Statistics for Environmental Engineers

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It is not necessary to compare entire variance-covariance matrices for different experimental designs. All we need to do is minimize the determinant of the [XX ]-1 matrix or the equivalent of this, which is to maximize the determinant of [XX ]. This determinant design criterion, presented by Box and Lucas (1959), is written as:

max A = max | XX |

where the vertical bars indicate the determinant. Maximizing A minimizes the size of the approximate joint confidence region, which is inversely proportional to the square root of the determinant, that is, proportional to A-1/2.

[XX]-1 is the variance-covariance matrix. It is obtained from X, an n row by p column (n x p) matrix, called the derivative matrix:

 X11 X21. • Xp! X Г2 X22 • • X p2 i = 1, 2,. — p; j= b 2— ,n X1n X2n X pn

where p and n are the number of parameters and observations as defined earlier.

The elements of the X matrix are partial derivatives of the model with respect to the parameters:

Xo =

dm,Xj)

дв,

i = 1, 2,…,p; j = 1, 2,…,n

For nonlinear models, however, the elements Xj are functions of both the independent variables Xj and the unknown parameters в. Thus, some preliminary work is required to compute the elements of the matrix in preparation for maximizing |XX|.

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