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Normalization uncertainty

Let us consider now the case where the calibration constant is the scale factor $ f$, known with a standard uncertainty $ \sigma_f$. Also in this case, for simplicity and without losing generality, let us suppose that the most probable value of $ f$ is 1. Then $ X_{n+1} = f$, i.e. $ x_{n+1} = 1$, and $ V_{X_{n+1,n+1}} = \sigma^2_f$. Then
$\displaystyle \sigma^2(Y_i)$ $\displaystyle =$ $\displaystyle \sigma_i^2 + \sigma_f^2\, x_i^2 \, ,$ (14.24)
Cov$\displaystyle (Y_i,Y_j)$ $\displaystyle =$ $\displaystyle \sigma_f^2\, x_i\, x_j
\hspace{1.3 cm}(i\ne j) \, ,$ (14.25)
$\displaystyle \rho_{ij}$ $\displaystyle =$ $\displaystyle \frac{x_i\, x_j}
{\sqrt{x_i^2+\frac{\sigma_i^2}{\sigma_f^2}}
\,\sqrt{x_j^2+\frac{\sigma_j^2}{\sigma_f^2}}}\, ,$ (14.26)
$\displaystyle \vert\rho_{ij}\vert$ $\displaystyle =$ $\displaystyle \frac{1}
{\sqrt{1+\left(\frac{\sigma_i}{\sigma_f\, x_i}\right)^2}
\,\sqrt{1+\left(\frac{\sigma_j}{\sigma_f\, x_j}\right)^2}
}\, .$ (14.27)

To verify the results let us consider two independent measurements $ X_1$ and $ X_2$; let us calculate the correlated quantities $ Y_1$ and $ Y_2$, and finally their product ( $ P\equiv Z_1$) and their ratio ( $ R\equiv Z_2$):
$\displaystyle {\bf V}_Y$ $\displaystyle =$ $\displaystyle \left( \begin{array}{cc}
\sigma_1^2+\sigma_f^2\, x_1^2
& \sigma_f...
...
& \\
\sigma_f^2\, x_1\, x_2
& \sigma_2^2+\sigma_f^2\, x_2
\end{array}\right),$ (14.28)
       
       
$\displaystyle {\bf V}_Z$ $\displaystyle =$ $\displaystyle \left( \begin{array}{cc}
\sigma_1^2\, x_2^2 +
\sigma_2^2\, x_1^2 ...
...rac{\sigma_1^2}{x_2^2} +
\sigma_2^2\,\frac{x_1^2}{x_2^4}
\end{array}\right)\, .$ (14.29)

It follows that
$\displaystyle \sigma^2(P)$ $\displaystyle =$ $\displaystyle \sigma_1^2\, x_2^2 +
\sigma_2^2\, x_1^2 +
(2\,\sigma_f\, x_1\, x_2)^2 \, ,$ (14.30)
$\displaystyle \sigma^2(R)$ $\displaystyle =$ $\displaystyle \frac{\sigma_1^2}{x_2^2} +
\sigma_2^2\,\frac{x_1^2}{x_2^4} \, .$ (14.31)

Just as an unknown common offset error cancels in differences and is enhanced in sums, an unknown normalization error has a similar effect on the ratio and the product. It is also interesting to calculate the standard uncertainty of a difference in the case of a normalization error:
$\displaystyle \sigma^2(D)$ $\displaystyle =$ $\displaystyle \sigma_1^2+\sigma_2^2
+\sigma_f^2\,(x_1-x_2)^2\, .$ (14.32)

The contribution from an unknown normalization error vanishes if the two values are equal.
next up previous contents
Next: General case Up: Matrice di covarianza di Previous: Offset uncertainty   Indice
Giulio D'Agostini 2001-04-02