[6296] in s-news-athena
CI for derivatives
daemon@ATHENA.MIT.EDU (Trevor Hastie)
Fri Jan 27 21:33:05 1995
Date: Fri, 27 Jan 1995 18:22:52 -0800
From: trevor@mallet.Stanford.EDU (Trevor Hastie)
To: s-news@utstat.toronto.edu
>
> I am trying to estimate confidence intervals of the derivitives obtained by
> smooth.spline(). There is no information about this in any of the S-PLUS
> manuals, nor in the monograph on this technique cited in the S-PLUS manual
> (Spline Models for Observational Data, by Grace Wahba). The monograph only
> gives Wahba's address as: University of Wisconsin at Madison. Does anyone
> know of her full address, phone number or E-MAIL (preferably) address? If I
> find the answer to my problem, I will post a summary on this list.
> Bill Shipley
> Departement de Biologie
> Universite de Sherbrooke
> Sherbrooke (Quebec)
> CANADA J1K 2R1
> bshipley@courrier.USherb.ca
>
One simple approach is to acknowledge that the derivative operator
is linear (for a given value of the smoothing parameter) in the
response y, hence it can be written as d=Gy for some matrix G.
Given G, and appropriate iid error asumptions, one can compute
the covariance matrix: Cov(d) = GG^T.sigma. If you prefer the
Bayesian style posterior intervals, these will also involve G and
perhaps some funtion of the smoothing matrix S itself.
How to get G? and S?
Well, G=GI where I is the identity matrix (also S=SI), so apply
smooth.spline() and predict.smooth.spline(, deriv=1) repeatedly
to the columns of I, and then paste the results together.
Voila --- you have G and S. (See page 127 of Hastie and Tibshirani,
"Generalized Additive Models", 1990, for slightly more details)
Trevor Hastie trevor@playfair.stanford.edu
Phone: 415-725-2231 Fax: 415-725-8977
Statistics Department, Sequoia Hall, Stanford University, Ca94305
