Empirical Density Estimation based on Spline Quasi-Interpolation with
  applications to Copulas clustering modeling
Cristiano Tamborrino Antonella Falini Francesca Mazzia
Density estimation is a fundamental technique employed in various fields to
model and to understand the underlying distribution of data. The primary
objective of density estimation is to estimate the probability density function
of a random variable. This process is particularly valuable when dealing with
univariate or multivariate data and is essential for tasks such as clustering,
anomaly detection, and generative modeling. In this paper we propose the
mono-variate approximation of the density using spline quasi interpolation and
we applied it in the context of clustering modeling. The clustering technique
used is based on the construction of suitable multivariate distributions which
rely on the estimation of the monovariate empirical densities (marginals). Such
an approximation is achieved by using the proposed spline quasi-interpolation,
while the joint distributions to model the sought clustering partition is
constructed with the use of copulas functions. In particular, since copulas can
capture the dependence between the features of the data independently from the
marginal distributions, a finite mixture copula model is proposed. The
presented algorithm is validated on artificial and real datasets.