`vignettes/mvrt.Rmd`

*This document is part of the the mvrt package. See the README.md for more information on download and installation.*

The objective of this R package vignette is to investigate the various ways of creating multivariate normal/t distributions. An algorithm using Cholesky decomposition is used; first in R code, then in C++ code. This is compared to existing methods for creating multivariate random matrices, such as `MASS::mvrnorm`

and `mvtnorm::rmvt`

.

The matrices are compared using summary statistics and graphics (histograms). The distributions are assessed for “believability”. Methods are also explored for ensuring that the specified correlation/covariance structure is preserved, despite the random process of number generation. Finally, benchmarking is perfromed to determine performance differences.

We will specify the sample n pairs, means, correlation matrix, and variances. The correlation matrix and variances will be used to generate a covariance matrix, which will be used by the random number generator.

```
n <- 30
mu <- c(5,4)
R <- matrix(c(1,.9,.9,1),2,2)
var <- c(2.5,2)
S <- mvrt::convert_R2S(R, var)
```

First we will define the random number generator functions in R. We will compare the performance of R code with- and without `for`

loops. Well-optimized R code may be sufficiently performant as to not warrant the implementation of a compiled alternative.

```
# With a for loop
mvrt_R_a <- function(n, mu, S, df = n - 1) {
g.t <- t(chol(S))
bivMat <- matrix(0,nrow = length(mu), ncol = n)
for (i in seq_len(n)) {
bivMat[,i] <- mu + g.t %*% rt(length(mu), df)
}
t(bivMat)
}
# Without a for loop
mvrt_R_b <- function(n, mu, S, df = n - 1) {
g <- chol(S)
random_matrix <- matrix(rt(n*length(mu), df),nrow = length(mu))
deviation <- t(g) %*% random_matrix
t(mu + deviation)
}
```

Next we will load in a function written in C++. The expectation is that the C++ code will be faster than the R code, possibly by several orders of magnitude.

`Rcpp::sourceCpp("../src/mvrt.cpp")`

The code that was loaded is below.

```
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp;
//' @rdname mvrt
//' @export
// [[Rcpp::export]]
arma::mat mvrt(int n, arma::vec mu, arma::mat S, int df=1)
{
// Cholesky decomp and transpose covariance matrix
arma::mat g = chol(S).t();
// Generate the random data
arma::vec x_vec = as< arma::vec >( rt(mu.size() * n, df) );
arma::mat x = arma::mat( (const double*)x_vec.begin(), mu.size(), n );
// Give the random data covariance structure and add mean offset
x = g * x;
x.each_col() += mu;
return x.t();
}
```

This function will take a function and its arguments and run it a specified number of times, performing a correlation on the output and extracting the off-diagonal element. Essentially, we can use this to run each random number generator an arbitrary number of times and get the Pearson’s correlation for each pair of vectors created. This can be used to examine the distribution of correlation coefficients that can be expected from the random generation processes. Note the random seed parameter which can be set for reproducibility.

```
get_cor_dist <- function(FUN, times, ..., seed = 999) {
set.seed(seed)
FUN <- match.fun(FUN)
my_call <- as.call(list(FUN, ...))
out <- numeric(times)
for (i in seq_len(times)) {
out[i] <- cor(eval(my_call))[2]
}
out
}
```

Using the function above, we will create similar multivariate t-distributed (or normally-distributed) random matrices using several functions.

```
mvrtRa_dist <- get_cor_dist(mvrt_R_a, 100, n, mu, S)
mvrtRb_dist <- get_cor_dist(mvrt_R_b, 100, n, mu, S)
mvrt_dist <- get_cor_dist(mvrt, 100, n, mu, S, n - 1)
MASS_dist <- get_cor_dist(MASS::mvrnorm, 100, n, mu, S)
mvtnorm_dist <- get_cor_dist(mvtnorm::rmvt, 100, n, S, n - 1)
```

First let’s check that our three home-grown functions produce identical results.

```
if (identical(mvrtRa_dist, mvrt_dist) && identical(mvrtRb_dist, mvrt_dist)) {
message("Distributions of correlation coefficients are identical")
} else {
message("Distributions of correlation coefficients are NOT identical")
}
```

`## Distributions of correlation coefficients are identical`

```
if (
identical(
(mvrt_sample <- {set.seed(999); mvrt(n, mu, S, n - 1)}),
{set.seed(999); mvrt_R_a(n, mu, S)}
) &&
identical(
mvrt_sample,
{set.seed(999); mvrt_R_b(n, mu, S)}
)
) {
message("Random samples generated match")
} else {
message("Random samples generated do NOT match")
}
```

`## Random samples generated match`

It appears that we are successful in that the three functions produce identical output. Therefore, we must only examine one function’s output when doing distribution analysis, rather than all three.

Let’s run a couple of checks to make sure that the algorithm is at least giving us data with approximately correct means and variances.

```
# Mean mu
colMeans(t(sapply(lapply(rep(30,100), mvrt, mu, S, n - 1),colMeans)))
```

`## [1] 4.943320 3.951612`

```
# Mean var
colMeans(t(sapply(lapply(rep(30,100), mvrt, mu, S, n - 1), apply, 2, var)))
```

`## [1] 2.607499 2.103790`

Means are close and variances are just a little higher than those specified in the covariance matrix, which we might expect from a relatively small sample size.

Let’s pull a single sample of n = 30. We can check the mean, variance, and correlation.

```
mvrt_sample <- mvrt(n, mu, S, n - 1)
colMeans(mvrt_sample)
```

`## [1] 5.261327 4.106697`

`apply(mvrt_sample, 2,var)`

`## [1] 2.157964 1.204207`

`cor(mvrt_sample)`

```
## [,1] [,2]
## [1,] 1.0000000 0.8381404
## [2,] 0.8381404 1.0000000
```

How does a plot of the data look?

```
plot(mvrt_sample)
# Regression line
abline(lm(mvrt_sample[,2] ~ mvrt_sample[,1]), col = "red")
# Center of points
points(mean(mvrt_sample[,1]), mean(mvrt_sample[,2]), col = "blue", pch = 22)
```

OK. So we are fairly confident that we are getting the data we requested. Now let’s look at how the correlation coefficients are distributed over several simulations.

```
# Summary of the distributions
summary(mvrt_dist)
```

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7809 0.8804 0.9084 0.9043 0.9316 0.9588
```

`summary(MASS_dist)`

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7500 0.8831 0.9082 0.9010 0.9219 0.9661
```

`summary(mvtnorm_dist)`

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7777 0.8816 0.9030 0.8980 0.9251 0.9570
```

```
# Graphical representations
my_hist <- function (data) {
hist(
data,
xlim = c(-1,1),
breaks = "fd",
main = paste("Histogram of",deparse(substitute(data))),
xlab = deparse(substitute(data))
)
abline(v = 0.9, col = 'red')
}
my_hist(mvrt_dist)
```

`my_hist(MASS_dist)`

`my_hist(mvtnorm_dist)`

What if we would like to specify that the matrix of values returned is within a reasonable margin to the specified correlation coefficient? This can be done through iterative generation of the random sample with acceptance checks at each step. A matrix norm of the difference in correlation matrices (between generated and specified) may be used as an intuitive check. We may specify that the maximum modulus of the difference between specified and returned correlation matrix elements is below a given threshold.

The determinant may also be used (and is potentially faster). However the norm may be more easily interpreted since the element-wise maximum modulus is a good way for the user to specify an acceptance criterion.

Why use the matrix norm rather than the determinant or some other measure? Because it is the most intuitive. The matrix norm is computed several ways. We will use the maximum modulus (absolute value) method, which is equivalent to performing `max(abs())`

on a matrix. This means that in we can specify the maximum devation of any element within the resultant correlation matrix generated by random sample versus the specified. For example, specifying a maximum matrix norm of 0.05 means that the Pearson’s correlation generated from outputted values will be within \(\pm\) 0.05 of the Pearson’s correlation used as the input correlation matrix (as derived form the input covariance matrix).

To demonstrate, we will create a reference correlation matrix and then compare it to a set of contender matrices using the determinant and norm.

```
# A single reference matrix
R_ref <- mvrt::make_cor_mat(.9)
R_ref
```

```
## [,1] [,2]
## [1,] 1.0 0.9
## [2,] 0.9 1.0
```

```
# A set of test matrices
R_test <- lapply(seq(.5,1,.05), mvrt::make_cor_mat)
R_test[[1]]
```

```
## [,1] [,2]
## [1,] 1.0 0.5
## [2,] 0.5 1.0
```

```
# What does the determinant look like for the difference matrices?
sapply(R_test, function(a,b) det(a - b), R_ref)
```

```
## [1] -0.1600 -0.1225 -0.0900 -0.0625 -0.0400 -0.0225 -0.0100 -0.0025
## [9] 0.0000 -0.0025 -0.0100
```

```
# And the matrix norm?
sapply(R_test, function(a,b) norm(a - b, "m"), R_ref)
```

`## [1] 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.05 0.10`

The user is more likely to understand how to specify a max norm than a max determinant.

Below we see two function definitions which perform the same task but implement different programming techniques. The first utilizes recursive function calls, while the second uses a `for`

loop to repeat sample generation until the acceptance criterion is met.

```
# Using recursive function calls
mvrt_R_c <- function(n, mu, S, df = n - 1, max.norm = 0.05, type.norm = "m") {
g <- chol(S)
random_matrix <- matrix(rt(n*length(mu), df),nrow = length(mu))
deviation <- t(g) %*% random_matrix
out <- t(mu + deviation)
if (norm(cov2cor(S) - cor(out), type = type.norm) <= max.norm) {
return(out)
} else {
eval(match.call())
}
}
# Using the for loop
mvrt_R_d <- function(
n,
mu,
S,
df = n - 1,
max_norm = 0.05,
max_iterations = 1000,
type_norm = "m")
{
g.t <- t(chol(S))
R_ref <- cov2cor(S)
for (iterations in seq_len(max_iterations)) {
random_matrix <- matrix(rt(n*length(mu), df),nrow = length(mu))
deviation <- g.t %*% random_matrix
if (norm(R_ref - cor(t(mu + deviation)), type = type_norm) <= max_norm)
return(t(mu + deviation))
}
stop(
"Correlation structure with max norm of ", max_norm,
" was not obtained in ", max_iterations, " iterations"
)
}
```

We may also include a `C++`

function to achieve this.

`Rcpp::sourceCpp("../src/mvrt2.cpp")`

The code that was loaded is below.

```
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp;
//' @rdname mvrt
//' @export
// [[Rcpp::export]]
arma::mat mvrt2(
int n,
arma::vec mu,
arma::mat S,
int df=1,
double max_norm=2,
int max_iterations=1000
)
{
// Cholesky decomp and transpose covariance matrix
arma::mat g = chol(S).t();
// Get correlation matrix of user-input S matrix
arma::mat V_sqrt_inv = diagmat(1 / sqrt(S.diag()));
arma::mat target_cor = V_sqrt_inv * S * V_sqrt_inv;
for (int iteration=0; iteration < max_iterations; iteration++)
{
// Generate the random data
arma::vec x_vec = as< arma::vec >( rt(mu.size() * n, df) );
arma::mat x = arma::mat( (const double*)x_vec.begin(), mu.size(), n );
// Give the random data covariance structure and add mean offset
x = g * x;
x.each_col() += mu;
// Compare to target and retrun if meets specification
arma::mat diff_matrix = abs(cor(x.t()) - target_cor);
if (diff_matrix.max() <= max_norm) return x.t();
}
stop(
"Did not generate matrix with max norm of %f in %i iterations",
max_norm,
max_iterations
);
}
```

Now lets see if the distributions tighten up with our new functions. We should now see that the correlation coefficient of each of our randomly generated samples will not stray more than 0.05 from the correlation of the correlation matrix used to specify the covariance structure of the data.

```
mvrtRc_dist <- get_cor_dist(mvrt_R_c, 100, n, mu, S)
mvrtRd_dist <- get_cor_dist(mvrt_R_d, 100, n, mu, S)
mvrt2_dist <- get_cor_dist(mvrt2, 100, n, mu, S, n - 1, 0.05)
if (identical(mvrtRc_dist, mvrtRd_dist) && identical(mvrtRc_dist, mvrt2_dist)) {
message("Distributions of correlation coefficients are identical")
} else {
message("Distributions of correlation coefficients are NOT identical")
}
```

`## Distributions of correlation coefficients are identical`

`summary(mvrtRc_dist)`

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.8545 0.8801 0.9070 0.9028 0.9257 0.9472
```

`my_hist(mvrtRc_dist)`

This appears to have worked. Compare with distributions explored in the previous section.

We can also specify a different covariance structure and check again. This demonstrates the usage of the various function definitions we have created for this exercise.

```
summary(
get_cor_dist(
mvrt_R_d,
100,
n,
mu,
mvrt::convert_R2S(mvrt::make_cor_mat(0.2),var)
)
)
```

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1502 0.1727 0.2059 0.2012 0.2254 0.2500
```

Not surprisingly, the correlation coefficients remain within [0.15,0.25].

Let’s see how the functions compare in speed.

```
microbenchmark::microbenchmark(
# No data checks
mvrt_R_a(n, mu, S), # 'for' loop
mvrt_R_b(n, mu, S), # Optimized R code
mvrt(n, mu, S, n - 1), # C++ code
# Iterative checking
mvrt_R_c(n, mu, S), # Recursive call
mvrt_R_d(n, mu, S), # 'for' loop
mvrt2(n, mu, S, n - 1, 0.05), # C++ code
# Available packages on CRAN
MASS::mvrnorm(n, mu, S),
mvtnorm::rmvt(n, S, n - 1) + mu
)
```

```
## Unit: microseconds
## expr min lq mean median
## mvrt_R_a(n, mu, S) 73.118 79.8330 85.53380 83.2860
## mvrt_R_b(n, mu, S) 20.265 23.6490 26.62688 25.5575
## mvrt(n, mu, S, n - 1) 9.516 11.9185 13.56137 13.7675
## mvrt_R_c(n, mu, S) 46.752 54.5880 70.53956 60.6215
## mvrt_R_d(n, mu, S) 52.844 58.7705 85.45690 64.5535
## mvrt2(n, mu, S, n - 1, 0.05) 10.755 14.4040 17.07289 15.9395
## MASS::mvrnorm(n, mu, S) 51.484 63.4195 67.71284 66.2470
## mvtnorm::rmvt(n, S, n - 1) + mu 204.635 216.7585 225.16570 221.8460
## uq max neval
## 87.9110 113.261 100
## 28.5385 59.406 100
## 14.7660 21.463 100
## 70.1820 174.048 100
## 71.3050 1533.552 100
## 18.9240 31.579 100
## 69.7340 124.292 100
## 229.6160 286.630 100
```

I know the `::`

operator has a performance cost for those CRAN packages. It may be interesting to try the benchmarking with loaded packages, too.

```
MASS_mvrnorm <- MASS::mvrnorm
mvtnorm_rmvt <- mvtnorm::rmvt
microbenchmark::microbenchmark(
MASS::mvrnorm(n, mu, S),
MASS_mvrnorm(n, mu, S),
mvtnorm::rmvt(n, S, n - 1) + mu,
mvtnorm_rmvt(n, S, n - 1) + mu
)
```

```
## Unit: microseconds
## expr min lq mean median
## MASS::mvrnorm(n, mu, S) 49.243 57.5085 73.79916 66.1565
## MASS_mvrnorm(n, mu, S) 42.783 52.5385 63.01251 56.5635
## mvtnorm::rmvt(n, S, n - 1) + mu 195.604 204.1195 249.19622 216.1355
## mvtnorm_rmvt(n, S, n - 1) + mu 186.588 198.6790 249.02270 211.5220
## uq max neval
## 81.1200 173.933 100
## 70.4090 107.845 100
## 267.6580 473.439 100
## 252.0095 1423.917 100
```

Clearly, rolling your own C++ code is superior. But given the performance gain from removing the `for`

loop from our original R function, optimization of R code cannot be underestimated. Given that the well-written R code is comparably performant within an order of magnitude, the switch to C++ is likely not worth the additional effort in this case.