arXiv subjects

Peide Liu

Publications and source records attributed to Peide Liu.

9 recordsLinked to original sources

arXiv2022

Strict Intuitionistic Fuzzy Distance/Similarity Measures Based on Jensen-Shannon Divergence

Being a pair of dual concepts, the normalized distance and similarity measures are very important tools for decision-making and pattern recognition under intuitionistic fuzzy sets framework. To be more effective for decision-making and pattern recognition applications, a good normalized distance measure should ensure that its dual similarity measure satisfies the axiomatic definition. In this paper, we first construct some examples to illustrate that the dual similarity measures of two nonlinear distance measures introduced in [A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems, \emph{IEEE Trans. Syst., Man, Cybern., Syst.}, vol.~51, no.~6, pp. 3980--3992, 2021] and [Intuitionistic fuzzy sets: spherical representation and distances, \emph{Int. J. Intell. Syst.}, vol.~24, no.~4, pp. 399--420, 2009] do not meet the axiomatic definition of intuitionistic fuzzy similarity measure. We show that (1) they cannot effectively distinguish some intuitionistic fuzzy values (IFVs) with obvious size relationship; (2) except for the endpoints, there exist infinitely many pairs of IFVs, where the maximum distance 1 can be achieved under these two distances; leading to counter-intuitive results. To overcome these drawbacks, we introduce the concepts of strict intuitionistic fuzzy distance measure (SIFDisM) and strict intuitionistic fuzzy similarity measure (SIFSimM), and propose an improved intuitionistic fuzzy distance measure based on Jensen-Shannon divergence. We prove that (1) it is a SIFDisM; (2) its dual similarity measure is a SIFSimM; (3) its induced entropy is an intuitionistic fuzzy entropy. Comparative analysis and numerical examples demonstrate that our proposed distance measure is completely superior to the existing ones.

math.GM↗

arXiv2022

A Monotonous Intuitionistic Fuzzy TOPSIS Method under General Linear Orders via Admissible Distance Measures

All intuitionistic fuzzy TOPSIS methods contain two key elements: (1) the order structure, which can affect the choices of positive ideal-points and negative ideal-points, and construction of admissible distance/similarity measures; (2) the distance/similarity measure, which is closely related to the values of the relative closeness degrees and determines the accuracy and rationality of decision-making. For the order structure, many efforts are devoted to constructing some score functions, which can strictly distinguish different intuitionistic fuzzy values (IFVs) and preserve the natural partial order for IFVs.This paper proves that such a score function does not exist, namely the application of a single monotonous and continuous function does not distinguish all IFVs. For the distance or similarity measure, some examples are given to show that classical similarity measures based on the normalized Euclidean distance and normalized Minkowski distance do not meet the axiomatic definition of intuitionistic fuzzy similarity measures. Moreover,some illustrative examples are given to show that classical intuitionistic fuzzy TOPSIS methods do not ensure the monotonicity with the natural partial order or linear orders, which may yield some counter-intuitive results. To overcome the limitation of non-monotonicity, we propose a novel intuitionistic fuzzy TOPSIS method,using three new admissible distances with the linear orders measured by a score degree/similarity function and accuracy degree, or two aggregation functions, and prove that the proposed TOPSIS method is monotonous under these three linear orders.} This is the first result with a strict mathematical proof on the monotonicity with the linear orders for the intuitionistic fuzzy TOPSIS method.

math.GM↗

arXiv2021

On the algebraic structures of the space of interval-valued intuitionistic fuzzy numbers

This study is inspired by those of Huang et al. (Soft Comput. 25, 2513--2520, 2021) and Wang et al. (Inf. Sci. 179, 3026--3040, 2009) in which some ranking techniques for interval-valued intuitionistic fuzzy numbers (IVIFNs) were introduced. In this study, we prove that the space of all IVIFNs with the relation in the method for comparing any two IVIFNs based on a score function and three types of entropy functions is a complete chain and obtain that this relation is an admissible order. Moreover, we demonstrate that IVIFNs are complete chains to the relation in the comparison method for IVIFNs on the basis of score, accuracy, membership uncertainty index, and hesitation uncertainty index functions.

cs.AI↗

arXiv2021

Topological and Algebraic Structures of Atanassov's Intuitionistic Fuzzy-Values Space

We prove that the space of intuitionistic fuzzy values (IFVs) with a linear order based on a score function and an accuracy function has the same algebraic structure as the one induced by a linear order based on a similarity function and an accuracy function. By introducing a new operator for IFVs via the linear order based on a score function and an accuracy function, we show that such an operator is a strong negation on IFVs. Moreover, we observe that the space of IFVs is a complete lattice and a Kleene algebra with the new operator. We also demonstrate that the topological space of IFVs with the order topology induced by the above two linear orders is not separable and metrizable but compact and connected. From some new perspectives,our results partially answer three open problems posed by Atanassov [Intuitionistic Fuzzy Sets: Theory and Applications, Springer, 1999] and [On Intuitionistic Fuzzy Sets Theory, Springer, 2012]. Furthermore, we construct an isomorphism between the spaces of IFVs and q-rung orthopedic fuzzy values (q-ROFVs) under the corresponding linear orders. To this end, we introduce the concept of admissible similarity measures with particular orders for IFSs, extending the existing definition of the similarity measure for IFSs, and construct an admissible similarity measure with a linear order based on a score function and an accuracy function, which is effectively applied to a pattern recognition problem about the classification of building materials.

cs.AI↗

arXiv2016

Martingale Inequalities in Variable Exponent Hardy spaces with $0<p^-\leq p^+<\infty$

We investigate the properties of the variable Lebesgue spaces with quasi-norm on a probability space, and give the atomic decompositions suited to the variable exponent martingale Hardy spaces. Using the decompositions and the harmonic mean of a variable exponent, we obtain several continuous embedding relations between martingale Hardy spaces with small exponent. Finally, we extend these results to the cases $0<p^-\leq p^+<\infty.$

math.PR↗

arXiv2015

Weighted norm inequalities for multisublinear maximal operator in martingale spaces

Let $v,~ω_1, ~ω_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(ω_1, ω_2)\in RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in martingale spaces, we characterize the weights for which $\mathfrak{M}$ is bounded from $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^{p,\infty}(v)\hbox{or}L^p(v).$ If $v=ω_2^{\frac{p}{p_2}}ω_2^{\frac{p}{p_2}},$ we partially give the bilinear version of one-weight theory.

math.CA↗

arXiv2014

Burkholder-Gundy-Davis Inequality in Martingale Hardy Spaces with Variable Exponent

In this paper, the classical Dellacherie's theorem about stochastic process is extended to variable exponent Lebesgue spaces. As its applications, we obtain variable exponent analogues of several famous inequalities in classical martingale theory, including convexity lemma, Burkholder-Gundy-Davis' inequality and Chevalier's inequality. Moreover, we investigate some other equivalent relations between variable exponent martingale Hardy spaces.

math.FA↗

arXiv2014

A Characterization of BMO^α-martingale spaces by fractional Carleson measures

We give a characterization of $BMO^α$-martingale spaces by using fractional Carleson measures. We get the boudedness of martingale transform and square function on $BMO^α$-martingale spaces easily by using this characterization. We also proved the martingale version of Carleson's inequality related with $BMO^α$-martingales.

math.PR↗

arXiv2010

Noncommutative weak Orlicz spaces and martingale inequalities

This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. In particular, we prove the weak type $Φ$-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type $Φ$-moment noncommutative Khintchine's inequality for Rademacher's random variables.

math.FA↗