Appendix A
Orthogonal polynomials
Let the dependent variable Y be associated with an independent variable x in the form of a polynomial, given by
Y(x)=β0+β1x+β2x2+...+βKxK,
(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 yT and xT where T = 1, …, n. Suppose that the variable xT progresses by constant intervals. It is then convenient to standardize the x-scale, written as
xT=T−(n+1)2 T=1,2,...,n,
and to fit Y(x) in terms of a weighted sum of orthogonal polynomial:
Y(x)=B0φ˜0(x)+B1φ˜1(x)+B2φ˜2(x)+...+BKφ˜K(x),
(A.2)
where
φ˜k(x) is the
kth degree of polynomial, where
k = 0, 1, …,
K.
Let any pair of these polynomials, φ˜k(x) and φ˜k′(x) where k ≠ k′, satisfy the orthogonal condition
∑nt=1φ˜k(x)φ˜k′(x)=0,
(A.3)
which can be expanded to determine all
φ˜k(x). Specifically, we have
φ˜0(x)=1, φ˜1(x)=λ˜1x, φ˜2(x)=λ˜2[x2−112(n2−1)],φ˜3(x)=λ˜3[x3−120(3n2−7)x], φ˜4(x)=λ˜4[x4−114(3n2−13)x2+3560(n2−1)(n2−9)],φ˜5(x)=λ˜5[x5−518(n2−7)x3+11008(15n4−230n2+407)x],φ˜6(x)=λ˜6[x6−544(3n2−31)x4+1176(5n4−110n2+329)x2−514784(n2−1)(n2−9)(n2−25)]⎫⎭⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪,
(A.4)
where the
λ˜k are selected such that the
φ˜k(x) are positive or negative integers throughout.
Suppose that the true regression law is a polynomial
η˜(x)=∑Kk=0β∗kφk(x),
from which the observed
yT differ by independent normal deviates or residuals
zT having a common variance
σ2. The orthogonal condition
(A.3) then implies that the least-square estimators
Bk of the
β∗k are given by
Bk=∑TyTφk(xt)∑T[φk(xT)]2,
(A.5)
where the coefficients
Bk are independent normal variates with mean
β∗k if
k ≤
K and mean 0 if
k >
K and with variance
s2=⟨∑nt=1y2t−∑Kk=0{B2k∑nt=1[φk(xt)2]}⟩n−K−1.
(A.6)
Thus, the ratios
(Bk−β∗k)∑t[φk(xt)2]√s,
follow Student’s
t-distribution for (
n −
K −1) degrees of freedom. As there are two terms involved in calculating
s, namely
Bk and
φ˜k(x), the null hypothesis for
β∗k=0 can be tested by using the
F-test.
The values of the functions φ˜k(xt) can be found in the orthogonal polynomials table in Pearson and Hartley (1976, Table 47).