The Cholesky decomposition

library(targets)
library(ggplot2)
library(tidyverse)
library(tidybayes)

code

plot(density(tan(runif(5000,min = 0, max = pi/2))*.3, from = 0), 
     xlim = c(0, 100))

curve(dexp(x, rate = 3), add = TRUE, col = "red", xlim = c(0, 100))

curve(tan(x), xlim = c(-5,5))

hist(-log(runif(200)), probability = TRUE)
curve(dexp(x), add = TRUE)

eg <- rethinking::rlkjcorr(1, 2, 1)
cc <- chol(eg)

purrr::rerun(30,{
  zz <- matrix(data = rnorm(500), ncol = 2)
  # plot(zz)
  rr <- t(cc) %*% t(zz)
  # plot(t(rr))
  cor(t(rr))[1,2]
}) |> 
  purrr::flatten_dbl() |> density() |> plot()
abline(v=eg[1,2])

#' modelling the plasticity slopes (xik - xbar) for each evironmental variable?
#' what's correlated here? slopes and intercepts

eg <- rethinking::rlkjcorr(1, 2, 5)
cc <- chol(eg)

t(cc) %*% cc

          [,1]      [,2]
[1,] 1.0000000 0.6450641
[2,] 0.6450641 1.0000000

mm <- matrix(rnorm(4000, mean = 0, sd = 1), ncol = 2)

plot(mm)

yy <- t(t(cc) %*% t(mm))

plot(yy)

cor(yy)

          [,1]      [,2]
[1,] 1.0000000 0.6548221
[2,] 0.6548221 1.0000000

purrr::rerun(50,{
  zz <- matrix(data = rnorm(500), ncol = 2)
  # plot(zz)
  rr <- t(cc) %*% t(zz)
  # plot(t(rr))
  cor(t(rr))[1,2]
}) |> 
  purrr::flatten_dbl() |> density() |> plot()
abline(v=eg[1,2])

p <- -.8
L <- matrix(c(1,0, p, sqrt(1 - p^2)), ncol = 2, byrow = TRUE)
zz <- matrix(data = rnorm(1000), ncol = 2)
yy <- t(L %*% t(zz))
plot(yy)

cor(yy)

           [,1]       [,2]
[1,]  1.0000000 -0.7913319
[2,] -0.7913319  1.0000000

Another way to make correlation matrices is here: https://www.rdatagen.net/post/2023-02-14-flexible-correlation-generation-an-update-to-gencorgen-in-simstudy/