Encyclopedia of Distances

The achievement of a 'consensual' solution in a group decision making problem depends on experts' ideas, principles, knowledge, experience, etc. The measurement of consensus has been widely studied from the point of view of different research areas, and consequently different consensus measures have been formulated, although a common characteristic of most of them is that they are driven by the implementation of either distance or similarity functions. In the present work though, and within the framework of experts' opinions modelled via reciprocal preference relations, a different approach to the measurement of consensus based on the Pearson correlation coefficient is studied. The new correlation consensus degree measures the concordance between the intensities of preference for pairs of alternatives as expressed by the experts. Although a detailed study of the formal properties of the new correlation consensus degree shows that it verifies important properties that are common either to distance or to similarity functions between intensities of preferences, it is also proved that it is different to traditional consensus measures. In order to emphasise novelty, two applications of the proposed methodology are also included. The first one is used to illustrate the computation process and discussion of the results, while the second one covers a real life application that makes use of data from Clinical Decision-Making.

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“…it does not verify d(x, x) = 0 [22]. Indeed, reflexivity property implies that CCD(P, P ) = 1 rather than CCD(P, P ) = 0.…”

Section: Reversibilitymentioning

confidence: 94%

“…Indeed, reflexivity property implies that CCD(P, P ) = 1 rather than CCD(P, P ) = 0. Secondly, a requirement for a similarity function [16,22] is that the similarity between two objects takes value 1 if and only if the two objects are equal, i.e. s(x, y) = 1 iff x = y.…”

Section: Reversibilitymentioning

confidence: 99%

A new measure of consensus with reciprocal preference relations: The correlation consensus degree

González-Arteaga

Calle

Chiclana

2016

Knowledge-Based Systems

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“…Let ℝ + be the set of all positive real numbers and ℝ * be the set of extended real numbers (Definition 2.1 page 4 [Blumenthal(1953)] pages 12-13, [Laos(1998)] pages 118-119 36 This is the method of "inscribed polygons" for calculating the length of a curve and goes back to Archimedes: [Brunschwig et al(2003) [Sherstnev(1962)], page 4, [Schweizer and Sklar(1983)] page 9 ⟨(1.6.1)-(1.6.4)⟩, [Bessenyei and Pales(2014) [Kirk and Shahzad(2014)] page 113 ⟨Definition 12.1⟩, [Deza and Deza(2014)] page 7, [Hoehn and Niven(1985)] page 151, [Gibbons et al(1977)Gibbons, Olkin, and Sobel] page 51 ⟨square-meanroot (SMR) (2.4.1)⟩, [Euclid(circa 300BC)] ⟨triangle inequality-Book I Proposition 20⟩ 39 metric space: [Dieudonné(1969)], page 28, [Copson(1968)], page 21, [Hausdorff(1937)] page 109, [Fréchet(1928)], [Fréchet(1906)] page 30 near metric space: [Czerwik(1993)] page 5 ⟨b-metric; (1),(2),(5)⟩, [Fagin et al(2003a)Fagin, Kumar, and Sivakumar], [Fagin et al(2003b) …”

Section: Examplesmentioning

confidence: 99%

“…[ Corazza(1999)] ⟨Proposition 2.3⟩, [Deza and Deza(2009) 3, 4, 6, 8, 9, 12-14, 20, 27, 27 boundary, 27 bounded, 2, 6, 9, 10, 21, 22, 36 bounded metric, 36…”

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confidence: 99%

Properties of distance spaces with power triangle inequalities

Greenhoe

2016

Preprint

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Metric spaces provide a framework for analysis and have several very useful properties.Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. Properties of distance spaces with power triangle inequalities DANIEL J. GREENHOEAbstract: Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.2010 Mathematics Subject Classification 54E25 (primary); 54A05,54A20 (secondary)

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“…A distance space is called semimetric provided…"⟩, Galvin and Shore (1984) page 67 ⟨"distance function"⟩, Laos (1998) page 118 ⟨"distance space"⟩, Khamsi and Kirk (2001) page 13 ⟨"semimetric space"⟩, Bessenyei and Pales (2014) page 2 ⟨"semimetric space"⟩, Deza and Deza (2014) The Lagrange arc distance is not a metric because in general the triangle inequality property does not hold (next theorem). Furthermore, the Lagrange arc distance does not induce a norm because it is not translation invariant (the translation invariant property is a necessary condition for a metric to induce a norm) and balls in a Lagrange arc distance space are in general not convex (balls are always convex in a normed linear space).…”

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confidence: 99%

Order and Metric Compatible Symbolic Sequence Processing

Greenhoe

2016

Preprint

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A traditional random variable X is a function that maps from a stochastic process to the real line (X,<=,d,+,.), where R is the set of real numbers, <= is the standard linear order relation on R, d(x,y)=|x-y| is the usual metric on R, and (R, +, .) is the standard field on R. has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that X maps from has structure that is dissimilar to that of the real line. Greenhoe(2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure X maps from and the structure X maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable.for example the expectation EX of X can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering), in making diagnostic measurements (using a post-transform metric space), or in making goptimal h decisions (based on gdistance h measurements in a metric space or more generally a distance space). Rather than mapping to a weighted graph, this paper proposes instead mapping to an ordered distance linear space Y=(R^n,<=,d,+,.,R,+,x), where (R,+,x) is a field, + is the vector addition operator on R^n Order and Metric Compatible Symbolic Sequence Processing DANIEL J. GREENHOEAbstract: A traditional random variable is a function that maps from a stochastic process to the "real line" (ℝ, ≤, , +, ⋅), where ℝ is the set of real numbers, ≤ is the standard linear order relation on ℝ, ( , ) ≜ | − | is the usual metric on ℝ, and (ℝ, +, ⋅) is the standard field on ℝ.Greenhoe (2015b) has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that maps from has structure that is dissimilar to that of the real line. Greenhoe (2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure maps from and the structure maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable-for example the expectation of can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis),...

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Encyclopedia of Distances

Cited by 298 publications

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A new measure of consensus with reciprocal preference relations: The correlation consensus degree

A new measure of consensus with reciprocal preference relations: The correlation consensus degree

Properties of distance spaces with power triangle inequalities

Order and Metric Compatible Symbolic Sequence Processing

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