对 "构建所需的类似距离度量的易懂方法 "的更正

IF 1.7 4区 工程技术 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Complexity Pub Date : 2024-08-30 DOI:10.1155/2024/9892058
Wen Qing Fu, Sheng Gang Li, Harish Garg, Heng Liu, Ahmed Mostafa Khalil, Jingjing Zhao
{"title":"对 \"构建所需的类似距离度量的易懂方法 \"的更正","authors":"Wen Qing Fu,&nbsp;Sheng Gang Li,&nbsp;Harish Garg,&nbsp;Heng Liu,&nbsp;Ahmed Mostafa Khalil,&nbsp;Jingjing Zhao","doi":"10.1155/2024/9892058","DOIUrl":null,"url":null,"abstract":"<p>Metrics and their weaker forms are used to measure difference between two data (or other things). There are many metrics that are available but not desired to a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain (or easy to coined by practitioners) pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to coined by practitioners et al). The simple reason to do this is that data for a real-world problem are sometimes full of multiagents. A distance-like notion, called weak interval-valued pseudometric (briefly, WIVP metric), is defined by using known notions of pseudo-semimetric, pseudometric, and metric; this notion is topological good and shows precision, flexibility, and compatibility than single pseudo-semimetric, pseudometric, or metric. Propositions and detailed examples are given to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems, and WIVP metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly.</p><p>In many cases, the measure values of true data are not unique (but two or more) for uncertainty or complexity. For example, there are several agents in China that value and order all periodicals published in China. Peking University Library and Nanjing University Library are generally thought to be the best two and incomparable to each other. For a journal <i>J</i>, assume the orders given by Peking University Library and the Nanjing University Library are <i>m</i>-th and <i>n</i>-th, respectively; then, <i>m</i> and <i>n</i> may be not the same in general. There are also many other examples. In 2012, breakthrough of the selected by the famous journal Science is different from those selected by the famous journal Nature; Gini coefficients in China in 2012 from two different agents are 0.481 and 0.61, respectively; the Chebyshev distance (resp., the Euclidean distance, the Manhattan distance or the city block distance, and the river distance) between two points (0,1) and (1,2) in the Euclidean plane <i>R</i><sup>2</sup> is 1 (resp., <span></span><math></math>, 2, 3). Please see Proposition 4 for definitions of these metrics; the effective distances used in cluster analysis are many and varied; a given asymptomatic infected people to corona virus disease (COVID-19 for short) is thought to be highly contagious (which can be represented by a fuzzy number <i>A</i>) by experts in one country but lowly contagious (which can also be represented by a fuzzy number <i>B</i> that is much different from <i>A</i>) by experts in another country.</p><p>In practice, most people choose just one of the measure values (or choose the arithmetic mean of these measure values) as the true data, such a kind of dispose can be accepted only in rare cases (e.g., the information loss cannot be avoided or make almost no difference). To make an improvement of disposal of these uncertain or complex data, at least two better theories (one is theoretically inspirational, and another is application-motivated; both are based mainly on the idea of fuzzy set) have been proposed which are mostly about measuring values of difference between two abstract “points” (precisely, two elements of a set) whose information or data can be provided by at least two different agents (but cannot be provided satisfactorily by one agent, see the following Example 1).</p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>In this section, we will define the notion of weak interval-valued pseudometric (shortly, WIVP metric) and exemplify in detail how to construct distance-like measures (including WIVP metrics) desired in practice by fusing easy-to-obtain or easy-to-coin pseudo-semimetrics, pseudometrics, or metrics based on operators ∧, ∨, and simple aggregation operators. We also characterize WIVP metric and its special forms intuitively so that practitioners can understand them easily.</p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>In this section, we will demonstrate how to construct by using some of these logic implication operators and some WIVP metrics which may be used in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]).</p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>Metric graph theory abounds in applications (e.g., it is applicable in such different areas as location theory, theoretical biology and chemistry, combinatorial optimization, and computational geometry, see [42], [p.99–121] for details). In this section, we extend the notion of metric on a graph to interval-valued metric on an interval-valued fuzzy graph (particularly, on a fuzzy graph) and give some related examples.</p><p>An <i>J</i>-graph (where <i>J</i> is a completely distributive complete lattice with the least element 0) is a triple <i>G</i> = (<i>V</i>, <i>σ</i>, <i>μ</i>) consisting of a nonempty finite set <i>V</i> and a pair of mappings <i>σ</i> : <i>V</i>⟶<i>J</i> and <i>μ</i> : <i>V</i> × <i>V</i>⟶<i>J</i> which satisfies supp. <i>σ</i> = <i>V</i> and <i>μ</i>(<i>x</i>, <i>y</i>) = <i>μ</i>(<i>y</i>, <i>x</i>) ≤ <i>σ</i>(<i>x</i>)∧<i>σ</i>(<i>y</i>) (∀(<i>x</i>, <i>y</i>) ∈ <i>V</i> × <i>V</i>). The underlying graph of <i>G</i> is defined as [<i>G</i>] = (<i>V</i>, <i>E</i>), where <i>E</i> = {{<i>x</i>, <i>y</i>}⊆<i>V</i>|<i>μ</i>(<i>x</i>, <i>y</i>) &gt; 0}. An <i>J</i>-graph <i>G</i> = (<i>V</i>, <i>σ</i>, <i>μ</i>) is said to be connected if its underlying graph [<i>G</i>] = (<i>V</i>, <i>E</i>) is connected, i.e., for any 2-element subset {<i>x</i>, <i>y</i>}⊆<i>V</i>, there exists an <i>m</i>(<i>x</i>, <i>y</i>)-element subset {<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>, …, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub>}⊆<i>V</i> (<i>m</i><sub><i>x</i><i>y</i></sub> ≥ 2) such that <i>x</i> = <i>z</i><sub>1</sub>, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub> = <i>y</i>, and {<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>}, {<i>z</i><sub>2</sub>, <i>z</i><sub>3</sub>}, ⋯, {<i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)−1</sub>, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub>} are all in <i>E</i>; the word <i>P</i> = <i>z</i><sub>1</sub><i>z</i><sub>2</sub> ⋯ <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub> is called a path from <i>x</i> to <i>y</i>, and the set of all paths from <i>x</i> to <i>y</i> is denoted by <span></span><math></math>.</p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>\n \n </p><p>The main results of this section are as follows:</p><p>\n \n </p><p>\n \n </p><p>Theorem 18 may be proved based on Lemma 17 and results on contractive type mappings satisfying (1), (4), (5), (7), (11), (12), (14), (18), (43) in [44], and Theorem 19 may be proved based on Lemma 17 and results on contractive type mappings satisfying (176), (179), (180), (182), (186), (187), (193) in [44].</p><p>Since data from many real-world problems are not only from multiagents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric do not meet the needs of some practical problems. Motivated by Polya’s plausible reasoning and artificial neural networks, this paper consider a distance-like notion, called weak interval-valued pseudometric (WIVP metric for short), which, as a generalization of the notion of metric, is still topological good. To benefit practitioners, easy-to-understand propositions and much detailed examples are given (in the first half of the paper) to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems. To show theoretical applications of WIVP metrics, we exemplify how to construct (by using some logic implication operators, some WIVP metrics which may be useful in quantitative logic [23] and quantitative reasoning [24]) and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. As these WIVP metrics are relatively precision, flexibility and compatibility than single pseudo-semimetric, pseudometric, and metric, more applications should be investigated (even put forward) based on plausible reasoning. Practitioners are also suggested to explore (in the plausible reasoning manner) other complex and more fitted methods to fabricate more desired distance-like measures. For examples, to fuse easy-to-obtain pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known or frequently used t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners or others; strategies also containing making full use of interval numbers and very special triangular fuzzy numbers.</p><p>Our future work includes completion of WIVP-metric spaces, interval-valued truth degrees of formulas based on deferent logic implication operators, interval-valued similarity degrees of formulas based on deferent logic implication operators, related approximate reasoning, dynamic systems on interval-valued metric spaces (even on interval-valued pseudometric spaces), and applications of weak interval-valued pseudometrics in medical diagnosis and decision-making problems (see related works [45, 46] for details).</p><p>The authors declare that there are no conflicts of interest regarding the publication of this paper.</p>","PeriodicalId":50653,"journal":{"name":"Complexity","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/2024/9892058","citationCount":"0","resultStr":"{\"title\":\"Corrigendum to “An Easy-to-Understand Method to Construct Desired Distance-Like Measures”\",\"authors\":\"Wen Qing Fu,&nbsp;Sheng Gang Li,&nbsp;Harish Garg,&nbsp;Heng Liu,&nbsp;Ahmed Mostafa Khalil,&nbsp;Jingjing Zhao\",\"doi\":\"10.1155/2024/9892058\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Metrics and their weaker forms are used to measure difference between two data (or other things). There are many metrics that are available but not desired to a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain (or easy to coined by practitioners) pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to coined by practitioners et al). The simple reason to do this is that data for a real-world problem are sometimes full of multiagents. A distance-like notion, called weak interval-valued pseudometric (briefly, WIVP metric), is defined by using known notions of pseudo-semimetric, pseudometric, and metric; this notion is topological good and shows precision, flexibility, and compatibility than single pseudo-semimetric, pseudometric, or metric. Propositions and detailed examples are given to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems, and WIVP metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly.</p><p>In many cases, the measure values of true data are not unique (but two or more) for uncertainty or complexity. For example, there are several agents in China that value and order all periodicals published in China. Peking University Library and Nanjing University Library are generally thought to be the best two and incomparable to each other. For a journal <i>J</i>, assume the orders given by Peking University Library and the Nanjing University Library are <i>m</i>-th and <i>n</i>-th, respectively; then, <i>m</i> and <i>n</i> may be not the same in general. There are also many other examples. In 2012, breakthrough of the selected by the famous journal Science is different from those selected by the famous journal Nature; Gini coefficients in China in 2012 from two different agents are 0.481 and 0.61, respectively; the Chebyshev distance (resp., the Euclidean distance, the Manhattan distance or the city block distance, and the river distance) between two points (0,1) and (1,2) in the Euclidean plane <i>R</i><sup>2</sup> is 1 (resp., <span></span><math></math>, 2, 3). Please see Proposition 4 for definitions of these metrics; the effective distances used in cluster analysis are many and varied; a given asymptomatic infected people to corona virus disease (COVID-19 for short) is thought to be highly contagious (which can be represented by a fuzzy number <i>A</i>) by experts in one country but lowly contagious (which can also be represented by a fuzzy number <i>B</i> that is much different from <i>A</i>) by experts in another country.</p><p>In practice, most people choose just one of the measure values (or choose the arithmetic mean of these measure values) as the true data, such a kind of dispose can be accepted only in rare cases (e.g., the information loss cannot be avoided or make almost no difference). To make an improvement of disposal of these uncertain or complex data, at least two better theories (one is theoretically inspirational, and another is application-motivated; both are based mainly on the idea of fuzzy set) have been proposed which are mostly about measuring values of difference between two abstract “points” (precisely, two elements of a set) whose information or data can be provided by at least two different agents (but cannot be provided satisfactorily by one agent, see the following Example 1).</p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>In this section, we will define the notion of weak interval-valued pseudometric (shortly, WIVP metric) and exemplify in detail how to construct distance-like measures (including WIVP metrics) desired in practice by fusing easy-to-obtain or easy-to-coin pseudo-semimetrics, pseudometrics, or metrics based on operators ∧, ∨, and simple aggregation operators. We also characterize WIVP metric and its special forms intuitively so that practitioners can understand them easily.</p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>In this section, we will demonstrate how to construct by using some of these logic implication operators and some WIVP metrics which may be used in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]).</p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>Metric graph theory abounds in applications (e.g., it is applicable in such different areas as location theory, theoretical biology and chemistry, combinatorial optimization, and computational geometry, see [42], [p.99–121] for details). In this section, we extend the notion of metric on a graph to interval-valued metric on an interval-valued fuzzy graph (particularly, on a fuzzy graph) and give some related examples.</p><p>An <i>J</i>-graph (where <i>J</i> is a completely distributive complete lattice with the least element 0) is a triple <i>G</i> = (<i>V</i>, <i>σ</i>, <i>μ</i>) consisting of a nonempty finite set <i>V</i> and a pair of mappings <i>σ</i> : <i>V</i>⟶<i>J</i> and <i>μ</i> : <i>V</i> × <i>V</i>⟶<i>J</i> which satisfies supp. <i>σ</i> = <i>V</i> and <i>μ</i>(<i>x</i>, <i>y</i>) = <i>μ</i>(<i>y</i>, <i>x</i>) ≤ <i>σ</i>(<i>x</i>)∧<i>σ</i>(<i>y</i>) (∀(<i>x</i>, <i>y</i>) ∈ <i>V</i> × <i>V</i>). The underlying graph of <i>G</i> is defined as [<i>G</i>] = (<i>V</i>, <i>E</i>), where <i>E</i> = {{<i>x</i>, <i>y</i>}⊆<i>V</i>|<i>μ</i>(<i>x</i>, <i>y</i>) &gt; 0}. An <i>J</i>-graph <i>G</i> = (<i>V</i>, <i>σ</i>, <i>μ</i>) is said to be connected if its underlying graph [<i>G</i>] = (<i>V</i>, <i>E</i>) is connected, i.e., for any 2-element subset {<i>x</i>, <i>y</i>}⊆<i>V</i>, there exists an <i>m</i>(<i>x</i>, <i>y</i>)-element subset {<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>, …, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub>}⊆<i>V</i> (<i>m</i><sub><i>x</i><i>y</i></sub> ≥ 2) such that <i>x</i> = <i>z</i><sub>1</sub>, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub> = <i>y</i>, and {<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>}, {<i>z</i><sub>2</sub>, <i>z</i><sub>3</sub>}, ⋯, {<i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)−1</sub>, <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub>} are all in <i>E</i>; the word <i>P</i> = <i>z</i><sub>1</sub><i>z</i><sub>2</sub> ⋯ <i>z</i><sub><i>m</i>(<i>x</i>, <i>y</i>)</sub> is called a path from <i>x</i> to <i>y</i>, and the set of all paths from <i>x</i> to <i>y</i> is denoted by <span></span><math></math>.</p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>\\n \\n </p><p>The main results of this section are as follows:</p><p>\\n \\n </p><p>\\n \\n </p><p>Theorem 18 may be proved based on Lemma 17 and results on contractive type mappings satisfying (1), (4), (5), (7), (11), (12), (14), (18), (43) in [44], and Theorem 19 may be proved based on Lemma 17 and results on contractive type mappings satisfying (176), (179), (180), (182), (186), (187), (193) in [44].</p><p>Since data from many real-world problems are not only from multiagents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric do not meet the needs of some practical problems. Motivated by Polya’s plausible reasoning and artificial neural networks, this paper consider a distance-like notion, called weak interval-valued pseudometric (WIVP metric for short), which, as a generalization of the notion of metric, is still topological good. To benefit practitioners, easy-to-understand propositions and much detailed examples are given (in the first half of the paper) to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems. To show theoretical applications of WIVP metrics, we exemplify how to construct (by using some logic implication operators, some WIVP metrics which may be useful in quantitative logic [23] and quantitative reasoning [24]) and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. As these WIVP metrics are relatively precision, flexibility and compatibility than single pseudo-semimetric, pseudometric, and metric, more applications should be investigated (even put forward) based on plausible reasoning. Practitioners are also suggested to explore (in the plausible reasoning manner) other complex and more fitted methods to fabricate more desired distance-like measures. For examples, to fuse easy-to-obtain pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known or frequently used t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners or others; strategies also containing making full use of interval numbers and very special triangular fuzzy numbers.</p><p>Our future work includes completion of WIVP-metric spaces, interval-valued truth degrees of formulas based on deferent logic implication operators, interval-valued similarity degrees of formulas based on deferent logic implication operators, related approximate reasoning, dynamic systems on interval-valued metric spaces (even on interval-valued pseudometric spaces), and applications of weak interval-valued pseudometrics in medical diagnosis and decision-making problems (see related works [45, 46] for details).</p><p>The authors declare that there are no conflicts of interest regarding the publication of this paper.</p>\",\"PeriodicalId\":50653,\"journal\":{\"name\":\"Complexity\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1155/2024/9892058\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Complexity\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1155/2024/9892058\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complexity","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1155/2024/9892058","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0

摘要

度量及其弱化形式用于衡量两个数据(或其他事物)之间的差异。有许多度量是可用的,但却不是从业者想要的。本文以一种似是而非的推理方式,推荐了一种易于理解的方法来构建所需的类距离度量:通过充分利用众所周知的 t-norms、t-conforms、聚合运算符和类似运算符(从业者等容易创造),融合易于获得(或从业者容易创造)的伪计量学、伪计量或度量。这样做的原因很简单,因为实际问题的数据有时充满了多重代理。一种类似距离的概念,称为弱区间值假计量(简称 WIVP 公制),是通过使用已知的假参量、假计量和公制概念定义的;这种概念是拓扑学上的好概念,比单一的假参量、假计量或公制概念更精确、灵活和兼容。命题和详细的例子说明了如何在实际问题中制造(包括使用什么 "材料")预期的或要求的 WIVP 度量(甚至是区间值度量),并用公理描述了 WIVP 度量及其特例。此外,还构建了一些与定量逻辑理论或区间值模糊图相关的 WIVP 度量,并介绍了弱区间值度量空间中的定点定理和常见定点定理。在许多情况下,真实数据的度量值并不是唯一的(而是两个或两个以上),这就是不确定性或复杂性。例如,中国有几家代理机构对中国出版的所有期刊进行估值和订购。一般认为北京大学图书馆和南京大学图书馆是最好的两家,相互之间没有可比性。对于期刊 J 来说,假设北京大学图书馆和南京大学图书馆给出的排序分别是第 m 次和第 n 次,那么一般情况下,m 和 n 可能不一样。这样的例子还有很多。2012 年,著名期刊《科学》和著名期刊《自然》所选文章的突破性不同;2012 年中国两个不同代理机构的基尼系数分别为 0.481 和 0.61;欧几里得平面 R2 中两点(0,1)和(1,2)之间的切比雪夫距离(即欧几里得距离、曼哈顿距离或城市街区距离和河流距离)为 1(即 ,2,3)。关于这些度量的定义,请参见命题 4;聚类分析中使用的有效距离多种多样;一个国家的专家认为某个无症状的电晕病毒感染者(简称 COVID-19)传染性很强(可用模糊数 A 表示),而另一个国家的专家则认为传染性很弱(也可用与 A 相差很大的模糊数 B 表示)。在实践中,大多数人只选择其中一个测量值(或选择这些测量值的算术平均值)作为真实数据、信息损失无法避免或几乎没有区别)。为了改进对这些不确定或复杂数据的处理,至少有两种更好的理论被提出来(一种是理论上的启发,另一种是应用上的启发;两者都主要基于模糊集的思想),它们主要是关于测量两个抽象 "点"(确切地说,一个集合的两个元素)之间的差值,其信息或数据至少可以由两个不同的代理提供(但不能由一个代理令人满意地提供,见下面的例 1)。 在本节中,我们将定义弱区间值伪计量(简称 WIVP 度量)的概念,并详细举例说明如何通过融合易得或易币的伪对称、伪计量或基于算子 ∧、∨ 和简单聚合算子的度量来构造实际中需要的类距离度量(包括 WIVP 度量)。我们还将直观地描述 WIVP 度量及其特殊形式的特征,以便实践者轻松理解。 在本节中,我们将演示如何利用其中一些逻辑蕴涵算子和一些 WIVP 度量来构造可用于定量逻辑(参见 [23])和定量推理(参见 [24])的 WIVP 度量。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Corrigendum to “An Easy-to-Understand Method to Construct Desired Distance-Like Measures”

Metrics and their weaker forms are used to measure difference between two data (or other things). There are many metrics that are available but not desired to a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain (or easy to coined by practitioners) pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to coined by practitioners et al). The simple reason to do this is that data for a real-world problem are sometimes full of multiagents. A distance-like notion, called weak interval-valued pseudometric (briefly, WIVP metric), is defined by using known notions of pseudo-semimetric, pseudometric, and metric; this notion is topological good and shows precision, flexibility, and compatibility than single pseudo-semimetric, pseudometric, or metric. Propositions and detailed examples are given to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems, and WIVP metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly.

In many cases, the measure values of true data are not unique (but two or more) for uncertainty or complexity. For example, there are several agents in China that value and order all periodicals published in China. Peking University Library and Nanjing University Library are generally thought to be the best two and incomparable to each other. For a journal J, assume the orders given by Peking University Library and the Nanjing University Library are m-th and n-th, respectively; then, m and n may be not the same in general. There are also many other examples. In 2012, breakthrough of the selected by the famous journal Science is different from those selected by the famous journal Nature; Gini coefficients in China in 2012 from two different agents are 0.481 and 0.61, respectively; the Chebyshev distance (resp., the Euclidean distance, the Manhattan distance or the city block distance, and the river distance) between two points (0,1) and (1,2) in the Euclidean plane R2 is 1 (resp., , 2, 3). Please see Proposition 4 for definitions of these metrics; the effective distances used in cluster analysis are many and varied; a given asymptomatic infected people to corona virus disease (COVID-19 for short) is thought to be highly contagious (which can be represented by a fuzzy number A) by experts in one country but lowly contagious (which can also be represented by a fuzzy number B that is much different from A) by experts in another country.

In practice, most people choose just one of the measure values (or choose the arithmetic mean of these measure values) as the true data, such a kind of dispose can be accepted only in rare cases (e.g., the information loss cannot be avoided or make almost no difference). To make an improvement of disposal of these uncertain or complex data, at least two better theories (one is theoretically inspirational, and another is application-motivated; both are based mainly on the idea of fuzzy set) have been proposed which are mostly about measuring values of difference between two abstract “points” (precisely, two elements of a set) whose information or data can be provided by at least two different agents (but cannot be provided satisfactorily by one agent, see the following Example 1).

In this section, we will define the notion of weak interval-valued pseudometric (shortly, WIVP metric) and exemplify in detail how to construct distance-like measures (including WIVP metrics) desired in practice by fusing easy-to-obtain or easy-to-coin pseudo-semimetrics, pseudometrics, or metrics based on operators ∧, ∨, and simple aggregation operators. We also characterize WIVP metric and its special forms intuitively so that practitioners can understand them easily.

In this section, we will demonstrate how to construct by using some of these logic implication operators and some WIVP metrics which may be used in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]).

Metric graph theory abounds in applications (e.g., it is applicable in such different areas as location theory, theoretical biology and chemistry, combinatorial optimization, and computational geometry, see [42], [p.99–121] for details). In this section, we extend the notion of metric on a graph to interval-valued metric on an interval-valued fuzzy graph (particularly, on a fuzzy graph) and give some related examples.

An J-graph (where J is a completely distributive complete lattice with the least element 0) is a triple G = (V, σ, μ) consisting of a nonempty finite set V and a pair of mappings σ : VJ and μ : V × VJ which satisfies supp. σ = V and μ(x, y) = μ(y, x) ≤ σ(x)∧σ(y) (∀(x, y) ∈ V × V). The underlying graph of G is defined as [G] = (V, E), where E = {{x, y}⊆V|μ(x, y) > 0}. An J-graph G = (V, σ, μ) is said to be connected if its underlying graph [G] = (V, E) is connected, i.e., for any 2-element subset {x, y}⊆V, there exists an m(x, y)-element subset {z1, z2, …, zm(x, y)}⊆V (mxy ≥ 2) such that x = z1, zm(x, y) = y, and {z1, z2}, {z2, z3}, ⋯, {zm(x, y)−1, zm(x, y)} are all in E; the word P = z1z2zm(x, y) is called a path from x to y, and the set of all paths from x to y is denoted by .

The main results of this section are as follows:

Theorem 18 may be proved based on Lemma 17 and results on contractive type mappings satisfying (1), (4), (5), (7), (11), (12), (14), (18), (43) in [44], and Theorem 19 may be proved based on Lemma 17 and results on contractive type mappings satisfying (176), (179), (180), (182), (186), (187), (193) in [44].

Since data from many real-world problems are not only from multiagents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric do not meet the needs of some practical problems. Motivated by Polya’s plausible reasoning and artificial neural networks, this paper consider a distance-like notion, called weak interval-valued pseudometric (WIVP metric for short), which, as a generalization of the notion of metric, is still topological good. To benefit practitioners, easy-to-understand propositions and much detailed examples are given (in the first half of the paper) to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems. To show theoretical applications of WIVP metrics, we exemplify how to construct (by using some logic implication operators, some WIVP metrics which may be useful in quantitative logic [23] and quantitative reasoning [24]) and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. As these WIVP metrics are relatively precision, flexibility and compatibility than single pseudo-semimetric, pseudometric, and metric, more applications should be investigated (even put forward) based on plausible reasoning. Practitioners are also suggested to explore (in the plausible reasoning manner) other complex and more fitted methods to fabricate more desired distance-like measures. For examples, to fuse easy-to-obtain pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known or frequently used t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners or others; strategies also containing making full use of interval numbers and very special triangular fuzzy numbers.

Our future work includes completion of WIVP-metric spaces, interval-valued truth degrees of formulas based on deferent logic implication operators, interval-valued similarity degrees of formulas based on deferent logic implication operators, related approximate reasoning, dynamic systems on interval-valued metric spaces (even on interval-valued pseudometric spaces), and applications of weak interval-valued pseudometrics in medical diagnosis and decision-making problems (see related works [45, 46] for details).

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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来源期刊
Complexity
Complexity 综合性期刊-数学跨学科应用
CiteScore
5.80
自引率
4.30%
发文量
595
审稿时长
>12 weeks
期刊介绍: Complexity is a cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with such methodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especially encouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries. Complexity is not meant to serve as a forum for speculation and vague analogies between words like “chaos,” “self-organization,” and “emergence” that are often used in completely different ways in science and in daily life.
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