# On the Role of Clustering and Visualization Techniques in Gene Microarray Data
## 2 Clustering for Microarray Gene Expression Data
### Hierarchical Clustering  
The primary characteristic of hierarchical techniques is that they gradually combine points or separate superclusters. Indeed, on these grounds, we can recognize two diverse kinds of hierarchical techniques: _agglomerative_ and _divisive_. In the following discussion, we focus mainly on the agglomerative strategy. The rationale is to begin with a number of clusters corresponding exactly to the cardinality of a data set, and then by merging continuously to reduce their number. Let us assume that the set of data points \(S\) consists of N samples, \(S=\{\mathbf{t}_{1},\mathbf{t}_{2},\ldots,\mathbf{t}_{N}\}\subset\mathbb{R}^{D}\). Notwithstanding their variety, any hierarchical clustering technique is conceptually the same and can be defined in a concise manner as follows: begin with \(N\) clusters by assigning each sample to a distinct cluster and continue with this original cluster setup by combining the nearest clusters. Namely, if \(H\) and \(T\) are the identified two closest clusters, consider a new cluster \(\{H,T\}\) and decrease by one the total number of clusters. The whole procedure repeats until a minimal number of the clusters has been reached. The result of these steps is a partition hierarchy of the data points. Despite the ease of the strategy, defining how to compute a distance between a pair of clusters in not immediate. Among the number of alternatives to choose from, the three most common policy employed in practice are [34; 35; 36]:  
* single linkage method: The closeness between \(H\) and \(T\) is calculated on the basis of the minimum distance between the samples that belong to the respective clusters. * complete linkage method: The closeness between \(H\) and \(T\) is calculated as the distance between the most two distant samples, one from each cluster. * average linkage method: The closeness between \(H\) and \(T\) is calculated as the average of all the distances between pairs of samples, one from each cluster. The criterion considers all possible pairs of distances between samples in the clusters, and is thus far more accurate and resistant to outliers. The computational burden depends on the size of the data. More importantly, significant memory demands exist as the consecutive outcomes (clusters) must be maintained at each stage of the technique. Unlike partition-based clustering, which tries to straightly decompose a data set into a collection of disjoint groups, hierarchical clustering produces a hierarchical sequence of nested groups that can be graphically depicted by a tree called a dendrogram. A dendrogram's branches not only keeps track of the clusters' development, but also shows the cluster closeness. One can get a given number of clusters by cutting the dendrogram at a desired level. An important important application of an agglomerative strategy to microarray data are the Unweighted Pair Group Method with Arithmetic Mean (UPGMA) [37], where the authors conceived a technique of representing the clustered data set graphically. In particular, each cell of the matrix of gene expression is colored based on the measured fluorescence ratio and the matrix rows are reordered based on the structure of the hierarchical dendrogram and a rule for the node ordering. Once the clustering procedure is performed, a colored table, i.e., an image of the clusters, represents the initial gene expression matrix where big adjacent color patches reflect clusters of genes that share an analogous pattern of expressions over multiple conditions. The method described in [37] was much favored by many biologists and has become a popular tool in gene expression data analysis [37; 38; 25; 39]. Although some effective variant was proposed as, for instance, in dynamic agglomerative clustering [40], standard agglomerative approaches are, however, not sufficiently robust [41], meaning that even little perturbations in the data set could significantly alter the hierarchical dendrogram structure.