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.
1 2 3 4 5 6 7 8 
x, y 
vectors giving the coordinates of the points to be
interpolated. Alternatively a single plotting structure can be
specified: see

m 
(for 
method 
specifies the type of spline to be used. Possible
values are 
n 
if 
xmin, xmax 
lefthand and righthand endpoint of the
interpolation interval (when 
xout 
an optional set of values specifying where interpolation is to take place. 
ties 
handling of tied 
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.
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.
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.
R Core Team.
Simon Wood for the original code for Hyman filtering.
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 convexitypreserving cubic and quintic Hermite interpolation. Mathematics of Computation, 52, 471–494. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1090/S00255718198909622091")}.
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. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1137/0717021")}.
Hyman, J. M. (1983). Accurate monotonicity preserving cubic interpolation. SIAM Journal on Scientific and Statistical Computing, 4, 645–654. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1137/0904045")}.
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99  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 < (x6)^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((x0.5)*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$x
u < seq(3, 6, by = 0.25)
(ii < findInterval(u, .x))
dx < u  .x[ii]
f.u < with(splinecoef,
y[ii] + dx*(b[ii] + dx*(c[ii] + dx* d[ii])))
stopifnot(all.equal(f(u), f.u))
## An example with ties (nonunique x values):
set.seed(1); x < round(rnorm(30), 1); y < sin(pi * x) + rnorm(30)/10
plot(x, y, main = "spline(x,y) when x has ties")
lines(spline(x, y, n = 201), col = 2)
## visualizes the nonunique ones:
tx < table(x); mx < as.numeric(names(tx[tx > 1]))
ry < matrix(unlist(tapply(y, match(x, mx), range, simplify = FALSE)),
ncol = 2, byrow = TRUE)
segments(mx, ry[, 1], mx, ry[, 2], col = "blue", lwd = 2)
## Another example with sorted x, but ties:
set.seed(8); x < sort(round(rnorm(30), 1)); y < round(sin(pi * x) + rnorm(30)/10, 3)
summary(diff(x) == 0) # > 7 duplicated xvalues
str(spline(x, y, n = 201, ties="ordered")) # all '$y' entries are NaN
## The default (ties=mean) is ok, but most efficient to use instead is
sxyo < spline(x, y, n = 201, ties= list("ordered", mean))
sapply(sxyo, summary)# all fine now
plot(x, y, main = "spline(x,y, ties=list(\"ordered\", mean)) for when x has ties")
lines(sxyo, col="blue")
## 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, 2.76429e5, 4.37498e2, 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.

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