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 and pcubature (where appropriate). We begin by creating a harness for these calls.

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

    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 0.005 0.244 6.09
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.002 0.041 1.00
Non-vectorized Pcubature 100 0.007 0.007 0.007 0.008 0.008 0.013 0.770 19.10
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.112 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 1.032 1.044 1.055 1.058 1.068 1.100 10.579 673.00
Vectorized Hcubature 10 0.002 0.002 0.002 0.003 0.003 0.004 0.027 1.57
Non-vectorized Pcubature 10 0.435 0.448 0.449 0.452 0.463 0.468 4.522 286.00
Vectorized Pcubature 10 0.001 0.002 0.002 0.002 0.002 0.002 0.016 1.00

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.

if (requireNamespace("mvtnorm", quietly = TRUE)) {
    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)
} else {
    cat("NOTE: Package mvtnorm not available for comparison\n")
}
expr n.eval min lw.qu median mean up.qu max total relative
g1() 20 0.001 0.002 0.002 0.002 0.002 0.004 0.044 1.02
g2() 20 0.001 0.001 0.002 0.002 0.002 0.004 0.041 1.00
g3() 20 0.001 0.001 0.002 0.002 0.002 0.004 0.041 1.02

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 0.212 2.70
Vectorized Hcubature 1000 0 0 0 0 0 0.002 0.080 1.00
Non-vectorized Pcubature 1000 0 0 0 0 0 0.034 0.341 3.86
Vectorized Pcubature 1000 0 0 0 0 0 0.002 0.149 1.91

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 0.014 0.015 0.016 0.016 0.016 0.017 0.155 84.70
Vectorized Hcubature 10 0.004 0.004 0.004 0.004 0.004 0.005 0.041 22.30
Non-vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.003 1.84
Vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.002 1.00

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 0.181 0.187 0.192 0.19 0.192 0.198 1.899 4.78
Vectorized Hcubature 10 0.039 0.040 0.040 0.04 0.040 0.041 0.400 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 0.002 0.020 4.23
Vectorized Hcubature 50 0 0 0 0 0 0.000 0.005 1.10
Non-vectorized Pcubature 50 0 0 0 0 0 0.000 0.016 3.47
Vectorized Pcubature 50 0 0 0 0 0 0.000 0.005 1.00

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 0.007 0.007 0.008 0.008 0.009 0.009 0.158 5.34
Vectorized Hcubature 20 0.001 0.001 0.001 0.001 0.001 0.003 0.030 1.00
Non-vectorized Pcubature 20 0.011 0.011 0.011 0.011 0.012 0.013 0.228 7.78
Vectorized Pcubature 20 0.002 0.002 0.002 0.002 0.002 0.003 0.042 1.43

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 0.016 0.017 0.018 0.020 0.018 0.062 0.396 6.77
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.005 0.073 1.38
Non-vectorized Pcubature 20 0.011 0.011 0.012 0.012 0.013 0.014 0.241 4.68
Vectorized Pcubature 20 0.002 0.003 0.003 0.003 0.003 0.004 0.053 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 0.009 0.010 0.010 0.011 0.012 0.012 0.214 5.62
Vectorized Hcubature 20 0.002 0.002 0.002 0.002 0.002 0.004 0.039 1.00
Non-vectorized Pcubature 20 0.050 0.054 0.056 0.055 0.057 0.060 1.099 29.90
Vectorized Pcubature 20 0.008 0.009 0.009 0.009 0.009 0.010 0.180 4.76

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 0.020 0.021 0.022 0.022 0.022 0.023 0.436 5.82
Vectorized Hcubature 20 0.003 0.004 0.004 0.004 0.004 0.005 0.077 1.00
Non-vectorized Pcubature 20 0.049 0.053 0.053 0.053 0.054 0.061 1.068 14.10
Vectorized Pcubature 20 0.008 0.008 0.009 0.009 0.010 0.011 0.179 2.29

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 0.003 0.109 5.28
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.000 0.022 1.12
Non-vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.002 0.038 1.78
Vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.000 0.020 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 0.033 0.035 0.036 0.037 0.037 0.080 3.663 56.60
Vectorized Hcubature 100 0.004 0.005 0.005 0.005 0.005 0.006 0.493 7.64
Non-vectorized Pcubature 100 0.003 0.003 0.003 0.003 0.003 0.005 0.308 4.65
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.007 0.070 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 is no longer available on CRAN. Fear not, for version 2.0 integrates the latest version (4.2) of the Cuba libraries (canonical web link is not secure) with R using Rcpp. In fact, you can install the development version from the master branch of my Github repo. Vectorization may be used with these routiness too.

Summary

My recommendations are:

  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 3.5.1 (2018-07-02)
## Platform: x86_64-apple-darwin17.7.0 (64-bit)
## Running under: macOS  10.14
## 
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.3/lib/libopenblasp-r0.3.3.dylib
## 
## locale:
## [1] en_US.UTF-8/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] cubature_1.4-1
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.12.19       mvtnorm_1.0-8      benchr_0.2.2      
##  [4] digest_0.6.17      rprojroot_1.3-2    backports_1.1.2   
##  [7] magrittr_1.5       evaluate_0.12      highr_0.7         
## [10] stringi_1.2.4      rmarkdown_1.10     tools_3.5.1       
## [13] stringr_1.3.1      RcppProgress_0.4.1 yaml_2.2.0        
## [16] compiler_3.5.1     htmltools_0.3.6    knitr_1.20