NB: linear splines, i.e. piecewise linear continuous functions.
Given the knots, use approx() or approxfun() however, the
automatic knots selection does not happen in the base R packages.
I'm sure there are several R packages doing this.
The best such package in my opinion is "earth" which does a
re-implementation (and extensive *generalization*) of the
famous MARS algorithm of Friedman.
==> https://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines
Note that their strengths and power is that they do their work
for multivariate x (MARS := Multivariate Adaptive Regression
Splines), but indeed do work for the simple 1D case.
In the following example, we always get 11 final knots,
but I'm sure one can tweak the many tuning paramters of earth()
to get more:
## Can we do knot-selection for simple (x,y) splines? === Yes, via earth() {using MARS}!
x <- (0:800)/8
f <- function(x) 7 * sin(pi/8*x) * abs((x-50)/20)^1.25 - (x-40)*(12-x)/64
curve(f(x), 0, 100, n = 1000, col=2, lwd=2)
set.seed(11)
y <- f(x) + 10*rnorm(x)
m.sspl <- smooth.spline(x,y) # base line "standard smoother"
require(earth)
fm1 <- earth(x, y) # default settings
summary(fm1, style = "pmax") #-- got 10 knots (x = 44 "used twice") below
## Call: earth(x=x, y=y)
## y =
## 175.9612
## - 10.6744 * pmax(0, x - 4.625)
## + 9.928496 * pmax(0, x - 10.875)
## - 5.940857 * pmax(0, x - 20.25)
## + 3.438948 * pmax(0, x - 27.125)
## - 3.828159 * pmax(0, 44 - x)
## + 4.207046 * pmax(0, x - 44)
## + 2.573822 * pmax(0, x - 76.5)
## - 10.99073 * pmax(0, x - 87.125)
## + 10.97592 * pmax(0, x - 90.875)
## + 9.331949 * pmax(0, x - 94)
## - 8.48575 * pmax(0, x - 96.5)
## Selected 12 of 12 terms, and 1 of 1 predictors
## Termination condition: Reached nk 21
## Importance: x
## Number of terms at each degree of interaction: 1 11 (additive model)
## GCV 108.6592 RSS 82109.44 GRSq 0.861423 RSq 0.86894
fm2 <- earth(x, y, fast.k = 0) # (more extensive forward pass)
summary(fm2)
all.equal(fm1, fm2)# they are identical (apart from 'call'):
fm3 <- earth(x, y, fast.k = 0, pmethod = "none", trace = 3) # extensive forward pass; *no* pruning
## still no change: fm3 "==" fm1
all.equal(predict(fm1, xx), predict(fm3, xx))
## BTW: The chosen knots and coefficients are
mat <- with(fm1, cbind(dirs, cuts=c(cuts), coef = c(coefficients)))
## Plots : fine grid for visualization: instead of xx <- seq(x[1], x[length(x)], length.out = 1024)
rnx <- extendrange(x) ## to extrapolate a bit
xx <- do.call(seq.int, c(rnx, list(length.out = 1200)))
cbind(f = f(xx),
sspl = predict(m.sspl, xx)$y,
mars = predict(fm1, xx)) -> fits
plot(x,y, xlim=rnx, cex = 1/4, col = adjustcolor(1, 1/2))
cols <- c(adjustcolor(2, 1/3),
adjustcolor(4, 2/3),
adjustcolor("orange4", 2/3))
lwds <- c(3, 2, 2)
matlines(xx, fits, col = cols, lwd = lwds, lty=1)
legend("topleft", c("true f(x)", "smooth.spline()", "earth()"),
col=cols, lwd=lwds, bty = "n")
title(paste("earth() linear spline vs. smooth.spline(); n =", length(x)))
mtext(substitute(f(x) == FDEF, list(FDEF = body(f))))