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Electronic Supplementary Material (Appendices A, B and C) for

Relative sensitivity of combined standard uncertainty to changes induced in uncertainty components

A C Baratto(*), I L Bezerra(*)

(*) Instituto Nacional de Metrologia, Qualidade e Tecnologia(Inmetro), Brazil

Appendix A. Coefficient of contributions to combined variance – correlated case

We derive here an expression (Eq.21) that gives the relative “contribution” of the uncertainty componentu(xi) to the variance uc2for correlated quantities. The derivation takes into account the specific weight of each component in each correlation term. In sequence we obtain an expression (Eq.22) thatexpresses the relative changecarriedout onuc2 by a relative variation in u(xi), a counterpart of Eq.11 for uc2. Equation 8 can be written as:

(16)

with(17)

According to (17) the contribution of the variableXi to each correlation term Wij in (16) is linear in ui. This contribution will be then proportional to its specific weight in the corresponding sum ׀ui׀+׀uj׀. So, its contribution P(xi,xj)to each pair (i,j) (that is, Wij+Wji), will be:

(18)

with(19)

If we consider all correlation terms Wij andWjiin which the variable Xi takes part, its contribution Q(xi) to the variance will be:

(20)

The overall relative contributionR(xi) of Xi to the combined variance uc2will be:

(21)

For independent input variables only the term j=i survives and we recover Eq.2. In Appendix Bwe show that.Equation21 is to be compared with h(y,xi) given in[4], thatwill result if we considerthe variables Xi and Xj contributing equally to the term Wij: P(xi,xj)=Wij=P(xj,xi).

Relative sensitivityof uc2 to a relative variationp in u(xi)

For a relative variationp in the component u(xi), the corresponding relative sensitivityof the combined variance, δuc2(p,i), will be, using the result in Eq.10:

(22)

This equation is to be compared with Eq.11. We get the integral relative sensitivity ofuc2 to u(xi) making p=–1:

(23)

From (22), for independent input variables:

(24)

For independent input variablesand p=–1, identically as for Eq.21 and except for the minus sign (expressing the decrease in the uc2 value), we have Eq.2again.

(25)

Equations 23 and 25may alternatively be interpreted as the relative variation that occur in uc2 when the component u(xi)is disregarded in the calculation, the other components remained constant, or (taking the absolute value) as the fraction of the combined variance that survives when all components except u(xi) are disregarded (the interpretation implicitly used in [4]).

Expressed in terms of the parameters u(xi) and r(xi,xj),equations22 and 23 are more readily interpretedthan are Eq.11 in [4], expressed in terms of u(xi,xj). Furthermore, Eq.23above and Eq.11 of[4], although slightly different, canboth beused for expressing the integral relativesensitivityof uc2to a component. Our approachis equivalent to that of [4], with the difference that it takes into account the relative weights of the components in each term Wij. But this is an academic discussion.

Comparing Eq.22 with Eq.11 we may write a general simple relation between the relative sensitivitiesof uc and uc2:

(26)

Appendix B. Demonstration that

=

(Rearranging the terms:)

Appendix C. Effect on ucof simultaneous changes in two uncertainty components

The overall effect on uc of simultaneous relative changes p and q in the components u(xi) and u(xj), respectively, is given, for the case of independent variables, by:

(27)

and for correlated input quantities, by:

(28)

Concerning the variance, the relative sensitivity of uc2 to a specific group of components is simply the sum of the individual sensitivities calculated independently. But this is valid only for the case of independent variables.

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