# [R] The explanation of ns() with df =2

John Fox jfox at mcmaster.ca
Tue Apr 15 15:17:39 CEST 2014

```Dear Xing Zhao,

To elaborate slightly on Michael's comments, a natural cubic spline with 2 df has one *interior* knot and two boundary knots (as is apparent in the output you provided). The linearity constraint applies beyond the boundary knots.

I hope this helps,
John

------------------------------------------------
John Fox, Professor
McMaster University
http://socserv.mcmaster.ca/jfox/

On Tue, 15 Apr 2014 08:18:40 -0400
Michael Friendly <friendly at yorku.ca> wrote:
> No, the curves on each side of the know are cubics, joined
> so they are continuous.  Se the discussion in \S 17.2 in
> Fox's Applied Regression Analysis.
>
> On 4/15/2014 4:14 AM, Xing Zhao wrote:
> > Dear all
> >
> > I understand the definition of Natural Cubic Splines are those with
> > linear constraints on the end points. However, it is hard to think
> > about how this can be implement when df=2. df=2 implies there is just
> > one knot, which, according the the definition, the curves on its left
> > and its right should be both be lines. This means the whole line
> > should be a line. But when making some fits. the result still looks
> > like 2nd order polynomial.
> >
> >
> > Thanks
> > Xing
> >
> > ns(1:15,df =2)
> >                1           2
> >   [1,] 0.0000000  0.00000000
> >   [2,] 0.1084782 -0.07183290
> >   [3,] 0.2135085 -0.13845171
> >   [4,] 0.3116429 -0.19464237
> >   [5,] 0.3994334 -0.23519080
> >   [6,] 0.4734322 -0.25488292
> >   [7,] 0.5301914 -0.24850464
> >   [8,] 0.5662628 -0.21084190
> >   [9,] 0.5793481 -0.13841863
> > [10,] 0.5717456 -0.03471090
> > [11,] 0.5469035  0.09506722
> > [12,] 0.5082697  0.24570166
> > [13,] 0.4592920  0.41197833
> > [14,] 0.4034184  0.58868315
> > [15,] 0.3440969  0.77060206
> > attr(,"degree")
> > [1] 3
> > attr(,"knots")
> > 50%
> >    8
> > attr(,"Boundary.knots")
> > [1]  1 15
> > attr(,"intercept")
> > [1] FALSE
> > attr(,"class")
> > [1] "ns"     "basis"  "matrix"
> >
>
>
> --
> Michael Friendly     Email: friendly AT yorku DOT ca
> Professor, Psychology Dept. & Chair, Quantitative Methods
> York University      Voice: 416 736-2100 x66249 Fax: 416 736-5814
> 4700 Keele Street    Web:   http://www.datavis.ca
> Toronto, ONT  M3J 1P3 CANADA
>
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```