Compute Jeffreys centroids of a set of normalized categorical distributions.

Method for the generic jcent() and the class igcmatcatdistn.

Usage

# S3 method for class 'igcmatcatdistn'
jcent(p, weight = NULL, base = exp(1), method = "gbic", tol = 1e-08, ...)

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.

method

A character string specifying the method to be used for computing the approximation of Jeffreys centroid. Possible values are:

"mag" for the mean of arithmetic and normalized geometric centroids;
"njc" for the normalized Jeffreys centroid of positive vectors;
"jfr" for the Jeffreys-Fisher-Rao centroid, i.e., the Fisher-Rao mean of arithmetic and normalized geometric centroids;
"bsa" for a single-value binary search alogrithm based on Lagrangian optimization;
"gbic" for the Gauss-Bregman inductive centroid. Default is "gbic". See Details section.

tol

For igcmatcatdistn objects only. A positive value, the tolerance in the approximation of Jeffreys centroid. Default is 1e-8.

...

Additionnal arguments for other methods.

Details

Let $p_1, \cdots, p_n$ denote the rows of p and $w_1, \cdots, w_n$ the components of the vector weight. The output of jcent() is a numerical approximation of the probability distribution $p^*$ (object of class igccatdistn) satisfying $$p^* \in \textrm{Argmin} \left\{ \sum_{i=1}^n w_i \mathbb{J}(p_i:q) \ | \ q >= 0 \sum_{x} q(x) =1 \right\},$$ where $\mathbb{J}((p_i:q)$ denotes the Jeffreys divergence between $p_i$ and $q$; see jdiv() for details. Nielsen (2013) establishes using Lagrangian multipliers method that $p^*$ is of the form $$p^*(x) = \frac{a(x)}{W( \exp( \log(b) (\lambda + 1) a(x)/g(x)))},$$ 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, $b >0$ is the logarithm basis used to compute the Jeffreys divergence and $\lambda$ is the unique real number such that $$\mathbb{K}(p^* | g) = - \lambda.$$ There is no known closed-form expression for $\lambda$.

The argument method identifies the method to be used to compute a numerical approximation of $\lambda$ or directly of $p^*$. More precisely, if:

method = "bsa", a numerical approximation of $\lambda$ satisfying the last equation is computed thanks to a binary search. The argument tol refers to the numerical tolerance for this approximation. Then, the corresponding probability distribution $p^*$ is computed. Due to the computational cost of evaluating the Lambert function, this method can be timec-consuming, particularly when the state space (i.e., the number of columns of p) is large;
method = "gbic", an iterative procedure called Gauss-Bregman inductive center is performed; see Nielsen (2024). It consists in iteratively computing arithmetic and geometric means of centroids until convergence. It is by far computationally less expansive than "bsa", with a similar quality of approximation. Its excellent performances make it the default method used by jcent();
method = "mag", the mean $(a+g)/2$ of the arithmetic and geometric centroids is returned; it corresponds to the first iteration of "gbic";
method = "jfr", the Fisher-Rao midpoint of the arithmetic and geometric centroids is returned, i.e., the normalized version of $(\sqrt{a} +\sqrt{g})^2$;
method = "njc", the normalized version of the positive vector Jeffreys centroid is returned.

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