Search

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Spatial capture–recapture (SCR) models are commonly used to estimate animal density from surveys on which detectors passively detect animals without physical capture, for example, using camera traps, hair snares, or microphones. An individual is more likely to be recorded by detectors close to its activity center, the centroid of its movement throughout the survey. Existing models to account for this spatial heterogeneity in detection probabilities rely on an assumption of independence between detection records at different detectors conditional on the animals' activity centers, which are treated as latent variables. In this paper, we show that this conditional independence assumption may be violated due to the way animals move around the survey region and encounter detectors, such that additional spatial correlation is almost inevitable. We highlight the links between the well‐studied issue of unmodeled temporal heterogeneity in nonspatial capture–recapture and this variety of unmodeled spatial heterogeneity in SCR, showing that the latter causes predictable bias in the same way as the former. We address this by introducing a latent detection field into the model, and illustrate the resulting approach with a simulation study and an application to a camera‐trap survey of snow leopards Panthera uncia. Our method is a unifying model for several existing SCR approaches, with special cases including standard SCR, models that account for nonspatial individual heterogeneity, and models with overdispersed detection counts. [ABSTRACT FROM AUTHOR]

We propose an independence test for random variables valued into metric spaces by using a test statistic obtained from appropriately centering and rescaling the squared Hilbert–Schmidt norm of the usual empirical estimator of normalized cross-covariance operator. We then get asymptotic normality of this statistic under independence hypothesis, so leading to a new test for independence of functional random variables. A simulation study that allows to compare the proposed test to existing ones is provided. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in ∏ n = 1 ∞ 1 + 1 F n , ∏ n = 3 ∞ 1 - 1 F n , ∏ n = 1 ∞ 1 + 1 L n , ∏ n = 2 ∞ 1 - 1 L n are algebraically independent over Q , where { F n } and { L n } are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series. [ABSTRACT FROM AUTHOR]

The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty arises, and it becomes necessary to build suitable approaches to overcome it. In this direction, this paper discusses a new approach of fuzzy binomial distributions (BDs) and their extensions. Four cases are considered: (1) When the elementary events are fuzzy. Based on this information, the probabilistic distribution of the corresponding fuzzy-random binomial variable is calculated. The conditions of restrictions on this distribution are obtained, and it is shown that these conditions depend on the ratio of success and failure of membership levels. The formulas for the generating function (GF) of the constructed distribution and the first and second order moments are also obtained. The Poisson distribution is calculated as the limit case of a fuzzy-random binomial experiment. (2) When the number of successes is of a fuzzy nature and is represented as a fuzzy subset of the set of possible success numbers. The formula for calculating the probability of convolution of binomial dependent fuzzy events is obtained, and the corresponding GF is built. As a result, the scheme for calculating the mathematical expectation of the number of fuzzy successes is defined. (3) When the spectrum of the extended distribution is fuzzy. The discussion is based on the concepts of a fuzzy-random event and its probability, as well as the notion of fuzzy random events independence. The fuzzy binomial upper distribution is specifically considered. In this case the fuzziness is represented by the membership levels of the binomial and non-binomial events of the complete failure complex. The GF of the constructed distribution and the first-order moment of the distribution are also calculated. Sufficient conditions for the existence of a limit distribution and a Poisson distribution are also obtained. (4) As is known, based on the analysis of lexical material, the linguistic spectrum of the statistical process of word-formation becomes two-component when switching to vocabulary. For this, two variants of the hybrid fuzzy-probabilistic process are constructed, which can be used in the analysis of the linguistic spectrum of the statistical process of word-formation. A fuzzy extension of standard Fuchs distribution is also presented, where the fuzziness is reflected in the growing numbers of failures. For better representation of the results, the examples of fuzzy BD are illustrated in each section. [ABSTRACT FROM AUTHOR]

A number of standard textbooks that are followed in a junior/senior level course or in a first-year graduate level course in mathematical statistics and probability, routinely include one single basic illustration, obviously in its variant forms, to highlight an important point: pairwise independence may not imply (mutual) independence. We earnestly believe that beginning students appreciate more examples to clarify these key issues. Hence, we hope that our new sets of nontrivial illustrations from Section 2 will help our audience. Next, in Section 3, we extend the notion to q-wise independence with a large set of illustrations using both discrete and continuous random variables showing that q-wise independence may not imply (mutual) independence. We reasonably assure that this discourse is immediately accessible to juniors/seniors and first-year graduate students. [ABSTRACT FROM AUTHOR]

The problem of hypothesis testing for the independence of two-dimensional random variables in the analysis of variables of multi-valued functions is considered. To solve it, we used a technique based on a nonparametric kernel-type pattern recognition algorithm corresponding to the maximum likelihood criterion. The technique made it possible to bypass the problem of decomposing the random variable domain of values into intervals. Based on the results of computational experiments, the effectiveness of the applied technique was estimated depending on the type of multi-valued functions, the level of random noise and the amount of initial statistical data. The results obtained are relevant for solving the problem of detecting natural and technical objects from remote sensing data. [ABSTRACT FROM AUTHOR]

We generalize the sign covariance introduced by Bergsma & Dassios (2014) to multivariate random variables and beyond. The new interpoint-ranking sign covariance is applicable to general types of random objects as long as a meaningful similarity measure can be defined, and it is shown to be zero if and only if the two random variables are independent. The test statistic is a |$U$| -statistic, whose large-sample behaviour guarantees that the proposed test is consistent against general types of alternatives. Numerical experiments and data analyses demonstrate the superior empirical performance of the proposed method. [ABSTRACT FROM AUTHOR]

With the rise of historical data availability and increased computational power, reinforcement learning (RL) has become more prominent. It has shown the ability to outperform human knowledge in certain areas. RL is particularly valuable in control applications due to its generalization, self-adjustment, and independence from a mathematical model. Despite its potential, there are limited studies and applications focusing on using RL for mining processes. This study aims to showcase the applicability of RL in the mining industry. An algorithm was developed using proximal policy optimization (PPO) to control a simulated conventional cylindrical-conical thickener. The controller's goal is to adjust the slurry density and solid interface height to produce a material with a high solid concentration. By regulating the thickener's output flow and the flocculant dosage in the incoming slurry, PPO achieves this objective. Through simulation, an RL agent was trained to control the thickener efficiently, managing flocculant usage and responding appropriately to system disturbances. The study concludes that applying reinforcement learning holds promise for enhancing mining process control. However, further research is needed to fine-tune algorithm parameters, enhance design structure, optimize control, and maximize the technique's benefits. • Demonstrate that reinforcement learning (RL) can be applied in the mining industry. • Control is performed using the proximal policy optimization (PPO). • Tests are conducted using a simulated conventional cylindrical-conical thickener. • RL agent capable of being applied to a real thickener. • Control presents efficient use of flocculant and rejection of disturbances. [ABSTRACT FROM AUTHOR]

This work addresses testing the independence of two continuous and finite-dimensional random variables from the design of a data-driven partition. The empirical log-likelihood statistic is adopted to approximate the sufficient statistics of an oracle test against independence (that knows the two hypotheses). It is shown that approximating the sufficient statistics of the oracle test offers a learning criterion for designing a data-driven partition that connects with the problem of mutual information estimation. Applying these ideas in the context of a data-dependent tree-structured partition (TSP), we derive conditions on the TSP’s parameters to achieve a strongly consistent distribution-free test of independence over the family of probabilities equipped with a density. Complementing this result, we present finite-length results that show our TSP scheme’s capacity to detect the scenario of independence structurally with the data-driven partition as well as new sampling complexity bounds for this detection. Finally, some experimental analyses provide evidence regarding our scheme’s advantage for testing independence compared with some strategies that do not use data-driven representations. [ABSTRACT FROM AUTHOR]