Statistics for Environmental Engineers

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dSdei = 0 = ?£( bx2 — xiyi)


This equation is called the normal equation. Note that this equation is linear with respect to b. The algebraic solution is:


b=


Ix?


Because xt and y, are known once the experiment is complete, this equation provides a generalized method for direct and exact calculation of the least squares parameter estimate. (Warning: This is not the equation for estimating the slope in a two-parameter model.)


If the linear model has two (or more) parameters to be estimated, there will be two (or more) normal equations. Each normal equation will be linear with respect to the parameters to be estimated and therefore an algebraic solution is possible. As the number of parameters increases, an algebraic solution is still possible, but it is tedious and the linear regression calculations are done using linear algebra (i.e., matrix operations). The matrix formulation was given in Chapter 30.


Unlike linear models, no unique algebraic solution of the normal equations exists for nonlinear models. For example, if n = exp(-Ox), the method of least squares requires that we find the value of в that minimizes S:


S(O) = X(у, — exp(- Ox,))2 = X[y? — 2y, exp (-Ox,) + (exp(-Ox,))?]


Example Data and the Sum of Squares Calculations for a One-Parameter Linear Model and a One-Parameter Nonlinear Model


TABLE 33.1


Linear Model:


-3


II


*


Nonlinear Model:


: n = exp(-exi)


Xi


.yobs,i

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