# A Study of OWA Operators Learned in Convolutional Neural Networks
## 2 Preliminaries
### Aggregation Functions and OWA Operators
**Orness**: This measure, also called _attitudinal character_, tells us how close a certain operator is to the max operator, that is, the preference it has for large argument values. It always lies in the range \([0,1]\). The orness measure of a weighting vector \(W\in[0,1]^{n}\) is defined as  
\[\text{\emph{orness}}(W)=\sum_{i=1}^{n}w_{i}\,\frac{n-i}{n-1}\,. \tag{2}\]  
As an example, the orness of the weighting vectors of some classical OWA operators are \(\text{\emph{orness}}(W_{max})=1\), \(\text{\emph{orness}}(W_{min})=0\) and \(\text{\emph{orness}}(W_{avg})=0.5\). **Dispersion**: This measure, explored in depth in [35], gives us an impression of how concentrated the weights of the operator are. For example, the maximum dispersion is obtained with a weighting vector corresponding to the average (\(w_{i}=\frac{1}{n}\) for every \(i=1,\dots,n\)), while the minimum is associated with those weighting vectors \(W\) where there exists \(i\in\{1,\dots,n\}\) with \(w_{i}=1\) (and \(w_{j}=0\) for every \(j\
eq i\)). In this work, we use the classic dispersion definition  
\[\text{\emph{dispersion}}(W)=-\sum_{i=1}^{n}w_{i}ln(w_{i})\,. \tag{3}\]  
There have been many proposals of OWA operator parameterizations. With these parameterizations, the weight distribution of the operator can be controlled with a few parameters. These methods guarantee that the operator maintains some interesting properties, but at the expense of constraining its expressivity. One of the latest proposals is the Stancu OWA operator [22], which at the same times generalizes an earlier proposal, the Binomial OWA operator [8]. **Binomial OWA**: The Binomial OWA operator [8] is based on a binomial distribution, and defined, for every \(i=1,\dots,n\), as  
\[w_{i,n}(t)=\binom{n-1}{n-i}t^{n-i}(1-t)^{i-1}\,. \tag{4}\]
**Stancu OWA**: The Stancu OWA operator [22] is instead based on Stancu polynomials and defined, for every \(i=1,\dots,n\), as  
\[w_{i,n}(t,\alpha)=\frac{\binom{n-1}{n-i}\prod_{s=0}^{n-i-1}(t+s\alpha)\prod_{ s=0}^{i-2}(1-t+s\alpha)}{\prod_{s=0}^{n-2}(1+s\alpha)}\,, \tag{5}\]  
where \(t\in[0,1]\), \(\alpha\geq 0\), and an empty product denotes \(1\). Another approach is the definition of weighting vectors guided by fuzzy quantifiers (also known as Regular Increasing Monotone quantifiers or RIM), widely used in the area of multi-criteria decision making [36]. In the literature many families of RIM quantifiers have been proposed [23]. In particular, we focus on exponential RIM quantifiers. **Exponential RIM OWA**: The exponential RIM OWA operator [23] is based on the generating function  
\[Q(x)=\left\{ \begin{array}{ll}\frac{e^{x}-1}{a-1}&\text{if $a>0$ and $a\
eq 1$}\,,\\ x&\text{if $a=1$}\,,\end{array} \right. \tag{6}\]  
with \(a=e^{\lambda}\), for a parameter \(\lambda\in\mathbb{R}\). From the generating function \(Q(x)\) of the quantifier, the weights of the associated OWA operator can be obtained as  
\[w_{i}=Q\bigg{(}\frac{i}{n}\bigg{)}-Q\bigg{(}\frac{i-1}{n}\bigg{)}\,. \tag{7}\]One of the main properties of the RIM quantifiers with exponential generating functions is that they maximize the entropy for a given orness. This property makes them a solution to the MEOWA operator proposed by [PERSON] [37], maybe the most well-known family of OWA operators.