## Introduction

Starting with version 1.2, cubature now uses Rcpp. Also, version 1.3 uses the newer version (1.0.2) of Steven G. Johnson’s hcubature and pcubature routines, including the vectorized interface.

Per the documentation, use of pcubature is advisable only for smooth integrands in dimesions up to three at most. In fact, the pcubature routines perform significantly worse than the vectorized hcubature in inappropriate cases. So when in doubt, you are better off using hcubature.

The main point of this note is to examine the difference vectorization makes. My recommendations are below in the summary section.

## A Timing Harness

Our harness will provide timing results for hcubature, pcubature (where appropriate) and R2Cuba calls. We begin by creating a harness for these calls.

loadedSuggested  <- c(benchr = FALSE, R2Cuba = FALSE)
if (requireNamespace("benchr", quietly = TRUE)) {
}
if (requireNamespace("R2Cuba", quietly = TRUE)) {
}

library(cubature)

harness <- function(which = NULL,
f, fv, lowerLimit, upperLimit, tol = 1e-3, times = 20, ...) {

fns <- c(hc = "Non-vectorized Hcubature",
hc.v = "Vectorized Hcubature",
pc = "Non-vectorized Pcubature",
pc.v = "Vectorized Pcubature")

fns <- c(fns, cc = "R2Cuba::cuhre")
cc <- function() R2Cuba::cuhre(ndim = ndim, ncomp = 1, integrand = f,
lower = lowerLimit, upper = upperLimit,
flags = list(verbose = 0, final = 1),
rel.tol = tol,
max.eval = 10^6,
...)
}

hc <- function() cubature::hcubature(f = f,
lowerLimit = lowerLimit,
upperLimit = upperLimit,
tol = tol,
...)

hc.v <- function() cubature::hcubature(f = fv,
lowerLimit = lowerLimit,
upperLimit = upperLimit,
tol = tol,
vectorInterface = TRUE,
...)

pc <- function() cubature::pcubature(f = f,
lowerLimit = lowerLimit,
upperLimit = upperLimit,
tol = tol,
...)

pc.v <- function() cubature::pcubature(f = fv,
lowerLimit = lowerLimit,
upperLimit = upperLimit,
tol = tol,
vectorInterface = TRUE,
...)

ndim = length(lowerLimit)

if (is.null(which)) {
fnIndices <- seq_along(fns)
} else {
fnIndices <- na.omit(match(which, names(fns)))
}
fnList <- lapply(names(fns)[fnIndices], function(x) call(x))

argList <- c(fnList, times = times, progress = FALSE)
result <- do.call(benchr::benchmark, args = argList)
d <- summary(result)[seq_along(fnIndices), ]
d$expr <- fns[fnIndices] d } else { d <- data.frame(expr = names(fns)[fnIndices], timing = NA) } } We reel off the timing runs. ## Example 1. func <- function(x) sin(x[1]) * cos(x[2]) * exp(x[3]) func.v <- function(x) { matrix(apply(x, 2, function(z) sin(z[1]) * cos(z[2]) * exp(z[3])), ncol = ncol(x)) } d <- harness(f = func, fv = func.v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = 1e-5, times = 100) knitr::kable(d, digits = 3, row.names = FALSE) expr n.eval min lw.qu median mean up.qu max total relative Non-vectorized Hcubature 100 0.002 0.003 0.003 0.003 0.003 0.005 0.283 6.17 Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.046 1.00 Non-vectorized Pcubature 100 0.008 0.008 0.009 0.009 0.009 0.043 0.928 20.00 Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.127 2.86 ## Multivariate Normal Using cubature, we evaluate $\int_R\phi(x)dx$ where $$\phi(x)$$ is the three-dimensional multivariate normal density with mean 0, and variance $\Sigma = \left(\begin{array}{rrr} 1 &\frac{3}{5} &\frac{1}{3}\\ \frac{3}{5} &1 &\frac{11}{15}\\ \frac{1}{3} &\frac{11}{15} & 1 \end{array} \right)$ and $$R$$ is $$[-\frac{1}{2}, 1] \times [-\frac{1}{2}, 4] \times [-\frac{1}{2}, 2].$$ We construct a scalar function (my_dmvnorm) and a vector analog (my_dmvnorm_v). First the functions. m <- 3 sigma <- diag(3) sigma[2,1] <- sigma[1, 2] <- 3/5 ; sigma[3,1] <- sigma[1, 3] <- 1/3 sigma[3,2] <- sigma[2, 3] <- 11/15 logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values))
my_dmvnorm <- function (x, mean, sigma, logdet) {
x <- matrix(x, ncol = length(x))
distval <- stats::mahalanobis(x, center = mean, cov = sigma)
exp(-(3 * log(2 * pi) + logdet + distval)/2)
}

my_dmvnorm_v <- function (x, mean, sigma, logdet) {
distval <- stats::mahalanobis(t(x), center = mean, cov = sigma)
exp(matrix(-(3 * log(2 * pi) + logdet + distval)/2, ncol = ncol(x)))
}

Now the timing.

d <- harness(f = my_dmvnorm, fv = my_dmvnorm_v,
lowerLimit = rep(-0.5, 3),
upperLimit = c(1, 4, 2),
tol = 1e-5,
times = 10,
mean = rep(0, m), sigma = sigma, logdet = logdet)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.924 1.011 1.075 1.078 1.114 1.329 10.781 594.0
Vectorized Hcubature 10 0.002 0.003 0.003 0.003 0.003 0.004 0.029 1.6
Non-vectorized Pcubature 10 0.398 0.475 0.487 0.497 0.521 0.590 4.969 270.0
Vectorized Pcubature 10 0.001 0.001 0.002 0.002 0.002 0.003 0.018 1.0

The effect of vectorization is huge. So it makes sense for users to vectorize the integrands as much as possible for efficiency.

Furthermore, for this particular example, we expect mvtnorm::pmvnorm to do pretty well since it is specialized for the multivariate normal. The good news is that the vectorized versions of hcubature and pcubature are quite competitive if you compare the table above to the one below.

library(mvtnorm)
g1 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
alg = Miwa(), abseps = 1e-5, releps = 1e-5)
g2 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
alg = GenzBretz(), abseps = 1e-5, releps = 1e-5)
g3 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
alg = TVPACK(), abseps = 1e-5, releps = 1e-5)

knitr::kable(summary(benchr::benchmark(g1(), g2(), g3(), times = 20, progress = FALSE)),
digits = 3, row.names = FALSE)
expr n.eval min lw.qu median mean up.qu max total relative
g1() 20 0.001 0.003 0.003 0.003 0.003 0.006 0.057 1.03
g2() 20 0.001 0.002 0.003 0.003 0.003 0.005 0.052 1.01
g3() 20 0.001 0.002 0.003 0.003 0.003 0.004 0.052 1.00

## Product of cosines

testFn0 <- function(x) prod(cos(x))
testFn0_v <- function(x) matrix(apply(x, 2, function(z) prod(cos(z))), ncol = ncol(x))

d <- harness(f = testFn0, fv = testFn0_v,
lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 1000)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 1000 0 0 0 0 0 0.001 0.217 2.84
Vectorized Hcubature 1000 0 0 0 0 0 0.001 0.076 1.00
Non-vectorized Pcubature 1000 0 0 0 0 0 0.003 0.307 4.07
Vectorized Pcubature 1000 0 0 0 0 0 0.001 0.146 1.93

## Gaussian function

testFn1 <- function(x) {
val <- sum(((1 - x) / x)^2)
scale <- prod((2 / sqrt(pi)) / x^2)
exp(-val) * scale
}

testFn1_v <- function(x) {
val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x))
scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x))
exp(-val) * scale
}

d <- harness(f = testFn1, fv = testFn1_v,
lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 10)

knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.015 0.016 0.018 0.020 0.019 0.047 0.205 95.70
Vectorized Hcubature 10 0.004 0.004 0.004 0.004 0.004 0.005 0.042 22.10
Non-vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.004 2.12
Vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.002 1.00

## Discontinuous function

testFn2 <- function(x) {
}

testFn2_v <- function(x) {
matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x))
}

d <- harness(which = c("hc", "hc.v", "cc"),
f = testFn2, fv = testFn2_v,
lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 10)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.202 0.248 0.251 0.246 0.251 0.258 2.458 4.96
Vectorized Hcubature 10 0.043 0.050 0.051 0.050 0.051 0.057 0.503 1.00

## A Simple Polynomial (product of coordinates)

testFn3 <- function(x) prod(2 * x)
testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x))

d <- harness(f = testFn3, fv = testFn3_v,
lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 50)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 50 0 0 0 0 0 0.001 0.024 4.40
Vectorized Hcubature 50 0 0 0 0 0 0.000 0.006 1.07
Non-vectorized Pcubature 50 0 0 0 0 0 0.000 0.019 3.62
Vectorized Pcubature 50 0 0 0 0 0 0.000 0.005 1.00

## Gaussian centered at

testFn4 <- function(x) {
a <- 0.1
s <- sum((x - 0.5)^2)
((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a))
}

testFn4_v <- function(x) {
a <- 0.1
r <- apply(x, 2, function(z) {
s <- sum((z - 0.5)^2)
((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a))
})
matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn4, fv = testFn4_v,
lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.007 0.008 0.008 0.008 0.008 0.013 0.167 5.92
Vectorized Hcubature 20 0.001 0.001 0.001 0.001 0.001 0.002 0.028 1.00
Non-vectorized Pcubature 20 0.011 0.012 0.012 0.014 0.013 0.049 0.286 8.75
Vectorized Pcubature 20 0.002 0.002 0.002 0.002 0.002 0.003 0.045 1.49

## Double Gaussian

testFn5 <- function(x) {
a <- 0.1
s1 <- sum((x - 1 / 3)^2)
s2 <- sum((x - 2 / 3)^2)
0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
}
testFn5_v <- function(x) {
a <- 0.1
r <- apply(x, 2, function(z) {
s1 <- sum((z - 1 / 3)^2)
s2 <- sum((z - 2 / 3)^2)
0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
})
matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn5, fv = testFn5_v,
lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.017 0.018 0.019 0.020 0.021 0.023 0.392 6.84
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.005 0.081 1.46
Non-vectorized Pcubature 20 0.012 0.013 0.013 0.014 0.015 0.016 0.275 4.89
Vectorized Pcubature 20 0.002 0.003 0.003 0.003 0.003 0.004 0.057 1.00

## Tsuda’s Example

testFn6 <- function(x) {
a <- (1 + sqrt(10.0)) / 9.0
prod( a / (a + 1) * ((a + 1) / (a + x))^2)
}

testFn6_v <- function(x) {
a <- (1 + sqrt(10.0)) / 9.0
r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2))
matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn6, fv = testFn6_v,
lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.010 0.011 0.011 0.011 0.012 0.013 0.225 6.21
Vectorized Hcubature 20 0.002 0.002 0.002 0.002 0.002 0.006 0.041 1.00
Non-vectorized Pcubature 20 0.053 0.055 0.055 0.060 0.060 0.092 1.192 31.80
Vectorized Pcubature 20 0.007 0.008 0.008 0.008 0.009 0.011 0.167 4.52

## Morokoff & Calflish Example

testFn7 <- function(x) {
n <- length(x)
p <- 1/n
(1 + p)^n * prod(x^p)
}
testFn7_v <- function(x) {
matrix(apply(x, 2, function(z) {
n <- length(z)
p <- 1/n
(1 + p)^n * prod(z^p)
}), ncol = ncol(x))
}

d <- harness(f = testFn7, fv = testFn7_v,
lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.022 0.022 0.024 0.025 0.026 0.030 0.497 6.89
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.006 0.079 1.00
Non-vectorized Pcubature 20 0.053 0.055 0.057 0.060 0.065 0.074 1.198 16.10
Vectorized Pcubature 20 0.007 0.008 0.009 0.009 0.010 0.015 0.187 2.48

## Wang-Landau Sampling 1d, 2d Examples

I.1d <- function(x) {
sin(4 * x) *
x * ((x * ( x * (x * x - 4) + 1) - 1))
}
I.1d_v <- function(x) {
matrix(apply(x, 2, function(z)
sin(4 * z) *
z * ((z * ( z * (z * z - 4) + 1) - 1))),
ncol = ncol(x))
}
d <- harness(f = I.1d, fv = I.1d_v,
lowerLimit = -2, upperLimit = 2, times = 100)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.126 5.54
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.027 1.13
Non-vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.042 1.85
Vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.023 1.00
I.2d <- function(x) {
x1 <- x[1]; x2 <- x[2]
sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2)
}
I.2d_v <- function(x) {
matrix(apply(x, 2,
function(z) {
x1 <- z[1]; x2 <- z[2]
sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2)
}),
ncol = ncol(x))
}
d <- harness(f = I.2d, fv = I.2d_v,
lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), times = 100)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 100 0.040 0.044 0.046 0.047 0.048 0.079 4.707 56.30
Vectorized Hcubature 100 0.005 0.006 0.006 0.006 0.007 0.009 0.644 7.70
Non-vectorized Pcubature 100 0.003 0.004 0.004 0.004 0.004 0.005 0.399 4.75
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.087 1.00

## An implementation note

About the only real modification we have made to the underlying cubature-1.0.2 library is that we use M = 16 rather than the default M = 19 suggested by the original author for pcubature. This allows us to comply with CRAN package size limits and seems to work reasonably well for the above tests. Future versions will allow for such customization on demand.

### Apropos the Cuba library

The package R2Cuba provides a suite of cubature and other useful Monte Carlo integration routines linked against version 1.6 of the C library. The authors of R2Cuba have obviously put a lot of work has into it since it uses C-style R API. This approach also means that it is harder to keep the R package in sync with new versions of the underlying C library. In fact, the Cuba C library has marched on now to version 4.2.

In a matter of a couple of hours, I was able to link the latest version (4.2) of the Cuba libraries with R using Rcpp; you can see it on the Cuba branch of my Github repo. This branch package builds and installs in R on my Mac and Ubuntu machines and gives correct answers at least for cuhre. The 4.2 version of the Cuba library also has vectorized versions of the routines that can be gainfully exploited (not implemented in the branch). As of this writing, I have also not yet carefully considered the issue of parallel execution (via fork()) which might be problematic in the Windows version. In addition, my timing benchmarks showed very disappointing results.

For the above reasons, I decided not to bother with Cuba for now, but if there is enough interest, I might consider rolling Cuba-4.2+ into this cubature package in the future.

## Summary

The following is therefore my recommendation.

1. Vectorize your function. The time spent in so doing pays back enormously. This is easy to do and the examples above show how.

2. Vectorized hcubature seems to be a good starting point.

3. For smooth integrands in low dimensions ($$\leq 3$$), pcubature might be worth trying out. Experiment before using in a production package.