10 Polynomial regression

Many relationships are curved: yield rises with fertiliser and then falls off, cost per unit falls with output and then rises, a growth curve bends. When the curvature is smooth and the theory gives no particular non-linear form, a polynomial in the regressors is the natural first model to try. A polynomial such as \[ y = \beta _0 + \beta _1x + \beta _2x^2 + \varepsilon \] is curved in \(x\) but linear in the parameters: it is the multiple regression model with regressors \(x_1 = x\) and \(x_2 = x^2\). So everything from Sections 3 to 9 applies: least squares, the tests, the residual analysis. This section is about what is special to polynomials: how to choose the order, the ill-conditioning of the powers of \(x\) and how to remove it, orthogonal polynomials, piecewise polynomials, models in several regressors, and the stationary points of a fitted surface.

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