Appendix A

Let the dependent variable Y be associated with an independent variable x in the form of a polynomial, given by

$Y\left(x\right)={\beta}_{0}+{\beta}_{1}x+{\beta}_{2}{x}^{2}+\mathrm{...}+{\beta}_{K}{x}^{K}\text{,}$

(A.1)

where K is the highest order of polynomials specified in the model.

Equation (A.1) is a convenient curvilinear expression of y as a function of x and is fitted to observed pairs of associated values y_{T} and x_{T} where T = 1, …, n. Suppose that the variable x_{T} progresses by constant intervals. It is then convenient to standardize the x-scale, written as

${x}_{T}=T-\frac{\left(n+1\right)}{2}\text{\hspace{1em}}\text{\hspace{1em}}T=\mathrm{1,2,...},n\text{,}$

and to fit Y(x) in terms of a weighted sum of orthogonal polynomial:

$Y\left(x\right)={B}_{0}{\stackrel{~}{\phi}}_{0}\left(x\right)+{B}_{1}{\stackrel{~}{\phi}}_{1}\left(x\right)+{B}_{2}{\stackrel{~}{\phi}}_{2}\left(x\right)+\mathrm{...}+$

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