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Journal ArticleDOI

A cardinal dissensus measure based on the Mahalanobis distance

TL;DR: A new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values, is proposed.
About: This article is published in European Journal of Operational Research.The article was published on 2016-06-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Mahalanobis distance.

Summary (3 min read)

1. Introduction

  • In Decision Making Theory and its applications, consensus measurement and its reaching in a society (i.e., a group of agents or experts) are relevant research issues.
  • Notwithstanding the use of different ordinal preference frameworks, the problem of how to measure consensus is an open-ended question in several research areas.
  • They evaluate two public goods with monetary amounts.
  • The Mahalanobis distance plays an important role in Statistics and Data Analysis.
  • In Section 3, the authors set forth the class of the Mahalanobis dissensus measures and their main properties.

2. Notation and definitions

  • This section is devoted to introduce some notation and a new concept in order to compare group cohesiveness: namely, dissensus measures.
  • The authors partially borrow notation and definitions from Alcantud et al. (2013b).
  • The authors consider that each expert evaluates each alternative by means of a quantitative value.
  • N×k A profile M ∈ MN×k is unanimous if the evaluations for all the alternatives are the same across experts.
  • The terms consensus and dissensus should not be taken as formal antonyms, especially because a universally accepted definition of consensus is not available and the authors do not intend to give an absolute concept of dissensus.

3. The class of Mahalanobis dissensus measures and its properties

  • The authors interest is to cover the specific characteristics in cardinal profiles, like possible differences in scales, and correlations among the issues.
  • Before providing their main definition, the authors recover the definition of the Mahalanobis distance on which their measure is based.
  • The off-diagonal elements of Σ permit to account for cross relations among the issues or alternatives.
  • The authors have only used the fact that the permutation matrix Pπ is orthogonal.

3.1. Some particular specifications

  • Some special instances of Mahalanobis dissensus measures have specific interpretations.
  • This ex- pression uses the square of the Euclidean distance between real-valued vectors, thus it recovers a version of the consensus measure for ordinal preferences based on this distance (Cook and Seiford (1982)).
  • Henceforth δI is called the Euclidean dissensus measure.
  • This particular specification of the dissensus measure allows to incorporate different weights to the alternatives.
  • This fact increases the richness of the analysis in comparison with the (square of the) Euclidean distance.

3.2. Some properties of the class of Mahalanobis dissensus measures

  • Measuring cohesiveness by means of the Mahalanobis dissensus measure ensures some interesting operational features.
  • The following properties hold true: 1. Neutrality .
  • If for a particular size N of a society the Mahalanobis dissensus measures associated with two matrices coincide for all possible profiles, then the corresponding dissensus measures are equal.
  • So an important question arises about if the scale choice disturbs the cohesiveness measures.
  • If the assessments of the new agent coincide with the average of the original agents’ evaluations for each alternative, then the minimal increment of the dissensus measure is obtained.

4. Comparison of Mahalanobis dissensus measures

  • In practical situations the authors could potentially use various Mahalanobis dissensus measures for profiles of cardinal information.
  • Theorems 1 and 2 below identify conditions on matrices that ensure consistent comparisons between Mahalanobis dissensus measures, whatever the number of agents.
  • Nevertheless, Theorem 2 below proves that a partial converse of Theorem 1 holds true under a technical restriction on the definite matrices.
  • Therefore it is not true that δΣ1(M) ≥ δΣ2(M) holds throughout.
  • Moreover, distance dΣ is always between the values of the corresponding distances dλ1I and dλkI .

5. Discussion on practical application using a real example

  • The authors are interested in assessing the cohesiveness of the forecasts of various magnitudes for the Spanish Economy in 2014: GDP (Gross Domestic Product), Unemployment Rate, Public Deficit, Public Debt and Inflation.
  • These forecasts have been published by different institutions and organizations, and each one was made at around the same time.
  • Next, the authors select a suitable reference matrix and finally they make the computations of the Mahalanobis dissensus measures.

5.1. Reference matrix

  • Once the profiles have been fixed, the following step to compute their Mahalanobis dissensus measures is to avail oneself of a suitable reference matrix Σ.
  • This matrix contains the variances and covariances among the statistical variables, therefore, those characteristics are brought into play in this distance.
  • One exception is the unlikely case when the data are generated by a known multivariate probability distribution.
  • It seems natural to produce such a matrix from historical macroeconomic data corresponding to the issues under inspection.
  • The ellipses slant upward (resp., downward) show a positive (resp., negative) correlation.

5.2. Computation of the dissensus

  • Now the authors calculate the Mahalanobis dissensus measures associated with Σ for the profiles of the forecasts for the Spanish Economy, namely, M (S), M (A) and M (lS).
  • Table 6 provides these items for comparison.

5.3. Other simpler approaches: Drawbacks or limitations

  • The choice of the reference matrix is a key point in the application of the Mahalanobis dissensus measure.
  • However the choice of the identity matrix as the reference matrix discards much relevant information.
  • The authors remove the effects of the interdependence among the economic magnitudes on the dissensus measure.
  • Table 7 shows the dissensus measures derived from the three matrices mentioned above, Σ, I and Σσ.

6. Concluding remarks

  • The authors use the general concept of dissensus measure and introduce one particular formulation based on the Mahalanobis distance for numerical vectors, namely the Mahalanobis dissensus measure.
  • The authors provide some properties which make their proposal appealing.
  • T. González-Arteaga acknowledges financial support by the Spanish Ministerio de Economı́a y Competitividad (Project ECO2012-32178).
  • The authors define the permutation matrix Pπ whose rows are eπ(i).

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Citations
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Journal ArticleDOI
TL;DR: 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, and it is proved that it is different to traditional consensus measures.
Abstract: Innovative methodology for measuring consensus based on the Pearson correlation coefficient is proposed.Experts express their opinions on alternatives or issues by means of reciprocal preference relations.We provide interesting properties for the new consensus measure proposed.An illustrative example with discussion is presented. 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.

55 citations

Journal ArticleDOI
TL;DR: A new procedure to codify ordinal information is provided and characterized and a new measurement of the degree of dissensus among individual preferences based on the Mahalanobis distance is defined.

25 citations

Journal ArticleDOI
TL;DR: The minimum deviation consensus ranking model (MDCRM), which seeks to minimize the ordinal information deviation between the original and adjusted preference orderings in the process of reaching consensus, is proposed and the properties of the optimal solution to MDCRM are studied.

24 citations

References
More filters
Journal ArticleDOI
TL;DR: It is shown that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation that is put forward to model the cardinal consistency of reciprocal preference relations.
Abstract: Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of (n-1) preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.

387 citations


"A cardinal dissensus measure based ..." refers background in this paper

  • ...(2008), Chiclana et al. (2013), Fu and Yang (2010, 2011), Palomares and Mart́ınez (2014), Gong et al....

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  • ...Modern convention applies the term cardinal to measurements that assign significance to differences (cf., Basu (1982), High and Bloch (1989), Chiclana et al. (2009))....

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Journal ArticleDOI
01 Oct 2015
TL;DR: A trust based estimation and aggregation methods as part of a visual consensus model for multiple criteria group decision making with incomplete linguistic information and it is proved that the implementation of the visual feedback mechanism guarantees the convergence of the consensus reaching process.
Abstract: Graphical abstract(A) Trust propagating aggregation and visual consensus model for MCGDM under incomplete information. (B) Visual feedback simulation: consensus levels before and after recommendations implemented by experts. Display Omitted HighlightsA theoretical framework to build consensus within a networked social group is presented.A novel trust propagation method is proposed to derive trust relationship from an incomplete connected trust network.A visual feedback process including a recommendation mechanism to provide individualised advice is implemented.The implementation of the visual feedback mechanism guarantees the convergence of the consensus reaching process. A theoretical framework to consensus building within a networked social group is put forward. This article investigates a trust based estimation and aggregation methods as part of a visual consensus model for multiple criteria group decision making with incomplete linguistic information. A novel trust propagation method is proposed to derive trust relationship from an incomplete connected trust network and the trust score induced order weighted averaging operator is presented to aggregate the orthopairs of trust/distrust values obtained from different trust paths. Then, the concept of relative trust score is defined, whose use is twofold: (1) to estimate the unknown preference values and (2) as a reliable source to determine experts' weights. A visual feedback process is developed to provide experts with graphical representations of their consensus status within the group as well as to identify the alternatives and preference values that should be reconsidered for changing in the subsequent consensus round. The feedback process also includes a recommendation mechanism to provide advice to those experts that are identified as contributing less to consensus on how to change their identified preference values. It is proved that the implementation of the visual feedback mechanism guarantees the convergence of the consensus reaching process.

314 citations


"A cardinal dissensus measure based ..." refers background in this paper

  • ...…R. de Andrés Calle ) Preprint submitted to European Journal of Operational Research October 16, 2015 Wu and Chiclana (2014b,a), Liu et al. (2015) and Wu et al. (2015) among others) and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al. (2007),…...

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Journal ArticleDOI
TL;DR: The proposed consensus operator provides an alternative consensus model for group decision making and preserves the original preference information given by the decision makers as much as possible, and supports consensus process automatically, without moderator.

311 citations


"A cardinal dissensus measure based ..." refers background in this paper

  • ...…been carried out under several frameworks (see Herrera-Viedma et al. (2002), Fedrizzi et al. (2007), Dong et al. (2008), Cabrerizo et al. (2010), Dong et al. (2010), Fu and Yang (2012), Dong and Zhang (2014), Palomares et al. (2014), ∗Corresponding author Email addresses: teresag@eio.uva.es (T.…...

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Journal ArticleDOI
TL;DR: A comparative study of the effect of the application of some different distance functions for measuring consensus in GDM, using the nonparametric Wilcoxon matched-pairs signed-ranks test, and concludes that differentdistance functions can produce significantly different results.

304 citations


"A cardinal dissensus measure based ..." refers methods in this paper

  • ...(2015) among others) and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al....

    [...]

  • ...…and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al. (2007), Fedrizzi et al. (2007), Eklund et al. (2008), Chiclana et al. (2013), Fu and Yang (2010, 2011), Palomares and Mart́ınez (2014), Gong et al. (2015) and Liu et al. (2015) among others)....

    [...]

  • ...(2015) among others) and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al. (2007), Fedrizzi et al....

    [...]

  • ...(2015) among others) and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al. (2007), Fedrizzi et al. (2007), Eklund et al. (2008), Chiclana et al....

    [...]

  • ...(2015) among others) and based on different methodologies (González-Pachón and Romero (1999), Cook (2006), Eklund et al. (2007), Fedrizzi et al. (2007), Eklund et al....

    [...]

Journal ArticleDOI
01 Mar 2010
TL;DR: The aim of this paper is to analyze the different consensus approaches in fuzzy group decision making problems and discuss their advantages and drawbacks.
Abstract: Two processes are necessary to solve group decision making problems: a consensus process and a selection process. The consensus process is necessary to obtain a final solution with a certain level of agreement between the experts, while the selection process is necessary to obtain such a final solution. Clearly, it is preferable that the set of experts reach a high degree of consensus before applying the selection process. In order to measure the degree of consensus, different approaches have been proposed. For example, we can use hard consensus measures, which vary between 0 (no consensus or partial consensus) and 1 (full consensus), or soft consensus measures, which assess the consensus degree in a more flexible way. The aim of this paper is to analyze the different consensus approaches in fuzzy group decision making problems and discuss their advantages and drawbacks. Additionally, we study the future trends.

288 citations


"A cardinal dissensus measure based ..." refers background in this paper

  • ...(2008), Cabrerizo et al. (2010), Dong et al. (2010), Fu and Yang (2012), Dong and Zhang (2014), Palomares et al....

    [...]

  • ...…subjects have been carried out under several frameworks (see Herrera-Viedma et al. (2002), Fedrizzi et al. (2007), Dong et al. (2008), Cabrerizo et al. (2010), Dong et al. (2010), Fu and Yang (2012), Dong and Zhang (2014), Palomares et al. (2014), ∗Corresponding author Email…...

    [...]

Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "A cardinal dissensus measure based on the mahalanobis distance" ?

In this paper the authors address the problem of measuring the degree of consensus/dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations. To this end the authors propose a new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values. Finally, an application over a real empirical example is presented and discussed.