dad (version 4.1.6)

ddlppar: Distance between discrete probability distributions given the probabilities on their common support

Description

\(L^p\) distance between two discrete probability distributions on the same support (which can be a Cartesian product of \(q\) sets) , given the probabilities of the states (which are \(q\)-tuples) of the support.

Usage

ddlppar(p1, p2, p = 1)

Arguments

p1

array (or table) the dimension of which is \(q\). The first probability distribution on the support.

p2

array (or table) the dimension of which is \(q\). The second probability distribution on the support.

p

integer. Parameter of the distance.

Author

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Details

The \(L^p\) distance \(||p_1 - p_2||\) between two discrete distributions \(p_1\) and \(p_2\) is given by the formula: $$||p_1 - p_2||^p = \sum_x{|p_1(x)-p_2(x)|^p}$$

If \(p=1\), it is the variational distance.

If \(p=2\), it is the Patrick-Fisher distance.

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddlp: \(L^p\) distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddhellingerpar, ddjeffreyspar, ddjensenpar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddlppar(p1, p2)
ddlppar(p1, p2, p=2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddlppar(p1, p2)