The R package **splines2** (version 0.4.1) provides functions to construct basis matrix of

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

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

You can install the released version from CRAN.

The latest version of 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")
```

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

Since v0.3.0, the implementation of the main functions has been rewritten in C++ with the help of the **Rcpp** and **RcppArmadillo** package. The computational performance has thus been boosted.

Some benchmarks are provided for reference as follows:

```
library(microbenchmark)
library(splines)
library(splines2)
x <- seq.int(0, 1, 0.001)
degree <- 3
ord <- degree + 1
knots <- seq.int(0.1, 0.9, 0.1)
b_knots <- range(x)
all_knots <- sort(c(knots, rep(b_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()`

.

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

```
Unit: microseconds
expr min lq mean median uq max neval cld
splines::bs 336.952 345.183 375.79 350.601 360.875 2505.0 1000 c
splines::splineDesign 205.412 208.695 234.77 210.150 214.097 2192.4 1000 b
splines2::bSpline 85.711 90.792 103.96 94.229 96.755 2143.7 1000 a
```

Similarly, for derivatives of B-splines, `splines2::dbs()`

provides equivalent results with `splines::splineDesign()`

, and is more than 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 = knots, degree = degree,
intercept = TRUE, Boundary.knots = b_knots),
check = my_check,
times = 1e3
)
```

```
Unit: microseconds
expr min lq mean median uq max neval cld
splines::splineDesign 274.138 277.414 315.71 278.56 285.50 2538.4 1000 b
splines2::dbs 95.243 97.353 115.49 100.96 103.44 2532.8 1000 a
```

The **splines** package does not provide function producing integrals of B-splines. So we instead performed a comparison with package **ibs** (version 1.4), 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 = knots, degree = degree,
intercept = TRUE, Boundary.knots = b_knots) %*% coef_sp
),
check = my_check,
times = 1e3
)
```

```
Unit: microseconds
expr min lq mean median uq max neval cld
ibs::ibs 2359.14 2533.36 3173.31 3155.07 3214.27 155132.5 1000 b
splines2::ibs 265.85 298.67 330.44 333.34 346.81 1267.2 1000 a
```

The function `ibs::ibs()`

returns the integrated B-splines instead of the integrals of spline bases. So we applied the same coefficients to the bases 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 QR decomposition to find the null space of the second derivatives of B-spline basis at boundary knots, while `splines2::naturalSpline()`

utilizes the close-form null space derived from the second derivatives of cubic B-splines, which produces nonnegative bases (within boundary) and is more computationally efficient.

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

```
Unit: microseconds
expr min lq mean median uq max neval cld
splines::ns 622.39 632.32 721.41 638.08 649.01 3409.0 1000 b
splines2::naturalSpline 120.08 129.01 144.23 140.33 143.45 2661.1 1000 a
```

The function `mSpline()`

produces periodic spline basis (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. So we performed a comparison with package **pbs** (version 1.1), 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 = knots, degree = degree, intercept = TRUE,
Boundary.knots = b_knots, periodic = TRUE),
"splines2::mSpline" = mSpline(
x, knots = knots, degree = degree, intercept = TRUE,
Boundary.knots = b_knots, periodic = TRUE
),
times = 1e3
)
```

```
Unit: microseconds
expr min lq mean median uq max neval cld
pbs::pbs 428.00 437.30 497.21 442.29 452.67 9996.2 1000 b
splines2::mSpline 122.13 126.25 155.73 135.47 138.67 2934.3 1000 a
```

```
R version 4.0.3 (2020-10-10)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Arch Linux
Matrix products: default
BLAS: /usr/lib/libopenblasp-r0.3.13.so
LAPACK: /usr/lib/liblapack.so.3.9.0
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_0.4.1 microbenchmark_1.4-7
loaded via a namespace (and not attached):
[1] Rcpp_1.0.5 knitr_1.30 magrittr_2.0.1 MASS_7.3-53 ibs_1.4
[6] lattice_0.20-41 rlang_0.4.10 multcomp_1.4-15 stringr_1.4.0 tools_4.0.3
[11] grid_4.0.3 xfun_0.19 TH.data_1.0-10 htmltools_0.5.0 yaml_2.2.1
[16] survival_3.2-7 digest_0.6.27 Matrix_1.3-0 codetools_0.2-16 evaluate_0.14
[21] rmarkdown_2.6 sandwich_3.0-0 stringi_1.5.3 compiler_4.0.3 pbs_1.1
[26] mvtnorm_1.1-1 zoo_1.8-8
```