Item weighted Kemeny distance for preference data

Preference data represent a particular type of ranking data where a group of people gives their preferences over a set of alternatives. The traditional metrics between rankings don’t take into account that the importance of elements can be not uniform. In this paper the item weighted Kemeny distance is introduced and its properties demonstrated

Measuring consensus in a preference-approval context

Producción CientíficaWe consider measuring the degree of homogeneity for preference-approval proles which include the approval information for the alternatives as well as the rankings of them. A distance-based approach is followed to measure the disagreement for any given two preference-approvals. Under the condition that a proper metric is used, we propose a measure of consensus which is robust to some extensions of the ordinal framework. This paper also shows that there exists a limit for increasing the homogeneity level in a group of individuals by simply replicating their preference approvals.Ministerio de Economía, Industria y Competitividad (ECO2009- 07332)Ministerio de Economía, Industria y Competitividad (ECO2008-03204-E/ECON

Measuring consensus in a preference-approval context

We consider measuring the degree of homogeneity for preference-approval profiles which include the approval information for the alternatives as well as the rankings of them. A distance-based approach is followed to measure the disagreement for any given two preference-approvals. Under the condition that a proper metric is used, we propose a measure of consensus which is robust to some extensions of the ordinal framework. This paper also shows that there exists a limit for increasing the homogeneity level in a group of individuals by simply replicating their preference-approvals

Element weighted Kemeny distance for ranking data

Preference data are a particular type of ranking data that arise when several individuals express their preferences over a finite set of items. Within this framework, the main issue concerns the aggregation of the preferences to identify a compromise or a “consensus”, defined as the closest ranking (i.e. with the minimum distance or maximum correlation) to the whole set of preferences. Many approaches have been proposed, but they are not sensitive to the importance of items: i.e. changing the rank of a highly-relevant element should result in a higher penalty than changing the rank of a negligible one. The goal of this paper is to investigate the consensus between rankings taking into account the importance of items (element weights). For this purpose, we present: i) an element weighted rank correlation coefficient as an extension of the Emond and Mason’s one, and ii) an element weighted rank distance as an extension of the Kemeny distance. The one-to-one correspondence between the weighted distance and the rank correlation coefficient is analytically proved. Moreover, a procedure to obtain the consensus ranking among several individuals is described and its performance is studied both by simulation and by the application to real datasets

Ensemble methods for ranking data with and without position weights

The main goal of this Thesis is to build suitable Ensemble Methods for ranking data with weights assigned to the items’positions, in the cases of rankings with and without ties. The Thesis begins with the definition of a new rank correlation coefficient, able to take into account the importance of items’position. Inspired by the rank correlation coefficient, τ x , proposed by Emond and Mason (2002) for unweighted rankings and the weighted Kemeny distance proposed by García-Lapresta and Pérez-Román (2010), this work proposes τ x w , a new rank correlation coefficient corresponding to the weighted Kemeny distance. The new coefficient is analized analitically and empirically and represents the main core of the consensus ranking process. Simulations and applications to real cases are presented. In a second step, in order to detect which predictors better explain a phenomenon, the Thesis proposes decision trees for ranking data with and without weights, discussing and comparing the results. A simulation study is built up, showing the impact of different structures of weights on the ability of decision trees to describe data. In the third part, ensemble methods for ranking data, more specifically Bagging and Boosting, are introduced. Last but not least, a review on a different topic is inserted in this Thesis. The review compares a significant number of linear mixed model selection procedures available in the literature. The review represents the answer to a pressing issue in the framework of LMMs: how to identify the best approach to adopt in a specific case. The work outlines mainly all approaches found in literature. This review represents my first academic training in making research.The main goal of this Thesis is to build suitable Ensemble Methods for ranking data with weights assigned to the items’positions, in the cases of rankings with and without ties. The Thesis begins with the definition of a new rank correlation coefficient, able to take into account the importance of items’position. Inspired by the rank correlation coefficient, τ x , proposed by Emond and Mason (2002) for unweighted rankings and the weighted Kemeny distance proposed by García-Lapresta and Pérez-Román (2010), this work proposes τ x w , a new rank correlation coefficient corresponding to the weighted Kemeny distance. The new coefficient is analized analitically and empirically and represents the main core of the consensus ranking process. Simulations and applications to real cases are presented. In a second step, in order to detect which predictors better explain a phenomenon, the Thesis proposes decision trees for ranking data with and without weights, discussing and comparing the results. A simulation study is built up, showing the impact of different structures of weights on the ability of decision trees to describe data. In the third part, ensemble methods for ranking data, more specifically Bagging and Boosting, are introduced. Last but not least, a review on a different topic is inserted in this Thesis. The review compares a significant number of linear mixed model selection procedures available in the literature. The review represents the answer to a pressing issue in the framework of LMMs: how to identify the best approach to adopt in a specific case. The work outlines mainly all approaches found in literature. This review represents my first academic training in making research

Element weighted Kemeny distance for ranking data

Preference data are a particular type of ranking data that arise when n individuals express their preferences over a finite set of items. Within this framework, the main issue concerns the aggregation of the preferences to identify a compromise or a “consensus”, defined as the closest ranking (i.e. with the minimum distance or maximum correlation) to the whole set of preferences.  Many approaches have been proposed, but they are not sensitive to the importance of items: i.e.  changing the rank of a highly-relevant element should result in a higher penalty than changing the rank of a negligible one. The goal of this paper is to investigate the consensus between rankings taking into account the importance of items (element weights).  For this purpose, we present:  i) an element weighted rank correlation coefficient tau_ew as an extension of the Emond and Mason’s tau, and ii) an element weighted rank distance d_ew as an extension of the Kemeny distance d. The one-to-one correspondence between the weighted distance and the rank correlation coefficient is analytically proved. Moreover, a procedure to obtain the consensus ranking among n individuals is described and its performance is studied both by simulation and by the application to real datasets

Quantifying consensus of rankings based on q-support patterns

Rankings, representing preferences over a set of candidates, are widely used in many information systems, e.g., group decision making and information retrieval. It is of great importance to evaluate the consensus of the obtained rankings from multiple agents. An overall measure of the consensus degree provides an insight into the ranking data. Moreover, it could provide a quantitative indicator for consensus comparison between groups and further improvement of a ranking system. Existing studies are insufficient in assessing the overall consensus of a ranking set. They did not provide an evaluation of the consensus degree of preference patterns in most rankings. In this paper, a novel consensus quantifying approach, without the need for any correlation or distance functions as in existing studies of consensus, is proposed based on a concept of q-support patterns of rankings. The q-support patterns represent the commonality embedded in a set of rankings. A method for detecting outliers in a set of rankings is naturally derived from the proposed consensus quantifying approach. Experimental studies are conducted to demonstrate the effectiveness of the proposed approach

Two algorithms for finding optimal solutions of the Kemeny rank aggregation problem for full rankings

The analysis of ranking data has recently received increasing attention in many fields (i.e. political sciences, computer sciences, social sciences, medical sciences, etc.).Typically when dealing with preference rankings one of the main issue is to find a ranking that best represents the set of input rankings.Among several measures of agreement proposed in the literature, the Kendall's distance is probably the most known. We propose a branch-and-bound algorithm to find the solution(s) even when we take into account a relatively large number of objects to be ranked. We also propose a heuristic variant of the branch-and-bound algorithm useful when the number of objects to rank is particularly high. We show how the solution(s) achieved by the algorithm can be employed in different analysis of rank data such as Mallow's phi model, mixtures of distance-based models, cluster analysis and so on