[R] Orthogonal Polynomial Regression Parameter Estimation

[Prof Brian Ripley]

You have sent this to both the S and R lists under different names:  who
are you and which system are you interested in?

You use poly(), as you did, but the internals of poly() are not the same 
on the two systems.  Do read the documentation to find out which 
orthogonal polynomials are used.

Hint: The S-PLUS description says

   Returns a matrix of orthonormal polynomials, which represents a basis
   for polynomial regression.

Notice the difference?  Orthogonal polynomials are not uniquely defined
(nor are orthonormal ones, BTW).


On Thu, 6 May 2004, WilD KID wrote:

> Dear all,
> 
> Can any one tell me how can i perform Orthogonal
> Polynomial Regression parameter estimation in R?
> 
> --------------------------------------------
> 
> Here is an "Orthogonal Polynomial" Regression problem
> collected from Draper, Smith(1981), page 269. Note
> that only value of alpha0 (intercept term) and signs
> of each estimate match with the result obtained from
> coef(orth.fit). What went wrong?
> 
> --------------------------------------------
> 
> Data:
> 
> > Z<-1957:1964
> > Y<-c(0.93,0.99,1.11,1.33,1.52,1.60,1.47,1.33)
> > X<-Z-1956
> > X
> [1] 1 2 3 4 5 6 7 8
> 
> --------------------------------------------
> 
> Using lm function to get orthogonal polynomial, we
> get-
> 
> > orth.fit<-lm(Y~poly(X,degree =6))
> > coef(orth.fit)
> 
> (Intercept)            poly(X, degree = 6)1   poly(X,
> degree = 6)2 
> 1.285000000                     0.529260490          
> -0.316321867 
> 
> poly(X, degree = 6)3   poly(X, degree = 6)4   poly(X,
> degree = 6)5 
>         -0.221564684            0.054795962           
> 0.062910192 
> 
> poly(X, degree = 6)6 
>          0.006154575 
> 
> --------------------------------------------
> 
> And using the solution procedure given in Draper,
> Smith(1981) is -
> 
> --------------------------------------------
> 
> The following values are coefficients of 0-6th order
> (for n=8) polynomial collected from Pearson, Hartley
> (1958) table, page 212:
> 
> > p0<-rep(1,8)
> > p1<-c(-7,-5,-3,-1,1,3,5,7)
> > p2<-c(7,1,-3,-5,-5,-3,1,7)
> > p3<-c(-7,5,7,3,-3,-7,-5,7)
> > p4<-c(7,-13,-3,9,9,-3,-13,7)
> > p5<-c(-7,23,-17,-15,15,17,-23,7)
> > p6<-c(1,-5,9,-5,-5,9,-5,1)
> 
> Now, the estimated parameters of the orthogonal
> polynomial is calculated by the following formula:
> 
> > alpha0<-sum(Y*p0)/sum(p0^2);
> alpha1<-sum(Y*p1)/sum(p1^2);
> alpha2<-sum(Y*p2)/sum(p2^2);
> alpha3<-sum(Y*p3)/sum(p3^2);
> alpha4<-sum(Y*p4)/sum(p4^2);
> alpha5<-sum(Y*p5)/sum(p5^2);
> alpha6<-sum(Y*p6)/sum(p6^2)
> > alpha0;alpha1;alpha2;alpha3;alpha4;alpha5;alpha6
> [1] 1.285
> [1] 0.04083333
> [1] -0.02440476
> [1] -0.01363636
> [1] 0.002207792
> [1] 0.001346154
> [1] 0.0003787879
> 
> --------------------------------------------
> 
> Any response / help / comment / suggestion / idea /
> web-link / replies will be greatly appreciated. 
> 
> Thanks in advance for your time.
> 
> _______________________
> 
> Mohammad Ehsanul Karim <wildscop at yahoo.com> 
> Institute of Statistical Research and Training 
> University of Dhaka, Dhaka- 1000, Bangladesh
> 
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> 

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595