Propagation of uncertainties

If present, Scipp propagates uncertainties (errors) as described in Wikipedia: Propagation of uncertainty. The implemented mechanism assumes uncorrelated data.

An overview for key operations is:

Operation \(f\)	Variance \(\sigma^{2}_{f}\)
\(-a\)	\(\sigma^{2}_{a}\)
\(|a|\)	\(\sigma^{2}_{a}\)
\(\sqrt{a}\)	\(\frac{1}{4} \frac{\sigma^{2}_{a}}{a}\)
\(a + b\)	\(\sigma^{2}_{a} + \sigma^{2}_{b}\)
\(a - b\)	\(\sigma^{2}_{a} + \sigma^{2}_{b}\)
\(a * b\)	\(\sigma^{2}_{a}b^{2} + \sigma^{2}_{b}a^{2}\)
\(a / b\)	\(\frac{\sigma^{2}_{a} + \sigma^{2}_{b} \frac{a^{2}}{b^{2}}}{b^{2}}\)
\(e^{a}\)	\(e^{2a} \sigma^{2}_{a}\)
\(\log(a)\)	\(\sigma^{2}_{a} / a^{2}\)
\(\log_{10}(a)\)	\(\sigma^{2}_{a} \log^{2}(10) / a^{2}\)
\(\sin(a)\) [1]	\(\sigma^{2}_{a} \cos^{2}(a)\)
\(\cos(a)\) [1]	\(\sigma^{2}_{a} \sin^{2}(a)\)
\(\tan(a)\) [1]	\(\sigma^{2}_{a} \left(\frac{1}{\cos^{2}(a)}\right)^{2}\)
\(\arcsin(a)\) [2]	\(\sigma^{2}_{a} \frac{1}{1 - a^{2}}\)
\(\arccos(a)\) [2]	\(\sigma^{2}_{a} \frac{1}{1 - a^{2}}\)
\(\arctan(a)\) [2]	\(\sigma^{2}_{a} \frac{1}{(1 - a^{2})^{2}}\)
\(\text{sinc}(a)\)	\(\sigma^{2}_{a} \left(\frac{a\cos(a) - \sin(a)}{a^2}\right)^{2}\)
\(\sinh(a)\)	\(\sigma^{2}_{a} \cosh^{2}(a)\)
\(\cosh(a)\)	\(\sigma^{2}_{a} \sinh^{2}(a)\)
\(\tanh(a)\)	\(\sigma^{2}_{a} \left(\frac{1}{\cosh^{2}(a)}\right)^{2}\)
\(\text{arsinh}(a)\)	\(\sigma^{2}_{a} \frac{1}{a^{2} + 1}\)
\(\text{arcosh}(a)\)	\(\sigma^{2}_{a} \frac{1}{a^{2} - 1}\)
\(\text{artanh}(a)\)	\(\sigma^{2}_{a} \frac{1}{(1 - a^{2})^{2}}\)

The expression for division is derived from \((\frac{\sigma_{a/b}}{a/b})^{2} = (\frac{\sigma_{a}}{a})^{2} + (\frac{\sigma_{b}}{b})^{2}\).

Footnotes

[1] (1,2,3)

The input to trigonometric functions is converted to radians before propagating the uncertainties. This means that, e.g., the uncertainty of the sine function for an input in degrees is \(\sigma^{2}_{\sin} = \sigma^{2}_{a} (\pi/180)^{2} \cos^{2}(a)\).

[2] (1,2,3)

Variances are not defined for inverse trigonometric functions at \(\pm 1\).