DescTools (version 0.99.53)

Entropy: Shannon Entropy and Mutual Information

Description

Computes Shannon entropy and the mutual information of two variables. The entropy quantifies the expected value of the information contained in a vector. The mutual information is a quantity that measures the mutual dependence of the two random variables.

Usage

Entropy(x, y = NULL, base = 2, ...)
MutInf(x, y, base = 2, ...)

Value

a numeric value.

Arguments

x

a vector or a matrix of numerical or categorical type. If only x is supplied it will be interpreted as contingency table.

y

a vector with the same type and dimension as x. If y is not NULL then the entropy of table(x, y, ...) will be calculated.

base

base of the logarithm to be used, defaults to 2.

...

further arguments are passed to the function table, allowing i.e. to set useNA.

Author

Andri Signorell <[e-mail]>

Details

The Shannon entropy equation provides a way to estimate the average minimum number of bits needed to encode a string of symbols, based on the frequency of the symbols.
It is given by the formula \(H = - \sum(\pi log(\pi))\) where \(\pi\) is the probability of character number i showing up in a stream of characters of the given "script".
The entropy is ranging from 0 to Inf.

References

Shannon, Claude E. (July/October 1948). A Mathematical Theory of Communication, Bell System Technical Journal 27 (3): 379-423.

Ihara, Shunsuke (1993) Information theory for continuous systems, World Scientific. p. 2. ISBN 978-981-02-0985-8.

See Also

package entropy which implements various estimators of entropy

Examples

Entropy(as.matrix(rep(1/8, 8)))

# http://r.789695.n4.nabble.com/entropy-package-how-to-compute-mutual-information-td4385339.html
x <- as.factor(c("a","b","a","c","b","c")) 
y <- as.factor(c("b","a","a","c","c","b")) 

Entropy(table(x), base=exp(1))
Entropy(table(y), base=exp(1))
Entropy(x, y, base=exp(1))

# Mutual information is 
Entropy(table(x), base=exp(1)) + Entropy(table(y), base=exp(1)) - Entropy(x, y, base=exp(1))
MutInf(x, y, base=exp(1))

Entropy(table(x)) + Entropy(table(y)) - Entropy(x, y)
MutInf(x, y, base=2)

# http://en.wikipedia.org/wiki/Cluster_labeling
tab <- matrix(c(60,10000,200,500000), nrow=2, byrow=TRUE)
MutInf(tab, base=2) 

d.frm <- Untable(as.table(tab))
str(d.frm)
MutInf(d.frm[,1], d.frm[,2])

table(d.frm[,1], d.frm[,2])

MutInf(table(d.frm[,1], d.frm[,2]))


# Ranking mutual information can help to describe clusters
#
#   r.mi <- MutInf(x, grp)
#   attributes(r.mi)$dimnames <- attributes(tab)$dimnames
# 
#   # calculating ranks of mutual information
#   r.mi_r <- apply( -r.mi, 2, rank, na.last=TRUE )
#   # show only first 6 ranks
#   r.mi_r6 <- ifelse( r.mi_r < 7, r.mi_r, NA) 
#   attributes(r.mi_r6)$dimnames <- attributes(tab)$dimnames
#   r.mi_r6