Notations

Notations	Explanation
$n$ or $N$	number of observations
$D$	number of variables
$d$	index of the variables in $[1, \dots, D]$
$K$	number of classes
$k$	index of the classes in $[1, \dots, K]$
$\mathcal{T}$	training set space
$\mathcal{X}$	space of variables
$\mathrm{X}$	random variable from $\mathcal{X}$
$x$	realization of $\mathrm{X}$
$\mathbf{X}$	matrix of variables in $\mathbb{R}^{n \times D}$
$\boldsymbol{x}_i$	column vector of the $i^{th}$ observation
$x_{id}$	scalar of the $i^{th}$ observation and the $d^{th}$ variable of $\mathbf{X}$
$\mathcal{Y}$	space of response
$\mathrm{Y}$	random variable from $\mathcal{Y}$
$y$	realization of $\mathrm{Y}$
$\mathbf{y}$	observed response vector in $\mathbb{R}^n$
$y_i$	scalar of the $i^{th}$ observation of $\mathbf{y}$
$\eta_i$	linear predictor of $i^{th}$ observation
$f : X \rightarrow Y$	some fixed but unknown function of $\mathbf{X}$
$s(x)$	smooth monotonic function of $x$
$||.||_1$	$\ell_1$ norm
$||.||_2^2$	$\ell_2$ squared norm
$\xi$	knot
$K_d$	number of knots of variable $d$
$\mathrm{B}(x)$	B-spline basis functions of $x$
$m$	polynomial order
$\mathbf{B}$	B-spline basis function matrix in $\mathbb{R}^{n \times (k + m +1)}$
$N(x)$	natural spline basis functions of $x$
$\mathbf{N}$	natural splines basis function matrix in $\mathbb{R}^{n \times n}$
$\mathbf{W}$	penalty matrix of natural splines in $\mathbb{R}^{n \times n}$
$\Omega$	training set to classify
$S$	number of levels in the hierarchical tree of $\Omega$
$s$	index of the levels of the hierarchy in $[1, \dots, S]$
$h_s$	height of the $s^{th}$ level of the hierarchy
$\mathcal{G}$	group partition of all $S$ levels of the hierarchy
$G_s$	number of groups at $s^{th}$ level of the hierarchy
$\hat{G}_s^*$	optimal number of groups
$g$	index of number of groups in $[1, \dots, G_s]$
$\mathcal{G}^s$	a partition of $\Omega$ at the $s^{th}$ level
$\mathcal{G}^s_g$	$g^{th}$ group of variables at the $s^{th}$ level
$\rho_s$	weights attributed to group partition $\mathcal{G}^s$
$\tilde{\mathbf{X}}^{(s)}$	matrix of supervariables at $s^{th}$ level of the hierarchy
$\tilde{\mathbf{X}}^{(s)}_{\mathcal{G}^s}$	matrix of supervariables for partition $\mathcal{G}^s$
$\mathbf{X}_{\mathcal{G}}$	variables matrix of concatenated group partition
$\mathbf{X}^s_{\mathcal{G}^s}$	variables matrix of group partition $\mathcal{G}^s$
$\mathit{G}$	genomic view
$\mathbf{G}$	matrix of genotype data
$\mathbf{Z}$	matrix of additively coded SNPs
$\mathit{M}$	metagenomic view
$\mathbf{M}$	matrix of metagenomic data
$\mathcal{M}$	group structure of metagenomic data
$\boldsymbol{\Delta}_{\mathit{G}\mathit{M}}$	matrix of interaction terms between genome and metagenome data