Plug-In Estimator of the Kullback-Leibler divergence and of the Chi-Squared Divergence
KL.plugin computes the Kullback-Leiber (KL) divergence between two discrete random variables x_1 and x_2. The corresponding probability mass functions are given by freqs1 and freqs2. Note that the expectation is taken with regard to x_1 using freqs1.
chi2.plugin computes the chi-squared divergence between two discrete random variables x_1 and x_2 with freqs1 and freqs2 as corresponding probability mass functions. Note that the denominator contains freqs2.
KL.plugin(freqs1, freqs2, unit=c("log", "log2", "log10"))
chi2.plugin(freqs1, freqs2, unit=c("log", "log2", "log10"))freqs1 |
frequencies (probability mass function) for variable x_1. |
freqs2 |
frequencies (probability mass function) for variable x_2. |
unit |
the unit in which entropy is measured.
The default is "nats" (natural units). For
computing entropy in "bits" set |
Kullback-Leibler divergence between the two discrete variables x_1 to x_2 is ∑_k p_1(k) \log (p_1(k)/p_2(k)) where p_1 and p_2 are the probability mass functions of x_1 and x_2, respectively, and k is the index for the classes.
The chi-squared divergence is given by ∑_k (p_1(k)-p_2(k))^2/p_2(k) .
Note that both the KL divergence and the chi-squared divergence are not symmetric in x_1 and x_2. The chi-squared divergence can be derived as a quadratic approximation of twice the KL divergence.
KL.plugin returns the KL divergence.
chi2.plugin returns the chi-squared divergence.
Korbinian Strimmer (http://www.strimmerlab.org).
# load entropy library
library("entropy")
# probabilities for two random variables
freqs1 = c(1/5, 1/5, 3/5)
freqs2 = c(1/10, 4/10, 1/2)
# KL divergence between x1 to x2
KL.plugin(freqs1, freqs2)
# and corresponding (half) chi-squared divergence
0.5*chi2.plugin(freqs1, freqs2)
## relationship to Pearson chi-squared statistic
# Pearson chi-squared statistic and p-value
n = 30 # sample size (observed counts)
chisq.test(n*freqs1, p = freqs2) # built-in function
# Pearson chi-squared statistic from Pearson divergence
pcs.stat = n*chi2.plugin(freqs1, freqs2) # note factor n
pcs.stat
# and p-value
df = length(freqs1)-1 # degrees of freedom
pcs.pval = 1-pchisq(pcs.stat, df)
pcs.pvalPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.