Dear John, Thanks for your patience, got your idea
Best, Xing On Wed, Apr 16, 2014 at 4:11 AM, John Fox <j...@mcmaster.ca> wrote: > Dear Xing, > > My point wasn't that you should add 1 df for a constant to each cubic but > that you need not subtract 1 for continuity. I'm sorry that I didn't make > that clear. When you use the natural spline in a regression, there's a > constant in the model as a separate term with 1 df, which in effect adjusts > the level of the regression curve (assuming one X for simplicity). If you add > a constant to each component cubic, the constant in the model is rendered > redundant, accounting for the extra df. > > I'm afraid that if this is still not sufficiently clear, I'll have to defer > to someone with greater powers of explanation. > > Best, > John > > ------------------------------------------------------------- > John Fox > McMaster University > Hamilton, Ontario, Canada > http://socserv.socsci.mcmaster.ca/jfox > >> On Apr 16, 2014, at 12:10 AM, Xing Zhao <zhaox...@uw.edu> wrote: >> >> 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. ______________________________________________ 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.
