R: Interpolating Splines

splinefun {stats} | R Documentation

## Interpolating Splines

### Description

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 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 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"`; note that the latter is the fastest for large `length(x)`.

### 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.

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.

### Author(s)

R Core Team.

Simon Wood for the original code for Hyman filtering.

### 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. doi: 10.1090/S0025-5718-1989-0962209-1.

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. doi: 10.1137/0717021.

Hyman, J. M. (1983). Accurate monotonicity preserving cubic interpolation. SIAM Journal on Scientific and Statistical Computing, 4, 645–654. doi: 10.1137/0904045.

### 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

    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.

[Package stats version 3.5.1 Index]