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

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.

You can install the released version from CRAN.

`install.packages("splines2")`

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")
::install_github("wenjie2wang/splines2", upgrade = "never") remotes
```

The Online document provides a 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** 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)
<- runif(1e3)
x <- 3
degree <- degree + 1
ord <- seq.int(0.1, 0.9, 0.1)
internal_knots <- c(0, 1)
boundary_knots <- sort(c(internal_knots, rep(boundary_knots, ord)))
all_knots
## check equivalency of outputs
<- function(values) {
my_check 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(
knots = internal_knots, degree = degree,
x, 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
<- 2
derivs 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)
<- rnorm(length(all_knots) - ord)
coef_sp microbenchmark(
"ibs::ibs" = ibs::ibs(x, knots = all_knots, ord = ord, coef = coef_sp),
"splines2::ibs" = as.numeric(
::ibs(x, knots = internal_knots, degree = degree,
splines2intercept = 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(
knots = internal_knots, intercept = TRUE,
x, 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(
knots = internal_knots, degree = degree, intercept = TRUE,
x, 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
```