one based on capitalization given above.

A large number of hierarchical clustering procedures can be found in the literature. For a review about the classical techniques see, for instance, Anderberg (1973). In this paper we focus our attention on the single linkage cluster analysis (SLCA) and average linkage cluster analysis (ALCA).\
The starting point of both the procedures is the empirical correlation matrix ${\bf C}$. The following procedure performs the ALCA giving as an output a hierarchical tree and a filtered correlation matrix ${\bf{C}}^<_{ALCA}$ :

1.  Set ${\bf{B}} = {\bf{C}}$.

2.  Select the maximum correlation $b_{hk}$ in the correlation matrix ${\bf{B}}$. Note that $h$ and $k$ can be simple elements (i.e. clusters of one element each) or clusters (sets of elements). $\forall \, i \in h$ and $\forall \, j \in k$ one sets the elements $\rho^<_{ij}$ of the matrix ${\bf{C}}^<_{ALCA}$ as $\rho^<_{ij}=\rho^<_{ji}=b_{hk}$.

3.  Merge cluster $h$ and cluster $k$ into a single cluster, say $q$. The merging operation identifies a node in the rooted tree connecting clusters $h$ and $k$ at the correlation $b_{hk}$.

4.  Redefine the matrix ${\bf{B}}$: $$\begin{aligned}
     \label{negspin}
    \left \{  \begin{aligned}
            &   b_{qj}= \frac{n_h\,b_{hj}+n_k \, b_{kj}}
                                   {n_h+n_k}  & 
                               ~~~~\text{ if } j\notin h \,{\rm{and}} \, j\notin 