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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