Cardinalities Of Classes Research Articles

A novel attribute reduction approach using coverage-credibility-based rough decision entropy for interval-valued data

Attribute reduction is an important method in data analysis and machine learning, and it usually relies on algebraic and informational measures. However, few existing informational measures have considered the relative information of decision class cardinality, and the fusion application of algebraic and informational measures is also limited, especially in attribute reductions for interval-valued data. In interval-valued decision systems, this paper presents a coverage-credibility-based condition entropy and an improved rough decision entropy, further establishes corresponding attribute reduction algorithms for optimization and applicability. Firstly, the concepts of interval credibility, coverage and coverage-credibility are proposed, and thus, an improved condition entropy is defined by virtue of the integrated coverage-credibility. Secondly, the fused rough decision entropy is constructed by the fusion of improved condition entropy and roughness degree. By introducing the coverage-credibility, the proposed uncertainty measurements enhance the relative information of decision classes. In addition, the nonmonotonicity of the improved condition entropy and rough decision entropy is validated by theoretical proofs and experimental counterexamples, with respect to attribute subsets and thresholds. Then, the two rough decision entropies drive monotonic and nonmonotonic attribute reductions, and the corresponding reduction algorithms are designed for heuristic searches. Finally, data experiments not only verify the effectiveness and improvements of the proposed uncertainty measurements, but also illustrate the reduction algorithms optimization through better classification accuracy than four comparative algorithms.

Dominator coloring of Kneser graph

A dominator coloring (briefly DC) of a graph G is a proper coloring such that each vertex dominates every vertex of at least one color class (possibly its own class). The dominator chromatic number χd(G) of G is the smallest cardinality of color classes in a DC of G. By using Steiner triple systems, in this paper, we establish that the dominator coloring of the Kneser graph KG(n, 2) is n for n ≥ 5.

Affinity and transformed class probability-based fuzzy least squares support vector machines

Inspired by the generalization efficiency of affinity and class probability-based fuzzy support vector machine (ACFSVM), a pair of class affinity and nonlinear transformed class probability-based fuzzy least squares support vector machine approaches is proposed. The proposed approaches handle the class imbalance problem by employing cost-sensitive learning, and by utilizing the samples' class probability determined using a novel nonlinear probability equation that adjusts itself with class size. Further, the sensitivity to outliers and noise is reduced with the help of each sample's affinity to its class obtained with the help of least squares one-class support vector machine. The first proposed approach incorporates fuzzy membership values, computed using transformed class probability and class affinity, into the objective function of LS-SVM type formulation, and introduces a new cost sensitive term based on the class cardinalities to normalize the effect of the class imbalance problem. The inherent noise and outlier sensitivity of the quadratic least squares loss function of the first approach is further reduced in the second proposed approach by truncating the quadratic growth of the loss function at a specified score. Thus, the concerns due to noise and outliers are further handled at the optimization level. However, the employed truncated loss function of the second approach takes a non-convex structure, which in turn, is resolved using ConCave-Convex Procedure (CCCP) for global convergence. Numerical experiments on artificial and real-world datasets of different imbalance ratio establish the effectiveness of the proposed approaches.

ОЦЕНКИ МОЩНОСТИ КЛАССОВ ОТОБРАЖЕНИЙ, ПРИМЕНЯЕМЫХ В ПРОТОКОЛАХ КВАНТОВОГО РАСПРЕДЕЛЕНИЯ КЛЮЧЕЙ

Introduction: In the near future, quantum cryptography will play an important role in maintaining a sufficient level of information security of modern telecommunication networks in the conditions of a quantum challenge, which refers to the emergence of quantum computers that will be able to effectively solve the mathematical problems on which, for example, modern key distribution systems are based. Now quantum cryptography is actively used by commercial and government agencies around the world and, in particular, in the Russian Federation. At the same time, a large amount of research is being carried out in the field of development and implementation of quantum key distribution systems, as the main part of quantum cryptography. In this regard, the task of developing new and refining existing protocols for quantum key distribution, as well as studying various mathematical and physical objects that are associated with these protocols, is an urgent task. In particular, one of the stages of the classical BB84 protocol implement ed in a noisy quantum channel is associated with the problem of studying correlation immune and stable mappings, part of which is the problem of estimating their number, which has not been completely solved. Purpose: to find mathematical expressions for exact and asymptotic estimates of the cardinalities of classes of (n,m,k) stable and correlation immune of order k boolean mappings. Results: The best currently asymptotic upper and lower bounds for the number of such classes of mappings with the number of outputs greater than or equal to five are obtained. Recurrent relations were also proved, which allow one to find the exact distribution of the cardinalities of classes of similar mappings for the case of small numbers n and m. Practical relevance: the results obtained allow us to estimate the probability that with a random choice of mapping to enhance secrecy at the stage of secondary processing of the BB84 protocol, the situation will be neutralized when the adversary has access to k photons sent over a communication chan nel of his choice.

Exploring the Sensitivity of Recurrent Neural Network Models for Forecasting Land Cover Change

Recurrent Neural Networks (RNNs), including Long Short-Term Memory (LSTM) architectures, have obtained successful outcomes in timeseries analysis tasks. While RNNs demonstrated favourable performance for Land Cover (LC) change analyses, few studies have explored or quantified the geospatial data characteristics required to utilize this method. Likewise, many studies utilize overall measures of accuracy rather than metrics accounting for the slow or sparse changes of LC that are typically observed. Therefore, the main objective of this study is to evaluate the performance of LSTM models for forecasting LC changes by conducting a sensitivity analysis involving hypothetical and real-world datasets. The intent of this assessment is to explore the implications of varying temporal resolutions and LC classes. Additionally, changing these input data characteristics impacts the number of timesteps and LC change rates provided to the respective models. Kappa variants are selected to explore the capacity of LSTM models for forecasting transitions or persistence of LC. Results demonstrate the adverse effects of coarser temporal resolutions and high LC class cardinality on method performance, despite method optimization techniques applied. This study suggests various characteristics of geospatial datasets that should be present before considering LSTM methods for LC change forecasting.

Enriched preference modeling and robustness analysis for\xa0the ELECTRE Tri-B method

We consider multiple criteria sorting problems with preference-ordered classes delimited by a set of boundary profiles. While significantly extending the ELECTRE Tri-B method, we present an integrated framework for modeling indirect preference information and conducting robustness analysis. We allow the Decision Maker (DM) to provide the following three types of holistic judgments: assignment examples, assignment-based pairwise comparisons, and desired class cardinalities. A diversity of recommendation that can be obtained given the plurality of outranking-based sorting models compatible with the DM’s preferences is quantified by means of six types of results. These include possible assignments, class acceptability indices, necessary assignment-based preference relation, assignment-based outranking indices, extreme class cardinalities, and class cardinality indices. We discuss the impact of preference information on the derived outcomes, the interrelations between the exact results computed with mathematical programming and stochastic indices estimated with the Monte Carlo simulations, and new measures for quantifying the robustness of results. The practical usefulness of the approach is illustrated on data from the Financial Times concerning MBA programs.

General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

Handling missing labels and class imbalance challenges simultaneously for facial action unit recognition

Facial action unit (AU) recognition has attracted great attention because of the applications in a wide range of fields. Missing labels and class imbalance (CIB) are both challenges for facial action unit recognition. Missing labels means that there are only apart label assignments for training samples. CIB is observed from two perspectives: firstly, the number of positive AUs is much smaller than that of negative AUs for each expressional image; secondly, the rate of positive samples of different AUs are significantly different. Both missing labels and CIB lead to performance degradation in AU recognition. In this work, we propose to handle these two challenges in AU recognition simultaneously. Specifically, we formulate AU recognition with missing labels as a multi label learning with missing labels (MLML) problem, which handles the missing label challenge naturally. However, different from most existing MLML approaches which usually employ same features from whole image for all classes, we select the most related features for each AU. To handle the CIB challenge, we further introduce class cardinality bounds which constrain the number of positive AUs for each data instance, as well as the number of positive labels for each AU in the overall dataset. The class cardinality bounds serve as linear constraints for the objective function, which turns the optimization NP-hard. Thus we present convex approximation based on the Lovasz extension, which leads to a linear program that can be efficiently solved by the alternative direction method of multipliers (ADMM). Experimental results on both posed and spontaneous facial expression datasets demonstrate the superiority of the proposed method compared to state-of-the-art.

The impact of the dependence structure in risk management: a focus on credit-risk

ABSTRACTThe aim of the paper is to discuss the important role of the dependence structure in risk management. Therefore, we focus on credit-risk and propose an innovative model to value the credit risk of a portfolio. This new approach (HYC for short) is based on a hierarchical hybrid copula and involves a clusterization of the portfolio in several risk's classes. The HYC model is classified as hybrid because the computation of the loss cdf depends on the class's cardinality: for large groups one is justified to apply a limiting approach, while for small ones one applies a procedure preserving the granularity of the group itself. In order to appreciate the impact of the dependence structure in credit-risk evaluation, a VaR analysis based on the HYC loss function is here compared to the CreditMetrics approach in an in-sample exercise and to the empirical VaR in an out-of sample exercise aimed to test the forecasting effectiveness of the model. This comparison allows us to appreciate over/under-valuation of the capital detained from the financial institution. Moreover, the impact of an enlargement of the dependence structure is discussed with respect to the systemic/contagious effects in the context of a portfolio optimisation with constraint on a sub-portfolio's risk.

Кодування і класифікація перестановок за спеціальним перетворенням з оцінками потужностей класів

Scientific articles investigating properties and estimates of the number of so-called complete permutations are surveyed and analyzed. The paper introduces a special S-transform on the set of permutations and determines the permutation properties according to this transform. Classification and coding of permutations by equivalence classes according to their properties with respect to S-transformation is proposed. This classification and permutation properties, in particular, generalize known results for complete permutations regarding determining certain cryptographic properties of substitutions that affect the cryptographic transformations security. The exact values of the number of permutations in equivalence classes for certain permutation sizes are calculated and the estimates of the cardinality of classes with various properties are constructed by statistical modeling. The complete list of permutation classes with the exact values of their sizes for permutations of order n = 11 is presented. The interval estimates for the size of classes with various characteristics for permutations of order n = 11, 26, 30, 31, 32, 33, 45, 55 are obtained. Monte Carlo estimates and bounds of confidence intervals used the approximation of the binomial distribution by the normal and Poisson distributions, as well as the Python programming language package Scipy. Statistical tables have been calculated that can be used for further conclusions and estimates. The classification of permutations by their properties with respect to the introduced transform can be used in constructing high-quality cryptographic transformations and transformations with special features. The classes of complete permutations with their properties are selected as the best for rotary cryptosystems applications. The obtained results can be used, in particular, to search for permutations with certain characteristics and properties, to find the probability that the characteristic of the generated permutation belongs to a collection of given characteristics, to estimate the complexity of finding permutations with certain properties. A statistical criterion of consent, which uses the characteristics of permutations by S-transformation to test the generators of random permutations and substitutions is proposed.

The independence and clique polynomial of the conjugacy class graph of dihedral group

The independence and clique polynomial are two types of graph polynomial that store combinatorial information of a graph. The independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of vertices that are not adjacent. The clique polynomial of a graph is the polynomial in which its coefficients are the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent. Meanwhile, a graph of group G is called conjugacy class graph if the vertices are non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The independence and clique polynomial of the conjugacy class graph of a group G can be obtained by considering the polynomials of complete graph or polynomials of union of some graphs. In this research, the independence and clique polynomials of the conjugacy class graph of dihedral groups of order 2n are determined based on three cases namely when n is odd, when n and n/2 are even, and when n is even and n/2 is odd. For each case, the results of the independence polynomials are of degree two, one and two, and the results of the clique polynomials are of degree (n-1)/2, (n+2)/2 and (n-2)/2, respectively.

Some bound of the edge chromatic surplus of certain cubic graphs

V.G. Vizing showed that any graph belongs to one of two classes: Class 1 if χ ʹ( G ) = Δ( G ) or in class 2 if χ ʹ( G ) = Δ( G ) + 1 , where χ ʹ( G ) and Δ( G ) denote the edge chromatic index of G and the maximum degree of G , respectively. This paper addresses the problem of finding the edge chromatic surplus of a cubic graph G in Class 2, namely the minimum cardinality of colour classes over all 4-edge chromatic colourings of E ( G ) . An approach to face this problem - using a new parameter q - is given in [1]. Computing q is difficult for the general case of graph G , so there is the need to find restricted classes of G , where q is easy to compute. Working in the same sense as in this paper we give an upper bound of the edge chromatic surplus for a special type of cubic graphs, that is the class of bridgeless non-planar cubic graphs in which in each pair of crossing edges, the crossing edges are adjacent to a third edge.

Tabu search with feasible and infeasible searches for equitable coloring

The equitable coloring problem is a variant of the classical graph coloring problem that arises from a number of real-life applications where the cardinality of color classes must be balanced. In this paper, we present a highly effective hybrid tabu search method for the problem. Based on three complementary neighborhoods, the algorithm alternates between a feasible local search phase where the search focuses on the most relevant feasible solutions and an infeasible local search phase where a controlled exploration of infeasible solutions is allowed by relaxing the equity constraint. A novel cyclic exchange neighborhood is also proposed in order to enhance the search ability of the hybrid tabu search algorithm. Experiments on a set of 73 benchmark instances in the literature indicate that the proposed algorithm is able to find improved best solutions for 15 instances (new upper bounds) and matches the best-known solutions for 57 instances. Additional analyses show the interest of the cyclic exchange neighborhood and the hybrid scheme combining both feasible and infeasible local searches.

Automatic Classification of Functional Gait Disorders.

This paper proposes a comprehensive investigation of the automatic classification of functional gait disorders (GDs) based solely on ground reaction force (GRF) measurements. The aim of this study is twofold: first, to investigate the suitability of the state-of-the-art GRF parameterization techniques (representations) for the discrimination of functional GDs; and second, to provide a first performance baseline for the automated classification of functional GDs for a large-scale dataset. The utilized database comprises GRF measurements from 279 patients with GDs and data from 161 healthy controls (N). Patients were manually classified into four classes with different functional impairments associated with the "hip", "knee", "ankle", and "calcaneus". Different parameterizations are investigated: GRF parameters, global principal component analysis (PCA) based representations, and a combined representation applying PCA on GRF parameters. The discriminative power of each parameterization for different classes is investigated by linear discriminant analysis. Based on this analysis, two classification experiments are pursued: distinction between healthy and impaired gait (N versus GD) and multiclass classification between healthy gait and all four GD classes. Experiments show promising results and reveal among others that several factors, such as imbalanced class cardinalities and varying numbers of measurement sessions per patient, have a strong impact on the classification accuracy and therefore need to be taken into account. The results represent a promising first step toward the automated classification of GDs and a first performance baseline for future developments in this direction.

Integrated framework for preference modeling and robustness analysis for outranking-based multiple criteria sorting with ELECTRE and PROMETHEE

We introduce an integrated framework for preference modeling and robustness analysis in outranking-based multiple criteria sorting. The preference information supplied by the Decision Maker (DM) is composed of parts of desired recommendation and imprecise requirements that the delivered outcomes should satisfy. Its allowed forms are: imprecise assignment examples, desired class cardinalities, and assignment-based pairwise comparisons. The exploitation of all instances of the outranking model compatible with these preferences results in three types of outcomes. These raise robustness concerns in terms of the stability of a suggested assignment for each alternative, an observed size of each class, and a comparison between recommendation delivered for pairs of alternatives. The correspondence between different types of inputs and outputs facilitates the dialogue with the DM and enhances her/his confidence in the suggested recommendation. While referring only to a semantic meaning of an outranking relation, we present the procedures for translating the preference information into parameters of compatible outranking model instances and for deriving various kinds of sorting results. Then, we implement this general framework in the context of outranking models specific for ELECTRE and PROMETHEE. Finally, we show how preference modeling and robustness analysis can be performed and greatly simplified with a set of preference model instances providing precise assignments for the alternatives. The framework proposed for this case is based on Mixed-Integer Linear Programming (MILP), being independent of the underlying model and method. Application of the approaches is demonstrated on the case of classifying Polish research units into four quality classes.

Constrained Submodular Minimization for Missing Labels and Class Imbalance in Multi-label Learning

In multi-label learning, there are two main challenges: missing labels and class imbalance (CIB). The former assumes that only a partial set of labels are provided for each training instance while other labels are missing. CIB is observed from two perspectives: first, the number of negative labels of each instance is much larger than its positive labels; second, the rate of positive instances (i.e. the number of positive instances divided by the total number of instances) of different classes are significantly different. Both missing labels and CIB lead to significant performance degradation. In this work, we propose a new method to handle these two challenges simultaneously. We formulate the problem as a constrained submodular minimization that is composed of a submodular objective function that encourages label consistency and smoothness, as well as, class cardinality bound constraints to handle class imbalance. We further present a convex approximation based on the Lovasz extension of submodular functions, leading to a linear program, which can be efficiently solved by the alternative direction method of multipliers (ADMM). Experimental results on several benchmark datasets demonstrate the improved performance of our method over several state-of-the-art methods.

Robustness analysis for decision under uncertainty with rule-based preference model

We consider decision under uncertainty as a multi-attribute classification problem where a set of acts is described by outcomes gained with given probabilities. The Decision Maker (DM) provides desired classification for a small subset of reference acts. Such preference information is structured using Dominance-based Rough Set Approach (DRSA), and the resulting lower approximations of the quality class unions are used as an input for construction of an aggregate preference model. We induce all minimal-cover sets of rules being compatible with the non-ambiguous assignment examples, and satisfying some additional requirements that may be imposed by the DM. Applying such compatible instances of the preference model on a set of all acts, we draw conclusions about the certainty of recommendation assured by different minimal-cover sets of rules. In particular, we analyze the diversity of class assignments, assignment-based preference relations, and class cardinalities. Then, we solve an optimization problem to get a univocal (precise) classification for all acts, taking into account the robustness concern. This optimization problem admits incorporation of additional indirect and imprecise preferences in form of desired class cardinalities and assignment-based pairwise comparisons. Finally, we extend the proposed approach to group decision under uncertainty. We present a set of indicators and outcomes giving an insight into the spaces of consensus and disagreement between the DMs.

Parametric evaluation of research units with respect to reference profiles

We introduce a method that jointly considers multiple criteria sorting and ranking. The method derives from a real-world problem of parametric evaluation of research units carried out by the Polish Ministry of Science and Higher Education. It assigns the units to three classes representing different qualities of both acquired effects and activities undertook in the evaluation period. Although units placed in the same class are guaranteed the same level of funding, they are not considered indifferent in the subsequent analysis and, thus, need to be ordered from the best to the worst in each class. A proposed outranking relation compares the units pairwise and the result is exploited so as to get a ranking of the units. The ranking is transformed to class assignments based on the attained comprehensive scores and ranks. To enhance interpretability of the results, we infer two reference profiles (artificial reference research units) separating the classes so that each class accumulates units ranked not worse than the corresponding lower profile and worse than the respective upper profile. The procedure takes into account desired cardinalities of classes, i.e., shares of units that are judged as leading, average, or weak. We discuss several procedures with different ways of inferring the reference profiles and scoring the units. We also analyze robustness of the results.

Labeled Rauzy classes and framed translation surfaces

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmuller flow. The second one is more recent and uses a labeling of the underlying intervals, and was used in the proof of some recent major results about the Teichmuller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of framed translation surfaces, which corresponds to surfaces where we label horizontal separatrices on the surface. We compute the number of connected component of these natural coverings of the moduli spaces of translation surfaces. Delecroix proved recently a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain a formula for labeled Rauzy classes.

Modeling assignment-based pairwise comparisons within integrated framework for value-driven multiple criteria sorting

We introduce a new preference disaggregation modeling formulations for multiple criteria sorting with a set of additive value functions. The preference information supplied by the Decision Maker (DM) is composed of: (1) possibly imprecise assignment examples, (2) desired class cardinalities, and (3) assignment-based pairwise comparisons. The latter have the form of imprecise statements referring to the desired assignments for pairs of alternatives, but without specifying any concrete class. Additionally, we account for preferences concerning the shape of the marginal value functions and desired comprehensive values of alternatives assigned to a given class or class range. The exploitation of all value functions compatible with these preferences results in three types of results: (1) necessary and possible assignments, (2) extreme class cardinalities, and (3) necessary and possible assignment-based preference relations. These outputs correspond to different types of admitted preference information. By exhibiting different outcomes, we encourage the DM in various ways to enrich her/his preference information interactively. The applicability of the framework is demonstrated on data involving the classification of cities into liveability classes.