Inference of evidence reasoning rule with Gaussian distribution reliability and its application in safety assessment

Abstract

In practical systems, observation attributes are often susceptible to noise perturbations. Evidential reasoning (ER) rule, by incorporating evidence reliability, is capable of discounting uncertainties such as perturbations to construct a joint reasoning model for obtaining reasonable evaluation results. However, in previous studies, evidence reliability has typically been treated as a quantitative value, which to some extent limits its effectiveness in handling uncertainties. In this paper, we propose for the first time the description of evidence reliability as a random variable following a Gaussian distribution and establish criteria for setting the Gaussian probability parameters (mean and variance), thereby constructing a new ER with Gaussian probability reliability (gpr), called ER-gpr. Firstly, we combine various reliability determination methods with game theory to calculate adaptive reliability, which serves as the principle for setting the mean. Secondly, two criteria are proposed to set the variance via this principle, with rigorous mathematical proofs provided. Thirdly, a multi-step Whale Optimization Algorithm (WOAMS) is proposed for optimizing the evidence parameters. Fourthly, a sensitivity analysis is conducted on the probability parameters. Finally, the practical utility of the ER-gpr model in safety assessment is demonstrated through a case study of the JRC-7M aerospace relay.

Appendix A

Derivation of sensitivity coefficient $S(R_{i} )$. Suppose $i = 1,2$, $E(U)$ can be rewritten as

$$E(U) = \frac{{\int_{{R_{1}^{ - } }}^{{R_{1}^{ + } }} {\int_{{R_{2}^{ - } }}^{{R_{2}^{ + } }} {\frac{{A_{1} R_{1} + B_{1} R_{2} + C_{1} }}{{A_{2} R_{1} + B_{2} R_{2} + C_{2} }}} } \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1} - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2} - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]dR_{1} dR_{2} }}{{(\rho_{1} \rho_{2} )^{ - 1} }}$$

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where $\rho_{1}$ and $\rho_{2}$ can be given by Eq. (35). Let

$$\Im = \int_{{R_{1}^{ - } }}^{{R_{1}^{ + } }} {\int_{{R_{2}^{ - } }}^{{R_{2}^{ + } }} {\frac{{A_{1} R_{1} + B_{1} R_{2} + C_{1} }}{{A_{2} R_{1} + B_{2} R_{2} + C_{2} }}} } \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1} - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2} - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]dR_{1} dR_{2}$$

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For evidence $e_{1}$, the partial derivatives of $\partial E(U)/\partial R_{1}^{ + }$ and $\partial E(U)/\partial R_{1}^{ - }$ can be calculated by

$$\left\{ \begin{gathered} \frac{\partial E(U)}{{\partial R_{1}^{ + } }} = \frac{{\int_{{R_{2}^{ - } }}^{{R_{2}^{ + } }} {\frac{{A_{1} R_{1}^{ + } + B_{1} R_{2} + C_{1} }}{{A_{2} R_{1}^{ + } + B_{2} R_{2} + C_{2} }} \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1}^{ + } - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2} - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]} dR_{2} }}{{(\rho_{1} \rho_{2} )^{ - 1} }} - \frac{{\Im \rho_{2}^{ - 1} \frac{1}{{\sqrt {2\pi \sigma_{1}^{2} } }}\exp - \left[ {\frac{{\left( {R_{1}^{ + } - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }}} \right]}}{{\left( {\rho_{1} \rho_{2} } \right)^{ - 2} }} \hfill \\ \frac{\partial E(U)}{{\partial R_{1}^{ - } }} = \frac{{\Im \rho_{2}^{ - 1} \frac{1}{{\sqrt {2\pi \sigma_{1}^{2} } }}\exp - \left[ {\frac{{\left( {R_{1}^{ - } - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }}} \right]}}{{\left( {\rho_{1} \rho_{2} } \right)^{ - 2} }} - \frac{{\int_{{R_{2}^{ - } }}^{{R_{2}^{ + } }} {\frac{{A_{1} R_{1}^{ - } + B_{1} R_{2} + C_{1} }}{{A_{2} R_{1}^{ - } + B_{2} R_{2} + C_{2} }} \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1}^{ - } - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2} - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]} dR_{2} }}{{(\rho_{1} \rho_{2} )^{ - 1} }} \hfill \\ \end{gathered} \right.$$

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For evidence $e_{2}$, the partial derivatives of $\partial E(U)/\partial R_{2}^{ + }$ and $\partial E(U)/\partial R_{2}^{ - }$ are calculated by Eq. (94).

Suppose $i = 1,2,...,L$, $E(U)$, we can replace Eq. (45) with $(A_{1} R_{1} + B_{1} R_{2} + C_{1} )/(A_{2} R_{1} + B_{2} R_{2} + C_{2} )$ and iteratively compute the partial derivatives of all outputs with respect to lower and upper bounds.

In view of this, the sensitivity coefficient $S(R_{i} )$ can be calculated.

$$\left\{ \begin{gathered} \frac{\partial E(U)}{{\partial R_{2}^{ + } }} = \frac{{\int_{{R_{1}^{ - } }}^{{R_{1}^{ + } }} {\frac{{A_{1} R_{1} + B_{1} R_{2}^{ + } + C_{1} }}{{A_{2} R_{1} + B_{2} R_{2}^{ + } + C_{2} }} \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1} - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2}^{ + } - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]} dR_{1} }}{{(\rho_{1} \rho_{2} )^{ - 1} }} - \frac{{\Im \rho_{1}^{ - 1} \frac{1}{{\sqrt {2\pi \sigma_{2}^{2} } }}\exp - \left[ {\frac{{\left( {R_{2}^{ + } - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]}}{{\left( {\rho_{1} \rho_{2} } \right)^{ - 2} }} \hfill \\ \frac{\partial E(U)}{{\partial R_{2}^{ - } }} = \frac{{\Im \rho_{1}^{ - 1} \frac{1}{{\sqrt {2\pi \sigma_{2}^{2} } }}\exp - \left[ {\frac{{\left( {R_{2}^{ - } - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]}}{{\left( {\rho_{1} \rho_{2} } \right)^{ - 2} }} - \frac{{\int_{{R_{1}^{ - } }}^{{R_{1}^{ + } }} {\frac{{A_{1} R_{1} + B_{1} R_{2}^{ + } + C_{1} }}{{A_{2} R_{1} + B_{2} R_{2}^{ + } + C_{2} }} \times \frac{1}{{2\pi \sigma_{1} \sigma_{2} }}\exp - \left[ {\frac{{\left( {R_{1} - \mu_{1} } \right)^{2} }}{{2\sigma_{1}^{2} }} + \frac{{\left( {R_{2}^{ + } - \mu_{2} } \right)^{2} }}{{2\sigma_{2}^{2} }}} \right]} dR_{1} }}{{\left( {\rho_{1} \rho_{2} } \right)^{ - 1} }} \hfill \\ \end{gathered} \right.$$

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