# Analysis and Evaluation of Clustering Techniques Applied to Wireless Acoustics Sensor Network Data
## 2 Materials and Methods
### Evaluation Metrics  
For the evaluation and comparison of the clustering algorithms, [PERSON] et al. [36] proposed two criteria for clustering evaluation and selection of an optimal clustering scheme: compactness and separation. Later, [PERSON] et al. [37] introduce a new criteria: connectedness. In this article, these three internal characteristics are chosen to be calculated and analyzed. Connectedness is related to what extent observations are placed in the same cluster as their nearest neighbors in the data space. To measure that connectivity [38], Equation (4) is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Family** & **Algorithms** & **Advantages** & **Disadvantages** \\ \hline \multirow{6}{*}{Hierarchical} & & \multicolumn{2}{c}{suitable for the data set} \\  & & \multicolumn{2}{c}{with arbitrary shape and} \\  & & \multicolumn{2}{c}{attribute of arbitrary type,} \\  & & \multicolumn{2}{c}{the hierarchical} \\  & & \multicolumn{2}{c}{relationship among} \\  & & \multicolumn{2}{c}{clusters easily detected,} \\  & & \multicolumn{2}{c}{and relatively high} \\  & & \multicolumn{2}{c}{scalability in general} \\ \hline \multirow{6}{*}{Partitional} & \multirow{6}{*}{KM, AM, CLARA} & \multicolumn{2}{c}{relatively low time} \\  & & complexity and high \\ \cline{1-1}  & & computing efficiency in \\ \cline{1-1}  & & general \\ \cline{1-1}  & & & result sensitive to the \\ \cline{1-1}  & & \multicolumn{2}{c}{number of clusters.} \\ \hline \multirow{6}{*}{Model-Based} & \multirow{6}{*}{SOM, SOTA, GAUSS} & diverse and well \\  & & developed models \\ \cline{1-1}  & & providing means to \\ \cline{1-1}  & & describe data adequately \\ \cline{1-1}  & & and each model having its \\ \cline{1-1}  & & own special characters \\ \cline{1-1}  & & that may bring about some \\ \cline{1-1}  & & significant advantages in \\ \cline{1-1}  & & some specific areas \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Advantages and disadvantages of clustering algorithms used in this work. applied. For each element \(i\), the \(n_{ij}\) represents the \(j\)-th nearest neighbor of \(i\) using a distance (often euclidean distance) and \(l_{i,n_{ij}}\) is a boolean function that takes value \(\frac{1}{j}\) when \(i\) and \(n_{ij}\) are not in the same cluster and zero otherwise. This metric is called the Connectivity metric. \[Connectivity=\sum_{i=1}^{N}\sum_{j=1}^{M}I_{i,n_{ij}} \tag{4}\]  
where \(N\) is the number of elements to group into \(K\) clusters and \(M\) is a parameter that determines the number of neighbors that contribute to the Connectivity measure, fixed to ten in this research as established in [39]. The Connectivity metric is equal to or higher than zero and the lower the value the better the clustering trained so must be minimized. Compactness is related to cluster cohesion or homogeneity, measuring how close are the objects within the same cluster, usually by looking at the intra-cluster or within-cluster variance. A lower within-cluster variation is an indicator of good compactness, and, hence, a good clustering. So compactness must be minimized. The different indices for evaluating the compactness of clusters are based on distance measures, such as the cluster-wise within average/median distances between observations. Separation measures how well-separated a cluster is from other clusters quantifying the degree of separation between clusters, usually by measuring the minimum distance between cluster centroids or the pairwise minimum distances between objects in different clusters. Therefore, separation must be maximized. When the number of clusters increases, by definition compactness and separation used decrease. To manage this trade-off, some methods combine the two measures into a single score. The Dunn index [40] and Silhouette width [41] are both examples of non-linear combinations of compactness and separation. The Dunn index aims to identify dense and well-separated clusters. It is defined as the ratio between the minimal inter-cluster distance to maximal intra-cluster distance.