Natural Cubic Spline Basis for Polynomial Splines

Generates the nonnegative natural cubic spline basis matrix, the corresponding integrals (from the left boundary knot), or derivatives of given order. Each basis is assumed to follow a linear trend for x outside of boundary.

Usage

naturalSpline(
  x,
  df = NULL,
  knots = NULL,
  intercept = FALSE,
  Boundary.knots = NULL,
  derivs = 0L,
  integral = FALSE,
  ...
)

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 returned matrix. One can specify df rather than knots, then the function chooses df - 1 - as.integer(intercept) internal knots at suitable quantiles of x ignoring missing values and those x outside of the boundary. Thus, df must be greater than or equal to 2. 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.
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.
derivs: A nonnegative integer specifying the order of derivatives of natural splines. The default value is 0L for the spline basis functions.
integral: A logical value. The default value is FALSE. If TRUE, this function will return the integrated natural splines from the left boundary knot.
...: Optional arguments that are not used.

Value

A numeric matrix of length(x) rows and df

columns if df is specified or length(knots) + 1 + as.integer(intercept) columns if knots are specified instead. Attributes that correspond to the arguments specified are returned for usage of other functions in this package.

Details

It is an implementation of the natural spline basis based on B-spline basis, which utilizes the close-form null space that can be derived from the recursive formula for the second derivatives of B-splines. The constructed spline basis functions are intended to be nonnegative within boundary with second derivatives being zeros at boundary knots.

A similar implementation is provided by splines::ns, which uses QR decomposition to find the null space of the second derivatives of B-spline basis at boundary knots. However, there is no guarantee that the resulting basis functions are nonnegative within boundary.

Examples

library(splines2)

x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)

## natural spline basis
nsMat0 <- naturalSpline(x, knots = knots, intercept = TRUE)
## integrals
nsMat1 <- naturalSpline(x, knots = knots, intercept = TRUE, integral = TRUE)
## first derivatives
nsMat2 <- naturalSpline(x, knots = knots, intercept = TRUE, derivs = 1)
## second derivatives
nsMat3 <- naturalSpline(x, knots = knots, intercept = TRUE, derivs = 2)

op <- par(mfrow = c(2, 2), mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
matplot(x, nsMat0, type = "l", ylab = "basis")
matplot(x, nsMat1, type = "l", ylab = "integral")
matplot(x, nsMat2, type = "l", ylab = "1st derivative")
matplot(x, nsMat3, type = "l", ylab = "2nd derivative")

par(op) # reset to previous plotting settings

## use the deriv method
all.equal(nsMat0, deriv(nsMat1), check.attributes = FALSE)
#> [1] TRUE
all.equal(nsMat2, deriv(nsMat0))
#> [1] TRUE
all.equal(nsMat3, deriv(nsMat2))
#> [1] TRUE
all.equal(nsMat3, deriv(nsMat0, 2))
#> [1] TRUE