of the other almost surely. However, more recent studies (Cao & Bickel 2020 ; Shi et al. 2022b) showed that independence tests based on Chatterjee's rank correlation are unfortunately rate inefficient against various local alternatives and they call for variants. We answer this call by proposing an improvement to Chatterjee's rank correlation that still consistently estimates the same dependence measure, but provably achieves near-parametric efficiency in testing against Gaussian rotation alternatives. This is possible by incorporating many right nearest neighbours in constructing the correlation coefficients. We thus overcome the ' only one disadvantage' of Chatterjee's rank correlation (Chatterjee, 2021 , § 7). [ABSTRACT FROM AUTHOR]

Sárközy's theorem states that dense sets of integers must contain two elements whose difference is a k th power. Following the polynomial method breakthrough of Croot, Lev and Pach [ 3 ], Green proved a strong quantitative version of this result for |$\mathbb{F}_{q}[T]$|⁠. In this paper we provide a lower bound for Sárközy's theorem in function fields by adapting Ruzsa's construction [ 17 ] for the analogous problem in |$\mathbb{Z}$|⁠. We construct a set A of polynomials of degree <  n such that A does not contain a k th power difference with |$|A|=q^{n-n/2k}$|⁠. Additionally, we prove a handful of results concerning the independence number of generalized Paley graphs, including a generalization of a claim of Ruzsa, which helps with understanding the limit of the method. [ABSTRACT FROM AUTHOR]

In block ciphers, the nonlinear components, also known as substitution boxes (S-boxes), are used with the purpose of inducing confusion in cryptosystems. For the last decade, most of the work on designing S-boxes over the points of elliptic curves has been published. The main purpose of these studies is to hide data and improve the security levels of crypto algorithms. In this work, we design pair of nonlinear components of a block cipher over the residue class of Gaussian integers (GI). The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters. But the prime field dependent on the Elliptic curve (EC) provides one S-box at a time by fixing three parameters a, b, and p. The newly designed pair of S-boxes are assessed by various tests like nonlinearity, bit independence criterion, strict avalanche criterion, linear approximation probability, and differential approximation probability. [ABSTRACT FROM AUTHOR]

Blind source separation (BSS) has been a great challenge in the field of signal processing due to the unknown distribution of the source signal and the mixing matrix. Traditional methods based on statistics and information theory use prior information such as source distribution independence, non-Gaussianity, sparsity, etc. to solve this problem. Generative adversarial networks (GANs) learn source distributions through games without being constrained by statistical properties. However, the current blind image separation methods based on GANs ignores the reconstruction of the structure and details of the separated image, resulting in residual interference source information in the generated results. This paper proposes a Transformer-guided GAN guided by an attention mechanism. Through the adversarial training of the generator and the discriminator, U-shaped Network (UNet) is used to fuse the convolutional layer features to reconstruct the structure of the separated image, and Transformer is used to calculate the position attention and guide the detailed information. We validate our method with quantitative experiments, showing that it outperforms previous blind image separation algorithms in terms of PSNR and SSIM. [ABSTRACT FROM AUTHOR]

We provide finite sample guarantees for the classical Chow--Liu algorithm [Chow and Liu, IEEE Trans. Inform. Theory, 14 (1968), pp. 462--467] to learn a tree-structured graphical model of a distribution. For a distribution P on \Sigma n and a tree T on n nodes, we say T is an \varepsilon -approximate tree for P if there is a T-structured distribution Q such that D(P | | Q) is at most \varepsilon more than the best possible tree-structured distribution for P. We show that if P itself is treestructured, then the Chow--Liu algorithm with the plug-in estimator for mutual information with \widetilde O (| \Sigma | 3n\varepsilon 1) independent and identically distributed samples outputs an \varepsilon -approximate tree for P with constant probability. In contrast, for a general P (which may not be tree-structured), \Omega (n2\varepsilon 2) samples are necessary to find an \varepsilon -approximate tree. Our upper bound is based on a new conditional independence tester that addresses an open problem posed by Canonne et al. [Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2018, pp. 735--748]: we prove that for three random variables X,Y,Z each over \Sigma, testing if I(X;Y | Z) is 0 or \geq \varepsilon is possible with \widetilde O (| \Sigma | 3/\varepsilon) samples. Finally, we show that for a specific tree T, with \widetilde O(| \Sigma | 2n\varepsilon 1) samples from a distribution P over \Sigma n, one can efficiently learn the closest T-structured distribution in KL divergence by applying the add-1 estimator at each node. [ABSTRACT FROM AUTHOR]

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In this paper we develop a likelihood ratio test for independence between multivariate discrete random vectors whose components may be dependent or may have different marginal distributions. A special case that tests independence between two discrete random variables is also discussed. The model under the null hypothesis is constructed based on the marginal distributions of the random variables, while the model under the alternative is built by adding a common random effect which is not restricted to follow the same distribution as those of the other variables. The test statistic is shown to admit an asymptotic distribution of a weighted sum of chi-square random variables, each with one degree of freedom. In order to avoid the high computational cost of finding the parameters of the asymptotic distribution, a permutation-based procedure is introduced. It is shown in simulations that the permutation-based realization yields empirical level close to the nominal level, while achieving power comparable to or higher than that of existing competitors in the literature. The application of the test is illustrated with a real data set. [ABSTRACT FROM AUTHOR]

Chowla's (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of L (1 , χ) = ∑ n = 1 ∞ χ (n) n . On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla's problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula. [ABSTRACT FROM AUTHOR]

It was conjectured by Jaeger et al. that all 5-edge-connected graphs are Z3-connected. In this paper, we confirm this conjecture for all 5-edge-connected graphs with independence number at most 3. [ABSTRACT FROM AUTHOR]

Pseudorandom number generators (PRNGs) have always been a central research topic in data science, and chaotic dynamical systems are one of the means to obtain scientifically proven data. Chaotic dynamical systems have the property that they have a seemingly unpredictable and random behavior obtained by making use of deterministic laws. The current paper will show how several notions used in the study of chaotic systems—statistical independence, singularity, and observability—can be used together as a suite of test methods for chaotic systems with high potential of being used in the PRNG or cryptography fields. In order to address these topics, we relied on the adaptation of the observability coefficient used in previous papers of the authors, we calculated the singularity areas for the chaotic systems considered, and we evaluated the selected chaotic maps from a statistical independence point of view. By making use of the three notions above, we managed to find strong correlations between the methods proposed, thus supporting the idea that the resulting test procedure is consistent. Future research directions consist of applying the proposed test procedure to other chaotic systems in order to gather more data and formalize the approach in a test suite that can be used by the data scientist when selecting the best chaotic system for a specific use (PRNG, cryptography, etc.). [ABSTRACT FROM AUTHOR]

We introduce a new codification on a time series that combines permutations and another symbol containing quantitative information. With these new codes, we construct a new test to determine whether a time series is independent and identically distributed. We compare our results with those obtained with tests based only on permutations. Our results show that the new test produces better results than those based only on permutations. [ABSTRACT FROM AUTHOR]

The ring of polyadic integers is the direct product of rings of p-adic integers over all primes p. Thus, elements of this ring can be treated as infinite-dimensional vectors, whose coordinates in the corresponding ring of p-adic integers are denoted by . The infinite linear independence of polyadic numbers means that, for any nonzero linear form with integer coefficients , there are infinitely many primes p such that in the field . At the same time, problems in which primes are considered only from some proper subsets of the set of primes are of interest. In this case, we talk about infinite linear independence with constraints on the specified set. The canonical representation of an element θ of the ring of polyadic integers has the form of a series Of course, a series with integer members that converges in all fields of p-adic numbers is a polyadic integer. We say that a polyadic number θ is a polyadic Liouville number (or a Liouville polyadic number) if for any numbers n and P there exists a positive integer A such that for all primes p satisfying the inequality it is true that < A–n. Here, we prove the infinite linear independence of polyadic numbers f0(1) = with constraints on a set of prime numbers in the aggregate of arithmetic progressions. An important apparatus for obtaining this result is Hermite–Padé approximations of generalized hypergeometric functions constructed by Nesterenko in [4]. Additionally, we use the approach from the work by Ernvall-Hytönen et al. [5]. [ABSTRACT FROM AUTHOR]

Arithmetic properties of series of the form with are studied. The concept of infinite algebraic independence of polyadic numbers is discussed. A theorem is proved concerning the infinite algebraic independence of polyadic numbers of the class that are connected by a system of linear differential equations of a certain type. [ABSTRACT FROM AUTHOR]

Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318-1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causalmodels, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the lattice conditional independence models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the latticewhich correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series and Shannon information flow. [ABSTRACT FROM AUTHOR]

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels. [ABSTRACT FROM AUTHOR]

Testing for dependence between pairs of random variables is a fundamental problem in statistics. In some applications, data are subject to selection bias that can create spurious dependence. An important example is truncation models, in which observed pairs are restricted to a specific subset of the X-Y plane. Standard tests for independence are not suitable in such cases, and alternative tests that take the selection bias into account are required. Here, we generalize the notion of quasi-independence with respect to the sampling mechanism, and study the problem of detecting any deviations from it. We develop two tests statistics motivated by the classic Hoeffding's statistic, and use two approaches to compute their distribution under the null: (i) a bootstrap-based approach, and (ii) a permutation-test with nonuniform probability of permutations. We also handle an important application to the case of censoring with truncation, by estimating the biased sampling mechanism from the data. We prove the validity of the tests, and show, using simulations, that they improve power compared to competing methods for important special cases. The tests are applied to four datasets, two that are subject to truncation, with and without censoring, and two to bias mechanisms related to length bias. [ABSTRACT FROM AUTHOR]

In this work, multiplicities of zeros of skew polynomials are studied. Two distinct definitions are considered: First, a is said to be a zero of F of multiplicity r if (x − a) r divides F on the right; second, a is said to be a zero of F of multiplicity r if some skew polynomial P = (x − a r) ⋯ (x − a 2) (x − a 1) , having a 1 = a as its only right zero, divides F on the right. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy's concept of P-independence does not behave naturally, whereas a union theorem still holds. In contrast, we show that P-independence behaves naturally for the second notion of multiplicities. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include: (1) The upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and (2) The equivalence of P-independence, Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives). [ABSTRACT FROM AUTHOR]

Correlations belong to the standard repertoire of ecologists for quantifying the strength of dependence between two random variables. Classical dependence measures are usually not capable of detecting non‐monotonic or non‐functional dependencies. Furthermore, they completely fail to detect asymmetry and direction in dependence, which exist in many situations and should not be ignored.In this paper, we present qad (short for quantification of asymmetric dependence), a nonparametric statistical method to quantify directed and asymmetric dependence of bivariate samples. Qad is applicable in general (e.g. linear, non‐linear, or non‐monotonic) situations, is sensitive to noise in data, exhibits a good small sample performance, detects asymmetry in dependence, shows high power in testing for independence, requires no assumptions regarding the underlying distribution of the data and reliably quantifies the information gain/predictability of quantity Y given knowledge of quantity X, and vice versa (i.e. q(X,Y) ≠q(Y,X)).Here, we briefly recall the methodology underlying qad, introduce the functions of the R‐package qad, which returns estimates for the measures qX,Y denoting the directed dependence of Y on X (or, equivalently, the influence of X on Y), qY,X the directed dependence of X on Y, aX,Y≔qX,Y−qY,X the asymmetry in dependence. Furthermore, qad can be used to predict Y given knowledge of X, and vice versa. Additionally, we compare empirical performance of qad with that of seven other well established measures and demonstrate the applicability of qad on ecological datasets.We illustrate that direction and asymmetry in dependence are universal properties of bivariate associations. Qad thus provides additional information gain and avoids model bias and will therefore advance and facilitate the understanding of ecological systems. [ABSTRACT FROM AUTHOR]