Top feature interactions visualization with rank and lambda penalty

Visualizes the most influential feature interactions (based on the L2 norm) from Ridge Redundancy Analysis (RRDA) as a heatmap.

Let the (rank-$r$ truncated) decomposition of $\hat{B}(\lambda)$ be $$\hat{B}(\lambda, r) = U_{\hat{B}(\lambda)} \, D_{\hat{B}(\lambda)} \, V_{\hat{B}(\lambda)}^{\prime}.$$

The following three biplot scalings are defined:

Symmetric scaling (default): $$\tilde{F} = U_{\hat{B}(\lambda)} \, D_{\hat{B}(\lambda)}^{1/2}, \qquad \tilde{G} = V_{\hat{B}(\lambda)} \, D_{\hat{B}(\lambda)}^{1/2}.$$

X scaling: $$\tilde{F} = U_{\hat{B}(\lambda)} \, D_{\hat{B}(\lambda)}, \qquad \tilde{G} = V_{\hat{B}(\lambda)}.$$

Y scaling: $$\tilde{F} = U_{\hat{B}(\lambda)}, \qquad \tilde{G} = V_{\hat{B}(\lambda)} \, D_{\hat{B}(\lambda)}.$$

In all three cases, $\hat{B}(\lambda, r) = \tilde{F} \, \tilde{G}^{\prime}.$

Variable importance is scored by the row-wise $\ell_2$-norms: $$s_i^{(\tilde{F})} = \| \tilde{F}_{i,\cdot} \|_2, \qquad s_j^{(\tilde{G})} = \| \tilde{G}_{j,\cdot} \|_2.$$

Selecting the top $m_x$ predictors and $m_y$ responses yields the submatrices of the scaled factor matrices (each with $r$ columns).

The reduced coefficient submatrix is then $$\hat{B}_{\mathrm{sub}}(\lambda, r) = \tilde{F}_{\mathrm{sub}} \, \tilde{G}_{\mathrm{sub}}^{\prime}.$$

The matrix $\hat{B}_{\mathrm{sub}}(\lambda, r)$ retains the dominant low-rank structure and is visualized as a heatmap (with $m_x = m_y = 20$ by default).

Usage

rrda.top(
  Y,
  X,
  nrank = NULL,
  lambda = NULL,
  mx = 20,
  my = 20,
  scaling = c("symmetric", "none", "x", "y"),
  title = TRUE
)

Arguments

Y: A numeric matrix of response variables.
X: A numeric matrix of explanatory variables.
nrank: Integer rank $r$ of $\hat{B}$ to visualize. If NULL (default), it is set to min(5, min(dim(X), dim(Y))).
lambda: A numeric vector of ridge penalty values. If NULL (default), it is set to 1.
mx: Integer; number of top $X$-features (predictors) to display. Defaults to 20.
my: Integer; number of top $Y$-features (responses) to display. Defaults to 20.
scaling: Character string specifying how to apply the singular values from the compositions of $\hat{B}(\lambda)$ when constructing the biplot factors. Options are: "symmetric" (default) distributes singular values evenly to both sides (balanced scaling), "x" applies them fully to the X (left) side, "y" applies them fully to the Y (right) side, and "none" removes them (no singular value weighting).
title: Figure title. If TRUE (default), a formatted title is used. If FALSE or NULL, no title is drawn. If a single string, it is passed through to the figure title.

Value

A list with elements: heatmap (pheatmap object), B_sub (mx x my matrix), top_x, top_y, b1_sub, b2_sub, fit, scaling.

Examples

set.seed(10)
simdata<-rdasim1(n = 10,p = 50,q = 50,k = 3) # data generation
X <- simdata$X
Y <- simdata$Y
rrda.top(Y=Y,X=X,nrank=5,lambda=1,mx=20,my=20)


if (FALSE) { # \dontrun{
### In practice, the parameters nrank and lambda should be selected by CV ###
cv_result<- rrda.cv(Y = Y, X = X, maxrank = 5, nfold = 5) # cv
best_lambda<-cv_result$opt_min$lambda
best_rank<-cv_result$opt_min$rank
rrda.summary(cv_result = cv_result)

rrda.top(Y=Y,X=X,nrank=best_rank,lambda=best_lambda,mx=20,my=20)
} # }