[R] smooth.spline
Spencer Graves
spencer.graves at pdf.com
Sun Jul 20 17:11:34 CEST 2008
Are you aware that there are many different kinds of splines?
With "spline" and "splinefun", you can use method = "fmm" (Forsyth,
Malcolm and Moler), "natural", or "periodic". I'm not familiar with
"fmm", but it seems to be adequately explained by the "Manual spline
evaluation" you quoted from the documentation.
Natural splines are perhaps the simplest: I(x-x0)*(x-x0)^j, where
x0 is a knot, and I(z) = 1 if z>0 and 0 otherwise.
However, computations using natural splines are numerically
unstable. The standard solution to this problem is to use B-splines,
which are 0 outside a finite interval.
Let's look at your example:
n <- 9
x <- 1:n
y <- rnorm(n)
plot(x, y, main = paste("spline[fun](.) through", n, "points"))
spl <- smooth.spline(x,y)
lines(spl)
The 'smooth.spline' function uses B-splines. To see what they
look like, let's do the following:
library(fda)
Bspl.basis <- create.bspline.basis(unique(spl$fit$knot))
# Check to make sure:
all.equal(knots(Bspl.basis, interior=FALSE), spl$fit$knot)
# TRUE
# What do B-splines look like?
plot(Bspl.basis)
abline(v=knots(Bspl.basis), lty='dotted', col='red')
# 7 interior knots, 2 end knots replicated 4 times each, for a spline
of order 4, degree 3 (cubic splines)
# total of 15 knots
# Each spline uses 5 consecutive knots, which means there will be 11
basis functions.
# NOTE: 'smooth.spline' rescaled the interval [1, 9] to [0, 1].
# Evaluate the 11 B-splines at 'x'
Bspl.basis.x <- eval.basis((x-1)/8, Bspl.basis)
round(Bspl.basis.x, 4)
# Now the manual computation:
y.spl <- Bspl.basis.x %*% spl$fit$coef
# Plot to confirm:
plot(x, y, main = paste("spline[fun](.) through", n, "points"))
spl.xy <- spline(x, y)
lines(spl.xy)
points(x, y.spl, pch=2, col='red')
Hope this helps.
Spencer
rkevinburton at charter.net wrote:
> Fair enough. FOr a spline interpolation I can do the following:
>
>
>> n <- 9
>> x <- 1:n
>> y <- rnorm(n)
>> plot(x, y, main = paste("spline[fun](.) through", n, "points"))
>> lines(spline(x, y))
>>
>
> Then look at the coefficients generated as:
>
>
>> f <- splinefun(x, y)
>> ls(envir = environment(f))
>>
> [1] "ties" "ux" "z"
>
>> splinecoef <- get("z", envir = environment(f))
>> slinecoef
>>
> $method
> [1] 3
>
> $n
> [1] 9
>
> $x
> [1] 1 2 3 4 5 6 7 8 9
>
> $y
> [1] 0.93571604 0.44240485 0.45451903 -0.96207396 -1.13246522 -0.60032698
> [7] -1.77506105 -0.09171419 -0.23262573
>
> $b
> [1] -1.53673409 0.22775629 -0.81788209 -1.16966436 0.73558677 -0.68744178
> [7] 0.08639287 1.86770869 -2.92992167
>
> $c
> [1] 1.3657783 0.3987121 -1.4443504 1.0925682 0.8126830 -2.2357115 3.0095462
> [8] -1.2282303 -3.5694000
>
> $d
> [1] -0.32235542 -0.61435416 0.84563953 -0.09329507 -1.01613149 1.74841922
> [7] -1.41259217 -0.78038989 -0.78038989
>
> WHen I look at ?spline there is even an example of "manually" using these coefficeients:
>
> ## Manual spline evaluation --- demo the coefficients :
> .x <- get("ux", envir = environment(f))
> u <- seq(3,6, by = 0.25)
> (ii <- findInterval(u, .x))
> dx <- u - .x[ii]
> f.u <- with(splinecoef,
> y[ii] + dx*(b[ii] + dx*(c[ii] + dx* d[ii])))
> stopifnot(all.equal(f(u), f.u))
>
>
> For the smooth.spline as
>
> spl <- smooth.spline(x,y)
>
> I can also look at the coefficients:
>
> spl$fit
> $knot
> [1] 0.000 0.000 0.000 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000
> [13] 1.000 1.000 1.000
>
> $nk
> [1] 11
>
> $min
> [1] 1
>
> $range
> [1] 8
>
> $coef
> [1] 0.90345898 0.73823276 0.40777431 -0.08046715 -0.54625461 -0.85205147
> [7] -0.96233408 -0.91373830 -0.66529714 -0.47674774 -0.38246971
>
> attr(,"class")
> [1] "smooth.spline.fit"
>
> But there isn't an example on how to "manual" use these coefficients. This is what I was asking about. Once I hae the coefficients how do I "manually" interpolate using the coefficients given and x.
>
> Thank you.
>
> Kevin
>
>
> ---- Spencer Graves <spencer.graves at pdf.com> wrote:
>
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html and provide commented,
>> minimal, self-contained, reproducible code.
>>
>> I do NOT know how to do what you want, but with a self-contained
>> example, I suspect many people on this list -- probably including me --
>> could easily solve the problem. Without such an example, there is a
>> high probability that any answer might (a) not respond to your need, and
>> (b) take more time to develop, just because we don't know enough of what
>> you are asking.
>>
>> Spencer
>>
>> rkevinburton at charter.net wrote:
>>
>>> Like I indicated. I understand the coefficients in a B-spline context. If I use the the 'spline' or 'splinefun' I can get the coefficients and they are grouped as 'a', 'b', 'c', and 'd' coefficients. But the coefficients for smooth.spline is just an array. I basically want to take these coefficients and outside of 'R' use them to form an interpolation. In other words I want 'R' to do the hard work and then export the results so they can be used else where.
>>>
>>> Thank you.
>>>
>>> Kevin
>>>
>>>
>> Spencer Graves wrote:
>>
>>> I believe that a short answer to your question is that the
>>> "smooth" is a linear combination of B-spline basis functions, and the
>>> coefficients are the weights assigned to the different B-splines in
>>> that basis.
>>> Before offering a much longer answer, I would want to know what
>>> problem you are trying to solve and why you want to know. For a brief
>>> description of B-splines, see
>>> "http://en.wikipedia.org/wiki/B-spline". For a slightly longer
>>> commentary on them I suggest the "scripts\ch01.R" in the DierckxSpline
>>> package: That script computes and displays some B-splines using
>>> "splineDesign", "spline.des" in the 'splines' package plus comparable
>>> functions in the 'fda' package. For more info on this, I found the
>>> first chapter of Paul Dierckx (1993) Curve and Surface Fitting with
>>> Splines (Oxford U. Pr.). Beyond that, I've learned a lot from the
>>> 'fda' package and the two companion volumes by Ramsay and Silverman
>>> (2006) Functional Data Analysis, 2nd ed. and (2002) Applied Functional
>>> Data Analysis (both Springer).
>>> If you'd like more help from this listserve, PLEASE do read the
>>> posting guide http://www.R-project.org/posting-guide.html and provide
>>> commented, minimal, self-contained, reproducible code.
>>> Hope this helps. Spencer Graves
>>>
>>> rkevinburton at charter.net wrote:
>>>
>>>> I like what smooth.spline does but I am unclear on the output. I can
>>>> see from the documentation that there are fit.coef but I am unclear
>>>> what those coeficients are applied to.With spline I understand the
>>>> "noraml" coefficients applied to a cubic polynomial. But these
>>>> coefficients I am not sure how to interpret. If I had a description
>>>> of the algorithm maybe I could figure it out but as it is I have this
>>>> question. Any help?
>>>>
>>>> Kevin
>>>>
>>>> ______________________________________________
>>>> R-help at r-project.org mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide
>>>> http://www.R-project.org/posting-guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>>
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
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