Statistics for Environmental Engineers

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We will develop the regression model and the derivation of the variance of parameter estimates in matrix notation. Our explanation is necessarily brief; for more details, one can consult almost any modern reference on regression analysis (e.g., Draper and Smith, 1998; Rawlings, 1988; Bates and Watts, 1988). Also see Chapter 30.

In matrix notation, a linear model is:

y = X в + e

where y is an n x 1 column vector of the observations, X is an n x p matrix of the independent variables (or combinations of them), в is a p x 1 column vector of the parameters, and e is an n x 1 column vector of the residual errors, which are assumed to have constant variance. n is the number of observations and p is the number of parameters in the model.

The least squares parameter estimates and their variances and covariances are given by:

b = [ XX ]-1Xy


Var( b) = [ XX ]V

The same equations apply for nonlinear models, except that the definition of the X matrix changes. A nonlinear model cannot be written as a matrix product of X and в, but we can circumvent this difficulty by using a linear approximation (Taylor series expansion) to the model. When this is done, the X matrix becomes a derivative matrix which is a function of the independent variables and the model parameters.

The variances and covariances of the parameters are given exactly by [XX ]-1ct2 when the model is linear. This expression is approximate when the model is nonlinear in the parameters. The minimum sized joint confidence region corresponds to the minimum of the quantity [XX ]-1ct2. Because the variance of random measurement error (a2) is a constant (although its value may be unknown), only the [XX ]-1 matrix must be considered.

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