Package 'kantorovich'

Kantorovich Distance Between Probability Measures

Description

Computes the Kantorovich distance between two probability measures on a finite set, also known as the earth mover's distance. The Kantorovich distance is not a "unique" distance: it is defined by a given distance on the two finite sets (generally equal). Note that the default distance is the 0-1 distance and with this choice the Kantorovich computation is totally useless (see the vignette). Computing the Kantoroich distance is a linear programming problem, and several methods are provided in the package. In particular there is an exact method available when the probability weights are rational numbers and when the distances are rational numbers as well. A benchmark suggests that the faster methods are those using the 'CVXR' package.

To learn more, start with the vignettes: browseVignettes(package="kantorovich").

If you encounter a bug, or if you have a suggestion to improve the package, please file an issue on the Github repo https://github.com/stla/kantorovich.

Details

Package:	kantorovich
Type:	Package
Version:	3.1.0
Date:	2023-08-22
License:	GPL-3

Author(s)

Stéphane Laurent

---

Extremal distances

Description

Compute the distances at the extreme joinings.

Usage

edistances(mu, nu, dist = NULL, ...)

Arguments

mu	(row margins) probability measure in numeric or bigq/character mode
nu	(column margins) probability measure in numeric or bigq/character mode
dist	function or matrix, the distance to be minimized on average. If NULL, the 0-1 distance is used.
...	arguments passed to dist

Value

A list with two components: the extreme joinings in a list and the distances in a vector.

Note

This function, called by kantorovich, is rather for internal purpose.

---

Extreme joinings

Description

Return extreme joinings between mu and nu.

Usage

ejoinings(mu, nu, zeros = FALSE)

Arguments

mu	(row margins) probability measure in numeric or bigq/character mode
nu	(column margins) probability measure in numeric or bigq/character mode
zeros	logical; in case when mu and nu have different lengths, set FALSE to remove lines or columns full of zeros

Value

A list containing the extreme joinings (matrices).

Examples

mu <- nu <- c(0.5, 0.5)
ejoinings(mu, nu)
# use exact arithmetic
library(gmp)
mu <- nu <- as.bigq(c(0.5,0.5))
ejoinings(mu, nu)
# different lengths example
mu <- setNames(as.bigq(c(1,2,4), 7), c("a", "b", "c"))
nu <- setNames(as.bigq(c(3,1), 4), c("b", "c"))
ejoinings(mu, nu)

---

Kantorovich distance

Description

Compute the Kantorovich distance between two probability measures on a finite set.

Usage

kantorovich(mu, nu, dist = NULL, details = FALSE, ...)

Arguments

mu	(row margins) probability measure in numeric or bigq/character mode
nu	(column margins) probability measure in numeric or bigq/character mode
dist	function or matrix, the distance to be minimized on average; if NULL, the 0-1 distance is used.
details	prints the joinings achieving the Kantorovich distance and returns them in the "joinings" attribute of the output
...	arguments passed to dist (only if it is a function)

Details

The function firstly computes all the extreme joinings of mu and nu, then evaluates the average distance for each of them, and then returns the minimal one.

Value

The Kantorovich distance between mu and nu.

Examples

mu <- c(1/7, 2/7, 4/7)
nu <- c(1/4, 1/4, 1/2)
kantorovich(mu, nu)
library(gmp)
mu <- as.bigq(c(1,2,4), 7)
nu <- as.bigq(c(1,1,1), c(4,4,2))
kantorovich(mu, nu)
mu <- c("1/7", "2/7", "4/7")
nu <- c("1/4", "1/4", "1/2")
kantorovich(mu, nu, details=TRUE)

---

Computes Kantorovich distance with CVX

Description

Kantorovich distance using the CVXR package

Usage

kantorovich_CVX(
  mu,
  nu,
  dist,
  solution = FALSE,
  stop_if_fail = TRUE,
  solver = "ECOS",
  ...
)

Arguments

mu	(row margins) probability measure in numeric mode
nu	(column margins) probability measure in numeric mode
dist	matrix defining the distance to be minimized on average
solution	logical; if TRUE the solution is returned in the "solution" attributes of the output
stop_if_fail	logical; if TRUE, an error is returned in the case when no solution is found; if FALSE, the output of psolve is returned with a warning
solver	the CVX solver, passed to psolve
...	other arguments passed to psolve

Examples

x <- c(1.5, 2, -3)
mu <- c(1/7, 2/7, 4/7)
y <- c(4, 3.5, 0, -2)
nu <- c(1/4, 1/4, 1/4, 1/4)
M <- outer(x, y, FUN = function(x, y) abs(x - y))
kantorovich_CVX(mu, nu, dist = M)

---

Computes Kantorovich distance with GLPK

Description

Kantorovich distance using the Rglpk package

Usage

kantorovich_glpk(mu, nu, dist, solution = FALSE, stop_if_fail = TRUE, ...)

Arguments

mu	(row margins) probability measure in numeric mode
nu	(column margins) probability measure in numeric mode
dist	matrix defining the distance to be minimized on average
solution	logical; if TRUE the solution is returned in the "solution" attributes of the output
stop_if_fail	logical; if TRUE, an error is returned in the case when no solution is found; if FALSE, the output of Rglpk_solve_LP is returned with a warning
...	arguments passed to Rglpk_solve_LP

Examples

x <- c(1.5, 2, -3)
mu <- c(1/7, 2/7, 4/7)
y <- c(4, 3.5, 0, -2)
nu <- c(1/4, 1/4, 1/4, 1/4)
M <- outer(x, y, FUN = function(x, y) abs(x - y))
kantorovich_glpk(mu, nu, dist = M)

---

Computes Kantorovich distance with lp_solve

Description

Kantorovich distance using the lpSolve package

Usage

kantorovich_lp(mu, nu, dist, solution = FALSE, lp.object = FALSE, ...)

Arguments

mu	(row margins) probability measure in numeric mode
nu	(column margins) probability measure in numeric mode
dist	matrix defining the distance to be minimized on average
solution	logical, to use only if lp.object=FALSE; if TRUE the solution is returned in the "solution" attributes of the output
lp.object	logical, if FALSE, the output is the Kantorovich distance; if TRUE, the output is a lp.object
...	arguments passed to lp

Examples

x <- c(1.5, 2, -3)
mu <- c(1/7, 2/7, 4/7)
y <- c(4, 3.5, 0, -2)
nu <- c(1/4, 1/4, 1/4, 1/4)
M <- outer(x, y, FUN = function(x, y) abs(x - y))
kantorovich_lp(mu, nu, dist = M)

---

Computes Kantorovich distance with 'ompr'

Description

Kantorovich distance using the ompr package

Usage

kantorovich_ompr(mu, nu, dist, solution = FALSE, stop_if_fail = TRUE)

Arguments

mu	(row margins) probability measure in numeric mode
nu	(column margins) probability measure in numeric mode
dist	matrix defining the distance to be minimized on average
solution	logical; if TRUE the solution is returned in the "solution" attributes of the output
stop_if_fail	logical; if TRUE, an error is returned in the case when no solution is found; if FALSE, the output of solve_model is returned with a warning

Note

The glpk solver is the one used to solve the problem.

Examples

x <- c(1.5, 2, -3)
mu <- c(1/7, 2/7, 4/7)
y <- c(-4, 3.5, 0)
nu <- c(1/4, 1/4, 1/2)
M <- outer(x, y, FUN = function(x, y) abs(x - y))
kantorovich_ompr(mu, nu, dist = M)

---

Names for bigq vectors

Description

Names for bigq vectors

Usage

## S3 method for class 'bigq'
names(x)

Arguments

x	a bigq vector

Value

the names of x

---