Orthogonal polynomials

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)

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

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

$Y\left(x\right)=B_0{\tilde{\phi }}_0\left(x\right)+B_1{\tilde{\phi }}_1\left(x\right)+B_2{\tilde{\phi }}_2\left(x\right)+\mathrm{...}+B_K{\tilde{\phi }}_K\left(x\right)\text{,}$

(A.2)

where ${\tilde{\phi }}_k(x)$ is the kth degree of polynomial, where k = 0, 1, …, K.

Let any pair of these polynomials, ${\tilde{\phi }}_k(x)$ and ${\tilde{\phi }}_{k^{\prime }}(x)$ where k ≠ k′, satisfy the orthogonal condition

$\sum _{t=1}^n{\tilde{\phi }}_k\left(x\right){\tilde{\phi }}_{k^{\prime }}\left(x\right)=0\text{,}$

(A.3)

which can be expanded to determine all ${\tilde{\phi }}_k(x)$. Specifically, we have

$\left.\begin{array}{l}{\tilde{\phi }}_0\left(x\right)=1,{\tilde{\phi }}_1\left(x\right)={\tilde{\lambda }}_{1}x,{\tilde{\phi }}_2\left(x\right)={\tilde{\lambda }}_2\left[x^2-\frac{1}{12}\left(n^2-1\right)\right],\\ {\tilde{\phi }}_3\left(x\right)={\tilde{\lambda }}_3\left[x^3-\frac{1}{20}\left(3n^2-7\right)x\right],{\tilde{\phi }}_4\left(x\right)={\tilde{\lambda }}_4\left[x^4-\frac{1}{14}\left(3n^2-13\right)x^2+\frac{3}{560}\left(n^2-1\right)\left(n^2-9\right)\right],\\ {\tilde{\phi }}_5\left(x\right)={\tilde{\lambda }}_5\left[x^5-\frac{5}{18}\left(n^2-7\right)x^3+\frac{1}{1008}\left(15n^4-230n^2+407\right)x\right],\\ {\tilde{\phi }}_6\left(x\right)={\tilde{\lambda }}_6\left[x^6-\frac{5}{44}\left(3n^2-31\right)x^4+\frac{1}{176}\left(5n^4-110n^2+329\right)x^2-\frac{5}{14784}\left(n^2-1\right)\left(n^2-9\right)\left(n^2-25\right)\right]\end{array}\right\}\text{,}$

(A.4)

where the ${\tilde{\lambda }}_k$ are selected such that the ${\tilde{\phi }}_k(x)$ are positive or negative integers throughout.

Suppose that the true regression law is a polynomial

$\tilde{\eta }\left(x\right)=\sum _{k=0}^K{\beta }_k^{*}{\phi }_k\left(x\right),$

from which the observed y_T differ by independent normal deviates or residuals z_T having a common variance σ². The orthogonal condition (A.3) then implies that the least-square estimators B_k of the ${\beta }_k^{*}$ are given by

$B_k=\frac{\sum _T{y}_T{\phi }_k\left(x_t\right)}{\sum _T{\left[{\phi }_k\left(x_T\right)\right]}^2}\text{,}$

(A.5)

where the coefficients B_k are independent normal variates with mean ${\beta }_k^{*}$ if k ≤ K and mean 0 if k > K and with variance

$s^2=\frac{〈\sum _{t=1}^n{y}_t^2-\sum _{k=0}^K\left\{B_k^2\sum _{t=1}^n\left[{\phi }_k{\left(x_t\right)}^2\right]\right\}〉}{n-K-1}\text{.}$

(A.6)

Thus, the ratios

$\frac{\left(B_k-{\beta }_k^{*}\right)\sqrt{\sum _t\left[{\phi }_k{\left(x_t\right)}^2\right]}}{s}\text{,}$

follow Student’s t-distribution for (n − K −1) degrees of freedom. As there are two terms involved in calculating s, namely B_k and ${\tilde{\phi }}_k(x)$, the null hypothesis for ${\beta }_k^{*}=0$ can be tested by using the F-test.

The values of the functions ${\tilde{\phi }}_k(x_t)$ can be found in the orthogonal polynomials table in Pearson and Hartley (1976, Table 47).