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) - 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.
