Approximation with Polynomials
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```{r echo=FALSE,message=FALSE,warning=FALSE,results="hide",label="options",echo=FALSE}
require(knitr, quietly=TRUE)
opts_chunk$set(fig.width=3,fig.height=3,out.width="3in")
library(mosaic,quietly=TRUE)
trellis.par.set(theme=col.mosaic())
```
#### Approximation with Polynomials
Representation of arbitrary functions (or patterns in data) with polynomials. Low order ... things work well.
* quadratic or higher runs off to $\pm \infty$ very quickly. Not useful for extrapolation.
* Estimation becomes hard with high-order polynomials, due to non-orthogonality.
* Selection of model order: use anova, so that you look at the incremental improvement of adding a new term
* Parameter estimation: variance inflation due to collinearity.
The general lessons of experience in science is:
* Use first or at most second order.
* Use multiple variables rather than high-order in a single variable.
* Try to make the multiple variables orthogonal (by randomization or orthogonal assignment) to avoid variance inflation.
The important concept is orthogonality, not convergence.