# Automated Exploratory Clustering to Democratize Clustering Analysis
## 3 Materials and Methods
### Theoretical Definition
For this reason, the Interestingness function will need to weight the novelty and the quality of a clustering. **Definition 3** (Interestingness function). : _The Interestingness function defines the likely interestingness of a set of clusterings to a practitioner as follows:_  
\[I(C):\mathcal{P}(\mathbb{U})\rightarrow\mathbb{R} \tag{4}\]  
In the end, the Interestingness function defines which sets of clusterings will be selected. For this reason, finding a good Interestingness function, most likely employing \(Q\), will be a main goal in the field of AEC. Before we propose and investigate different Interestingness functions, we will first provide a formal definition of the AEC problem. **Definition 4** (AEC Problem). : _The Automated Exploratory Clustering problem is to find an Interestingness function I such that the set \(t\) of clusterings, which is defined as_  
\[t=\operatorname*{argmax}_{C\in\mathcal{P}(\mathbb{U})}I(C), \tag{5}\]  
_satisfies_ minimality_, i.e., \(|t|\) is minimal, and_ coverage_, i.e., \(t\) contains all relevant clusterings._  
As can be seen in the definition above, the problem of Automated Exploratory Clustering is to find an appropriate Interestingness function that minimizes the set of relevant clusterings. While this problem definition is not tied to any specific application domain, it can accommodate various notions of interestingness and thus it is generally difficult to identify suitable characteristics. The following Corollary summarizes the characteristics for a well-chosen Interestingness function. **Corollary 1**. : _A well-chosen Interestingness function I epitomizes the following properties:_  
1. _All clusterings in_ t _are different._  
Figure 4: Three different clusterings of the same dataset with two properties, where the cluster label is depicted by the colour. The dataset has the form of two concentric circles with one connecting element. (**a**) The blue cluster contains all elements on the left side, the red cluster all elements on the right side. (**b**) The blue cluster contains all elements on the outer circle and the connecting element, the red cluster all elements on the inner circle. (**c**) The blue cluster contains all elements on the outer circle, the red cluster all elements on the inner circle and the connecting element. 2. _The set_ \(t\) _is much smaller than the set of all clusterings, i.e.,_ \(t\ll|\operatorname{\mathbb{U}}|\)_._
3. _Any two clusterings in_ \(t\) _exhibit a sufficient degree of dissimilarity._
4. _The set_ \(t\) _is the smallest set containing all relevant clusterings._  
Based on the definition of the AEC problem, we propose three different Interestingness functions which aim for (i) minimality, (ii) coverage, and (iii) tradeoff between both. The first Interestingness function, introduced in Section 3.3, optimizes minimality by returning a single clustering like in traditional Automated Clustering approaches, while the second function, presented in Section 3.4, optimizes coverage by returning all clusterings regarded important by several CVIs, regardless of their similarity among each other. Finally, the Interestingness function proposed in Section 3.5 balances the tradeoff between minimality and coverage by taking a skyline and pruning similar clusterings. Without loss of generality, we assume for all subsequent Interestingness functions that the Quality function \(Q(c)\) of a single clustering \(c\in\operatorname{\mathbb{U}}\) is to be maximized in all components, and all resulting values are non-negative (as a minimized value can be multiplied by \(-1\) in order to be maximized and normalization can ensure that all values are greater than \(0\) by an addition with a constant \(c\)).