Fuzzy Statistical Limits

Statistical limits are defined relaxing conditions on conventional convergence. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with other elements. At the same time, it is known that sequences that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they converge or statistically converge in the strict mathematical sense. To overcome these limitations, fuzzy convergence was introduced earlier in the context of neoclassical analysis and fuzzy statistical convergence is introduced and studied in this paper. We find relations between fuzzy statistical convergence of a sequence and fuzzy statistical convergence of its subsequences (Theorem 2.1), as well as between fuzzy statistical convergence of a sequence and conventional convergence of its subsequences (Theorem 2.2). It is demonstrated what operations with fuzzy statistical limits are induced by operations on sequences (Theorem 2.3) and how fuzzy statistical limits of different sequences influence one another (Theorem 2.4). In Section 3, relations between fuzzy statistical convergence and fuzzy convergence of statistical characteristics, such as the mean (average) and standard deviation, are studied (Theorems 3.1 and 3.2)

Characterizations of endograph metric and Γ\Gamma-convergence on fuzzy sets

This paper is devoted to the relationships and properties of the endograph metric and the $\Gamma$-convergence. The main contents can be divided into three closely related parts. Firstly, on the class of upper semi-continuous fuzzy sets with bounded $\alpha$-cuts, we find that an endograph metric convergent sequence is exactly a $\Gamma$-convergent sequence satisfying the condition that the union of $\alpha$-cuts of all its elements is a bounded set in $\mathbb{R}^m$ for each $\alpha > 0$. Secondly, based on investigations of level characterizations of fuzzy sets themselves, we present level characterizations (level decomposition properties) of the endograph metric and the $\Gamma$-convergence. It is worth mentioning that, using the condition and the level characterizations given above, we discover the fact: the endograph metric and the $\Gamma$-convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality, convexity or star-shapedness. Its subsets include common particular fuzzy sets such as fuzzy numbers (compact and noncompact), fuzzy star-shaped numbers (compact and noncompact), and general fuzzy star-shaped numbers (compact and noncompact). Thirdly, on the basis of the conclusions presented above, we study various subspaces of the space of upper semi-continuous fuzzy sets with bounded $\alpha$-cuts equipped with the endograph metric. We present characterizations of total boundedness, relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces. It is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric.Comment: This revised paper has been submitted to Fuzzy Sets and Systems at 2018/1/1

The FA Quantifier Fuzzification Mechanism: analysis of convergence and efficient implementations

The fuzzy quantification model FA has been identified as one of the best behaved quantification models in several revisions of the field of fuzzy quantification. This model is, to our knowledge, the unique one fulfilling the strict Determiner Fuzzification Scheme axiomatic framework that does not induce the standard min and max operators. The main contribution of this paper is the proof of a convergence result that links this quantification model with the Zadeh's model when the size of the input sets tends to infinite. The convergence proof is, in any case, more general than the convergence to the Zadeh's model, being applicable to any quantitative quantifier. In addition, recent revisions papers have presented some doubts about the existence of suitable computational implementations to evaluate the FA model in practical applications. In order to prove that this model is not only a theoretical approach, we show exact algorithmic solutions for the most common linguistic quantifiers as well as an approximate implementation by means of Monte Carlo. Additionally, we will also give a general overview of the main properties fulfilled by the FA model, as a single compendium integrating the whole set of properties fulfilled by it has not been previously published.Comment: 22 page

Characterizations of compact sets in fuzzy sets spaces with LpL_p metric

In this paper, we present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ equipped with $L_p$ metric, where $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ are two kinds of general fuzzy sets on $\mathbb{R}^m$ which do not have any assumptions of convexity or star-shapedness. Subsets of $F_B(\mathbb{R}^m)^p$ include common fuzzy sets such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and the general fuzzy star-shaped numbers introduced by Qiu et al. The existed compactness criteria are stated for three kinds of fuzzy sets spaces endowed with $L_p$ metric whose universe sets are the former three kinds of common fuzzy sets respectively. Constructing completions of fuzzy sets spaces with respect to $L_p$ metric is a problem which is closely dependent on characterizing totally bounded sets. Based on preceding characterizations of totally boundedness and relatively compactness and some discussions on convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy sets spaces mentioned in this paper can be obtained by using the $L_p$-extension. We also clarify relation among all the ten fuzzy sets spaces discussed in this paper, which consist of five pairs of original spaces and the corresponding completions. Then, we show that the subspaces of $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ mentioned in this paper have parallel characterizations of totally bounded sets, relatively compact sets and compact sets. At last, as applications of our results, we discuss properties of $L_p$ metric on fuzzy sets space and relook compactness criteria proposed in previous work.Comment: This paper is submitted to Fuzzy Sets and Systems at 29/08/201

On the convergence of the sparse possibilistic c-means algorithm

In this paper, a convergence proof for the recently proposed sparse possibilistic c-means (SPCM) algorithm is provided, utilizing the celebrated Zangwill convergence theorem. It is shown that the iterative sequence generated by SPCM converges to a stationary point or there exists a subsequence of it that converges to a stationary point of the cost function of the algorithm

A Convergence Theorem for the Graph Shift-type Algorithms

Graph Shift (GS) algorithms are recently focused as a promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the GS Algorithm. In this paper, we propose a generic theoretical framework consisting of three key GS components: simplex of generated sequence set, monotonic and continuous objective function and closed mapping. We prove that GS algorithms with such components can be transformed to fit the Zangwill's convergence theorem, and the sequence set generated by the GS procedures always terminates at a local maximum, or at worst, contains a subsequence which converges to a local maximum of the similarity measure function. The framework is verified by expanding it to other GS-type algorithms and experimental results

Representation of Uncertainty for Limit Processes

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve only approximate results. In some cases, this discrepancy between theoretical schemes and practical actions changes drastically outcomes of a research and decision-making resulting in uncertainty of knowledge. In the paper, a mathematical approach to such kind of uncertainty, which emerges in computation and measurement, is suggested on the base of the concept of a fuzzy limit. A mathematical technique is developed for differential models with uncertainty. To take into account the intrinsic uncertainty of a model, it is suggested to use fuzzy derivatives instead of conventional derivatives of functions in this model

On the Laws of Large Numbers in Possibility Theory

In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are based on the possibility measure and on the "maxitive" definitions for possibility expectation and possibility variance. Also, we show that in this frame, the weak form of the law of large numbers, implies the strong law of large numbers.Comment: 10 page

Some properties of Skorokhod metric on fuzzy sets

In this paper, we have our discussions on normal and upper semi-continuous fuzzy sets on metric spaces. The Skorokhod-type metric is stronger than the Skorokhod metric. It is found that the Skorokhod metric and the Skorokhod-type metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the Skorokhod-type metric need not be equivalent on $L_p$-integrable fuzzy sets. Based on this, we investigate relations between these two metrics and the $L_p$-type $d_p$ metric. It is found that the relations can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the $d_p$ metric. On $L_p$-integrable fuzzy sets, which take compact fuzzy sets as special cases, the Skorokhod metric is not necessarily stronger than the $d_p$ metric, but the Skorokhod-type metric is still stronger than the $d_p$ metric. On general fuzzy sets, even the Skorokhod-type metric is not necessarily stronger than the $d_p$ metric. We also show that the Skorokhod metric is stronger than the sendograph metric

Some characterizations on weighted αβ\alpha\beta-statistical convergence of fuzzy functions of order θ\theta

Based on the concept of new type of statistical convergence defined by Aktuglu, we have introduced the weighted $\alpha\beta$ - statistical convergence of order $\theta$ in case of fuzzy functions and classified it into pointwise, uniform and equi-statistical convergence. We have checked some basic properties and then the convergence are investigated in terms of their $\alpha$-cuts. The interrelation among them are also established. We have also proved that continuity, boundedness etc are preserved in the equi-statistical sense under some suitable conditions, but not in pointwise sense.Comment: Communicate