## 5 Experiments
### Creating Compact Maps through Clustering  
This section focuses on the evaluation of clustering methods to compact the information contained in a set of global appearance descriptors. To carry out the experiments, two clustering methods were studied for each environment, and three global appearance descriptors were considered. The first \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset Name** & \begin{tabular}{c} **Number** \\ **of Images** \\ \end{tabular} & \begin{tabular}{c} **Number** \\ **of Rooms** \\ \end{tabular} \\ \hline \begin{tabular}{c} _QuorumV\_training_ \\ \end{tabular} & 872 & 6 \\ \hline \begin{tabular}{c} _QuorumV\_test_ \\ \end{tabular} & 77 & \\ \hline \begin{tabular}{c} _Freiburg\_training_ \\ \end{tabular} & 519 & 9 \\ \hline \begin{tabular}{c} _Freiburg\_test_ \\ \end{tabular} & 52 & \\ \hline \
\begin{tabular}{c} _Saarbrucken\_training_ \\ \end{tabular} & 566 & 8 \\ \hline
\begin{tabular}{c} _Saarbrucken\_test_ \\ \end{tabular} & 57 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Datasets used to carry out the experiments. Figure 5: Bird’s eye view of the COsy Localization Database (COLD). (**a**) Freiburg and (**b**) Saarbrücken environment. Extracted from [[https://www.nada.kth.se/cas/COLD/](https://www.nada.kth.se/cas/COLD/)]([https://www.nada.kth.se/cas/COLD/](https://www.nada.kth.se/cas/COLD/)). method (Method 1) consists of spectral clustering along with k-means as was explained in Section 3.1. Other configurations were tested, such as to use of SOM instead of k-means to solve Step 5 of the spectral clustering, but the results were quite similar; thus, only the spectral clustering along with k-means to cluster the normalized matrix of the \(n_{c}\) eigenvectors is shown. The second method (Method 2) consists of the use of SOM, which was explained in Section 3.2. Therefore, for the two proposed methods, several experiments were carried out to study the influence of the parameters of the three global appearance descriptors. Table 2 summarizes the experiments developed. The values \(k_{1}\), \(k_{2}\), and \(k_{3}\) define the length of each descriptor, but their meaning is not the same (equal values of \(k_{1}\), \(k_{2}\), and \(k_{3}\) would not lead to the same descriptor size). Therefore, as our aim is to study the correct tuning of these values to use each descriptor as efficiently as possible, we do not apply the same values for all the descriptors in the experiments. Once the compact map has been produced, it may be interesting to provide some measures that permit quantifying the compactness of the map. In this context, the concept of the silhouette is commonly used. Silhouette values point out the degree of similarity between the instances within the same cluster and at the same time the dissimilarity with the instances that belong to other clusters. The silhouette takes values in the range \([-1,1]\), and it provides information about how compact the clusters are. Therefore, in order to quantify the goodness of each method, three parameters are considered:  
* The average moment of inertia of the cluster. * The average silhouette of the points. * The average silhouette of the descriptors. These values are collected after the clustering process. As for the moment of inertia, it measures the compactness of the clusters (if the clusters group images captured from geometrically-close points) and is calculated as:  
\[M=\sum_{i=1}^{n_{c}}\frac{\sum_{j=1}^{n_{i}}dist((x,y)_{r_{i}},(x_{j},y_{j}))^ {2}}{n_{i}} \tag{7}\]  
where \(dist((x,y)_{r_{i}},(x_{j},y_{j}))\) is the Euclidean distance between the coordinates of the representative \(\overrightarrow{r_{i}}\) and the position of the jth image that belongs to the cluster \(C_{i}\), and \(n_{i}\) is the number of images within this cluster. As for the silhouettes values, two types of silhouette are used: the average silhouette of points is defined as:  
\[S_{points}=\frac{\sum_{y=1}^{N}s_{w}}{N} \tag{8}\]