ddlppar function

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

$L^p$ 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.

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.

Details

The $L^p$ distance $||p_1 - p_2||$ between two discrete distributions $p_1$ and $p_2$ is given by the formula:

If $p=1$, it is the variational distance.

If $p=2$, it is the Patrick-Fisher distance.

Author(s)

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

See Also

ddlp: $L^p$ distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddhellingerpar, ddjeffreyspar, ddjensenpar.

References

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

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)