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Advances in Nuclear Science and Technology: Volume 14 by C. R. Weisbin (auth.), Jeffery Lewins, Martin Becker (eds.)

By C. R. Weisbin (auth.), Jeffery Lewins, Martin Becker (eds.)

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Again, most existing fitting codes do not permit use of non-diagonal weighting matrices, so the resulting inverse least-squares matrix is not generally a good approximation to the parameter uncertainty matrix (Equation 28 holds only if V is valid) . 41 UNCERTAINTY IN NUCLEAR DATA In curve or physical model fitting the A. elements give the sensitivity of the model output valu~alr at the ith value of some independent variable, to the parameter value b a . The equation (27) is much more general, and applies to any set of observations of quantities that can be represented by a sequence of linear equations in terms of a lesser number of parameters.

Note that the problem chosen was explicitly linear, that the minimum variance solution was boldly assumed to be the one desired, and that no assumptions at all were made concerning the shapes of the density functions for Yl and Y2 • b. A General Least-Squares Formulation. ) be considered, and let the associated variance-covari~nce matrix be (Vij). Suppose the physical quantities Yi that underlie the observations can be represented in terms of the parameter vector (b ) through the (possibly implicit) relation a G[ y (b) i b ] = o.

A la ob a + • • . (24) R. W. •. , (b na ) is to be sought. In this equation, the Yoi are the values of Yi computed from Equation (23), using the estimated parameter vector (boa)' The matrix (Aia) of derivatives is called the sensitivity or sometimes the design matrix, and initially is calculated at (boa)' The iterated solution is to be obtained by minia weighted quadratic form of the vector of residuals corresponding to the second approximation above. m~z~ng (25) Using the linearization of Equation (24) permits Equation (25) to be cast in terms of parameter increments (S ) _ - a (cb a ) and the reduced observation vector (~i) = (y.

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