Introduction

This R cubature package exposes both the hcubature and pcubature routines of the underlying C library, including the vectorized interfaces.

Per the documentation, use of pcubature is advisable only for smooth integrands in dimensions 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)) {
    loadedSuggested["benchr"] <- TRUE
}
if (requireNamespace("R2Cuba", quietly = TRUE)) {
    loadedSuggested["R2Cuba"] <- 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")

    if (loadedSuggested["R2Cuba"]) {
        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))

    if (loadedSuggested["benchr"]) {
        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 0.002 0.002 0.002 0.002 0.004 0.218 6.54
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.002 0.037 1.10
Non-vectorized Pcubature 100 0.006 0.006 0.007 0.007 0.007 0.013 0.704 20.90
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.103 3.13
R2Cuba::cuhre 100 0.000 0.000 0.000 0.000 0.000 0.002 0.034 1.00

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 0.814 0.839 0.846 0.844 0.852 0.868 8.436 634.0
Vectorized Hcubature 10 0.002 0.002 0.002 0.002 0.002 0.002 0.019 1.4
Non-vectorized Pcubature 10 0.346 0.360 0.362 0.365 0.373 0.383 3.651 272.0
Vectorized Pcubature 10 0.001 0.001 0.001 0.001 0.001 0.001 0.014 1.0
R2Cuba::cuhre 10 0.275 0.278 0.281 0.286 0.288 0.328 2.865 210.0

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 0.002 0.002 0.002 0.003 0.005 0.046 1.01
g2() 20 0.001 0.002 0.002 0.002 0.003 0.003 0.046 1.00
g3() 20 0.001 0.001 0.002 0.002 0.003 0.005 0.045 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 0.002 0.200 2.54
Vectorized Hcubature 1000 0 0 0 0 0 0.005 0.084 1.00
Non-vectorized Pcubature 1000 0 0 0 0 0 0.002 0.289 3.63
Vectorized Pcubature 1000 0 0 0 0 0 0.002 0.149 1.90
R2Cuba::cuhre 1000 0 0 0 0 0 0.002 0.231 2.89

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 0.012 0.012 0.013 0.013 0.014 0.014 0.130 85.70
Vectorized Hcubature 10 0.003 0.003 0.003 0.003 0.003 0.005 0.034 21.80
Non-vectorized Pcubature 10 0.000 0.000 0.000 0.001 0.000 0.005 0.008 1.91
Vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.002 1.00
R2Cuba::cuhre 10 0.007 0.007 0.007 0.007 0.008 0.008 0.074 46.60

Discontinuous function

testFn2 <- function(x) {
    radius <- 0.50124145262344534123412
    ifelse(sum(x * x) < radius * radius, 1, 0)
}

testFn2_v <- function(x) {
    radius <- 0.50124145262344534123412
    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 0.166 0.174 0.183 0.183 0.188 0.209 1.829 4.9
Vectorized Hcubature 10 0.035 0.035 0.037 0.038 0.040 0.042 0.380 1.0
R2Cuba::cuhre 10 0.431 0.464 0.487 0.493 0.526 0.545 4.928 13.1

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 0.002 0.019 3.83
Vectorized Hcubature 50 0 0 0 0 0 0.000 0.005 1.05
Non-vectorized Pcubature 50 0 0 0 0 0 0.000 0.015 3.17
Vectorized Pcubature 50 0 0 0 0 0 0.000 0.005 1.00
R2Cuba::cuhre 50 0 0 0 0 0 0.002 0.019 3.79

Gaussian centered at \(\frac{1}{2}\)

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 0.006 0.007 0.008 0.009 0.008 0.041 0.184 5.67
Vectorized Hcubature 20 0.001 0.001 0.001 0.001 0.001 0.003 0.029 1.00
Non-vectorized Pcubature 20 0.010 0.010 0.011 0.011 0.011 0.015 0.220 8.14
Vectorized Pcubature 20 0.002 0.002 0.002 0.002 0.002 0.002 0.038 1.44
R2Cuba::cuhre 20 0.001 0.001 0.002 0.002 0.002 0.003 0.034 1.14

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 0.017 0.018 0.018 0.018 0.019 0.353 7.05
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.005 0.073 1.40
Non-vectorized Pcubature 20 0.011 0.012 0.012 0.012 0.013 0.014 0.243 4.70
Vectorized Pcubature 20 0.002 0.002 0.003 0.003 0.003 0.004 0.051 1.00
R2Cuba::cuhre 20 0.004 0.004 0.004 0.004 0.005 0.006 0.089 1.66

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 0.009 0.010 0.010 0.010 0.011 0.011 0.200 5.69
Vectorized Hcubature 20 0.002 0.002 0.002 0.002 0.002 0.003 0.035 1.00
Non-vectorized Pcubature 20 0.049 0.051 0.052 0.052 0.052 0.059 1.037 30.30
Vectorized Pcubature 20 0.007 0.008 0.008 0.008 0.008 0.010 0.164 4.70
R2Cuba::cuhre 20 0.002 0.002 0.002 0.003 0.003 0.004 0.051 1.43

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 0.018 0.021 0.021 0.021 0.022 0.023 0.421 6.40
Vectorized Hcubature 20 0.003 0.003 0.003 0.003 0.004 0.004 0.068 1.00
Non-vectorized Pcubature 20 0.048 0.051 0.052 0.054 0.053 0.091 1.074 15.70
Vectorized Pcubature 20 0.007 0.008 0.008 0.008 0.008 0.010 0.158 2.31
R2Cuba::cuhre 20 0.021 0.022 0.023 0.023 0.024 0.025 0.465 7.01

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 0.097 5.92
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.021 1.27
Non-vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.002 0.034 2.00
Vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.000 0.018 1.17
R2Cuba::cuhre 100 0.000 0.000 0.000 0.000 0.000 0.000 0.016 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 0.030 0.032 0.034 0.034 0.035 0.075 3.391 54.80
Vectorized Hcubature 100 0.004 0.004 0.005 0.005 0.005 0.007 0.473 7.48
Non-vectorized Pcubature 100 0.002 0.003 0.003 0.003 0.003 0.004 0.284 4.56
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.063 1.00
R2Cuba::cuhre 100 0.001 0.001 0.001 0.001 0.001 0.002 0.077 1.20

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 (canonical web link now seems dead) 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.

Session Info

sessionInfo()
## R version 3.5.0 (2018-04-23)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.5
## 
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] mvtnorm_1.0-7 cubature_1.4 
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.12.17       benchr_0.2.0       digest_0.6.15     
##  [4] rprojroot_1.3-2    backports_1.1.2    magrittr_1.5      
##  [7] evaluate_0.10.1    highr_0.7          stringi_1.2.2     
## [10] R2Cuba_1.1-0       rmarkdown_1.10     tools_3.5.0       
## [13] stringr_1.3.1      RcppProgress_0.4.1 yaml_2.1.19       
## [16] compiler_3.5.0     htmltools_0.3.6    knitr_1.20