Dear Xing, > -----Original Message----- > From: r-help-boun...@r-project.org [mailto:r-help-bounces@r- > project.org] On Behalf Of Xing Zhao > Sent: Tuesday, April 15, 2014 1:18 PM > To: John Fox > Cc: r-help@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 <j...@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 <frien...@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 > >> > >> ______________________________________________ > >> R-help@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@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@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.
