Luis González-de la Fuente , Alicia Nieto-Reyes , Pedro Terán
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引用次数: 0
摘要
统计深度函数是非参数统计中的一种标准工具,用于将基于阶次的单变量方法扩展到多变量环境中。由于模糊数据(即使在单变量情况下)没有普遍接受的总阶,而且缺乏参数模型,因此基于深度方法的模糊扩展非常有趣。在本文中,我们将多元深度投影深度和 Lr 型深度函数调整为模糊设置,并对 Lr 型深度提出了不同的概括。我们证明了所提出的模糊深度函数具有非常好的特性,并得出模糊投影深度是文献中第二个同时满足半线性深度和几何深度概念的例子。这意味着模糊投影深度具有非常好的性能,可以对模糊随机变量的模糊集进行排序。此外,我们还以梯形模糊集的真实数据为例,说明了所提出的模糊深度函数的良好经验行为,以及模糊深度在基于深度的分类程序中的应用。最后,由于梯形模糊集可以用 R4 的元素表示,我们还通过经验证明了模糊深度比应用于模糊集的多元投影深度更优越,从而证明了我们的建议是正确的。
Projection depth and Lr-type depths for fuzzy random variables
Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt the multivariate depths projection depth and -type depth functions to the fuzzy setting, proposing different generalizations for the -type depths. We prove that the proposed fuzzy depth functions have very good properties, obtaining that the fuzzy projection depth is the second example in the literature to satisfy simultaneously the notion of semilinear and of geometric depth. This implies that the fuzzy projection depth is extremely well behave, to order fuzzy sets with respect to fuzzy random variables. Furthermore, we illustrate the good empirical behavior of the proposed fuzzy depth functions with a real data example of trapezoidal fuzzy sets and the used of fuzzy depths in depth-based classification procedures. Finally, as trapezoidal fuzzy sets can be represented by elements of , we justify our proposals by also showing empirically the superiority of the fuzzy depths over the multivariate projection depth applied to fuzzy sets.
期刊介绍:
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.