[R] ordinary polynomial coefficients from orthogonal polynomials?

[Prof Brian Ripley]

On Tue, 14 Jun 2005, James Salsman wrote:

> How can ordinary polynomial coefficients be calculated
> from an orthogonal polynomial fit?

Why would you want to do that?  predict() is perfectly happy with an
orthogonal polynomial fit and the `ordinary polynomial coefficients' are 
rather badly determined in your example since the design matrix has a very 
high condition number.

> I'm trying to do something like find a,b,c,d from
>  lm(billions ~ a+b*decade+c*decade^2+d*decade^3)
> but that gives:  "Error in eval(expr, envir, enclos) :
> Object "a" not found"

You could use

lm(billions ~ decade + I(decade^2) + I(decade^3))

except that will be numerically inaccurate, since

> m <- model.matrix(~ decade + I(decade^2) + I(decade^3))
> kappa(m)
[1] 3.506454e+16



> > decade <- c(1950, 1960, 1970, 1980, 1990)
> > billions <- c(3.5, 5, 7.5, 13, 40)
> > # source: http://www.ipcc.ch/present/graphics/2001syr/large/08.17.jpg
> >
> > pm <- lm(billions ~ poly(decade, 3))
> >
> > plot(decade, billions, xlim=c(1950,2050), ylim=c(0,1000),
> main="average yearly inflation-adjusted dollar cost of extreme weather
> events worldwide")
> > curve(predict(pm, data.frame(decade=x)), add=TRUE)
> > # output: http://www.bovik.org/storms.gif
> >
> > summary(pm)
>
> Call:
> lm(formula = billions ~ poly(decade, 3))
>
> Residuals:
>       1       2       3       4       5
>  0.2357 -0.9429  1.4143 -0.9429  0.2357
>
> Coefficients:
>                  Estimate Std. Error t value Pr(>|t|)
> (Intercept)        13.800      0.882  15.647   0.0406 *
> poly(decade, 3)1   25.614      1.972  12.988   0.0489 *
> poly(decade, 3)2   14.432      1.972   7.318   0.0865 .
> poly(decade, 3)3    6.483      1.972   3.287   0.1880
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
>
> Residual standard error: 1.972 on 1 degrees of freedom
> Multiple R-Squared: 0.9957,     Adjusted R-squared: 0.9829
> F-statistic: 77.68 on 3 and 1 DF,  p-value: 0.08317
>
> > pm
>
> Call:
> lm(formula = billions ~ poly(decade, 3))
>
> Coefficients:
>      (Intercept)  poly(decade, 3)1  poly(decade, 3)2  poly(decade, 3)3
>           13.800            25.614            14.432             6.483
>
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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