A2 The Linearization Method

Students familiar with calculus will recall Taylor's theorem, which states that any arbitrary function f (X) that is continuous and has a continuous nth-order derivative can be approximated around point X = X0 by a polynomial function and a remainder as follows:

™ f (Xo) f'(X0)(X — Xq) f"(Xo)(X — X0)2 , I (X) = rn + -71- +

where f (X0) is the first derivative of f (X) evaluated at X = X0, f"(X0) is the second derivative of f (X) evaluated at X = X0 and so on, where n! (read n factorial) stands for n(n — 1)(n — 2)... 1 with the convention that 0! = 1, and R stands for the remainder. If we take n = 1, we get a linear approximation; choosing n = 2, we get a second-degree polynomial approximation. As you can expect, the higher the order of the polynomial, the better the approximation to the original function. The series given in (1) is called Taylor's series expansion of f(X) around the point X = X0. As an example, consider the function:

Y = f (X) = a1 + a2 X + a3 X2 + a4 X3 Suppose we want to approximate it at X = 0. We now obtain:

f (0) = ax f'(Q) = a2 f "(0) = 2a3 f'"(0) = 6a4 Hence we can obtain the following approximations:

First order: Y = ai + f 1(Q) = ai + a2 X + remainder (= a3 X2 + a4 X3)

= a1 + a2 X + a3 X2 + remainder (= a4 X3) Third order: Y = a1 + a2 X + a3 X2 + a4 X3

The third-order approximation reproduces the original equation exactly.

CHAPTER FOURTEEN: NONLINEAR REGRESSION MODELS 577

The objective of Taylor series approximation is usually to choose a lower-order polynomial in the hope that the remainder term will be inconsequential. It is often used to approximate a nonlinear function by a linear function, by dropping the higher-order terms.

The Taylor series approximation can be easily extended to a function containing more than one X. For example, consider the following function:

and suppose we want to expand it around X = a and Z = b. Taylor's theorem shows that f (x, z) = f (a, b) + fx (a, b)(x - a)

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