Penalty-based functions defined by pre-aggregation functions

Abstract

Pre-aggregation function (PAF) is an important concept that has emerged in the context of directional monotonicity functions. Such functions satisfy the same boundary conditions of an aggregation functions, but it is not required the monotone increasingness in all the domain, just in some fixed directions. On the other hand, penalty functions is another important concept for decision making applications, since they can provide a measure of deviation from the consensus value given by averaging aggregation functions, or a penalty for not having such consensus. This paper studies penalty-based functions defined by PAFs. We analyse some properties (e.g.: idempotency, averaging behavior and shift-invariance), providing a characterization of idempotent penalty-based PAFs and a weak characterization of averaging penalty-based PAFs. The use of penalty-based PAFs in spatial/tonal filters is outlined.