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| ==Confidence Bounds for Competing Failure Modes==
| | #REDIRECT [[Competing Failure Modes Analysis]] |
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| The method available in Weibull++ for estimating the different types of confidence bounds, for competing failure modes analysis, is the Fisher matrix method, and is presented in this section.
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| ===Variance/Covariance Matrix===
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| The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:
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| <math>\begin{align}
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| & \left( \begin{matrix}
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| Var({{{\hat{a}}}_{1}}) & Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\
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| Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & Var({{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & \cdot & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & Var({{{\hat{a}}}_{n}}) & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) \\
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| 0 & 0 & 0 & 0 & 0 & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) & Var({{{\hat{b}}}_{n}}) \\
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| \end{matrix} \right) \\
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| & =\left( \begin{matrix}
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| -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & 0 & 0 & 0 & 0 & 0 \\
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| -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{1}^{2}} & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & \cdot & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} \\
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| 0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{n}^{2}} \\
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| \end{matrix} \right) \\
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| \end{align}</math>
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| where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in Chapter [[Parameter Estimation]].
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| ===Bounds on Reliability===
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| The competing failure modes reliability function is given by:
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| ::<math>\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}</math>
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| where:
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| ::• <math>{{R}_{i}}</math> is the reliability of the <math>{{i}^{th}}</math> mode,
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| ::• <math>n</math> is the number of failure modes.
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| The upper and lower bounds on reliability are estimated using the logit transformation:
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| ::<math>\begin{align}
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| & {{R}_{U}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}} \\
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| & {{R}_{L}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}}
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| \end{align}</math>
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| where <math>\widehat{R}</math> is calculated using the reliability equation for competing failure modes.
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| <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\underset{{{K}_{\alpha }}}{\overset{\infty }{\mathop \int }}\,{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| (If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.)
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| The variance of <math>\widehat{R}</math> is estimated by:
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| ::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial R}{\partial {{R}_{i}}} \right)}^{2}}Var({{\hat{R}}_{i}})</math>
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| ::<math>\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}}</math>
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| Thus:
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| ::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})</math>
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| ::<math>Var({{\hat{R}}_{i}})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial {{R}_{i}}}{\partial {{a}_{i}}} \right)}^{2}}Var({{\hat{a}}_{i}})</math>
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| where <math>\widehat{{{a}_{i}}}</math> is an element of the model parameter vector.
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| Therefore, the value of <math>Var({{\hat{R}}_{i}})</math> is dependent on the underlying distribution.
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| For the Weibull distribution:
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| ::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})</math>
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| where:
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| ::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>
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| and <math>Var(\widehat{{{u}_{i}}})</math> is given in Chapter [[The Weibull Distribution]].
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| For the exponential distribution:
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| ::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>
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| where <math>Var(\widehat{{{\lambda }_{i}}})</math> is given in Chapter [[The Exponential Distribution]].
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| For the normal distribution:
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| ::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\hat{\sigma } \right)}^{2}}Var({{\hat{z}}_{i}})</math>
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| ::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>
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| where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter [[The Normal Distribution]].
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| For the lognormal distribution:
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| ::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})</math>
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| ::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>
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| where <math>Var(\widehat{{{z}_{i}}})</math> is given in Chapter [[The Lognormal Distribution]].
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| ===Bounds on Time===
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| The bounds on time are estimate by solving the reliability equation with respect to time. From the reliabilty equation for competing faiure modes, we have that:
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| ::<math>\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})</math>
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|
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| ::<math>i=1,...,n</math>
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| where:
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| :• <math>\varphi </math> is inverse function for Eqn. (CFMReliability)
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| :• for the Weibull distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\beta }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\eta }}_{i}}</math>
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| :• for the exponential distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\lambda }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> =0
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| :• for the normal distribution <math>{{\hat{a}}_{i}}</math> is <math>{{\hat{\mu }}_{i}}</math> , and <math>{{\hat{b}}_{i}}</math> is <math>{{\hat{\sigma }}_{i}}</math> , and
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| :• for the lognormal distribution <math>{{\hat{a}}_{i}}</math> is <math>\hat{\mu }_{i}^{\prime }</math> , and <math>{{\hat{b}}_{i}}</math> is <math>\hat{\sigma }_{i}^{\prime }</math>
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| Set:
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| ::<math>u=\ln (t)</math>
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| The bounds on <math>u</math> are estimated from:
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| and:
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| Then the upper and lower bounds on time are found by using the equations
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| ::<math>{{t}_{U}}={{e}^{{{u}_{U}}}}</math>
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| and:
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| ::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>
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| <math>{{K}_{\alpha }}</math> is calculated using Eqn. (ka) and <math>Var(\widehat{u})</math> is computed as:
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| ::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>
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