Statistics for Environmental Engineers

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The criterion is written as a combined sum of squares function augmented by the covariances between responses. For the example reaction A ^ B ^ C, with three responses measured, the Box-Draper determinant criterion has the following form:

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X(Уа — yA)2 X( Уа — y A)( Ув — У в) X(Уа — yA)(Ус — yc)

X( Уа — У a )( Ув — У в ) X(Ув yв)2 X( Ув — У в )( Ус — У c )

X Уа У a )( Ус У c ) X Ув У в )( Ус У c ) X(Ус — yc)2

where X indicates summation over all observations. We assume that each response has been measured the same number of times. The vertical lines indicate the determinant of the matrix. The best parameter estimates, analogous to the least squares estimates, are those which minimize the determinant of this matrix.

The diagonal elements correspond to the residual (error) sum of squares for each of the three responses. The off-diagonal terms account for measurements on the different responses being correlated. If the residual errors of the three responses are independent of each other, the expected values of the off-diagonal terms will be zero and the parameter estimation criterion simplifies to:

min ^( Уа — у a )2 + (Ув — У в)2 + (Ус У с )2

For the special case of a single response, the determinant criterion simplifies to the method of least squares: minimize the sum of the squared residuals for the response of interest.

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