Y Po 2 M e23

> = i n n n—1 n y = Ao + 2 A*, + 2 Puxf + 22 PijXiXj + e 1=1 1=1 1 = 1 j = i + l

where /?0, ftb ftJ are constant coefficients usually determined by least-squares methods and e is the error involved in estimating the coefficients /? from experimental data. At times, polynomials of higher order are used.

Polynomials are popular approximating functions for several reasons. They provide simple curvilinear relationships that can approximate practically any true continuous function within a specified range, and they usually possess a clearly defined optimum. They can be expanded to include any number of decision variables (x;). Finally, a number of transformations such as logarithmic, exponential, inverse, power, and trigonometric, may be applied to the independent (x,;) or dependent (y) variables, which add some asymmetry and flexibility desirable in many cases of biological and biochemical systems. The weakness associated with the use of polynomials is that they are "smooth" functions without any biological or biochemical justification. Therefore, extrapolation beyond the experimental space (the region where data were collected to estimate /?'s) is usually not allowed.

The first step in a RSM study is to select an appropriate experimental design with a limited number of experimental runs k (where k > n), which will allow estimation of the coefficients by minimizing the sum of squares of errors E

Ready-made designs already exist in terms of coded variables, which are linear functions of the actual decision variables, and minimize the overall error e

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