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View Help V ALI Normal No Spacing Heading 1 Heading 2 V Paragraph Styles 12345678910 1 10 11 12 11 12 13 14 15 1 1 1 2.5.5 Consider the exponential function model y - y +a (exp[b(x -x)]-1). The parameters y, a, and b have to be optimized. a) Develop a numerical scheme for using the formulas in Sect. 2.5.2 to find the yea, and b values that minimize the least-squares error of this model. b) Calculate the optimal yo, a, and b for the data set (1. 1.95), (2. 1.91). (3, 1.87), and (4. 1.84). Use x 0. NS Errer Definition. As a second example, let us address the optimization of the exponential function (2916). For this function, the least squares crror reads *-*-*--*-.' -** -2,{-", -1)+((x-ate**) - (299) -603-7)+60** (*)--))-(-a)"). Here, we used the wbbreviation - expl)-1. This error formula is equal to 9 (292) Thus, the optimal values of sand and the resulting expression for are equal to the formulas given in Sect. 23.1 2.5 Optimal Power and Exponential Models 67 0.5* 12 (a) 10 (b) 0.4 e E 0.3 y 6 JP) 0.2 4 0.1- 2 0.0 O 0.0 0.2 0.4 0.6 0.8 1.0 1.00 216 0 2 4 6 8 10 12 5 Fig. 2.12. An optimal exponential function model, (a) The error E/<=> given by Eq. (2.97), this function has a minimum at b - 0,5043; (b) the dors show the data pivon Table 2.2, the line shows the optimal exponential function model 12.100) Application. The following example illustrates the use of this approach. The data shown in Table 22 and Fig 2.12 suggest the use of an exponential function for the modeling of these data. We may use the first point for the definition of 4-1 The error E1<! determined by F4 (297) is shown in Fig. 2.12a. The minimum of this function is pven at b-05043. The optimal values of a and ure then plen by 00963 and - 0.9959. With these model parameters we find the exponential model -0.9959.00263-1) (2.100) +