Gaussian graphical modeling for spectrometric data analysis

Learning block structured graphs in Gaussian graphical models

“…However, the peculiarity of a graphical model is the possibility of investigating long-term interactions and we indeed found out some off-diagonal blocks. Our estimated structure of dependencies and the one reported in Codazzi et al (2022) are similar, even though there are some differences due to the different modelling assumptions. The work of Codazzi et al (2022) relied on a BDgraph method, which does not look for block structured graph.…”

“…The problem has already been addressed in Codazzi et al (2022), were data were firstly analyzed. From a mathematical point of view, a spectrum is a continuous function of Finally, note that, in this application, the support of the spline basis functions coincide with bands of the spectrum.…”

“…Therefore the problem of studying interactions between different substances simply translates in studying the dependencies between the basis expansion coefficients which can be read off from the graph. Codazzi et al (2022) used independent Bernoulli prior distributions on the graph edges and a BDgraph method to provide posterior inference on the graph and the precision matrix. As already discussed, this is a general non-informative setting that does not allow to include further prior information even when those are available.…”

“…Clearly, the nodes representing the basis expansion coefficients are ordered and so the groups are contiguous in the functional domain. The figure also shows the support of 𝑝 = 40 spline functions; the number of basis was chosen following Codazzi et al (2022), as a trade-off between good fitting of the smoothed curves and limited computational burden.…”

“…For example, in spectrometric data analysis, the goal is usually to investigate relationships among the substances within the compound. To this end, a nonparametric regression coupled with a Gaussian graphical model on basis expansion coefficients can be employed, providing an estimate of their conditional independence structure (Codazzi et al 2022). Since the elements of a B-Spline basis have compact support, the independence structure of the smoothing coefficients is reflected on portions of the spectrum, that is known a prior to be grouped in intervals of chemical interest.…”