[Rd] Rd] Numerics behind splineDesign

[Nathaniel Smith]

On Thu, Aug 2, 2012 at 2:09 PM, Terry Therneau <therneau at mayo.edu> wrote:
>
>
> On 08/02/2012 05:00 AM, r-devel-request at r-project.org wrote:
>>
>> Now I just have to grovel over the R code in ns() and bs() to figure
>> out how exactly they pick knots and handle boundary conditions, plus
>> there is some code that I don't understand in ns() that uses qr() to
>> postprocess the output from spline.des. I assume this is involved
>> somehow in imposing the boundary conditions...
>>
>> Thanks again everyone for your help,
>> -- Nathaniel
>
> The ns and bs function post-process the spline bases to get an orthagonal
> basis matrix, this is the use of qr.  I think this causes much more grief
> than it is worth, for the sake of a small increase in numeric stability.
> For instance when you plot the spline bases, they don't look anything like
> the basis functions one would expect.  (Perhaps my background in numerical
> analysis was a hindrance here, since I know something about splines and thus
> have an expectation).

You know, the white book does claim that S orthogonalizes the spline
basis functions, but R doesn't seem to actually do that:

> crossprod(bs(1:10, 4, intercept=TRUE))
           1         2         3          4
1 1.84104162 0.6154211 0.2570069 0.06430817
2 0.61542109 0.7710207 0.5787736 0.25700689
3 0.25700689 0.5787736 0.7710207 0.61542109
4 0.06430817 0.2570069 0.6154211 1.84104162
> crossprod(ns(1:10, 4, intercept=TRUE))
           1         2           3           4
1  1.0659929 0.5314089  0.37125505 -0.17838443
2  0.5314089 1.0681360  0.23122018  0.14422236
3  0.3712550 0.2312202  1.30226657 -0.03809082
4 -0.1783844 0.1442224 -0.03809082  1.10144173

and they look quite reasonable when plotted to me.

ns is doing something more complicated; I think it's computing the
second derivatives of each basis function at the position of the
boundary knots, and then using that to somehow transform the original
basis. Since the key feature of ns is that the returned basis has a
zero second derivative at the boundary knots, I'm sure this makes
sense if one stares at it for long enough.

-n