splines2

The R package splines2 is intended to be a user-friendly supplementary package to the base package splines.

Features

The package splines2 provides functions to construct basis matrices of

• B-splines
• M-splines
• I-splines
• convex splines (C-splines)
• periodic splines
• natural cubic splines
• generalized Bernstein polynomials
• their integrals (except C-splines) and derivatives of given order by closed-form recursive formulas

In addition to the R interface, splines2 provides a C++ header-only library integrated with Rcpp, which allows the construction of spline basis functions directly in C++ with the help of Rcpp and RcppArmadillo. Thus, it can also be treated as one of the Rcpp* packages. A toy example package that uses the C++ interface is available here.

Installation of CRAN Version

You can install the released version from CRAN.

install.packages("splines2")

Development

The latest version of the package is under development at GitHub. If it is able to pass the automated package checks, one may install it by

if (! require(remotes)) install.packages("remotes")
remotes::install_github("wenjie2wang/splines2", upgrade = "never")

Getting Started

The Online document provides a reference for all functions and contains the following vignettes:

• Demonstration of the common usages in R through examples.
• Introduction to the usage with Rcpp

Performance

Since v0.3.0, the implementation of the main functions has been rewritten in C++ with the help of the Rcpp and RcppArmadillo packages. The computational performance has thus been boosted and comparable with the function splines::splineDesign().

Some quick micro-benchmarks are provided for reference as follows:

library(microbenchmark)
options(microbenchmark.unit="relative")
library(splines)
library(splines2)

set.seed(123)
x <- runif(1e3)
degree <- 3
ord <- degree + 1
internal_knots <- seq.int(0.1, 0.9, 0.1)
boundary_knots <- c(0, 1)
all_knots <- sort(c(internal_knots, rep(boundary_knots, ord)))

## check equivalency of outputs
my_check <- function(values) {
    all(sapply(values[- 1], function(x) {
        all.equal(unclass(values[[1]]), x, check.attributes = FALSE)
    }))
}

For B-splines, function splines2::bSpline() provides equivalent results with splines::bs() and splines::splineDesign(), and is about 3x faster than bs() and 2x faster than splineDesign() for this example.

## B-splines
microbenchmark(
    "splines::bs" = bs(x, knots = internal_knots, degree = degree,
                       intercept = TRUE, Boundary.knots = boundary_knots),
    "splines::splineDesign" = splineDesign(x, knots = all_knots, ord = ord),
    "splines2::bSpline" = bSpline(
        x, knots = internal_knots, degree = degree,
        intercept = TRUE, Boundary.knots = boundary_knots
    ),
    check = my_check,
    times = 1e3
)

Unit: relative
                  expr    min     lq   mean median     uq    max neval
           splines::bs 3.9005 3.6551 3.6253 3.5585 3.6014 1.1993  1000
 splines::splineDesign 2.3123 2.1286 2.2174 2.0487 2.0755 1.1118  1000
     splines2::bSpline 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  1000

Similarly, for derivatives of B-splines, splines2::dbs() provides equivalent results with splines::splineDesign(), and is about 2x faster.

## Derivatives of B-splines
derivs <- 2
microbenchmark(
    "splines::splineDesign" = splineDesign(x, knots = all_knots,
                                           ord = ord, derivs = derivs),
    "splines2::dbs" = dbs(x, derivs = derivs, knots = internal_knots,
                          degree = degree, intercept = TRUE,
                          Boundary.knots = boundary_knots),
    check = my_check,
    times = 1e3
)

Unit: relative
                  expr    min    lq   mean median     uq    max neval
 splines::splineDesign 2.7543 2.618 2.6206 2.4983 2.5032 1.1626  1000
         splines2::dbs 1.0000 1.000 1.0000 1.0000 1.0000 1.0000  1000

The splines package does not contain an implementation for integrals of B-splines. Thus, we performed a comparison with package ibs (version r packageVersion("ibs")), where the function ibs::ibs() was also implemented in Rcpp.

## integrals of B-splines
set.seed(123)
coef_sp <- rnorm(length(all_knots) - ord)
microbenchmark(
    "ibs::ibs" = ibs::ibs(x, knots = all_knots, ord = ord, coef = coef_sp),
    "splines2::ibs" = as.numeric(
        splines2::ibs(x, knots = internal_knots, degree = degree,
                      intercept = TRUE, Boundary.knots = boundary_knots) %*%
        coef_sp
    ),
    check = my_check,
    times = 1e3
)

Unit: relative
          expr    min     lq   mean median     uq   max neval
      ibs::ibs 22.695 21.034 20.532 20.701 20.847 13.81  1000
 splines2::ibs  1.000  1.000  1.000  1.000  1.000  1.00  1000

The function ibs::ibs() returns the integrated B-splines instead of the integrals of spline basis functions. Thus, we applied the same coefficients to the basis functions from splines2::ibs() for equivalent results, which was still much faster than ibs::ibs().

For natural cubic splines (based on B-splines), splines::ns() uses the QR decomposition to find the null space of the second derivatives of B-spline basis functions at boundary knots, while splines2::naturalSpline() utilizes the closed-form null space derived from the second derivatives of cubic B-splines, which produces nonnegative basis functions (within boundary) and is more computationally efficient.

microbenchmark(
    "splines::ns" = ns(x, knots = internal_knots, intercept = TRUE,
                       Boundary.knots = boundary_knots),
    "splines2::naturalSpline" = naturalSpline(
        x, knots = internal_knots, intercept = TRUE,
        Boundary.knots = boundary_knots
    ),
    times = 1e3
)

Unit: relative
                    expr    min     lq   mean median    uq   max neval
             splines::ns 5.7796 5.4612 5.3913 5.1387 5.051 1.594  1000
 splines2::naturalSpline 1.0000 1.0000 1.0000 1.0000 1.000 1.000  1000

The function mSpline() produces periodic spline basis functions (based on M-splines) when periodic = TRUE is specified. The splines::periodicSpline() returns a periodic interpolation spline (based on B-splines) instead of basis matrix. Thus, we performed a comparison with package pbs (version r packageVersion("pbs")), where the function pbs::pbs() produces a basis matrix of periodic B-spline by using splines::spline.des() (a wrapper function of splines::splineDesign()).

microbenchmark(
    "pbs::pbs" = pbs::pbs(x, knots = internal_knots, degree = degree,
                          intercept = TRUE, periodic = TRUE,
                          Boundary.knots = boundary_knots),
    "splines2::mSpline" = mSpline(
        x, knots = internal_knots, degree = degree, intercept = TRUE,
        Boundary.knots = boundary_knots, periodic = TRUE
    ),
    times = 1e3
)

Unit: relative
              expr    min     lq   mean median     uq    max neval
          pbs::pbs 3.6138 3.4382 3.4608 3.2618 3.2852 1.3793  1000
 splines2::mSpline 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  1000

sessionInfo()

R version 4.2.3 (2023-03-15)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Arch Linux

Matrix products: default
BLAS:   /usr/lib/libopenblas.so.0.3
LAPACK: /usr/lib/liblapack.so.3.11.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C               LC_TIME=en_US.UTF-8       
 [4] LC_COLLATE=en_US.UTF-8     LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                  LC_ADDRESS=C              
[10] LC_TELEPHONE=C             LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] splines   stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] splines2_0.4.8       microbenchmark_1.4.9

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.10      codetools_0.2-19 ibs_1.4          digest_0.6.31    evaluate_0.20   
 [6] rlang_1.1.0      cli_3.6.1        rmarkdown_2.20   tools_4.2.3      xfun_0.38       
[11] yaml_2.3.7       fastmap_1.1.1    compiler_4.2.3   pbs_1.1          htmltools_0.5.5 
[16] knitr_1.42      

License

GNU General Public License (≥ 3)