The R package **splines2** is intended to be a
user-friendly supplementary package to the base package
**splines**. It provides functions to construct a variety
of regression spline basis functions that are not available from
**splines**. Most functions have a very similar user
interface with the function `splines::bs()`

. More
specifically, **splines2** allows users to construct the
basis functions of

- B-splines
- M-splines
- I-splines
- C-splines
- periodic splines
- natural cubic splines
- generalized Bernstein polynomials

along with their integrals (except C-splines) and derivatives of given order by closed-form recursive formulas.

Compared to **splines**, the package
**splines2** provides convenient interfaces for spline
derivatives with consistent handling on `NA`

’s. Most of the
implementations are in *C++* with the help of
**Rcpp** and **RcppArmadillo** since v0.3.0,
which boosted the computational performance.

In the remainder of this vignette, we illustrate the basic usage of most functions in the package through examples. We refer readers to Wang and Yan (2021) for a more formal introduction to the package with applications to shape-restricted regression. See the package manual for more details about function usage.

`## [1] '0.5.2'`

The `bSpline()`

function generates the basis matrix for
B-splines and extends the function `bs()`

of the package
**splines** by providing 1) the piece-wise constant basis
functions when `degree = 0`

, 2) the derivatives of basis
functions for a positive `derivs`

, 3) the integrals of basis
functions if `integral = TRUE`

, 4) periodic basis functions
based on B-splines if `periodic = TRUE`

.

One example of linear B-splines with three internal knots is as follows:

```
knots <- c(0.3, 0.5, 0.6)
x <- seq(0, 1, 0.01)
bsMat <- bSpline(x, knots = knots, degree = 1, intercept = TRUE)
plot(bsMat, mark_knots = "all")
```

For convenience, the package also provides functions
`ibs()`

and `dbs()`

for constructing the B-spline
integrals and derivatives, respectively. Two toy examples are as
follows:

```
ibsMat <- ibs(x, knots = knots, degree = 1, intercept = TRUE)
op <- par(mfrow = c(1, 2))
plot(bsMat, mark_knots = "internal")
plot(ibsMat, mark_knots = "internal")
abline(h = c(0.15, 0.2, 0.25), lty = 2, col = "gray")
```

```
bsMat <- bSpline(x, knots = knots, intercept = TRUE)
dbsMat <- dbs(x, knots = knots, intercept = TRUE)
plot(bsMat, mark_knots = "internal")
plot(dbsMat, mark_knots = "internal")
```

We may also obtain the derivatives easily by the `deriv()`

method as follows:

The function `bSpline()`

produces periodic spline basis
functions following Piegl and Tiller (1997, chap.
12) when `periodic = TRUE`

is specified. Different
from the regular basis functions, the `x`

is allowed to be
placed outside the boundary and the `Boundary.knots`

defines
the cyclic interval. For instance, one may obtain the periodic cubic
B-spline basis functions with cyclic interval (0, 1) as follows:

```
px <- seq(0, 3, 0.01)
pbsMat <- bSpline(px, knots = knots, Boundary.knots = c(0, 1),
intercept = TRUE, periodic = TRUE)
ipMat <- ibs(px, knots = knots, Boundary.knots = c(0, 1),
intercept = TRUE, periodic = TRUE)
dp1Mat <- deriv(pbsMat)
dp2Mat <- deriv(pbsMat, derivs = 2)
par(mfrow = c(1, 2))
plot(pbsMat, ylab = "Periodic B-splines", mark_knots = "boundary")
plot(ipMat, ylab = "The integrals", mark_knots = "boundary")
```

```
plot(dp1Mat, ylab = "The 1st derivatives", mark_knots = "boundary")
plot(dp2Mat, ylab = "The 2nd derivatives", mark_knots = "boundary")
```

For reference, the corresponding integrals and derivatives are also visualized.

M-splines (Ramsay 1988) can be
considered the normalized version of B-splines with unit integral within
boundary knots. An example given by Ramsay
(1988) was a quadratic M-splines with three internal knots placed
at 0.3, 0.5, and 0.6. The default boundary knots are the range of
`x`

, and thus 0 and 1 in this example.

```
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
par(op)
plot(msMat, mark_knots = "all")
```

The derivative of the given order of M-splines can be obtained by
specifying a positive integer to argument `dervis`

of
`mSpline()`

. Similarly, for an existing `mSpline`

object generated by `mSpline()`

, one can use the
`deriv()`

method for derivaitives. For example, the first
derivative of the M-splines given in the previous example can be
obtained equivalently as follows:

The `mSpline()`

function produces periodic splines based
on M-spline basis functions when `periodic = TRUE`

is
specified. The `Boundary.knots`

defines the cyclic interval,
which is the same with the periodic B-splines.

```
pmsMat <- mSpline(px, knots = knots, intercept = TRUE,
periodic = TRUE, Boundary.knots = c(0, 1))
plot(pmsMat, ylab = "Periodic Basis", mark_knots = "boundary")
```

We may still specify the argument `derivs`

in
`mSpline()`

or use the corresponding `deriv()`

method to obtain the derivatives when `periodic = TRUE`

.

Furthermore, we can obtain the integrals of the periodic M-splines by
specifying `integral = TRUE`

. The integral is integrated from
the left boundary knot.

```
ipmsMat <- mSpline(px, knots = knots, intercept = TRUE,
periodic = TRUE, Boundary.knots = c(0, 1), integral = TRUE)
plot(ipmsMat, ylab = "Integrals", mark_knots = "boundary")
abline(h = seq.int(0, 3), lty = 2, col = "gray")
```

I-splines (Ramsay 1988) are simply the
integral of M-splines and thus monotonically nondecreasing with unit
maximum value. A monotonically nondecreasing (nonincreasing) function
can be fitted by a linear combination of I-spline basis functions with
nonnegative (nonpositive) coefficients *plus a constant*, where
the coefficient of the constant is unconstrained.

The example given by Ramsay (1988) was the I-splines corresponding to the quadratic M-splines with three internal knots placed at 0.3, 0.5, and 0.6. Notice that the degree of I-splines is defined from the associated M-splines instead of their polynomial degree.

```
isMat <- iSpline(x, knots = knots, degree = 2, intercept = TRUE)
plot(isMat, mark_knots = "internal")
```

The corresponding M-spline basis matrix can be obtained easily as the
first derivatives of the I-splines by the `deriv()`

method.

We may specify `derivs = 2`

in the `deriv()`

method for the second derivatives of the I-splines, which are equivalent
to the first derivatives of the corresponding M-splines.

Convex splines (Meyer 2008) called C-splines are scaled integrals of I-splines with unit maximum value at the right boundary knot. Meyer (2008) applied C-splines to shape-restricted regression analysis. The monotone (nondecreasing) property of I-spines ensures the convexity of C-splines. A convex regression function can be estimated using linear combinations of the C-spline basis functions with nonnegative coefficients, plus an unconstrained linear combination of a constant and an identity function \(g(x)=x\). If the underlying regression function is both increasing and convex, the coefficient on the identity function is restricted to be nonnegative as well.

We may specify the argument `scale = FALSE`

in the
function `cSpline()`

to disable the scaling of the integrals
of I-splines. Then the actual integrals of the corresponding I-splines
will be returned. If `scale = TRUE`

(by default), each
C-spline basis is scaled to have unit height at the right boundary
knot.

```
csMat1 <- cSpline(x, knots = knots, degree = 2, intercept = TRUE)
plot(csMat1)
abline(h = 1, v = knots, lty = 2, col = "gray")
```

Similarly, the `deriv()`

method can be used to obtain the
derivatives. A nested call of `deriv()`

is supported for
derivatives of a higher order. However, the argument `derivs`

of the `deriv()`

method can be specified directly for better
computational performance. For example, the first and second derivatives
can be obtained by the following equivalent approaches,
respectively.

The Bernstein polynomials are equivalent to B-splines without internal knots and have also been applied to shape-constrained regression analysis (e.g., Wang and Ghosh 2012). The \(i\)-th basis of the generalized Bernstein polynomials of degree \(n\) over \([a, b]\) is defined as follows: \[ B_i^n(x)=\frac{1}{(b-a)^n}{n\choose i}(x-a)^i (b-x)^{n-i},~i\in\{0,\ldots,n\}, \] where \(a\le x\le b\). It reduces to regular Bernstein polynomials defined over \([0, 1]\) when \(a = 0\) and \(b = 1\).

We may obtain the basis matrix of the generalized using the function
`bernsteinPoly()`

. For example, the Bernstein polynomials of
degree 4 over \([0, 1]\) and is
generated as follows:

```
x1 <- seq.int(0, 1, 0.01)
x2 <- seq.int(- 1, 1, 0.01)
bpMat1 <- bernsteinPoly(x1, degree = 4, intercept = TRUE)
bpMat2 <- bernsteinPoly(x2, degree = 4, intercept = TRUE)
par(mfrow = c(1, 2))
plot(bpMat1)
plot(bpMat2)
```

In addition, we may specify `integral = TRUE`

or
`derivs = 1`

in `bernsteinPoly()`

for their
integrals or first derivatives, respectively.

```
ibpMat1 <- bernsteinPoly(x1, degree = 4, intercept = TRUE, integral = TRUE)
ibpMat2 <- bernsteinPoly(x2, degree = 4, intercept = TRUE, integral = TRUE)
dbpMat1 <- bernsteinPoly(x1, degree = 4, intercept = TRUE, derivs = 1)
dbpMat2 <- bernsteinPoly(x2, degree = 4, intercept = TRUE, derivs = 1)
par(mfrow = c(2, 2))
plot(ibpMat1, ylab = "Integrals")
plot(ibpMat2, ylab = "Integrals")
plot(dbpMat1, ylab = "Derivatives")
plot(dbpMat2, ylab = "Derivatives")
```

Similarly, we may also use the `deriv()`

method to get
derivatives of an existing `bernsteinPoly`

object.

```
stopifnot(is_equivalent(dbpMat1, deriv(bpMat1)))
stopifnot(is_equivalent(dbpMat2, deriv(bpMat2)))
stopifnot(is_equivalent(dbpMat1, deriv(ibpMat1, 2)))
stopifnot(is_equivalent(dbpMat2, deriv(ibpMat2, 2)))
```

The package provides two variants of the natural cubic splines that
can be constructed by `naturalSpline()`

and
`nsk()`

, respectively, both of which are different from
`splines::ns()`

.

The `naturalSpline()`

function returns nonnegative basis
functions (within the boundary) for natural cubic splines by utilizing a
closed-form null space derived from the second derivatives of cubic
B-splines. When `integral = TRUE`

, the function
`naturalSpline()`

returns the integral of each natural spline
basis.

```
nsMat <- naturalSpline(x, knots = knots, intercept = TRUE)
insMat <- naturalSpline(x, knots = knots, intercept = TRUE, integral = TRUE)
par(mfrow = c(1, 2))
plot(nsMat, ylab = "Basis")
plot(insMat, ylab = "Integrals")
```

Similarly, one may directly specify the argument `derivs`

in `naturalSpline()`

or use the corresponding
`deriv()`

method to obtain the derivatives of spline basis
functions.

```
d1nsMat <- naturalSpline(x, knots = knots, intercept = TRUE, derivs = 1)
d2nsMat <- deriv(nsMat, 2)
matplot(x, d1nsMat, type = "l", ylab = "The 1st derivatives")
matplot(x, d2nsMat, type = "l", ylab = "The 2nd derivatives")
```

The function `nsk()`

produces another variant of natural
cubic splines, where only one of the spline basis functions is nonzero
with unit height at every boundary and internal knot. As a result, the
coefficients of the basis functions are the values of the spline
function at the knots, which makes it more straightforward to interpret
the coefficient estimates. This idea originated from the function
`nsk()`

of the **survival** package (introduced
in version 3.2-8). The implementation of the `nsk()`

of
**splines2** essentially follows the
`survival::nsk()`

function. One noticeable argument for
`nsk()`

is `trim`

(equivalent to the argument
`b`

of `survival::nsk()`

). One may specify
`trim = 0.05`

to exclude 5% of the data from both sides when
setting the boundary knots, which can be a more sensible choice in
practice due to possible outliers. The `trim`

argument is
also available for `naturalSpline()`

, which however is zero
by default for backward compatibility. An illustration of the basis
functions generated by `nsk()`

is as follows: