Responsive surfaces

One of the most effective ways to think about optimization is to visualize how a system s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one factor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is... 

For a two-factor system, such as the quantitative analysis for vanadium described earlier, the response surface is a flat or curved plane plotted in three dimensions. For example. Figure 14.2a shows the response surface for a system obeying the equation... 

The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 < A < 10, 0 < B < 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most response surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural constraint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. 

Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. 

Mountain-climbing analogy to using a searching algorithm to find the optimum response for a response surface. The path on the left leads to the global optimum, and the path on the right leads to a local optimum. 

Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as... 

The resulting response surface for equation 14.1 is shown in Figure 14.5. 

The initial simplex is determined by choosing a starting point on the response surface and selecting step sizes for each factor. Ideally the step sizes for each factor should produce an approximately equal change in the response. For two factors a convenient set of factor levels is (a, b), a + s, h), and (a + 0.5sa, h + 0.87sb), where sa and sb are the step sizes for factors A and B. Optimization is achieved using the following set of rules ... 

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

Progress of Fixed-Sized Simplex Optimization for Response Surface in Figure 14.10... 

Progress of a fixed-sized simplex optimization for the response surface of Example 14.1. The optimum response at (3, 7) corresponds to vertex 25. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca... 

Empirical Models of the Response Surface In many cases the underlying theoretical relationship between the response and its factors is unknown, making impossible a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response surface. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors... 

The terms Po, Pa, Pt, Pat, Paa, and Pt,t, are adjustable parameters whose values are determined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theoretical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected. 

To develop an empirical model for a response surface, it is necessary to collect the right data using an appropriate experimental design. Two popular experimental designs are considered in the following sections. 

Equation 14.9 gives the empirical model of the response surface for the data in Table 14.4 when the factors are in coded form. Convert the equation to its uncoded form. 

Substituting these equations into equation 14.9 gives, after simplifying, the uncoded equation for the response surface. 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

In optimizing a method, we seek to find the combination of experimental parameters producing the best result or response. We can visualize this process as being similar to finding the highest point on a mountain, in which the mountain s topography, called a response surface, is a plot of the system s response as a function of the factors under our control. 

Another approach to optimizing a method is to develop a mathematical model of the response surface. Such models can be theoretical, in that they are derived from a known chemical and... 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

Tor each of the following equations, determine the optimum response, using the one-factor-at-a-time searching algorithm. Begin the search at (0, 0) with factor A, and use a step size of 1 for both factors. The boundary conditions for each response surface are 0 < A < 10 and 0 < B < 10. Continue the search through as many cycles as necessary until the optimum response is found. Compare your optimum response for each equation with the true optimum. 

Note These equations are from Doming, S. N. Morgan, S. L. Experimental Design A Chemometric Approach. Elsevier Amsterdam, 1987, and pseudo-three-dimensional plots of the response surfaces can be found in their figures 11.4, 11.5, and 11.14. The response surface for problem (a) also is shown in Color Plate 13. 

A 2 factorial design was used to determine the equation for the response surface in problem lb. The uncoded levels, coded levels, and the responses are shown in the following table. 

Determine the coded and uncoded equation for the response surface. 

Hendrix, C. D. Through the Response Surface with Test Tube and Pipe Wrench, Chemtech 1980, 10, 488-497. 

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