Consensus-Based Agglomerative Hierarchical Clustering

Producción CientíficaIn this contribution, we consider that a set of agents assess a set of alternatives through numbers in the unit interval. In this setting, we introduce a measure that assigns a degree of consensus to each subset of agents with respect to every subset of alternatives. This consensus measure is defined as 1 minus the outcome generated by a symmetric aggregation function to the distances between the corresponding individual assessments. We establish some properties of the consensus measure, some of them depending on the used aggregation function. We also introduce an agglomerative hierarchical clustering procedure that is generated by similarity functions based on the previous consensus measuresMinisterio de Economía, Industria y Competitividad (ECO2012-32178)Junta de Castilla y León (programa de apoyo a proyectos de investigación – Ref. VA066U13

Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index

Approximations of Lovasz extensions and their induced interaction index

The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is defined on each simplex of the standard triangulation of $[0,1]^n$ as the unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the $n+1$ vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses $f$. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension $\hat f$ by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of $\hat f$ and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page

Capacities and Games on Lattices: A Survey of Result

We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.capacity, fuzzy measure, game, lattice, Choquet integral,Shapley value

Approval consensus measures

In many realistic group decision making problems where a “representative” collective output must be produced, it is relevant to measure how much consensus this solution conveys to the group. Many aspects influence the final decision in group decision making problems. Two key issues are the experts’ individual opinions and the methodology followed to compute such a final decision (aggregation operators, voting systems, etc.). In this paper we consider situations where each member of a population decides upon approving or not approving each of a set of options. The experts express their opinions in a dichotomous way, e.g., because they intend to use approval voting. In order to measure the consensus or cohesiveness that the expression of the individual preferences conveys we propose the concept of approval consensus measure (ACM), which does not refer to any priors of the agents like preferences or other decision-making processes. Then we give axiomatic characterizations of two generic classes of ACMs

Approval consensus measures

In many realistic group decision making problems where a “representative” collective output must be produced, it is relevant to measure how much consensus this solution conveys to the group. Many aspects influence the final decision in group decision making problems. Two key issues are the experts’ individual opinions and the methodology followed to compute such a final decision (aggregation operators, voting systems, etc.). In this paper we consider situations where each member of a population decides upon approving or not approving each of a set of options. The experts express their opinions in a dichotomous way, e.g., because they intend to use approval voting. In order to measure the consensus or cohesiveness that the expression of the individual preferences conveys we propose the concept of approval consensus measure (ACM), which does not refer to any priors of the agents like preferences or other decision-making processes. Then we give axiomatic characterizations of two generic classes of ACMs