polynomial regression in R: how to add hard constraints (go through specified points)?

Question

I'd like to perform polynomial regression on my data (which lies in [0,10]), but I need to ensure that the endpoints of the range are fixed, i.e. that the curve goes through (0,0) and (10,10). So the interpolant

$y = a + bx + cx^2 + \epsilon$

becomes

$y = bx + cx^2 + \epsilon$ with $b = 1 - 10c$.

Assuming just quadratic for now, although I might go to a cubic if the data warrants it. But how do I express this constraint during fitting?

Do you want any random components in your data? If yes, the description is needed. — user158565, Dec 10 '18 at 15:21
Ah, it's only the interpolant that I need, thinking about this... — Paul Miller, Dec 10 '18 at 15:28

score 1 · Accepted Answer · answered Dec 10 '18 at 15:19

1

It would be easiest just to plug your constraint into the model and rewrite it:

$y = bx + cx^2 = (1 - 10c)x + cx^2 = x + c(x^2 - 10x)$

Regress $y-x$ against $x^2 - 10x$, without an intercept, to get $c$. Then you can compute $b$ from that.

answered Dec 10 '18 at 15:19

The Laconic

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polynomial regression in R: how to add hard constraints (go through specified points)?

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