Producción CientíficaThe main purpose of this contribution is to measure the dispersion associated with the different results of a variable when they form a set of linguistic terms obtained after examining different qualities of people, services, etc, ordered, but with a non-homogeneous and non-quantifiable distance between the linguistic terms.Ministerio de Economía, Industria y Competitividad (project ECO2016-77900-P

Borge González, Luis Moisés

García Lapresta, José Luis

English

Repositorio Documental de la Universidad de Valladolid

Chapter 1MEASURINGDISPERSION IN THE CONTEXTOF ORDERED QUALITATIVE SCALESJose´ Luis Garc´ıa-Lapresta1 and Luis Borge21PRESAD Research Group, BORDA Research Unit, IMUVA, Departamento de Economı´aAplicada, Universidad de Valladolid, Spainlapresta@eco.uva.es2Departamento de Economı´a Aplicada, Universidad de Valladolid, Spainborge@eco.uva.esAbstract In this contribution, we introduce a family of dispersion measures in the contextof ordered qualitative scales.Keywords: Qualitative scales; Dispersion; Ordinal proximity measures.1. IntroductionThe main purpose of this contribution is to measure the dispersion associ-ated with the different results of a variable when they form a set of linguisticterms obtained after examining different qualities of people, services, etc, or-dered, but with a non-homogeneous and non-quantifiable distance between thelinguistic terms.There is in the literature a great variety of dispersion measures associatedwith a data set, being the most used the standard deviation. For a very longtime, the normality assumption was present in most of the statistical studiesand its estimator in combination with the sample mean provide excellent effi-ciency from the practical side due to its precision to adjust the data and fromthe capacity of giving rise to theoretical results in the sample distributions.However, most of the estimators of dispersion presents in literature do not hold“This is a post-peer-review, pre-copyedit version of an article published in Gil, Eduardo; Gil, Eva; Gil, Juan; Gil, María Ángeles. The Mathematics of the Uncertain. Cham: Spinger, 2018, p. 605-614.The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-319-73848-2_56”.2 Measuring Dispersion in the Context of Ordered Qualitative Scalesfor our purposes, because they are based on numerical values obtained in thesample.When the data that we want to study are obtained from a statistical behaviourdifferent from the normal one, there are atypical observations and losses ofefficiency on the classical estimators. The growth of statistical methods andtheir increasing use in different areas of research poses the study of alternativeestimators which are less sensible to deviations to the normal. In the middleof the last century there was an important revival of the robust statistics. Theconcept of robustness is related to the fact that a estimator is less sensibleto small changes in the statistical hypothesis about the models or about thepresence of atypical observations, being the breakdown point one of the waysto measure the robustness of a estimator which is related to the percentage ofpolluted observations of a sample.The most well known measure of localization of the robust statistical is themedian which is estimated ordering the values obtained in a sample, whilethe sample median is obtained as the central value of the ordered data, if thenumber of data is odd, and as the average of the central data, if it is even.When the data of a sample is like the ones we are interested in, we may usethis localization measurement when the number of data n is odd. When weare dealing with the robust dispersion the most used measurement is the in-terquartile range. This method is not the best one because a estimator is morerobust if it is obtained as a median of the absolute deviation of the median ofthe sample, also known as MAD. If we have an ordered sample of a variable(x(1), . . . ,x(n)), its median, medi(xi) is definedMAD = med j|x j−medi(xi)|.To obtain this estimator, it is necessary to evaluate the median two times.The first one between all the linguistic terms of the data. The second median,after ordering the different proximities between the linguistic terms, we choosethe median of these proximities.Rousseeuw and Croux [8] presented two alternative robust estimators to theMAD and, as the interquartile range, they do not require to center the data torespect to the sample median. The first of them isSn = medi{med j|xi− x j|},which for any xi value we obtain the median of the differences of the form{|xi− x j|, j = 1, . . . ,n}, which gives us n medians, being the estimator Sn themedian of this set of medians.The second estimator proposed by Rousseeuw and Croux, which they callQn, is based only on the differences between data values. If {x(1), . . . ,x(n)}is the ordered sample, let D = {x(i)− x( j), i > j} be the set of interpoint dis-tances in which the number of elements is m =(n2). Then, the estimate isNotation and Basic Notions 3defined as the k-th order statistic of D , where k =([n/2]+12)and that is roughlyhalf the number of observations. These estimator has attractive properties (seeMaronna, et al.**)**Some absolute dispersion measures used in Statistics are the range, thevariance, the mean deviation, the standard deviation, the absolute Gini index,etc. (see Martı´nez-Panero et al. [7]).The rest of the contribution is organized as follows. Section 2 is devoted tointroduce the notation and some basic notions that are necessary for definingthe two families of dispersion measures we propose in the setting of orderedqualitative scales. Section 3 present these two families of dispersion measuresand includes some illustrative examples. Section 4 contains some properties.Finally, Section 5 concludes with some remarks.2. Notation and Basic NotionsFirst we present the notion of ordinal proximity measure.2.1 Ordinal Proximity MeasuresLet L = {l1, . . . , lg} be an ordered qualitative scale arranged from the low-est to the highest linguistic terms, l1 < l2 < · · ·< lg, with g≥ 3.In order to recall the notion of ordinal proximity measure on L , intro-duced by Garcı´a-Lapresta and Pe´rez-Roma´n [4], we shall use a linear order∆= {δ1, . . . ,δh}, with δ1 ·· ·  δh, for representing different degrees of prox-imity among the terms of L , being δ1 and δh the maximum and minimumdegrees, respectively.It is important noticing that the elements of ∆ have no meaning and theyonly represent different degrees of proximity.As usual in the setting of linear orders, δr  δs means δr  δs or δr = δs;δr ≺ δs means δs  δr; and δr  δs means δr ≺ δs or δr = δs.Given a weak order E onL n, with C we denote the asymmetric part of E,i.e., xCy ⇔ not yEx.Definition 1.1 ([4]) An ordinal proximity measure onL with values in ∆is a mapping pi :L 2−→∆, where pi(lr, ls) = pirs means the degree of proximitybetween lr and ls, satisfying the following conditions:1 Exhaustiveness: For every δ ∈ ∆, there exist lr, ls ∈L such that δ =pirs.2 Symmetry: pisr = pirs, for all r,s ∈ {1, . . . ,g}.3 Maximum proximity: pirs = δ1 ⇔ r = s, for all r,s ∈ {1, . . . ,g}.4 Monotonicity: pirs  pirt and pist  pirt , for all r,s, t ∈ {1, . . . ,g} suchthat r < s < t.4 Measuring Dispersion in the Context of Ordered Qualitative ScalesThe previous conditions are independent (see Garcı´a-Lapresta and Pe´rez-Roma´n [4, Prop. 1]).Every ordinal proximity measure pi : L 2 −→ ∆ can be represented by ag×g symmetric matrix with coefficients in ∆, where the elements in the maindiagonal are pirr = δ1, for every r = 1, . . . ,g:pi11 · · · pi1s · · · pi1g· · · · · · · · · · · · · · ·pir1 · · · pirs · · · pirg· · · · · · · · · · · · · · ·pig1 · · · pigs · · · pigg= (pirs) .This matrix is called the proximity matrix associated with pi .Taking into account the conditions appearing in Definition 1.1, it is onlynecessary to show the upper half proximity matrixδ1 pi12 pi13 · · · pi1(g−1) pi1gδ1 pi23 · · · pi2(g−1) pi2g· · · · · · · · ·δ1 pi(g−1)gδ1 .As shown in Garcı´a-Lapresta and Pe´rez-Roma´n [4, Prop. 2], the minimumproximity between linguistic terms is only reached when comparing the ex-treme linguistic terms: pirs = δh ⇔ (r,s) ∈ {(1,g),(g,1)}.The cardinality of ∆ is located between the cardinality of L and a polyno-mial of degree 2 of that cardinality (see Garcı´a-Lapresta and Pe´rez-Roma´n [4,Prop. 4]):g≤ h≤ g · (g−1)2+ 1.2.2 MediansFollowing Garcı´a-Lapresta and Pe´rez-Roma´n [5], we now introduce the me-dian operator in the setting of ordinal degrees of proximity.Given a vector of ordinal degrees of proximity δ = (δ1, . . . ,δp) ∈ ∆p, wearrange its components in a decreasing fashion, from the highest to the lowestdegrees. If p is odd, then the median of δ is unique, say δr ∈ ∆. However, ifp is even, then δ has two medians, say δs,δt ∈ ∆ such that s≤ t, i.e., δs  δt .In order to unify the assignment of medians, we consider the pair of medians(δr,δr) and (δs,δt) whenever p is odd and even, respectively.Dispersion Measures 5More formally, given the set of feasible medians ∆2 = {(δr,δs)∈∆2 | r≤ s},the median operator is the mappingM :∞⋃p=1∆p −→ ∆2that assigns the corresponding pair of medians to each vector of ordinal de-grees.For ordering the pairs of medians of ordinal proximities, consider the linearorder 2 on ∆2 defined as(δr,δs)2 (δt ,δu) ⇔r + s < t + uorr + s = t + u and s− r ≤ u− t,(1)for all (δr,δs),(δt ,δu) ∈ ∆2.It is easy to see that if r + s = t + u, then s− r ≤ u− t ⇔ r ≥ t ⇔ s≤ u.3. Dispersion MeasuresIn this section we present two families of dispersion measures in the settingof ordered qualitative scales equipped with ordinal proximity measures. Thefirst family generalizes the most basic dispersion measure, the range.3.1 Range-based dispersion measuresGiven n≥ 2, let DR :L n −→ ∆ be the mapping defined asDR(x) = pi(minx,maxx), (2)for every x ∈L n.Based on the linear order  on ∆, we introduce the weak order ER on L ndefined asxER y ⇔ DR(x) DR(y),with the meaning of the dispersion in x is lower than or equal to in y (withrespect to DR).Example 1.2 Consider the ordered qualitative scale L = {l1, l2, l3, l4} andthe vectors x= (l1, l2, l2, l3), y = (l3, l3, l4, l4) ∈L 4. We want to compare thedispersion in these vectors with respect to three different ordinal proximitymeasures.1 If L is equipped with the ordinal proximity measurepi :L 2 −→ ∆ = {δ1, . . . ,δ7}6 Measuring Dispersion in the Context of Ordered Qualitative Scaleswith associated proximity matrixδ1 δ2 δ4 δ7δ1 δ3 δ6δ1 δ5δ1 ,we have DR(x) = pi13 = δ4  δ5 = pi34 = DR(y). Thus, xCR y.2 If L is equipped with the ordinal proximity measurepi :L 2 −→ ∆ = {δ1, . . . ,δ7}with associated proximity matrixδ1 δ4 δ6 δ7δ1 δ3 δ5δ1 δ2δ1 ,we have DR(x) = pi13 = δ6 ≺ δ2 = pi34 = DR(y). Thus, yCRx.3 If L is equipped with the ordinal proximity measurepi :L 2 −→ ∆ = {δ1, . . . ,δ4}with associated proximity matrixδ1 δ2 δ3 δ4δ1 δ2 δ3δ1 δ2δ1 ,we have DR(x) = pi13 = δ3 ≺ δ2 = pi34 = DR(y). Thus, yCRx.3.2 Gini-based dispersion measuresWe now introduce a new family of dispersion measures in the mentionedframework. It is based on the Gini index ([6]) and it is closely related to thescale estimator appearing in Shamos [9, p. 260] and Bickel and Lehmann [3,p. 38] in the setting of real numbers (see Rousseeuw and Croux [8, p. 1277]).Given n≥ 2, let DG :L n −→ ∆2 be the mapping defined asDG(x) = M(pi(xi,x j)i< j), (3)for every x= (x1, . . . ,xn) ∈L n.Properties 7Based on the linear order 2 on ∆2 defined in (1), we introduce the weakorder EG onL n defined asxEG y ⇔ DG(x)2 DG(y),with the meaning of the dispersion in x is lower than or equal to in y (withrespect to DG).Since some vectors can share the same pair of medians, it is necessary todevise a tie-breaking process for ordering the vectors. We propose to use asequential procedure based on Balinski and Laraki [1] (see Balinski and Laraki[2] for practical examples). It consists of withdrawing the pair of medians ofthe vectors that are in a tie, and then selecting the new pairs of medians ofthe remaining proximity degrees for the corresponding vectors. The processcontinues until the ties are broken. It is important to note that different vectorsnever are in a final tie.Example 1.3 Consider Example 1.2 and the same three ordinal proximitymeasures.1 We haveDG(x) = M(pi12,pi12,pi13,pi22,pi23,pi23) = M(δ2,δ2,δ4,δ1,δ3,δ3) = (δ2,δ3)andDG(y) = M(pi33,pi34,pi34,pi34,pi34,pi44) = M(δ1,δ5,δ5,δ5,δ5,δ1) = (δ5,δ5).Since (δ2,δ3)2 (δ5,δ5), we have xCG y.2 We haveDG(x) = M(pi12,pi12,pi13,pi22,pi23,pi23) = M(δ4,δ4,δ6,δ1,δ3,δ3) = (δ3,δ4)andDG(y) = M(pi33,pi34,pi34,pi34,pi34,pi44) = M(δ1,δ2,δ2,δ2,δ2,δ1) = (δ2,δ2).Since (δ3,δ4)≺2 (δ2,δ2), we have yCGx.3 We haveDG(x) = M(pi12,pi12,pi13,pi22,pi23,pi23) = M(δ2,δ2,δ3,δ1,δ2,δ2) = (δ2,δ2)andDG(y) = M(pi33,pi34,pi34,pi34,pi34,pi44) = M(δ1,δ2,δ2,δ2,δ2,δ1) = (δ2,δ2).Consequently, in x and y the dispersion is the same. If we apply the tie-breaking procedure, then we have DG(x) = M(δ1,δ2,δ2,δ3) = (δ2,δ2)and DG(y) = M(δ1,δ1,δ2,δ2) = (δ1,δ2). Since (δ2,δ2)≺2 (δ1,δ2), wefinally have yCGx.8 Measuring Dispersion in the Context of Ordered Qualitative Scales4. PropertiesLet L be an ordered qualitative scale equipped with an ordinal proximitymeasure pi :L 2 −→ ∆. We say that pi is totally uniform if pir (r+t) = pis(s+t)for all r,s, t ∈ {1, . . . ,g−1} such that r + t ≤ g and s + t ≤ g.Let N :L −→L be the negation operator defined as N(lr) = lg+1−r, forevery r ∈ {1, . . . ,g}.Given k ∈ {1−g, . . . ,g−1}, let Tk :L −→L be the translation operatordefined as Tk(lr) = lr+k, for every r ∈ {1, . . . ,g} such that r + k ≤ g.In the following proposition we establish some properties of the mappingsintroduced in (2) and (3). They are related to the ones considered in Martı´nez-Panero et al. [7] in a quantitative context.Proposition 1.1 Consider the mappings DR :L n −→ ∆ and DG :L n −→∆2 defined in (2) and (3), respectively, and its extensions D˜R :∞⋃n=2L n −→ ∆and D˜G :∞⋃n=2L n −→ ∆2. The following properties hold:1 Symmetry: DR(xσ(1), . . . ,xσ(n)) = DR(x1, . . . ,xn) and DG(xσ(1), . . . ,xσ(n)) =DG(x1, . . . ,xn), for every permutation σ : {1, . . . ,n} −→ {1, . . . ,n} andevery (x1, . . . ,xn) ∈L n.2 Invariance for replications: D˜R(m︷ ︸︸ ︷x, . . . ,x) = DR(x) and D˜G(m︷ ︸︸ ︷x, . . . ,x) =DG(x), for every x ∈L n and any number m ∈N of replications of x.3 Minimum dispersion: DR(x1, . . . ,xn) = δ1 ⇔ x1 = · · ·= xn; and DG(lr, . . . , lr) =(δ1,δ1) , for every lr ∈L .4 Anti-self-duality: if pi is totally uniform, then DR(N(x1), . . . ,N(xn)) =DR(x1, . . . ,xn) and DG(N(x1), . . . ,N(xn)) = DG(x1, . . . ,xn), for every (x1, . . . ,xn)∈L n.5 Invariance for translations: if pi is totally uniform, then DR(Tk(x1), . . . ,Tk(xn)) =DR(x1, . . . ,xn) and DG(Tk(x1), . . . ,Tk(xn)) = DG(x1, . . . ,xn), for every(x1, . . . ,xn)∈L n and every k∈{1, . . . ,g−1} such that (Tk(x1), . . . ,Tk(xn))∈L n.5. Concluding RemarksAcknowledgmentsThis contribution is dedicated to the memory of Pedro Gil. The first au-thor gratefully acknowledges the funding support of the Spanish Ministerio deEconomı´a y Competitividad (project ECO2016-77900-P) and ERDF.Concluding Remarks 9References[1] Balinski, M., Laraki, R. (2007). A theory of measuring, electing andranking. Proceedings of the National Academy of Sciences of the UnitedStates of America 104, pp. 8720–8725.[2] Balinski, M., Laraki, R. (2013). How best to rank wines: Majority Judg-ment. In: Wine Economics: Quantitative Studies and Empirical Observa-tions, pp. 149–172, Palgrave-MacMillan.[3] Bickel, P.J., Lehmann, E.L. (1979). Descriptive Statistics for nonparamet-ric models IV: Spread. In: J. Jurecˇkova´ (ed.) Contributions to Statistics,Ha´jek Memorial Volume, pp. 33–40, Academia, Prague.[4] Garcı´a-Lapresta, J.L., Pe´rez-Roma´n, D. (2015). Ordinal proximity mea-sures in the context of unbalanced qualitative scales and some applica-tions to consensus and clustering. Applied Soft Computing 35, pp. 864–872.[5] Garcı´a-Lapresta, J.L., Pe´rez-Roma´n, D. (2017). A consensus reachingprocess in the context of non-uniform ordered qualitative scales. FuzzyOptimization and Decision Making. Forthcoming, DOI: 10.1007/s10700-016-9256-6.[6] Gini, C. (1912). Variabilita` e Mutabilita`, Tipografia di Paolo Cuppini,Bologna.[7] Martı´nez-Panero, M., Garcı´a-Lapresta, J.L., Meneses, L.C. (2016). Mul-tidistances and dispersion measures. In: T. Calvo Sa´nchez, J. Torrens Sas-tre (eds.) Fuzzy Logic and Information Fusion, Studies in Fuzziness andSoft Computing, pp. 123–134, Springer.[8] Rousseeuw, P.J., Croux, C. (1993). Alternatives to the median absolutedeviation. Journal of the American Statistical Association 88, pp. 1273–1283.[9] Shamos, M.I. (1976). Geometry and Statistics: Problems at the interface.In: J.F. Traub (ed.) New Directions and Recent Results in Algorithms andComplexity, pp. 251–280, New York Academic Press.

Measuring dispersion in the context of ordered qualitative scales

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