Very much looking forward to attending the IPMU conference (Information Processing and Management of Uncertainty) in Eindhoven, Netherlands later in June this year. Over January I had worked with Marek Gagolewski on two papers and then also submitted a paper based on the work done with Dale Nimmo and honours student Andrew Geschke in ecological optimisation last year.

**Title:** Fitting aggregation functions to data: Part I – Linearization and regularization

**Authors:** M. Bartoszuk, G. Beliakov, M. Gagolewski and S. James

**Abstract**

The use of supervised learning techniques for fitting weights and/or generator functions of weighted quasi-arithmetic means – a special class of idempotent and nondecreasing aggregation functions – to empirical data has already been considered in a number of papers. Nevertheless, there are still some important issues that have not been dis- cussed in the literature yet. In the first part of the two-part contribution we deal with the concept of regularization, a quite standard technique from machine learning applied so as to increase the fit quality on test and validation data samples. Due to the constraints on the weighting vector, it turns out that many more methods can be used in the cur- rent framework, as compared to regression models. Moreover, it is worth noting that so far fitting weighted quasi-arithmetic means to empirical data has only been performed approximately, via the so-called lineariza- tion technique. In this paper we consider exact solutions to such special optimization tasks and indicate cases where linearization leads to much worse solutions.

**Title:** Fitting aggregation functions to data: Part II – Idempotization

**Authors:** M. Bartoszuk, G. Beliakov, M. Gagolewski and S. James

**Abstract**

The use of supervised learning techniques for fitting weights and/or generator functions of weighted quasi-arithmetic means – a special class of idempotent and nondecreasing aggregation functions – to empirical data has already been considered in a number of papers. Nevertheless, there are still some important issues that have not been discussed in the literature yet. In the second part of the two-part contribution we deal with a quite common situation in which we have inputs coming from different sources, describing a similar phenomenon, but which have not been properly normalized. In such a case, idempotent and nonde- creasing functions cannot be used to aggregate them unless proper pre- processing is performed. The proposed idempotentization method, based on the notion of B-splines, allows for an automatic calibration of independent variables. The introduced technique is applied in an R source code plagiarism detection system.

**Title:** Linear optimization for ecological indices based on aggregation functions

**Authors:** G. Beliakov, A. Geschke, S. James and D. Nimmo.

**Abstract**

We consider an optimization problem in ecology where our objective is to maximize biodiversity with respect to different land-use allocations. As it turns out, the main problem can be framed as learning the weights of a weighted arithmetic mean where the objective is the geometric mean of its outputs. We propose methods for approximating solutions to this and similar problems, which are non-linear by nature, using linear and bilevel techniques.