# 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 for`method = "hyman"`

. - m
- (for
`splinefunH()`

): vector of*slopes*$m[i]$ at the points $(x[i],y[i])$; these together determine the**H**ermite “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 at`n`

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.
```

*Documentation reproduced from package stats, version 3.2.5, License: Part of R 3.2.5*