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Jeffreys Divergence

Source: R/divergences_generic.R
jdiv.Rd

jdiv() is an S3 generic for computing the Jeffreys divergence – or symmetrized Kullback-Leibler divergence – between R objects, with methods for the following classes:

  • igccatdist

  • igcmatcatdist

Usage

jdiv(p, q, base = exp(1))

Arguments

p, q

Two R objects of same class. Currently, there are methods for igccatdist and igcmatcatdist objects.

base

A positive value specifying the base of logarithm used for computing Jeffreys divergence. Default is exp(1) (natural logarithm).

Value

  • If p and q are igccatdist objects, returns the value of the Jeffreys divergence (or symmetrized Kullback-Leibler) between p and q;

  • if p and q are igcmatcatdist objects, returns a matrix of nrow(p) rows and nrow(q) columns whose coefficients are the Jeffreys divergence from rows of p to rows of q.

Details

The Jeffreys divergence or symmetric Kullback-Leibler divergence is a symmetric and bounded discrimination measure between positive measures defined on a measurable space. It is formally defined, for \(p\) and \(q\) two positive measures with same support, by $$\mathbb{J}(p,q) = \mathbb{K}\left(p: q \right) + \mathbb{K}\left(q:p \right),$$ where \(\mathbb{K}\) denotes the Kullback-Leibler divergence; see kldiv().

See also

kldiv(), jsdiv(), catdist

Examples

# Compute the Jeffreys divergence
# between two probability distributions with common support.
p <- igccatdist(dbinom(0:4, size = 4, prob = 0.5))
q <- igccatdist(rep(1/5, times = 5))
jdiv(p, q)
#> [1] 0.4521873
# Compute the (squared) matrix of Jeffreys divergences between
# the elements of set of probabilily distributions
p <- igccatdist(dbinom(0:4, size = 4, prob = 0.5))
q <- igccatdist(rep(1/5, times = 5))
r <- igccatdist(c(0.3, 0.4, 0.1, 0.1, 0.1))
matp <- matrix(c(p, q, r), nrow = 3, byrow = TRUE)
matptest <- igcmatcatdistn(p = matp)
jdiv(p = matptest)
#>           [,1]      [,2]      [,3]
#> [1,] 0.0000000 0.4521873 0.9615984
#> [2,] 0.4521873 0.0000000 0.3871201
#> [3,] 0.9615984 0.3871201 0.0000000
# Compute the matrix of Jeffreys divergences
# between two sets of probability distributions
matp2 <- matrix(c(dbinom(0:4, size = 5, prob = 0.3),
                      c(0.1, 0.1, 0.2, 0.2, 0.4)),
                    nrow = 2, byrow = TRUE)
matptest2 <- igcmatcatdist(matp2)
jdiv(p = matptest, q = matptest2)
#>           [,1]      [,2]
#> [1,] 0.2594416 0.9027330
#> [2,] 0.5102637 0.2772589
#> [3,] 0.4152278 1.1901285