# Statistics for Environmental Engineers

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S(в) = (0.15 — 2в)2 + (0.461 — 4в)2 + (0.559 — 6в)2

+ (1.045 — 10в)2 + (1.361 — 14в)2 + (1.919 — 19в)2

FIGURE 33.3 The values of the sum of squares plotted as a function of the trial parameter values. The least squares estimates are b = 0.1 and k = 0.2. The sum of squares function is symmetric (parabolic) for the linear model (left) and asymmetric for the nonlinear model (right).

в

Nonlinear Model

For the nonlinear model it is:

S(0) = (0.62 — e~20)2 + (0.51 — e^e)2 + (0.26 — e60)2

+ (0.18 — e~100)2 + (0.025 — e~14e)2 + (0.041 — e~1<90)2

An algebraic solution exists for the linear model, but to show the essential similarity between linear and nonlinear parameter estimation, the least squares parameter estimates of both models will be determined by a straightforward numerical search of the sum of squares functions. We simply plot S (в) over a range of values of в, and do the same for S(0) over a range of 0.

Two iterations of this calculation are shown in Table 33.1. The top part of the table shows the trial calculations for initial parameter estimates of b = 0.115 and k = 0.32. One clue that these are poor estimates is that the residuals are not random; too many of the linear model regression residuals are negative and all the nonlinear model residuals are positive. The bottom part of the table is for b = 0.1 and k = 0.2, the parameter values that give the minimum sum of squares.

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