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.