By P. Ciarlini, M. G. Cox, F. Pavese, G. B. Rossi

ISBN-10: 9812389040

ISBN-13: 9789812389046

This quantity collects refereed contributions in response to the displays made on the 6th Workshop on complex Mathematical and Computational instruments in Metrology, held on the Istituto di Metrologia "G. Colonnetti" (IMGC), Torino, Italy, in September 2003. It presents a discussion board for metrologists, mathematicians and software program engineers that might motivate a more advantageous synthesis of talents, services and assets, and promotes collaboration within the context of european programmes, EUROMET and EA tasks, and MRA necessities. It includes articles by means of a huge, around the globe workforce of metrologists and mathematicians concerned with dimension technology and, including the 5 earlier volumes during this sequence, constitutes an authoritative resource for the mathematical, statistical and software program instruments essential to smooth metrology.

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**Sample text**

Our concluding remarks are given in section 5. 2. Least squares analysis Suppose we have a linear model in which the responses r] = (71,.. ,Q ~ through a of a system are determined by parameters a = ( ( ~ 1 , .. fixed, known m x n matrix C so that r] = C a . We assume that m 2 n and C is of rank n. Suppose the measurement model is Y=v+E (1) with E(E) = 0 , Var(E) = V, and that we observe measurements y of Y . If the uncertainty matrix V has full rank, the least squares estimate a of a, given y, minimizes x2 = (y - Ca)TV-l(y- Ca).

For a general (full rank) uncertainty matrix V with a factorization V = LLT (cf. 4), where L is an m x m matrix, also necessarily full rank, the least squares estimate is given by a = Cty, C = L-~c, 6 = ~ - l y , where C t is the pseudo-inverse of C. For well conditioned V and L, this approach is satisfactory. , can be expected to introduce numerical errors. The generalized QR f a c t ~ r i z a t i o n ~ ~approach ' ~ ~ ~ ' avoids this potential numerical instability. Suppose V = LLT, where L is m x p .

The parameters of the scratch, namely the initial column kl and the width w’,are computed from k8, w and the length of the low pass filter, used in the first step. 1” (MATLAB notation) imply w’ = 3. The basic idea is now to replace the corrupted data in the columns of A with the values of a suitable bidimensional approximating function, say s, defined in the domain D = [kl - w’, ki + 2 ~ -’ 11 x [l,2561. It has horizontal dimension equal to 3w’ and is centered on the scratch. The function s must be able to preserve the spatial correlation and the features, present in a neighborhood of the scratch.

### Advanced Mathematical and Computational Tools in Metrology VI by P. Ciarlini, M. G. Cox, F. Pavese, G. B. Rossi

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