# Aggregation functions and dependence structures

In its most general meaning, aggregation denotes the process of determining a representative value or structure from a given set of several inputs or input structures. Typical aggregation processes are for example:

• in data analysis the computation of the mean or median value of a data set;
• in multi-criteria decision making the determination of a final evaluation of some alternative based on evaluations w.r.t. single objectives;
• in social choice theory or voting the determination of a group consensus or preference based on individual preferences;
• in many-valued logics the determination of the truth degree of a compound expression based on the truth degrees of the single terms;
• in risk analysis the determination of a coefficient of riskiness based on singular risky factors (not necessarily independent).

In many cases aggregation processes are represented by real-valued functions, called aggregation functions, which, depending on the actual context, have to fulfill characteristic properties. The determination and characterization of such functions is therefore of interest. Another challenging focus of our basic research work is related to the preservation of properties in aggregation processes most often leading to functional equations (like, e.g., bisymmetry) and inequalities (like, e.g., dominance). The (in)equalities in turn allow to define binary relations on the set of aggregation functions whose properties reveal insight in the structure of (sub)sets of aggregation functions.

Especially in risk analysis, the aggregation process deals with random variables and related distribution functions, which can exhibit some dependencies. Due to Sklar's theorem, most of these problems can be reduced to the study of copulas, which allow to capture the dependence structure of multi-dimensional random variables (see also CopulaWiki). The investigation, classification, construction and determination of such functions (and their generalizations) with respect to a given set of properties demanded by actual applications is therefore of interest and part of the research work.