Aggregation functions based on penalties

This article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs, or equivalently, minimizing a penalty imposed for deviations of individual inputs from the aggregated value. We provide a systematization of various types of penalty based aggregation functions, and show how many special cases arise as the result. We show how new aggregation functions can be constructed either analytically or numerically and provide many examples. We establish connection with the maximum likelihood principle, and present tools for averaging experimental noisy data with distinct noise distributions.