Marginal distribution of a single correlation from an LKJ distribution

Source: R/lkjcorr_marginal.R

Marginal distribution for the correlation in a single cell from a correlation matrix distributed according to an LKJ distribution.

dlkjcorr_marginal(x, K, eta, log = FALSE)

plkjcorr_marginal(q, K, eta, lower.tail = TRUE, log.p = FALSE)

qlkjcorr_marginal(p, K, eta, lower.tail = TRUE, log.p = FALSE)

rlkjcorr_marginal(n, K, eta)

Arguments

x, q

vector of quantiles.

K

Dimension of the correlation matrix. Must be greater than or equal to 2.

eta

Parameter controlling the shape of the distribution

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Value

  • dlkjcorr_marginal gives the density

  • plkjcorr_marginal gives the cumulative distribution function (CDF)

  • qlkjcorr_marginal gives the quantile function (inverse CDF)

  • rlkjcorr_marginal generates random draws.

The length of the result is determined by n for rlkjcorr_marginal, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Details

The LKJ distribution is a distribution over correlation matrices with a single parameter, \(\eta\). For a given \(\eta\) and a \(K \times K\) correlation matrix \(R\):

$$R \sim \textrm{LKJ}(\eta)$$

Each off-diagonal entry of \(R\), \(r_{ij}: i \ne j\), has the following marginal distribution (Lewandowski, Kurowicka, and Joe 2009):

$$\frac{r_{ij} + 1}{2} \sim \textrm{Beta}\left(\eta - 1 + \frac{K}{2}, \eta - 1 + \frac{K}{2}\right) $$

In other words, \(r_{ij}\) is marginally distributed according to the above Beta distribution scaled into \((-1,1)\).

References

Lewandowski, D., Kurowicka, D., & Joe, H. (2009). Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis, 100(9), 1989--2001. doi:10.1016/j.jmva.2009.04.008 .

See also

parse_dist() and marginalize_lkjcorr() for parsing specs that use the LKJ correlation distribution and the stat_slabinterval() family of stats for visualizing them.

Examples


library(dplyr)
library(ggplot2)
library(forcats)

theme_set(theme_ggdist())

expand.grid(
  eta = 1:6,
  K = 2:6
) %>%
  ggplot(aes(y = fct_rev(ordered(eta)), dist = "lkjcorr_marginal", arg1 = K, arg2 = eta)) +
  stat_slab() +
  facet_grid(~ paste0(K, "x", K)) +
  labs(
    title = paste0(
      "Marginal correlation for LKJ(eta) prior on different matrix sizes:\n",
      "dlkjcorr_marginal(K, eta)"
    ),
    subtitle = "Correlation matrix size (KxK)",
    y = "eta",
    x = "Marginal correlation"
  ) +
  theme(axis.title = element_text(hjust = 0))