Monotone Quantifier Research Articles

Learning the weights using attribute order information for multi-criteria decision making tasks

AbstractIn multi-criteria decision making, the importance of decision criteria (decision attributes) plays a crucial role. Ranking is a useful technique for expressing the importance of decision criteria in a decision-makers’ preference system. Since weights are commonly utilized for characterizing the importance of criteria, weight determination and assessment are important tasks in multi-criteria decision making and in voting systems as well. In this study, we concentrate on the connection between the preference order of decision criteria and the decision weights. Here, we present an easy-to-use procedure that can be used to produce a sequence of weights corresponding to a decision-makers’ preference order of decision criteria. The proposed method does not require pairwise comparisons, which is an advantageous property especially in cases where the number of criteria is large. This method is based on the application of a class of regular increasing monotone quantifiers, which we refer to as the class of weighting generator functions. We will show that the derivatives of these functions can be used for approximating the criteria weights. Also, we will demonstrate that using weighting generator functions, weights can be inverted in a consistent way. We will deduce the generators for arithmetic and geometric weight sequences, and we will present a one-parameter generator function known as the tau function in continuous-valued logic. We will show that using these weighting generator functions, the weight learning task can be turned into a simple, one-parameter optimization problem.

Wasserstein distance for OWA operators

We suggest a distance measure for OWA operators. First we associate an OWA operator with a unique regular increasing monotone quantifier and then define the distance between two OWA operators as the Wasserstein-1 distance between their associated quantifiers.

Quantifiers satisfying semantic universals have shorter minimal description length

Despite wide variation among natural languages, there are linguistic properties thought to be universal to all or nearly all languages. Here, we consider universals at the semantic level, in the domain of quantifiers, which are given by the properties of monotonicity, quantity, and conservativity, and we investigate whether these universals might be explained by differences in complexity. First, we use a minimal pair methodology and compare the complexities of individual quantifiers using approximate Kolmogorov complexity. Second, we use a simple yet expressive grammar to generate a large collection of quantifiers and we investigate their complexities at an aggregate level in terms of both their minimal description lengths and their approximate Kolmogorov complexities. For minimal description length we find that quantifiers satisfying semantic universals are simpler: they have a shorter minimal description length. For approximate Kolmogorov complexity we find that monotone quantifiers have a lower Kolmogorov complexity than non-monotone quantifiers and for quantity and conservativity we find that approximate Kolmogorov complexity does not scale robustly. These results suggest that the simplicity of quantifier meanings, in terms of their minimal description length, partially explains the presence of semantic universals in the domain of quantifiers.

Monotone Quantifiers Emerge via Iterated Learning.

Natural languages exhibit many semantic universals, that is, properties of meaning shared across all languages. In this paper, we develop an explanation of one very prominent semantic universal, the monotonicity universal. While the existing work has shown that quantifiers satisfying the monotonicity universal are easier to learn, we provide a more complete explanation by considering the emergence of quantifiers from the perspective of cultural evolution. In particular, we show that quantifiers satisfy the monotonicity universal evolve reliably in an iterated learning paradigm with neural networks as agents.

A hesitant fuzzy wind speed forecasting system with novel defuzzification method and multi-objective optimization algorithm

Owing to the nondeterministic nature of wind speed, the conventional fuzzy time series forecasting model has difficulty in establishing a common membership level. Therefore, in this study, the fuzzy series forecasting model was improved based on hesitant fuzzy sets. A hesitant fuzzy wind speed forecasting system with a novel defuzzification method and multiobjective optimization algorithm was developed. First, an advanced decomposition model is employed to extract the effective feature and remove the noise component from the raw wind speed series. Then, the universe of discourse is partitioned into equal and unequal intervals by multifuzzification methods and merged by aggregating hesitant information. A multiobjective intelligent optimization algorithm is applied to determine the optimal weights of different intervals accurately and stably. Furthermore, a novel defuzzification model based on an ordered weighted averaging operator and a regular increasing monotone quantifier is proposed to calculate the final forecasting results. The crucial strengths of the developed system are verifying the possibility of enhancing the performance of wind speed forecasting models by improving conventional fuzzy time series forecasting models and integrating them with decomposition models and artificial-intelligence models. Typical wind speed series datasets with different resolutions were selected to evaluate the performance of the proposed system, and experimental results prove that the proposed system outperforms other comparison models with high forecasting accuracy and computing efficiency.

Two-level multi-criteria comprehensive evaluation for preference vectors in online shopping platform evaluation

The evaluation for online shopping platform is the basis for further decision and policy taking. The collected individual opinion and evaluation information are often represented by some linguistic/preference vectors. Further aggregating those vector needs to simultaneously consider two contradictory factors: the original weights assigned and the inconsistencies involved which requires some new weights assigned. Around those weights allocation factors, to mitigate the negative effect of inconsistency in the collected information, we propose an integrated evaluation model. The model uses the scatter degree as a main indicator, and extends some weights allocation methods such as regular increasing monotone (RIM) quantifier based weights allocation in a new environment, and applies the three sets expression based paradigm and formulation. The proposed model is able to simultaneously give emphasis on those input data with high consistency and to consider the preferences of decision makers. Some detailed evaluation processes and numerical examples are also provided for practitioners to refer to.

The General Model for Least Convex Disparity RIM Quantifier Problems

Hong (Mathematics 2019, 7, 326) recently introduced the general least squares deviation (LSD) model for ordered weighted averaging (OWA) operator weights. In this paper, we propose the corresponding generalized least square disparity model for regular increasing monotone (RIM) quantifier determination under a given orness level. We prove this problem mathematically. Using this result, we provide the full solution of the least square disparity RIM quantifier model as an illustrative example.

The emergence of monotone quantifiers via iterated learning

Natural languages exhibit many semantic universals: properties of meaning shared across all languages. In this paper, we develop an explanation of one very prominent semantic universal: that all simple determiners denote monotone quantifiers. While existing work has shown that monotone quantifiers are easier to learn, we provide a complete explanation by considering the emergence of quantifiers from the perspective of cultural evolution. In particular, in an iterated learning paradigm,with neural networks as agents, monotone quantifiers regularly evolve.

Measuring the cognitive cost of downward monotonicity by controlling for negative polarity

Our goal in this study was to behaviorally characterize the property (or properties) that render negative quantifiers more complex in processing compared to their positive counterparts (e.g. the pair few/many). We examined two sources: (i) negative polarity; (ii) entailment reversal (aka downward monotonicity). While negative polarity can be found in other pairs in language such as dimensional adjectives (e.g. the pair small/large), only in quantifiers does negative polarity also reverse the entailment pattern of the sentence. By comparing the processing traits of negative quantifiers with those of non-monotone expressions that contain negative adjectives, using a verification task and measuring reaction times, we found that negative polarity is cognitively costly, but in downward monotone quantifiers it is even more so. We therefore conclude that both negative polarity and downward monotonicity contribute to the processing complexity of negative quantifiers.

An Evidential Aggregation Method of Intuitionistic Fuzzy Sets Based on Belief Entropy

Intuitionistic fuzzy sets (IFSs) are essential in the multi-criteria decision making (MCDM) under uncertain environment. However, how to reasonably aggregate them with considering the uncertainty contained in the IFSs is still an open issue. In this paper, a new method is proposed to solve such a problem based on the Dempster-Shafer evidence theory, belief entropy, and the weighted ordered weighted averaging (WOWA) operator. One of the advantages of the presented model is that the uncertainty contained in the IFSs is effectively modeled based on belief entropy and the conversion from the IFS to Dempster-Shafer evidence theory. In the framework of evidence theory, the uncertain information contained in the IFSs can be embodied effectively. Then, the belief entropy is calculated to determine the certainty weights for each IFS. With the various definitions of the regular increasing monotone (RIM) quantifier Q function, the preference relationship of a decision maker is considered. A numerical example is shown to illustrate the feasibility and effectiveness of the proposed method.

Elliptical distribution-based weight-determining method for ordered weighted averaging operators

The ordered weighted averaging (OWA) operators play a crucial role in aggregating multiple criteria evaluations into an overall assessment supporting the decision makers’ choice. One key point steps is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then we propose a new approach for determining OWA weights by using the regular increasing monotone quantifier. Motivated by the idea of normal distribution-based method to determine the OWA weights, we develop a method based on elliptical distributions for determining the OWA weights, and some of its desirable properties have been investigated.

Comparing Apples and Oranges: The Weighted OWA Function

This paper advocates the use of weighted ordered weighted averaging (WOWA) functions in decision-making processes, where the alternatives are not directly comparable. In particular, WOWA allows one to compare the strongest points of each alternative, also weighted by the importance of each criterion. Four different approaches to applying weights in OWA functions are reviewed. Torra's method based on interpolating regular increasing monotone quantifier and the pruned n-ary tree are compared to the WOWA obtained from recently proposed implicit averaging. Computationally efficient algorithms are outlined. The use of WOWA is illustrated in several examples.

Quantum Coherence Quantifiers Based on Rényi α-Relative Entropy**Supported by by National Natural Science Foundation of China under Grant Nos. 11271237, 11671244, 61671280, and the Higher School Doctoral Subject Foundation of Ministry of Education of China under Grant No. 20130202110001, and Fundamental Research Funds for the Central Universities (GK201502004 and 2016CBY003), and the Academic Leaders and Academic Backbones, Shaanxi Normal University

The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived.

Classification of Denver System of Chromosomes Using Similarity Classifier Guided by OWA Operators

This paper proposes a Similarity Classifier guided by Ordered Weighted Averaging (OWA) operators to classify ‘Denver System’ of human chromosomes. A chromosome is assigned to one of the seven groups from A to G, based on the ‘Denver System’ of classification of chromosomes. Chromosomes within the group are difficult to identify, possessing almost identical characteristics including the length and centromere position. The proposed method is applied to the chromosome dataset and the similarity values using the generalized Lukasiewicz structure are calculated. The similarity values are aggregated with the help of Ordered Weighted Averaging (OWA) operators with a Regular Increasing Monotone (RIM) quantifier and the chromosome is assigned to that group for which it has the highest value of similarity. This work evaluates the performance of classifiers including Naive Bayes, Quadratic classifier and a novel, unsupervised, Fuzzy Similarity Classifier guided by OWA Operators, in classifying the Denver Group of chromosomes. A fundamental review on fuzzy similarity based classification is also presented. Experimental results clearly demonstrates that the proposed Fuzzy Similarity Classifier guided by OWA Operators with a RIM quantifier, produces the best classification results with a mean classification accuracy of 96.0028% and variance of 5.0871e-005. Statistical testing with one-way Analysis of Variance (ANOVA) at 95% and 99% level of confidence and Tukey’s post-hoc analysis is performed to validate the selection of the classifier. The proposed Fuzzy Similarity Classifier gives the most promising classification results and can be applied to any large scale biomedical data and other applications. Keywords: Analysis of variance, chromosomes, classification, denver system, fuzzy similarity classifier, lukasiewicz structure, ordered weighted averaging, operators, regular increasing monotone, quantifier, Tukey's post-hoc.

The orness measures for two compound quasi-arithmetic mean aggregation operators

The paper first summarizes the orness measures and their common characteristics of some averaging operators: the quasi-arithmetic mean, the ordered weighted averaging (OWA) operator, the regular increasing monotone (RIM) quantifier and the weighted function average operator, respectively. Then it focuses on the aggregation properties and operator determination methods for two kinds of quasi-arithmetic mean-based compound aggregation operators: the quasi-OWA (ordered weighted averaging) operator and the Bajraktarević mean. The former is the combination of the quasi-arithmetic mean and the OWA operator, while the latter is the combination of the quasi-arithmetic mean and the weighted function average operator. Two quasi-OWA operator forms are given, where the OWA operator is assigned directly or generated from a RIM (regular increasing monotone) quantifier indirectly. The orness indexes to reflect the or-like level of the quasi-OWA operator and Bajraktarević mean are proposed. With generating function techniques, the properties of the quasi-OWA operator and Bajraktarević mean are discussed to show the rationality of these orness definitions. Based on these properties, two families of parameterized quasi-OWA operator and Bajraktarević mean with exponential and power function generators are proposed and compared. It shows that the method of this paper can also be applied to other function-based aggregation operators.

On the properties of regular increasing monotone (RIM) quantifiers with maximum entropy††

Compared with the large number of research papers dealing with the ordered weighted averaging (OWA) operators, the research on relative quantifiers has been relatively sparse so far. With the quantifier guided aggregation method of OWA operators and the generating function representation of regular increasing monotone (RIM) quantifiers, a parameterized RIM quantifier family with the maximum entropy principle is proposed. This quantifier family can be regarded as the continuous case of the commonly used maximum entropy OWA (MEOWA) operator. Like the MEOWA operator, the RIM quantifier with maximum entropy generating function also possesses some interesting properties. One of them is that the aggregation value for any aggregated set always monotonically increases with the orness level. This can be used to represent the decision maker's preference information, but with free dimension.

Some Quantifier Functions From Weighting Functions With Constant Value of Orness

The quantifier-guided aggregation is used for aggregating the multiple-criteria input. Therefore, the selection of appropriate quantifiers is crucial in multicriteria aggregation since the weights for the aggregation are generated from the selected quantifier. Since Yager proposed a method for obtaining the ordered weighted averaging (OWA) vector via the three relative quantifiers used for the quantifier-guided aggregation, limited efforts have been devoted to developing new quantifiers that are suitable for use in multicriteria aggregation. In this correspondence, we propose some new quantifier functions that are based on the weighting functions characterized by showing a constant value of orness independent of the number of criteria aggregated. The proposed regular increasing monotone and regular decreasing monotone quantifiers produce the same orness as the weighting functions from which each quantifier function originates. Further, the quantifier orness rapidly converges into the value of orness of the weighting functions having a constant value of orness. This result indicates that a quantifier-guided OWA aggregation will result in a similar aggregate in case the number of criteria is not too small.

On the equivalence of some approaches to the OWA operator and RIM quantifier determination

The ordered weighted averaging (OWA) operator is a widely used aggregation method, and its determination is usually a prerequisite step in many related applications. The regular increasing monotone (RIM) quantifier can be seen as the continuous case of the OWA operator with the quantifier aggregation method. Some approaches with optimization criteria for the determination of OWA operator and RIM quantifier were proposed. Although these problems look different at the first sight, a deeper investigation can reveal the equivalence of solutions between them. Inspired by the solution equivalence of minimum variance problems and minimax disparity problem for OWA operator, we propose the minimax disparity RIM quantifier problem and two minimax ratio problems for OWA operator and RIM quantifier, respectively. We investigate the equivalence of solutions for the maximum entropy and minimax ratio problems, and solutions for the minimum variance and minimax disparity problems of OWA operator and RIM quantifier, respectively, by a theoretical point of view.

A general model of parameterized OWA aggregation with given orness level

The paper proposes a general optimization model with separable strictly convex objective function to obtain the consistent OWA (ordered weighted averaging) operator family. The consistency means that the aggregation value of the operator monotonically changes with the given orness level. Some properties of the problem are discussed with its analytical solution. The model includes the two most commonly used maximum entropy OWA operator and minimum variance OWA operator determination methods as its special cases. The solution equivalence to the general minimax problem is proved. Then, with the conclusion that the RIM (regular increasing monotone quantifier) can be seen as the continuous case of OWA operator with infinite dimension, the paper further proposes a general RIM quantifier determination model, and analytically solves it with the optimal control technique. Some properties of the optimal solution and the solution equivalence to the minimax problem for RIM quantifier are also proved. Comparing with that of the OWA operator problem, the RIM quantifier solutions are usually more simple, intuitive, dimension free and can be connected to the linguistic terms in natural language. With the solutions of these general problems, we not only can use the OWA operator or RIM quantifier to obtain aggregation value that monotonically changes with the orness level for any aggregated set, but also can obtain the parameterized OWA or RIM quantifier families in some specific function forms, which can incorporate the background knowledge or the required characteristic of the aggregation problems.

Orness and parameterized RIM quantifier aggregation with OWA operators: A summary

A necessary and sufficient condition for the ordered weighted average (OWA) aggregation value of an arbitrary aggregated set to consistently increase with the orness level is proposed. The OWA operator properties associated with the orness level are extended. Then, with the generating function representation of Regular Increasing Monotone (RIM) quantifier, all these conditions and properties are extended to the case of RIM quantifiers, which can be seen as the continuous OWA operator with free dimension. Some families of consistency RIM quantifiers and their corresponding OWA operators are summarized. Some existing linguistic term RIM quantifiers are collected and two parameterized generalization forms of them are proposed, which can be useful for the selection and comparison of the linguistic quantifier in theory and applications.