## 3 Results and Discussion
### HMCL Clustering Results  
Fifty runs of the NMF algorithm varying the rank value in the interval 2-10 were executed [42; 43], and both the consensus matrices and the cophenetic correlation coefficient were collected from the analysis of the encoding matrix \(H\). Whilst the consensus matrix gives a qualitative measurement of the cluster quality, the cophenetic correlation coefficient is a quantitative measure, and its trend is used to choose the optimal rank (i.e., the number of clusters). Figure 4 shows the cophenetic correlation coefficient trend in the interval 2-10 (Each point represents the mean value of the cophenetic correlation coefficient on the fifty runs). From value five we observed a significant reduction of this trend, so it has been chosen as factorization rank. Figure 4: Cophenetic index values in 2–10, as a function of the factorization rank. Figure 3: Visual representation of the data matrix. Different colors are used to represent the gene expression level: yellow grades for high expression, green grades for medium expression, blue grades for low or no expression. The consensus matrix in Figure 5 represents the HMCLs (rows and columns) grouped according to the metagene expression profile similarities. Five clusters (yellow squares) are depicted showing some regularities within the groups. The algorithm proved to be stable and it converged to the same solution in all 50 runs. This result is shown in Figure 5 where the squares are clearly separated and only yellow and blue values are present. It is known that NMF allows a part-based representation of the original data by a linear additive composition of parts that can be interpreted as building blocks. In this scenario, the 'blocks' are the metagenes. Figure 6 shows the HMCLs composition in terms of metagenes in each of the five groups that have been returned from the consensus matrix. The whole bar is given by the sum of the single metagene bars, expressed as a percentage. The emerging patterns are now clearer. The dimensionality reduction and the clustering have made it possible. They allow the study of both the metagene expression profiles through the HMCLs and the different behaviors of the metagenes among the clusters. This simplified representation helps the domain experts in better understanding the relationships hidden in data. It is a starting point for conducting the biological analysis. Figure 6 shows that the HMCLs in the first cluster are characterized by a high expression of the first metagene (dark blue bars), and the HMCLs in the second cluster from high expression of the third metagene (green). The HMCLs in the third cluster are characterized by a high expression of the fourth metagene (yellow) bars, and the bars in clusters four and five have very high expression of the metagenes two (light blue) and five (dark yellow), respectively. Furthermore, it is noteworthy that, even if the metagenes two and four are highly expressed in clusters four and three, they are always expressed in all the HMCLs. Note that even if this representation has allowed the identification of the most expressed metagenes per HMCLs, the analysis has to take into account the less expressed metagenes. Indeed, all the extracted metagenes contribute to defining each HMCL, thus they cannot be removed as noise. In fact, the dimensionality reduction has already  
Figure 5: Consensus matrix obtained with 50 runs of non-negative matrix factorization (NMF) and \(k=5\). removed the noise from data, whilst it has highlighted the relationships among genes and HMCLs. The emerged patterns are then used by the domain experts, involved in the data analysis process (biologists and physicians in this case), to better understand their role in arachidonic acid metabolism, as it is described in Section 3.4. The expression profiles of the single metagenes are depicted (same colors code used before) in Appendix A, reporting the same behavior summarized in Figure 6.  
Moreover, visual analysis allows us to detect some subgroups within the groups.