Generates the spline basis matrix for B-splines representing the family of
piecewise polynomials with the specified interior knots, degree, and
boundary knots, evaluated at the values of `x`

.

## Usage

```
bSpline(
x,
df = NULL,
knots = NULL,
degree = 3L,
intercept = FALSE,
Boundary.knots = NULL,
periodic = FALSE,
derivs = 0L,
integral = FALSE,
warn.outside = getOption("splines2.warn.outside", TRUE),
...
)
ibs(
x,
df = NULL,
knots = NULL,
degree = 3,
intercept = FALSE,
Boundary.knots = NULL,
...
)
dbs(
x,
derivs = 1L,
df = NULL,
knots = NULL,
degree = 3,
intercept = FALSE,
Boundary.knots = NULL,
...
)
bsp(
x,
df = NULL,
knots = NULL,
degree = 3L,
intercept = FALSE,
Boundary.knots = NULL,
periodic = FALSE,
derivs = 0L,
integral = FALSE,
warn.outside = getOption("splines2.warn.outside", TRUE),
...
)
```

## Arguments

- x
The predictor variable. Missing values are allowed and will be returned as they are.

- df
Degree of freedom that equals to the column number of the returned matrix. One can specify

`df`

rather than`knots`

, then the function chooses`df - degree - as.integer(intercept)`

internal knots at suitable quantiles of`x`

ignoring missing values and those`x`

outside of the boundary. For periodic splines,`df - as.integer(intercept)`

internal knots will be chosen at suitable quantiles of`x`

relative to the beginning of the cyclic intervals they belong to (see Examples) and the number of internal knots must be greater or equal to the specified`degree - 1`

. If internal knots are specified via`knots`

, the specified`df`

will be ignored.- knots
The internal breakpoints that define the splines. The default is

`NULL`

, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots. For periodic splines, the number of knots must be greater or equal to the specified`degree - 1`

. Duplicated internal knots are not allowed.- degree
A nonnegative integer specifying the degree of the piecewise polynomial. The default value is

`3`

for cubic splines. Zero degree is allowed for piecewise constant basis functions.- intercept
If

`TRUE`

, the complete basis matrix will be returned. Otherwise, the first basis will be excluded from the output.- Boundary.knots
Boundary points at which to anchor the splines. By default, they are the range of

`x`

excluding`NA`

. If both`knots`

and`Boundary.knots`

are supplied, the basis parameters do not depend on`x`

. Data can extend beyond`Boundary.knots`

. For periodic splines, the specified bounary knots define the cyclic interval.- periodic
A logical value. If

`TRUE`

, the periodic splines will be returned. The default value is`FALSE`

.- derivs
A nonnegative integer specifying the order of derivatives of splines basis function. The default value is

`0`

.- integral
A logical value. If

`TRUE`

, the corresponding integrals of spline basis functions will be returned. The default value is`FALSE`

. For periodic splines, the integral of each basis is integrated from the left boundary knot.- warn.outside
A logical value indicating if a warning should be thrown out when any

`x`

is outside the boundary. This option can also be set through`options("splines2.warn.outside")`

after the package is loaded.- ...
Optional arguments that are not used.

## Value

A numeric matrix of `length(x)`

rows and `df`

columns if

`df`

is specified. If `knots`

are specified instead, the
output matrix will consist of ```
length(knots) + degree +
as.integer(intercept)
```

columns if `periodic = FALSE`

, or

`length(knots) + as.integer(intercept)`

columns if ```
periodic =
TRUE
```

. Attributes that correspond to the arguments specified are
returned for usage of other functions in this package.

## Details

This function extends the `bs()`

function in the `splines`

package
for B-spline basis functions by allowing piecewise constant (left-closed and
right-open except on the right boundary) spline basis of degree zero. In
addition, the function provides derivatives or integrals of the B-spline
basis functions when one specifies the arguments `derivs`

or
`integral`

appropriately. The function constructs periodic B-splines
when `periodic`

is `TRUE`

. All the implementations are based on
the closed-form recursion formula following De Boor (1978) and Wang and Yan
(2021).

The functions `ibs()`

and `dbs()`

are provided for convenience.
The former provides the integrals of B-splines and is equivalent to
`bSpline()`

with `integral = TRUE`

. The latter produces the
derivatives of given order of B-splines and is equivalent to
`bSpline()`

with default `derivs = 1`

. The function `bsp()`

is an alias of to encourage the use in a model formula.

## References

De Boor, Carl. (1978). *A practical guide to splines*.
Vol. 27. New York: Springer-Verlag.

Wang, W., & Yan, J. (2021). *Shape-restricted regression splines with R
package splines2*. Journal of Data Science, 19(3),498--517.

## See also

`knots`

for extracting internal and boundary knots.

## Examples

```
library(splines2)
set.seed(1)
x <- runif(100)
knots <- c(0.3, 0.5, 0.6) # internal knots
## cubic B-splines
bsMat <- bSpline(x, knots = knots, degree = 3, intercept = TRUE)
ibsMat <- update(bsMat, integral = TRUE) # the integrals
d1Mat <- deriv(bsMat) # the 1st derivaitves
d2Mat <- deriv(bsMat, 2) # the 2nd derivaitves
op <- par(mfrow = c(2, 2), mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
plot(bsMat, ylab = "Cubic B-splines")
plot(ibsMat, ylab = "The integrals")
plot(d1Mat, ylab = "The 1st derivatives")
plot(d2Mat, ylab = "The 2nd derivatives")
## evaluate at new values
predict(bsMat, c(0.125, 0.801))
#> 1 2 3 4 5 6 7
#> [1,] 0.2276364 0.5596808 0.1956893 0.01699357 0.0000000 0.0000000 0.0000000
#> [2,] 0.0000000 0.0000000 0.0000000 0.05216125 0.3495278 0.4634015 0.1349094
## periodic B-splines
px <- seq(0, 3, 0.01)
pbsMat <- bSpline(px, knots = knots, Boundary.knots = c(0, 1),
intercept = TRUE, periodic = TRUE)
ipMat <- update(pbsMat, integral = TRUE)
dpMat <- deriv(pbsMat)
dp2Mat <- deriv(pbsMat, 2)
plot(pbsMat, ylab = "Periodic B-splines", mark_knots = "b")
plot(ipMat, ylab = "The integrals", mark_knots = "b")
plot(dpMat, ylab = "The 1st derivatives", mark_knots = "b")
plot(dp2Mat, ylab = "The 2nd derivatives", mark_knots = "b")
par(op) # reset to previous plotting settings
```