# metrics: RMSE, Expected Log Likelihood and KL Divergence Between Two... In monomvn: Estimation for MVN and Student-t Data with Monotone Missingness

## 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

 ```1 2 3``` ```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:

0.5 (log(|S1|/|S2|) + tr(inv(S1)S2) + t(mu1-m2)inv(S2)(mu1-mu2) - N)

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

-0.5 ln((2 pi e)^N |S2|) - kl.norm(mu1, S1, mu2, S2).

## 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 rbg@vt.edu

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```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) ```

### Example output

```Loading required package: pls

Attaching package: 'pls'

The following object is masked from 'package:stats':