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

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).

Figures

Table 7: Dissensus for several profiles of economic forecasts for the Spanish Economy for the year 2014. Data published in Spring of 2013, Autumn of 2013 and in Spring of 2014.

Table 4: Past data for the Spanish Economy (2001 − 2012). Source: Spanish National Statistics Institute (INE) and Bank of Spain.

Table 3: Forecasts for several magnitudes for the Spanish Economy for the year 2014 published in Spring of 2014.

Figure 2: A depiction of the correlation matrix of the Spanish macroeconomic data from 2001 to 2012.

Table 5: Correlations between macroeconomic magnitudes for historial data.

Figure 1: Curves of equidistance to point A with dΣ (ellipse), dλ1I and dλkI (circumferences).

Table 6: Dissensus between pairs of agents for the profiles of forecasts published in Spring of 2013 (in descending order), Autumn of 2013 and Spring of 2014.

Table 1: Forecasts for several magnitudes for the Spanish Economy for the year 2014 published in Spring of 2013.

Table 2: Forecasts for sevaral magnitudes for the Spanish Economy for the year 2014 published in Autumn of 2013.

Frequently asked questions

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.