Jeffreys centroid of positive vectors and categorical probability distributions.

jcent() is an S3 generic for computing the Jeffreys centroid of R objects, with methods for the following classes:

igcmatcatdist
igcmatcatdistn

Usage

jcent(p, weight = NULL, base = exp(1), ...)

Arguments

p: An object of classes igcmatcatdist or igcmatcatdistn, i.e., a positive or stochastic matrix (for the former, rows sum to one).
weight: A vector of weights for the rows of p.
base: A positive value, the base of the logarithm to be used to compute Jeffreys divergence.
...: Additionnal arguments for other methods.

Value

A positive vector (object of class igccatdist) if p is of class igcmatcatdist or a categorical probability distribution (object of class igccatdistn), approximation of the Jeffreys centroid of the rows of p.

Details

If input p is a positive matrix (object of class igcmatcatdist), jcent()returns the positive vector minimizing the weighted mean of Jeffreys divergences with the rows ofp. More precisely, by denoting \eqn{p_1, \cdots, p_n} the rows of pand \eqn{w_1, \cdots, w_n} the components of the vectorweight, the output of jcent() is the positive vector $p^*$ satisfying $$p^* \in \textrm{Argmin} \left\{ \sum_{i=1}^n w_i \mathbb{J}(p_i:q) \ | \ q \textrm{ pos. vector} \right\},$$ where $\mathbb{J}((p_i:q)$ denotes the Jeffreys divergence between $p_i$ and $$q$$; see jdiv() for details. The minimizer $p^*$ has a closed-from expression given by $$p^*(x) = \frac{a(x)}{W( \exp( \log(b) a(x)/g(x)))},$$ where $W$ denotes the Lambert function, $a = \sum w_i p_i$ is the arithmetic centroid of the vectors $p_i$, $g = \prod p_i^{w_i}$ is their geometric centroid and $b >0$ is the logarithm basis used to compute the Jeffreys divergence.

If input p is a stochastic matrix (object of class igcmatcatdistn), the output is the probability distribution minimizing the weighted mean of Jeffreys divergences with the rows of p; formally, $$p^* \in \textrm{Argmin} \left\{ \sum_{i=1}^n w_i \mathbb{J}(p_i:q) | q >= 0 \sum_{x} q(x) =1 \right\}.$$ No closed-form expression exists for $p^*$ is this case. Several methods are implemented so as to compute numerical approximations; see Details section in jcent.igcmatcatdistn().

References

Nielsen, F. (2013). Jeffreys Centroids: A Closed-From Expresion for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms. IEEE Signal Processing Letters, 20(7), 657–660. doi:10.1109/LSP.2013.2260538

Nielsen, F. (2024). Fast Proxy Centers for the Jeffreys Centroid: The Jeffreys-Fisher-Rado Center and the Gauss-Bregman Inductive Center. Entropy, 26, 1008-1037. doi:10.3390/e26121008

Examples

u <- rep(1/3, 3)
prob <- c(0.9, 0.05, 0.05)
p <- matrix(c(u, prob),
             nrow = 2, ncol =3, byrow = TRUE)
p <- igcmatcatdistn(p)
meths <- c("mag", "njc", "jfr", "bsa", "gbic")
t(sapply(meths, function(m) jcent(p, method = m)))
#>           [,1]      [,2]      [,3]
#> mag  0.6481447 0.1759276 0.1759276
#> njc  0.6468378 0.1765811 0.1765811
#> jfr  0.6484677 0.1757661 0.1757661
#> bsa  0.6484927 0.1757537 0.1757537
#> gbic 0.6484681 0.1757659 0.1757659