# Statistics for Environmental Engineers

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0.041

0.0017

10

1.045

1.000

0.045

0.0020

10

0.180

0.135

0.045

0.0020

14

1.364

1.400

0.036

0.0013

14

0.025

0.061

0.036

0.0013

19

1.919

1.900

0.019

0.0004

19

0.041

0.022

0.019

0.0003

Minimum sum of

squares =

0.0116

Minimum sum

of squares

= 0.0115

FIGURE    33.2    Plots    of    data    to be fitted to linear    (left) and nonlinear (right) models and the    curves    generated from the

initial parameter estimates    of    b = 0.115 and k = 0.32 and the minimum least squares values (b    = 0.1    and k =    0.2).

The least squares estimate of в still satisfies dS/дв = 0, but the resulting derivative does not have an algebraic solution. The value of в that minimizes S is found by iterative numerical search.

Examples

The similarities and differences of linear and nonlinear regression will be shown with side-by-side examples using the data in Table 33.1. Assume there are theoretical reasons why a linear model (П = fix) fitted to the data in Figure 33.2 should go through the origin, and an exponential decay model (n = exp(-вх)) should have y = 1 at t = 0. The models and their sum of squares functions are:

Уi    = fa, + e,    min S(в) = (y, — fix,)2

yt    = exp(-ext) + e,    min S( в) = ^(y, — exp(-вх{))2

For the linear model, the sum of squares function expanded in terms of the observed data and the parameter в is:

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