Effect of Epistemic Uncertainty in Markovian Reliability Models

Abstract

This chapter introduces the moment-based epistemic uncertainty propagation in Markov models. The epistemic uncertainty in Markov models introduces the uncertainty of model parameters, and it can be propagated by regarding parameters as random variables. The idea behind the moment-based approach is to approximate the multiple integration with a series expansion of model parameters. This leads to the efficient computation of the uncertainty in the expected output measure. The expected output measure is represented by the expected value and the variance of model parameters and the first and second derivatives of output measure with respect to model parameters. In this chapter, we introduce the formulation of moment-based epistemic uncertainty propagation and the concrete methods to obtain the first and second derivatives of output measures in Markov models.

Appendix

Let $M(\boldsymbol{\theta })$ be an output measure of a dependability model, where $\boldsymbol{\theta }= (\boldsymbol{\theta }_1, \ldots , \boldsymbol{\theta }_l)^T$ is a column vector representing a parameter vector. Also, $\boldsymbol{\Theta }$ is a parameter random vector. Define $f_{\boldsymbol{\Theta }}(\boldsymbol{\theta })$ is the joint epistemic density of the parameter vector $\boldsymbol{\Theta }$.

Suppose that the point estimate of $\hat{\boldsymbol{\theta }}$ is given by the expected value of $f_{\boldsymbol{\Theta }}(\boldsymbol{\theta })$, i.e.,

$$\begin{aligned} \hat{\boldsymbol{\theta }} = \text {E}[\boldsymbol{\Theta }] = \int \boldsymbol{\theta }f_{\boldsymbol{\Theta }}(\boldsymbol{\theta }) d\boldsymbol{\theta }. \end{aligned}$$

(41)

First we consider the approximation of the expectation of $M(\boldsymbol{\Theta })$:

$$\begin{aligned} \text {E}[M(\boldsymbol{\Theta })] = \int M(\boldsymbol{\theta }) f_{\boldsymbol{\Theta }}(\boldsymbol{\theta }) d\boldsymbol{\theta }. \end{aligned}$$

(42)

By taking Taylor series expansion of $M(\boldsymbol{\theta })$ at $\hat{\boldsymbol{\theta }}$, we have:

$$\begin{aligned} \text {E}[M(\boldsymbol{\Theta })] =&M(\hat{\boldsymbol{\theta }}) + \text {E}[\boldsymbol{M}'(\hat{\boldsymbol{\theta }})^T (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})] \nonumber \\&+ \frac{1}{2} \text {E}\left[ (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})^T \boldsymbol{M}''(\hat{\boldsymbol{\theta }}) (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }}) \right] + \cdots \end{aligned}$$

(43)

where

$$\begin{aligned} \boldsymbol{M}'(\hat{\boldsymbol{\theta }})&= \frac{\partial M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} \nonumber \\&= \begin{pmatrix} \displaystyle \frac{\partial M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_1} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}}&\cdots&\displaystyle \frac{\partial M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_l} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} \end{pmatrix}^T \end{aligned}$$

(44)

and

$$\begin{aligned} \boldsymbol{M}''(\hat{\boldsymbol{\theta }})&= \frac{\partial ^2}{\partial \boldsymbol{\theta }^2} M(\boldsymbol{\theta }) \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} \nonumber \\&= \begin{pmatrix} \displaystyle \frac{\partial ^2 M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_1^2} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} &{} \cdots &{} \displaystyle \frac{\partial ^2 M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_1 \partial \boldsymbol{\theta }_l} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} \\ \vdots &{} \ddots &{} \vdots \\ \displaystyle \frac{\partial ^2 M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_l \partial \boldsymbol{\theta }_1} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} &{} \cdots &{} \displaystyle \frac{\partial ^2 M(\boldsymbol{\theta })}{\partial \boldsymbol{\theta }_l^2} \bigg |_{\boldsymbol{\theta }=\hat{\boldsymbol{\theta }}} \end{pmatrix}. \end{aligned}$$

(45)

Since $\hat{\boldsymbol{\theta }} = \text {E}[\boldsymbol{\Theta }]$, the second term of Taylor series expansion becomes 0. We have the following approximation

$$\begin{aligned} \text {E}[M(\boldsymbol{\Theta })]&\approx M(\hat{\boldsymbol{\theta }}) + \frac{1}{2} \text {E}\left[ (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})^T \boldsymbol{M}''(\hat{\boldsymbol{\theta }}) (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})\right] \nonumber \\&= M(\hat{\boldsymbol{\theta }}) + \frac{1}{2} \Bigg (\sum _{i=1}^l M''_{i,i}(\hat{\boldsymbol{\theta }}) \text {Var}[\Theta _i] + 2 \sum _{i=1}^l \sum _{j=1}^{i-1} M''_{i,j}(\hat{\boldsymbol{\theta }}) \text {Cov}[\Theta _i, \Theta _j]\Bigg ), \end{aligned}$$

(46)

where $M''_{i,j}(\hat{\boldsymbol{\theta }})$ is an (i, j)-element of $\boldsymbol{M}''(\hat{\boldsymbol{\theta }})$.

Similar to the approximation of $\text {E}[M(\boldsymbol{\Theta })]$, we consider Taylor series expansion of the second moment of $M(\boldsymbol{\Theta })$, i.e.,

$$\begin{aligned}&\text {E}\left[ M(\boldsymbol{\Theta })^2\right] = M(\hat{\boldsymbol{\theta }})^2 + \text {E}[2 M(\hat{\boldsymbol{\theta }}) \boldsymbol{M}'(\hat{\boldsymbol{\theta }})^T (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})] \nonumber \\&+ \text {E}\left[ (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})^T \left( \boldsymbol{M}'(\hat{\boldsymbol{\theta }}) \boldsymbol{M}'(\hat{\boldsymbol{\theta }})^T + M(\hat{\boldsymbol{\theta }}) \boldsymbol{M}''(\hat{\boldsymbol{\theta }}) \right) (\boldsymbol{\Theta }- \hat{\boldsymbol{\theta }})\right] + \cdots . \end{aligned}$$

(47)

The approximation is given by

$$\begin{aligned}&\text {E}\left[ M(\boldsymbol{\Theta })^2\right] \approx M(\hat{\boldsymbol{\theta }})^2 + \sum _{i=1}^l \left( M'_i(\hat{\boldsymbol{\theta }})^2 + M(\hat{\boldsymbol{\theta }}) M''_{i,i}(\hat{\boldsymbol{\theta }}) \right) \text {Var}[\Theta _j] \nonumber \\&{} + 2 \sum _{i=1}^l \sum _{j=1}^{i-1} \left( M'_i(\hat{\boldsymbol{\theta }})M'_j(\hat{\boldsymbol{\theta }}) + M(\hat{\boldsymbol{\theta }}) M''_{i,j}(\hat{\boldsymbol{\theta }}) \right) \text {Cov}[\Theta _i, \Theta _j]. \end{aligned}$$

(48)