Introduction
This follows up on the previous posting, using data provided by Akima (1972). The four panels of the plot produced by the following code compare the original Akima spline formula with an improved version, and three styles of spline()
fits.
As noted in the code, two spline()
methods are useless for general tasks and are ignored here. The "period"
method sets the endpoint yvalues equal, which will be useless in most applications. The "hyman"
method requires the y values to be increasing, which again is useless in most applications.
Methods
The code first creates the data, the MD=1 case of Akima (1972), and defines xx
as a finerresolution version of x
for plotting. Then it defines a function start
for common operations on each panel, and then it displays comparisons. The idea of showing each in reference to the original akima method is to guide the eye, since plotting all in one panel makes a tangle that is difficult to grasp.

if (!interactive()) png("splines2.png", width=7, height=7, unit="in", res=150, pointsize=8)
## Data from "MD=1" set of akima1972iasc
x < c(0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9)
y < c(0, 0, 0, 0, 0, 0, 1, 10, 10, 80, 100, 150)
## A mesh for plotting
xx < seq(min(x), max(x), length.out=200)
library(akima) # for aspline()
start < function()
{
plot(x, y, pch=20)
grid()
akima < aspline(x, y, xx, method="original")
lines(akima$x, akima$y, lty=1)
}
par(mar=c(3, 3, 1, 1), mgp=c(2, 0.7, 0), mfrow=c(2,2))
start()
akima < aspline(x, y, xx, method="improved")
lines(akima$x, akima$y, col=2)
legend("topleft", lwd=1, col=1:2, legend=c("aspline/original", "aspline/improved"), bg="white")
start()
fun < splinefun(x, y, method="fmm")
lines(xx, fun(xx), col=2)
legend("topleft", lwd=1, col=1:2, legend=c("aspline/original", "spline/fmm"), bg="white")
start()
fun < splinefun(x, y, method="natural")
lines(xx, fun(xx), col=2)
legend("topleft", lwd=1, col=1:2, legend=c("aspline/original", "spline/natural"), bg="white")
start()
fun < splinefun(x, y, method="monoH.FC")
lines(xx, fun(xx), col=2)
legend("topleft", lwd=1, col=1:2, legend=c("aspline/original", "spline/monoH.FC"), bg="white")
if (!interactive()) dev.off()

Results
Conclusions

spline()
withmethod="natural"
or"fmm"
produces undesirable wiggles (for x between 5 and 6). 
spline()
withmethod="monoH.FC"
produces results that re very similar toaspline()
withmethod="original"
. 
Arguably,
aspline()
withmethod="original"
produces results most similar to what one might draw by hand; as just noted,spline()
withmethod="monoH.FC"
produces essentially equivalent results. 
It would be a bad idea to base general selection on just this one test, but the code provided here is easy to modify for real data (alter lines 5 and 6), so this posting may be useful to readers.
Citations
Hiroshi Akima, 1972. Algorithm 433 Interpolation and smooth curve fitting based on local procedures (E2). Communications of the Association for Computing Machinery, 15(10): 914918.
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