Standardization of Quantitative Variables in ClustOfVar-Based Variable Clustering for Mixed Data

In many real-world datasets, individuals are described by both quantitative
and qualitative variables. Clustering variables rather than individuals
can provide valuable insight into the underlying structure of such
mixed datasets. ClustOfVar is a technique for clustering variables proposed
by [2] and widely used for mixed data, with both hierarchical and
partitioning algorithms. The method aims to maximize cluster homogeneity,
which measures the strength of association between each variable and a
synthetic numerical variable representing the cluster. This synthetic variable
corresponds to the first principal component obtained from Principal
Component Analysis for Mixed Data (PCAMIX) applied to the variables
within the cluster [1, 3].
As in many clustering procedures, data preprocessing plays a crucial
role. In ClustOfVar, the algorithm operates on normalized data.
Quantitative variables are standardized by centering and scaling, while
qualitative variables are standardized through the normalization of their
indicator matrix. This step ensures that variables contribute equally to
the analysis and prevents variables with larger scales from dominating the
clustering process. However, categorical variables may implicitly induce
dependencies between quantitative variables. In particular, two standardized
quantitative variables may appear correlated simply because they are
both associated with the same categorical variable. As a result, the observed
correlation may reflect the influence of the categorical grouping
rather than a genuine relationship between the quantitative variables. To address this issue, we propose two local standardization strategies
within the ClustOfVar framework, to reduce spurious correlations
induced by categorical variables. The first approach performs conditional
standardization within the categories of a predefined qualitative variable.
The second approach relies on a consensus partition derived from all qualitative
variables. To construct this consensus partition, we use the Clusterbased
Similarity Partitioning Algorithm (CSPA) [4]. This algorithm combines
multiple partitions into a single consensus clustering by building a
co-association similarity matrix between individuals. Once the consensus
partition is obtained, quantitative variables are standardized locally
within each consensus group. The impact of these local standardization
strategies on the ClustOfVar clustering results is evaluated through a comparative
experimental study on several datasets using multiple clustering
quality metrics. These strategies aim to mitigate spurious correlations induced
by categorical variables and to better capture intrinsic relationships
between quantitative variables.
Preliminary experiments indicate that the clusters obtained may depend
on the chosen standardization procedure. In particular, local standardization
can lead to improved clustering quality. As an illustrative
example, on the well-known mtcars dataset, the cophenetic coefficient is
consistently higher when local standardization is applied.