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Table 2 Sensitivity coefficients of the 1st order (\(W_{{S_{z} j}}\)) and 2nd order (\(W_{{S_{z} jk}}^{(2)}\)) and their values for the nominal set of parameters

From: Computing Methods in the Analysis of Road Accident Reconstruction Uncertainty

j 1st-order sensitivity coefficients k 2nd-order sensitivity coefficientsa
\(W_{{S_{z} j}}\) Value for the set of nominal data \(W_{{S_{z} jk}}^{(2)}\) Value for the set of nominal data
1 \(\frac{{\partial S_{z} }}{{\partial V_{0} }} = t_{r} + \frac{{t_{n} }}{2} + \frac{{V_{0} }}{\mu \cdot g}\) 3.43 s 1 \(\frac{{\partial^{2} S_{z} }}{{\partial V_{0}^{2} }} = \frac{1}{\mu \cdot g}\) 0.146 s2/m
    2 \(\frac{{\partial^{2} S_{z} }}{{\partial V_{0} \partial \mu }} = - \frac{{V_{0} }}{{\mu^{2} \cdot g}}\) − 2.89 s
    3 \(\frac{{\partial^{2} S_{z} }}{{\partial V_{0} \partial t_{r} }} = 1\) 1
    4 \(\frac{{\partial^{2} S_{z} }}{{\partial V_{0} \partial t_{n} }} = \frac{1}{2}\) 0.5
2 \(\frac{{\partial S_{z} }}{\partial \mu } = - \frac{{V_{0}^{2} }}{{2 \cdot \mu^{2} \cdot g}}\) − 20.1 m 2 \(\frac{{\partial^{2} S_{z} }}{{\partial \mu^{2} }} = \frac{{V_{0}^{2} }}{{\mu^{3} \cdot g}}\) − 40.1 m
    3 \(\frac{{\partial^{2} S_{z} }}{{\partial \mu \partial t_{r} }} = 0\) 0
    4 \(\frac{{\partial^{2} S_{z} }}{{\partial \mu \partial t_{n} }} = 0\) 0
3 \(\frac{{\partial S_{z} }}{{\partial t_{r} }} = V_{0}\) 13.9 m/s 3 \(\frac{{\partial^{2} S_{z} }}{{\partial t_{r}^{2} }} = 0\) 0
    4 \(\frac{{\partial^{2} S_{z} }}{{\partial t_{r} \partial t_{n} }} = 0\) 0
4 \(\frac{{\partial S_{z} }}{{\partial t_{n} }} = \frac{{V_{0} }}{2}\) 6.94 m/s 4 \(\frac{{\partial^{2} S_{z} }}{{\partial t_{n}^{2} }} = 0\) 0
  1. aAccording to Schwarz’s theorem, mixed partial derivatives do not depend on the differentiation order (they have an identical form): 2f/∂xj∂xk= 2f/∂xk∂xj. Therefore, they are only determined for the pairs j, k such that k > j. The occurrence of such a pair of derivatives is reflected in Eq. (10) by multiplier “2” in the terms with mixed derivatives