Use and analysis of new optimization techniques for decision theory and data mining

This dissertation addresses important problems in decision theory anddata mining. In particular, we focus on problems of the form: Each ofseveral information sources provides evaluations or measurements ofthe objects in a universal set, and the objective is to aggregatethese, possibly conflicting, evaluations into a consensusevaluation of each object in the universal set. In addition, weconcentrate on the scenario where each source provides evaluations ofonly a strict subset of the objects; that is, each source provides an incomplete evaluation . In order to define the consensus evaluation from a given set ofincomplete evaluations, two distances are developed: the first is adistance between incomplete rankings (ordinal evaluations) and thesecond is a distance between incomplete ratings (cardinalevaluations). These two distances generalize Kemeny and Snellrs distance between complete rankings and Cook and Kressr distance between complete ratings, respectively. Specifically, we introduce a set of natural axioms that must be satisfied by a distance between two incomplete rankings (ratings) and prove the uniqueness and existence of a distance satisfying such axioms. Given a set of incomplete rankings (ratings), the consensus ranking (rating) isdefined as the complete ranking (rating) that minimizes the sum ofdistances to each of the given rankings (ratings). We provide severalexamples that show that the consensus ranking (rating) obtained bythis approach is more intuitive than that obtained by other approaches.Finding the consensus ranking is NP-hard, thus we develop twooptimization methodologies to find the consensus ranking: oneefficient approximation algorithm based on the separation-deviationmodel and one exact algorithm based on the implicit hitting setapproach. In addition, we show that the optimization problem thatneeds to be solved in order to find the consensus rating is a specialcase of the separation-deviation model (hereafter SD model), which issolvable in polynomial time. In this sense, the herein developedtheory (described in the previous paragraph) can be thought of anaxiomatization of the SD model.Three applications of the SD model are presented: rating thecredit-risk of countries; customer segmentation; and ranking theparticipants in a student paper competition. In the credit-riskrating study, it is shown that the SD model leads to an improvedaggregate rating with respect to several criteria. We compare the SDmodel with other aggregation methods and show the following: Althoughthe SD model is a method to aggregate cardinal evaluations, theaggregate credit-risk ratings obtained by the SD model are also goodwith respect to lordinal criteriar. Several properties of the SD model are proven, including the property that the aggregate ratingobtained by the SD model agrees with the majority of agencies orreviewers, regardless of the scale used.The customer segmentation study shows how to use the SD model toprocess data on customer purchasing timing. The outcome of the SDmodel provides insights on the rate of new product adoption by thecompanyrs consumers. In particular, the SD model is used as follows: given the purchase dates for each customer of several products, this information is aggregated in order to rate the customers with regardto their promptness to adopt new technology. We show that thisapproach outperforms unidimensional scalingma widely used data mining methodology. We analyze the results with respect to variousdimensions of the customer base and report on the generated insights.The last presented application illustrates our aggregation methods inthe context of the 2007 MSOMrs student paper competition. Theaggregation problem in this competition poses two challenges. First,each paper was reviewed only by a very small fraction of the judges;thus the aggregate evaluation is highly sensitive to the subjectivescales chosen by the judges. Second, the judges provided bothcardinal and ordinal evaluations (ratings and rankings) of the papersthey reviewed. This chapter develops the first known methodology tosimultaneously aggregate ordinal and cardinal evaluations into aconsensus evaluation.Although the content of this dissertation is framed in terms ofdecision theory, Hochbaum showed that data mining problems can beviewed as special cases of decision theory problems. In particular,the customer segmentation study is a classic data mining problem.