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