Aggregation of data is almost an omnipresent process in our lives and in decision making. One can consider, for instance, when, given a list of items that we bought, each at a different price, we use the sum to synthesize all the prices into the total, a single representative number. When, at the high school, you were given a single final evaluation, on the ground of several single past evaluations, your professors aggregated your past evaluations into a single final one.
Aggregation functions are function whose aim is that of synthesizing numerical evaluations into a single representative real number. Indeed, the choice of the proper aggregation method is a crucial step and it deeply affects the outcomes of decision making processes. Hence, when making decisions we want to use a suitable aggregation method. In this course we shall discuss the main properties of aggregation methods, explore some important families and their generalizations, and look at some case studies.
The student will be aware that there are many methods for aggregating data in decision making, and that the weighted arithmetic mean is only a special case in a much wider framework. The choice of the method for aggregating data should reflect the properties that we want the aggregation process to respect and it is highly context dependent. In the course, the most important families of aggregation functions will be presented, together with their main properties. Finally, the student shall be able to choose the right method for different real-world problems.
Real world examples of aggregation of information (student evaluation - aggregation of objective functions in multicriteria optimization - ...)
Information fusion problem, from Cauchy on.
Formal definition of aggregation function, and justification of
minimal properties (boundedness and monotonicity)
further properties (continuity, associativity, commutativity, idempotency,...)
List of basic functions: arithmetic mean, product, max, min, median, but also copulas and others
Curiosities and dangers: the arithmetic mean is not associative, the product is not idempotent, and so forth.
Averaging functions (=means)
Pythagorean means (Arithmetic/geometric/harmonic), geometric interpretation and inequalities.
Generalization of the arithmetic mean to quasi-means à la Kolmogorov. Deduction that there are infinitely many quasi-means
Interpretation of the means as arguments minimizing distance functions.
Introductive examples: astronaut selection, and student selection.
Indices of "orness” saying how much an aggregation function tends to max or min. Indices of entropy for weight vectors associated to the aggregation functions.
Weight determination methods
Other examples: OWA functions as indices of inequality.
Discrete Choquet integral
The incapacity of the weighted mean to describe some non-additive cases. Example: student evaluation.
Choquet integral as a distorted generalization of the arithmetic mean and a generalization of weighted mean (additive measure based integral) and OWA operators (symmetric measure based integral).
Non-averaging aggregation functions:
disjuntive and conjunctive
t-norms and t-conorms
something on copulas
mixed non-averaging aggregation functions (uninorms)
Practical issues on how to derive weights:
Optimization based methods for learning weights
Optimization of measures of entropy and orness
Learning from past observations.
Quantifier-based methods for learning weights