Statistics for Environmental Engineers

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The partial derivatives of the model with respect to each parameter are:


v 3^(1 — e-j    t    -в,,,    . t


X1j = —в—= 1e 2j j =1 2


X 2 j =



,]]1(1-ев2У)]


дв2



= вр^, j = 1,2


The derivative matrix X for n = 2 experiments is 2 x 2:


X=


X11


X21


=


-1 в-ltl


1 — e 21


врх e вл


X12


X22


1 — е~в22


в1г2е^вг2

where 1 and 2 are the times at which observations will be made.


Premultiplying X by its transpose gives:


X’X =



X+ X


X11X 21 + X12 X 22



X11X 21 + X12 X 22 X221 + X222


The objective now is to maximize the determinant of the X’X matrix:


max A = |x’x|


The vertical bars indicate a determinant. In a complicated problem, the matrix multiplication and the minimization of the determinant of the matrix would be done using numerical methods. The analytical solution for this example, and several other interesting models and provided by Box (1971), and can be derived algebraically. For the case where n = p = 2:


A = (X11X22 — X12 X 21 )2


This expression is maximized when the absolute value of the quantity (XnX22 — X12X21) is maximized. Therefore, the design criterion becomes:


A = max | X11X22X12X21 |


Here, the vertical bars designate the absolute value. The asterisk on delta merely indicates this redefinition of the A criterion.

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