splinefun
Interpolating Splines
Perform cubic (or Hermite) spline interpolation of given data points, returning either a list of points obtained by the interpolation or a function performing the interpolation.
Usage
splinefun(x, y = NULL, method = c("fmm", "periodic", "natural", "monoH.FC", "hyman"), ties = mean)
spline(x, y = NULL, n = 3*length(x), method = "fmm", xmin = min(x), xmax = max(x), xout, ties = mean)
splinefunH(x, y, m)
Arguments
- x, y
- vectors giving the coordinates of the points to be
interpolated. Alternatively a single plotting structure can be
specified: see
xy.coords
.y
must be increasing or decreasing formethod = "hyman"
. - m
- (for
splinefunH()
): vector of slopes $m[i]$ at the points $(x[i],y[i])$; these together determine the Hermite spline which is piecewise cubic, (only) once differentiable continuously. - method
- specifies the type of spline to be used. Possible
values are
"fmm"
,"natural"
,"periodic"
,"monoH.FC"
and"hyman"
. Can be abbreviated. - n
- if
xout
is left unspecified, interpolation takes place atn
equally spaced points spanning the interval [xmin
,xmax
]. - xmin, xmax
- left-hand and right-hand endpoint of the
interpolation interval (when
xout
is unspecified). - xout
- an optional set of values specifying where interpolation is to take place.
- ties
- Handling of tied
x
values. Either a function with a single vector argument returning a single number result or the string"ordered"
.
Details
The inputs can contain missing values which are deleted, so at least
one complete (x, y)
pair is required.
If method = "fmm"
, the spline used is that of Forsythe, Malcolm
and Moler (an exact cubic is fitted through the four points at each
end of the data, and this is used to determine the end conditions).
Natural splines are used when method = "natural"
, and periodic
splines when method = "periodic"
.
The method "monoH.FC"
computes a monotone Hermite spline
according to the method of Fritsch and Carlson. It does so by
determining slopes such that the Hermite spline, determined by
$(x[i],y[i],m[i])$, is monotone (increasing or
decreasing) iff the data are.
Method "hyman"
computes a monotone cubic spline using
Hyman filtering of an method = "fmm"
fit for strictly monotonic
inputs. (Added in R 2.15.2.)
These interpolation splines can also be used for extrapolation, that is
prediction at points outside the range of x
. Extrapolation
makes little sense for method = "fmm"
; for natural splines it
is linear using the slope of the interpolating curve at the nearest
data point.
Value
spline
returns a list containing components x
and
y
which give the ordinates where interpolation took place and
the interpolated values.splinefun
returns a function with formal arguments x
and
deriv
, the latter defaulting to zero. This function
can be used to evaluate the interpolating cubic spline
(deriv
= 0), or its derivatives (deriv
= 1, 2, 3) at the
points x
, where the spline function interpolates the data
points originally specified. It uses data stored in its environment
when it was created, the details of which are subject to change.
Warning
The value returned by splinefun
contains references to the code
in the current version of R: it is not intended to be saved and
loaded into a different R session. This is safer in R >= 3.0.0.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Dougherty, R. L., Edelman, A. and Hyman, J. M. (1989) Positivity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation. Mathematics of Computation 52, 471--494.
Forsythe, G. E., Malcolm, M. A. and Moler, C. B. (1977) Computer Methods for Mathematical Computations. Wiley.
Fritsch, F. N. and Carlson, R. E. (1980) Monotone piecewise cubic interpolation, SIAM Journal on Numerical Analysis 17, 238--246.
Hyman, J. M. (1983) Accurate monotonicity preserving cubic interpolation. SIAM J. Sci. Stat. Comput. 4, 645--654.
See Also
approx
and approxfun
for constant and
linear interpolation.
Package splines, especially interpSpline
and periodicSpline
for interpolation splines.
That package also generates spline bases that can be used for
regression splines.
smooth.spline
for smoothing splines.
Examples
library(stats)
require(graphics)
op <- par(mfrow = c(2,1), mgp = c(2,.8,0), mar = 0.1+c(3,3,3,1))
n <- 9
x <- 1:n
y <- rnorm(n)
plot(x, y, main = paste("spline[fun](.) through", n, "points"))
lines(spline(x, y))
lines(spline(x, y, n = 201), col = 2)
y <- (x-6)^2
plot(x, y, main = "spline(.) -- 3 methods")
lines(spline(x, y, n = 201), col = 2)
lines(spline(x, y, n = 201, method = "natural"), col = 3)
lines(spline(x, y, n = 201, method = "periodic"), col = 4)
legend(6, 25, c("fmm","natural","periodic"), col = 2:4, lty = 1)
y <- sin((x-0.5)*pi)
f <- splinefun(x, y)
ls(envir = environment(f))
splinecoef <- get("z", envir = environment(f))
curve(f(x), 1, 10, col = "green", lwd = 1.5)
points(splinecoef, col = "purple", cex = 2)
curve(f(x, deriv = 1), 1, 10, col = 2, lwd = 1.5)
curve(f(x, deriv = 2), 1, 10, col = 2, lwd = 1.5, n = 401)
curve(f(x, deriv = 3), 1, 10, col = 2, lwd = 1.5, n = 401)
par(op)
## Manual spline evaluation --- demo the coefficients :
.x <- splinecoef$x
u <- seq(3, 6, by = 0.25)
(ii <- findInterval(u, .x))
dx <- u - .x[ii]
f.u <- with(splinecoef,
y[ii] + dx*(b[ii] + dx*(c[ii] + dx* d[ii])))
stopifnot(all.equal(f(u), f.u))
## An example with ties (non-unique x values):
set.seed(1); x <- round(rnorm(30), 1); y <- sin(pi * x) + rnorm(30)/10
plot(x, y, main = "spline(x,y) when x has ties")
lines(spline(x, y, n = 201), col = 2)
## visualizes the non-unique ones:
tx <- table(x); mx <- as.numeric(names(tx[tx > 1]))
ry <- matrix(unlist(tapply(y, match(x, mx), range, simplify = FALSE)),
ncol = 2, byrow = TRUE)
segments(mx, ry[, 1], mx, ry[, 2], col = "blue", lwd = 2)
## An example of monotone interpolation
n <- 20
set.seed(11)
x. <- sort(runif(n)) ; y. <- cumsum(abs(rnorm(n)))
plot(x., y.)
curve(splinefun(x., y.)(x), add = TRUE, col = 2, n = 1001)
curve(splinefun(x., y., method = "monoH.FC")(x), add = TRUE, col = 3, n = 1001)
curve(splinefun(x., y., method = "hyman") (x), add = TRUE, col = 4, n = 1001)
legend("topleft",
paste0("splinefun( \"", c("fmm", "monoH.FC", "hyman"), "\" )"),
col = 2:4, lty = 1, bty = "n")
## and one from Fritsch and Carlson (1980), Dougherty et al (1989)
x. <- c(7.09, 8.09, 8.19, 8.7, 9.2, 10, 12, 15, 20)
f <- c(0, 2.76429e-5, 4.37498e-2, 0.169183, 0.469428, 0.943740,
0.998636, 0.999919, 0.999994)
s0 <- splinefun(x., f)
s1 <- splinefun(x., f, method = "monoH.FC")
s2 <- splinefun(x., f, method = "hyman")
plot(x., f, ylim = c(-0.2, 1.2))
curve(s0(x), add = TRUE, col = 2, n = 1001) -> m0
curve(s1(x), add = TRUE, col = 3, n = 1001)
curve(s2(x), add = TRUE, col = 4, n = 1001)
legend("right",
paste0("splinefun( \"", c("fmm", "monoH.FC", "hyman"), "\" )"),
col = 2:4, lty = 1, bty = "n")
## they seem identical, but are not quite:
xx <- m0$x
plot(xx, s1(xx) - s2(xx), type = "l", col = 2, lwd = 2,
main = "Difference monoH.FC - hyman"); abline(h = 0, lty = 3)
x <- xx[xx < 10.2] ## full range: x <- xx .. does not show enough
ccol <- adjustcolor(2:4, 0.8)
matplot(x, cbind(s0(x, deriv = 2), s1(x, deriv = 2), s2(x, deriv = 2))^2,
lwd = 2, col = ccol, type = "l", ylab = quote({{f*second}(x)}^2),
main = expression({{f*second}(x)}^2 ~" for the three 'splines'"))
legend("topright",
paste0("splinefun( \"", c("fmm", "monoH.FC", "hyman"), "\" )"),
lwd = 2, col = ccol, lty = 1:3, bty = "n")
## --> "hyman" has slightly smaller Integral f''(x)^2 dx than "FC",
## here, and both are 'much worse' than the regular fmm spline.