Graphical Lasso for Coloured Gaussian Graphical Models for Paired Data

Implements methods for coloured Gaussian graphical models with equality "R"estrictions on "CON"centration values (RCON models; Højsgaard and Lauritzen (2008) ) for paired data (pdRCON models). Provides an Alternating Direction Method of Multipliers (ADMM) algorithm for solving the penalized likelihood problem introduced by Ranciati and Roverato (2024) . Provides functions for computing maximum likelihood estimates and generating simulated pdRCON models and datasets.

Graphical Lasso for Coloured Gaussian Graphical Models for Paired Data

How to install the package:

library(devtools)
install_github("savranciati/pdglasso", build_vignettes=FALSE)
library(pdglasso)

Relevant papers for the package

• Ranciati, S., Roverato, A., (2024). "On the application of Gaussian graphical models to paired data problems", Statistics and Computing, doi: https://doi.org/10.1007/s11222-024-10513-6
• Ranciati, S., Roverato, A., (2024). "On the application of Gaussian graphical models to paired data problems", arXiv pre-print, https://arxiv.org/abs/2307.14160 [please check latest version]

Code reproducibility and data availability

Updated version

The .zip folder `additional simulations' contains ten subfolders with further results concerning 10 different independent replicated runs from the same simulation scheme. These are related to an ongoing review process for a submitted version of the paper so they might slightly differ from simulations of the arXiv version.

arXiv version

Inside this repository we made publicly available a .zip file containing:

• raw_data, a folder containing the original data and some metadata information;
• breast_cancer.RData, the dataset used in Section 6 of the manuscript
• breast_cancer_analysis.r, the R script to reproduce the analysis of Section 6 of the manuscript.

Also, for replicability, we deposit a .zip version of the entire package and environment 'freezed' at the submission of the arXiv version of the paper.

Package overview

An RCON model for paired data (pdRCON model) is a coloured Gaussian Graphical Model (GGM) where the $p$ variables are partitioned into a Left block $L$ and a right block $R$. Every variable in the left block has an homologous variable in the right block and certain types of equality Restrictions on the entries of the CONcentration matrix $K$ are allowed. Every pdRCON model is uniquely represented by a Coloured Graph for Paired Data (pdColG) implemented in the form of a $p\times p$ symmetric matrix, where every entry is one of the values 0, 1 or 2, as follows:

• The diagonal entries of the pdColG matrix are all equal to either 1 or 2. More specificaly, every variable in $L$ has an homologous variable in $R$ and the corresponding diagonal entries of $K$ can be constrained to have equal value, and in this case the corresponding entries of the pdColG matrix are set to 2. These are referred to as coloured vertices whereas the unconstrained diagonal entries of $K$ are encoded by the value 1 and are referred to as uncoloured vertices.

• The off-diagonal entries of the pdColG matrix are equal to 0 for missing edges and either 1 or 2 for present edges, where the value 2 is used to encode coloured edges as detailed below.

• For every pair of variables in $L$ there exists an homologous pair of variables in $R$, thereby identifying a pair of homologous edges. If both edges are present in the graph the corresponding off-diagonal entries of $K$ can be constrained to have equal value. These type of edges are referred to as coloured symmetric inside-block edges and the corresponding entries of the pdColG matrix are set to 2. On the other hand, if both edges are present but the corresponding parameters are unconstrained then they form a pair of uncoloured symmetric inside-block edges.

• We say that two variables are across-block if one variable belongs to $L$ and the other to $R$. For every pair of non-homologous across-block variables there exists an homologous pair across-block variables, thereby identifying a pair of homologous edges. If both edges are present in the graph the corresponding off-diagonal entries of $K$ can be constrained to have equal value. These type of edges are referred to as coloured symmetric across-block edges and the corresponding entries of the pdColG matrix are set to 2. On the other hand, if both edges are present but the corresponding parameters are unconstrained then they form a pair of uncoloured symmetric across-block edges.

The functions of this package make it possible to specify different types of pdRCON submodels of interest through the arguments type and force.symm which can both take as value any subvector of the character vector c("vertex", "inside.block.edge", "across.block.edge"); note that the names of the components can be abbreviated down, up to the first letter only, and are not case-sensitive. The argument type cannot be NULL and:

• if type contains the string "vertex" then coloured vertex symmetries are allowed and, if in addition also force.symm contains the string "vertex", then all vertices are coloured.

• if type contains the string "inside.block.edge" then coloured inside- block edge symmetries are allowed and, if in addition also force.symm contains the string "inside.block.edge", then only coloured edges are allowed inside blocks.

• if type contains the string "across.block.edge" then coloured across-block edge symmetries are allowed and, if in addition also force.symm contains the string "across.block.edge", then only coloured edges are allowed across blocks, with the exception of edges joining across-block homologous variables.

For the in-package documentation please check:

help("pdglasso-package")