二元下半线性联与星积

IF 3 3区 计算机科学 Q2 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE
Lea Maislinger, Wolfgang Trutschnig
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In particular we prove that the star product (also known as Markov product) <span><math><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>⁎</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> of two LSL copulas <span><math><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> is again an LSL copula, i.e., that the family <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>L</mi><mi>S</mi><mi>L</mi></mrow></msup></math></span> is closed with respect to the star product. 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引用次数: 0

摘要

我们重新审视了Durante等人在2008年首次引入的所有双变量下半线性(LSL)联结的CLSL族,并利用具有特定性质的对角线对LSL联结进行表征,得出了一些新颖且部分出乎意料的结果。特别地,我们证明了两个LSL共轭子Sδ1,Sδ2的星积(也称为马尔可夫积)Sδ1 Sδ2也是一个LSL共轭子,即族CLSL相对于星积是闭合的。此外,我们证明了将星积平移到对应对角线DLSL的类可以确定序列Sδ,Sδ Sδ,Sδ Sδ,…对于每一个对角线δ∈DLSL的极限。事实上,对于每一个LSL copula Sδ,序列(Sδ n)n∈n收敛于某个LSL copula Sδ,极限Sδ是幂等的,并且所有幂等LSL copula的类允许一个简单的描述。补充这些结果,然后我们将重点放在LSL联结的一致性上。在回顾Kendall τ和Spearman ρ的简单公式后,我们研究了CLSL中所有元素的这两个一致性度量所确定的精确区域ΩLSL,得出了一个明显的下界,并最终证明ΩLSL是凸紧的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On bivariate lower semilinear copulas and the star product
We revisit the family CLSL of all bivariate lower semilinear (LSL) copulas first introduced by Durante et al. in 2008 and, using the characterization of LSL copulas in terms of diagonals with specific properties, derive several novel and partially unexpected results. In particular we prove that the star product (also known as Markov product) Sδ1Sδ2 of two LSL copulas Sδ1,Sδ2 is again an LSL copula, i.e., that the family CLSL is closed with respect to the star product. Moreover, we show that translating the star product to the class of corresponding diagonals DLSL allows to determine the limit of the sequence Sδ,SδSδ,SδSδSδ, for every diagonal δDLSL. In fact, for every LSL copula Sδ the sequence (Sδn)nN converges to some LSL copula Sδ, the limit Sδ is idempotent, and the class of all idempotent LSL copulas allows for a simple characterization.
Complementing these results we then focus on concordance of LSL copulas. After recalling simple formulas for Kendall's τ and Spearman's ρ we study the exact region ΩLSL determined by these two concordance measures of all elements in CLSL, derive a sharp lower bound and finally show that ΩLSL is convex and compact.
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来源期刊
International Journal of Approximate Reasoning
International Journal of Approximate Reasoning 工程技术-计算机:人工智能
CiteScore
6.90
自引率
12.80%
发文量
170
审稿时长
67 days
期刊介绍: The International Journal of Approximate Reasoning is intended to serve as a forum for the treatment of imprecision and uncertainty in Artificial and Computational Intelligence, covering both the foundations of uncertainty theories, and the design of intelligent systems for scientific and engineering applications. It publishes high-quality research papers describing theoretical developments or innovative applications, as well as review articles on topics of general interest. Relevant topics include, but are not limited to, probabilistic reasoning and Bayesian networks, imprecise probabilities, random sets, belief functions (Dempster-Shafer theory), possibility theory, fuzzy sets, rough sets, decision theory, non-additive measures and integrals, qualitative reasoning about uncertainty, comparative probability orderings, game-theoretic probability, default reasoning, nonstandard logics, argumentation systems, inconsistency tolerant reasoning, elicitation techniques, philosophical foundations and psychological models of uncertain reasoning. Domains of application for uncertain reasoning systems include risk analysis and assessment, information retrieval and database design, information fusion, machine learning, data and web mining, computer vision, image and signal processing, intelligent data analysis, statistics, multi-agent systems, etc.
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