# [R] Orthogonal polynomials used by R

Fox, John j|ox @end|ng |rom mcm@@ter@c@
Wed Nov 27 18:11:04 CET 2019

```Dear Ashim,

Orthogonal polynomials are used because they tend to produce more accurate numerical computations, not because their coefficients are interpretable, so I wonder why you're interested in the coefficients.

The regressors produced are orthogonal to the constant regressor and are orthogonal to each other (and in fact are orthonormal), as it's simple to demonstrate:

------- snip -------

> x <- 1:93
> y <- 1 + x + x^2 + x^3 + x^4 + rnorm(93)
> (m <- lm(y ~ poly(x, 4)))

Call:
lm(formula = y ~ poly(x, 4))

Coefficients:
(Intercept)  poly(x, 4)1  poly(x, 4)2  poly(x, 4)3  poly(x, 4)4
15574516    172715069     94769949     27683528      3429259

> X <- model.matrix(m)
(Intercept) poly(x, 4)1 poly(x, 4)2 poly(x, 4)3 poly(x, 4)4
1           1  -0.1776843   0.2245083  -0.2572066  0.27935949
2           1  -0.1738216   0.2098665  -0.2236579  0.21862917
3           1  -0.1699589   0.1955464  -0.1919525  0.16390514
4           1  -0.1660962   0.1815482  -0.1620496  0.11487597
5           1  -0.1622335   0.1678717  -0.1339080  0.07123722
6           1  -0.1583708   0.1545171  -0.1074869  0.03269145

> zapsmall(crossprod(X))# X'X
(Intercept) poly(x, 4)1 poly(x, 4)2 poly(x, 4)3 poly(x, 4)4
(Intercept)          93           0           0           0           0
poly(x, 4)1           0           1           0           0           0
poly(x, 4)2           0           0           1           0           0
poly(x, 4)3           0           0           0           1           0
poly(x, 4)4           0           0           0           0           1

------- snip -------

If for some not immediately obvious reason you're interested in the regression coefficients, why not just use a "raw" polynomial:

------- snip -------

> (m1 <- lm(y ~ poly(x, 4, raw=TRUE)))

Call:
lm(formula = y ~ poly(x, 4, raw = TRUE))

Coefficients:
(Intercept)  poly(x, 4, raw = TRUE)1  poly(x, 4, raw = TRUE)2  poly(x, 4, raw = TRUE)3
1.5640                   0.8985                   1.0037                   1.0000
poly(x, 4, raw = TRUE)4
1.0000

------- snip -------

These coefficients are simply interpretable but the model matrix is more poorly conditioned:

------- snip -------

(Intercept) poly(x, 4, raw = TRUE)1 poly(x, 4, raw = TRUE)2 poly(x, 4, raw = TRUE)3
1           1                       1                       1                       1
2           1                       2                       4                       8
3           1                       3                       9                      27
4           1                       4                      16                      64
5           1                       5                      25                     125
6           1                       6                      36                     216
poly(x, 4, raw = TRUE)4
1                       1
2                      16
3                      81
4                     256
5                     625
6                    1296
> round(cor(X1[, -1]), 2)
poly(x, 4, raw = TRUE)1 poly(x, 4, raw = TRUE)2 poly(x, 4, raw = TRUE)3
poly(x, 4, raw = TRUE)1                    1.00                    0.97                    0.92
poly(x, 4, raw = TRUE)2                    0.97                    1.00                    0.99
poly(x, 4, raw = TRUE)3                    0.92                    0.99                    1.00
poly(x, 4, raw = TRUE)4                    0.87                    0.96                    0.99
poly(x, 4, raw = TRUE)4
poly(x, 4, raw = TRUE)1                    0.87
poly(x, 4, raw = TRUE)2                    0.96
poly(x, 4, raw = TRUE)3                    0.99
poly(x, 4, raw = TRUE)4                    1.00

------- snip -------

The two parametrizations are equivalent, however, in that they represent the same regression surface, and so, e.g., produce the same fitted values:

------- snip -------

> all.equal(fitted(m), fitted(m1))
 TRUE

------- snip -------

Because one is usually not interested in the individual coefficients of a polynomial there usually isn't a reason to prefer one parametrization to the other on the grounds of interpretability, so why do you need to interpret the regression equation?

I hope this helps,
John

-----------------------------
John Fox, Professor Emeritus
McMaster University
Web: http::/socserv.mcmaster.ca/jfox

> On Nov 27, 2019, at 10:17 AM, Ashim Kapoor <ashimkapoor using gmail.com> wrote:
>
> Dear Petr,
>
> Many thanks for the quick response.
>
> https://en.wikipedia.org/wiki/Discrete_orthogonal_polynomials
>
> Also I read  in ?poly:-
>     The orthogonal polynomial is summarized by the coefficients, which
>     can be used to evaluate it via the three-term recursion given in
>     Kennedy & Gentle (1980, pp. 343-4), and used in the ‘predict’ part
>     of the code.
>
>
> Out of curiosity, what is the name of the discrete orthogonal polynomial
> used by R ?
> What discrete measure is it orthogonal with respect to ?
>
> Many thanks,
> Ashim
>
>
>
>
> On Wed, Nov 27, 2019 at 6:11 PM PIKAL Petr <petr.pikal using precheza.cz> wrote:
>
>> You could get answer quickly by searching net.
>>
>>
>> https://stackoverflow.com/questions/39031172/how-poly-generates-orthogonal-p
>> olynomials-how-to-understand-the-coefs-ret/39051154#39051154
>> <https://stackoverflow.com/questions/39031172/how-poly-generates-orthogonal-polynomials-how-to-understand-the-coefs-ret/39051154#39051154>
>>
>> Cheers
>> Petr
>>
>>> -----Original Message-----
>>> From: R-help <r-help-bounces using r-project.org> On Behalf Of Ashim Kapoor
>>> Sent: Wednesday, November 27, 2019 12:55 PM
>>> To: R Help <r-help using r-project.org>
>>> Subject: [R] Orthogonal polynomials used by R
>>>
>>> Dear All,
>>>
>>> I have created a time trend by doing x<-1:93 because I have a time series
>>> with 93 data points. Next I did :-
>>>
>>> y = lm(series ~ poly(x,4))\$residuals
>>>
>>> to detrend series.
>>>
>>> I choose this 4 as the order of my polynomial using cross validation/
>>> checking the absence of trend in the residuals so I think I have not
>> overfit
>>> this series.
>>>
>>> I wish to document the formula of poly(x,4). I am not able to find it in
>> ?poly
>>>
>>> Can someone please tell me what the formula for the orthogonal
>>> polynomial used by R is ?
>>>
>>> Thank you,
>>> Ashim
>>>
>>>      [[alternative HTML version deleted]]
>>>
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>>
>
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>
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