Dear John Sorry I use 3 degree of freedom for cubic spline. After using 4, it is still not 2. I may make some naive mistake, but I cannot figure out. Where is the problem?
4 (cubic on the right side of the *interior* knot 8) + 4 (cubic on the left side of the *interior* knot 8) - 1 (two curves must be continuous at the *interior* knot 8) - 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) = 3, not 2 Thanks Xing On Tue, Apr 15, 2014 at 10:54 AM, John Fox <j...@mcmaster.ca> wrote: > 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.
