2.4.4. TAB “Uncertainty Budget”¶
Under this being selected by mouse click the uncertainty budget is presented as a table. The output quantity is indicated which this uncertainty budget is referred to.
The table “Uncertainty budget” again contains the three columns “Symbols”, “Type” and “Unit” already known.
The columns “Value” and “Std. uncertainty” contain the input values of the measured value and its standard uncertainty of each of the independent measurement quantities as well as of each dependent quantity (auxiliary and output quantity).
In the column “Sensitivity coefficient” partial derivatives of the function of the output quantity y (i.e., the function determined by the equations with which the value of the output quantity is calculated) are given for each independent quantity.
From the products Uncertainty x Sensitivity coefficient, the values of which are shown in the next column, the uncertainty budget is deduced in two different ways.
The column “relat. contribut(%)“ shows for each independent symbol - in case its uncertainty is non-zero - the relative contribution (in %) of its variance to the total variance of the output quantity. This column gives the information which of the (independent) quantities contributes at most to the combined uncertainty of the output quantity. The indication of 100% in this column for the output quantity is only that the control sum of the individual relative contributions. Using the button “Change budget type” allows to display absolute uncertainty contributions in this column, given in the unit of the output quantity.
The preceding definition of the “relat. contribut(%)“ to the uncertainty budget **has got a further plausible meaning by the recent publication by** Kessel, Kacker and Berglund (2006) with the title “Coefficient of contribution to the combined standard uncertainty“:
That relative contribution of an input quantity \(x_{i}\), divided by 100, is in the case of un-correlated input quantities identical with the square of the correlation coefficient between \(x_{i}\) and the output quantity y! This quantity is now called “coefficient of contribution” and is represented by the symbol \(h\left( y,x_{i} \right)\).
The generalised definition of the “coefficient of contribution” is now:
As we have for non-correlated input quantities:
from Eq. (2.4.2) then follows Eq. (2.4.4), which was already known - for non-correlated input quantities - as the positive relative contribution to the variance of the output quantity:
If correlations \(r\left( x_{i},x_{j} \right)\) between input quantities exist, they are inserted into Eq. (2.4.2) in the factor \(r\left( y,x_{i} \right)\) defined as follows:
Note that this may lead now in some cases to negative values of \(h\left( y,x_{i} \right)\), the coefficient of contribution.
Notes on effects from covariances:
If covariances are considered for the calculation of uncertainties negative values may occur in the column “relat. contribut(%)“; this is not a program error.
According to the mentioned paper by Kessel et al. correlations (covariances) between input quantities are considered according to Eq. (2.4.2) in combination with Eq. (2.4.5) for calculating the “coefficient of contribution” and resulting values presented in the column “relat. contribut(%) in the uncertainty budget table. Eq. (2.4.4) then is no longer valid.
With UncertRadio this procedure according to Kessel et al. is implemented since version 0.05 (2007/11), i.e. the values shown in the column “relat. contribut(%)“ of the uncertainty budget correspond to this new definition. This may be demonstrated with the example projects Kessel-2a-2006.txp and Kessel-2b-2006.txp, which were prepared from two examples from that publication.
