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

John Fox jfox at mcmaster.ca
Tue Apr 15 19:54:35 CEST 2014

```Dear Xing,

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
> project.org] On Behalf Of Xing Zhao
> Sent: Tuesday, April 15, 2014 1:18 PM
> To: John Fox
> Cc: r-help at r-project.org; Michael Friendly
> Subject: Re: [R] The explanation of ns() with df =2
>
> Dear Michael and Fox
>
> Thanks for your elaboration. Combining your explanations would, to my
> understanding, lead to the following  calculation of degree of
> freedoms.
>
> 3 (cubic on the right side of the *interior* knot 8)
> + 3 (cubic on the left side of the *interior* knot 8)
> - 1 (two curves must be continuous at the *interior* knot 8)

You shouldn't subtract 1 for continuity since you haven't allowed a
different level on each side of the knot (that is your initial counting of 3
parameters for the cubic doesn't include a constant).

Best,
John

> - 1 (two curves must have 1st order derivative continuous at the
> *interior* knot 8)
> - 1 (two curves must have 2nd order derivative continuous at the
> *interior* knot 8)
> - 1 (right side cubic curve must have 2nd order derivative = 0 at the
> boundary knot 15 due to the linearity constraint)
> - 1 (similar for the left)
> = 1, not 2
>
> Where is the problem?
>
> Best,
> Xing
>
> On Tue, Apr 15, 2014 at 6:17 AM, John Fox <jfox at mcmaster.ca> wrote:
> > 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
> > Hamilton, Ontario, Canada
> > 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.
> >> >
> >> > How to think about this problem?
> >> >
> >> > 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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> guide.html
> >> and provide commented, minimal, self-contained, reproducible code.
> >
> >
> >
> >
> >
>
> ______________________________________________
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> guide.html
> and provide commented, minimal, self-contained, reproducible code.

```

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