metrics {monomvn}

RMSE, Expected Log Likelihood and KL Divergence Between Two Multivariate Normal Distributions

Description

These functions calculate the root-mean-squared-error, the expected log likelihood, and Kullback-Leibler (KL) divergence (a.k.a. distance), between two multivariate normal (MVN) distributions described by their mean vector and covariance matrix

Usage

rmse.muS(mu1, S1, mu2, S2)
Ellik.norm(mu1, S1, mu2, S2, quiet=FALSE)
kl.norm(mu1, S1, mu2, S2, quiet=FALSE, symm=FALSE)

Arguments

mu1

mean vector of first (estimated) MVN

S1

covariance matrix of first (estimated) MVN

mu2

mean vector of second (true, baseline, or comparator) MVN

S2

covariance matrix of second (true, baseline, or comparator) MVN

quiet

when FALSE (default)

symm

when TRUE a symmetrized version of the KL divergence is used; see the note below

Details

The root-mean-squared-error is calculated between the entries of the mean vectors, and the upper-triangular part of the covariance matrices (including the diagonal).

The KL divergence is given by the formula:

D_{\mbox{\tiny KL}}(N_1 \| N_2) = \frac{1}{2} \left( \log \left( \frac{|\Sigma_1|}{|\Sigma_2|} \right) + \mbox{tr} \left( \Sigma_1^{-1} \Sigma_2 \right) + \left( \mu_1 - \mu_2\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_2 ) - N \right)

where N is length(mu1), and must agree with the dimensions of the other parameters. Note that the parameterization used involves swapped arguments compared to some other references, e.g., as provided by Wikipedia. See note below.

The expected log likelihood can be formulated in terms of the KL divergence. That is, the expected log likelihood of data simulated from the normal distribution with parameters mu2 and S2 under the estimated normal with parameters mu1 and S1 is given by

-\frac{1}{2} \ln \{(2\pi e)^N |\Sigma_2|\} - D_{\mbox{\tiny KL}}(N_1 \| N_2).

Value

In the case of the expected log likelihood the result is a real number. The RMSE is a positive real number. The KL divergence method returns a positive real number depicting the distance between the two normal distributions

Note

The KL-divergence is not symmetric. Therefore

kl.norm(mu1,S1,mu2,S2) != kl.norm(mu2,S2,mu1,S1).

But a symmetric metric can be constructed from

0.5 * (kl.norm(mu1,S1,mu2,S2) + kl.norm(mu2,S2,mu1,S1))

or by using symm = TRUE. The arguments are reversed compared to some other references, like Wikipedia. To match those versions use kl.norm(mu2, S2, mu1, s1)

Author(s)

Robert B. Gramacy [e-mail]

References

https://bobby.gramacy.com/r_packages/monomvn/

Examples

mu1 <- rnorm(5)
s1 <- matrix(rnorm(100), ncol=5)
S1 <- t(s1) %*% s1

mu2 <- rnorm(5)
s2 <- matrix(rnorm(100), ncol=5)
S2 <- t(s2) %*% s2

## RMSE
rmse.muS(mu1, S1, mu2, S2)

## expected log likelihood
Ellik.norm(mu1, S1, mu2, S2)

## KL is not symmetric
kl.norm(mu1, S1, mu2, S2)
kl.norm(mu2, S2, mu1, S1)

## symmetric version
kl.norm(mu2, S2, mu1, S1, symm=TRUE)