## 5 Knowledge Validation and Visualization  
Evaluating data models using unlabeled data is a complex and time-consuming task. ESCAPE includes both quantitative indices and visualization techniques. Quantitative metrics include, for the joint-approach, the silhouette-based indices, while for the probabilistic model, they include (i) the perplexity and (ii) the entropy. The silhouette-based indices could be summarized as follows:  
* the weighted-silhouette (WS) [1] is an index that can take values between \(0\) and \(1\) and represents the percentage of documents in each positive bin, suitably weighted with an integer value \(w\) between \(1\) and \(10\) (the highest weight is associated with the first bin [1; 2; 3; 4; 5; 6; 7; 8; 9], and so on) and normalized within the sum of all the weights. The higher the silhouette index, the better the identified partition is;
* The average silhouette index (ASI) [48] is expressed as \[ASI=\frac{1}{N}\sum_{k=1}^{K}\sum_{i\in\mathbb{C}_{k}}s_{i};\]
* The global silhouette index (GSI) [48] is expressed as \[GSI=\frac{1}{K}\sum_{k=1}^{K}\frac{1}{|\mathbb{C}_{k}|}\sum_{i\in\mathbb{C}_{ k}}s_{i}.\]  
On the other hand, for the probabilistic model, ESCAPE integrates (i) the perplexity and (ii) the entropy. * The perplexity is a measure of the quality of probabilistic models which describes how well a model predicts a sample (i.e., how much it is perplexed by a sample from the observed data). Perplexity is monotonic, decreasing in the likelihood of the data, and is equivalent to the inverse of the per-word likelihood. It is defined as: \[Perplexity(D)=\exp\bigg{\{}-\frac{\sum_{d=1}^{D}\log p(w_{d})}{\sum_{d=1}^{D}N _{d}}\bigg{\}}\] Here, \(D\) is the number of documents (the corpus under analysis), \(w_{d}\) represents the words in document \(d\), and \(N_{d}\) is the number of words in document \(d\). Given a calculated model, the lower the general perplexity, the better the model performance and the probability estimate of the corpus [52];
* The entropy, when applied to the modeling context, measures how uncertain the model is: the lower the entropy of the model, the more certain it is that the model is describing the corpus under analysis. Specifically, for each \(d\) document in the corpus \(D\), we have calculated that entropy must belong to one of \(K^{\prime}\) s topics, and it is calculated as follows: \[H(d)=-\sum_{k=1}^{K}p(d=k)\log(p(d=k)),\] where \(p(d=k)\) is the probability that the considered document will be assigned to the topic \(k\). To compute the entropy of the whole clustering model, we averaged the entropy of each document on the whole corpus: \(H(model)=\frac{\sum_{d=1}^{D}H(d)}{D}\). To compare the different solutions found by ESCAPE, the adjusted Rand index (ARI) metric has been integrated in ESCAPE. The ARI is the corrected-for-chance version of the Rand index [53; 54; 55]. The Rand index can assume values between 0 and 1. When there is a perfect agreement between two partitions, the Rand index reaches the value 1 (its maximum). A limitation of the Rand index is that its expected value in the comparison of two randomly formed classifications is not always the same, as it should be. This problem is solved using the adjusted Rand index [54], which assumes a generalized hyper-geometric distribution as the model of randomness. The adjusted Rand index is ensured to have a value close to 0 in the case of random labeling and, differently from the Rand index, it can assume negative values if the index is less than the expected index. Even if the partitions do not have the same number of clusters, it is recommendable to use the adjusted Rand index. To this end, ESCAPE reported the ARI between solutions using the same strategy (i.e., joint-approach or probabilistic approach) in order to compare the different weighting schemes' impact. Such choice also enables us to determine and analyze the main differences between the two approaches.