# Approximation with Polynomials

#### 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.