---
title: "Ridge networks: effective penalty and edge selection"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Ridge networks: effective penalty and edge selection}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

The ridge precision estimator $(S + \lambda I)^{-1}$ has no zero entries.
For Gaussian data its partial correlations estimate those of
$(\Sigma + \kappa I)^{-1}$, where the effective penalty $\kappa$ exceeds
$\lambda$ (Arashi and Ebrahim, 2026, submitted). This vignette shows how to
estimate $\kappa$ and how to select edges. These functions come from the package
ridgegraph by Mohammad Arashi.

```{r}
library(ebrahim.gof)
set.seed(1)
p <- 30; n <- 20
theta <- diag(p)
for (i in 1:(p - 1)) theta[i, i + 1] <- theta[i + 1, i] <- 0.4
x <- matrix(rnorm(n * p), n, p) %*% chol(solve(theta))
effective_penalty(x, c(0.05, 0.5, 5))
effective_penalty_floor(x)
```

With $p > n$ the effective penalty does not fall below the floor however
small $\lambda$ is. Edges can be selected with the permutation threshold.

```{r}
sel <- mps_select(x, B = 99)
sum(sel$adjacency[upper.tri(sel$adjacency)])
```