Template:Confidence bounds for competing failure modes

Confidence Bounds for Competing Failure Modes

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.

Variance/Covariance Matrix

The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:

[math]\displaystyle{ \begin{align} & \left( \begin{matrix} Var({{{\hat{a}}}_{1}}) & Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\ Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & Var({{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdot & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & Var({{{\hat{a}}}_{n}}) & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) \\ 0 & 0 & 0 & 0 & 0 & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) & Var({{{\hat{b}}}_{n}}) \\ \end{matrix} \right) \\ & =\left( \begin{matrix} -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{1}^{2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdot & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} \\ 0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{n}^{2}} \\ \end{matrix} \right) \\ \end{align} }[/math]

where [math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the failure distribution, described in Chapter Parameter Estimation.

Bounds on Reliability

The competing failure modes reliability function is given by:

[math]\displaystyle{ \widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}} }[/math]

where:

• [math]\displaystyle{ {{R}_{i}} }[/math] is the reliability of the [math]\displaystyle{ {{i}^{th}} }[/math] mode,

• [math]\displaystyle{ n }[/math] is the number of failure modes.

The upper and lower bounds on reliability are estimated using the logit transformation:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}} \\ & {{R}_{L}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}} \end{align} }[/math]

where [math]\displaystyle{ \widehat{R} }[/math] is calculated using the reliability equation for competing failure modes. [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\underset{{{K}_{\alpha }}}{\overset{\infty }{\mathop \int }}\,{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

(If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.)

The variance of [math]\displaystyle{ \widehat{R} }[/math] is estimated by:

[math]\displaystyle{ Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial R}{\partial {{R}_{i}}} \right)}^{2}}Var({{\hat{R}}_{i}}) }[/math]

[math]\displaystyle{ \frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}} }[/math]

Thus:

[math]\displaystyle{ 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]

[math]\displaystyle{ 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]

where [math]\displaystyle{ \widehat{{{a}_{i}}} }[/math] is an element of the model parameter vector.

Therefore, the value of [math]\displaystyle{ Var({{\hat{R}}_{i}}) }[/math] is dependent on the underlying distribution.

For the Weibull distribution:

[math]\displaystyle{ Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}}) }[/math]

where:

[math]\displaystyle{ {{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}}) }[/math]

and [math]\displaystyle{ Var(\widehat{{{u}_{i}}}) }[/math] is given in Chapter 6.

For the exponential distribution:

[math]\displaystyle{ Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}}) }[/math]

where [math]\displaystyle{ Var(\widehat{{{\lambda }_{i}}}) }[/math] is given in Chapter 7.

For the normal distribution:

[math]\displaystyle{ Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\hat{\sigma } \right)}^{2}}Var({{\hat{z}}_{i}}) }[/math]

[math]\displaystyle{ {{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}} }[/math]

where [math]\displaystyle{ Var(\widehat{{{z}_{i}}}) }[/math] is given in Chapter 8.

For the lognormal distribution:

[math]\displaystyle{ Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}}) }[/math]

[math]\displaystyle{ {{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }} }[/math]

where [math]\displaystyle{ Var(\widehat{{{z}_{i}}}) }[/math] is given in Chapter 9.

Bounds on Time

The bounds on time are estimate by solving the reliability equation with respect to time. From Eqn. (CFMReliability) we have that:

[math]\displaystyle{ \hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}}) }[/math]

[math]\displaystyle{ i=1,...,n }[/math]

where:

• [math]\displaystyle{ \varphi }[/math] is inverse function for Eqn. (CFMReliability)

• for the Weibull distribution [math]\displaystyle{ {{\hat{a}}_{i}} }[/math] is [math]\displaystyle{ {{\hat{\beta }}_{i}} }[/math] , and [math]\displaystyle{ {{\hat{b}}_{i}} }[/math] is [math]\displaystyle{ {{\hat{\eta }}_{i}} }[/math]

• for the exponential distribution [math]\displaystyle{ {{\hat{a}}_{i}} }[/math] is [math]\displaystyle{ {{\hat{\lambda }}_{i}} }[/math] , and [math]\displaystyle{ {{\hat{b}}_{i}} }[/math] =0

• for the normal distribution [math]\displaystyle{ {{\hat{a}}_{i}} }[/math] is [math]\displaystyle{ {{\hat{\mu }}_{i}} }[/math] , and [math]\displaystyle{ {{\hat{b}}_{i}} }[/math] is [math]\displaystyle{ {{\hat{\sigma }}_{i}} }[/math] , and

• for the lognormal distribution [math]\displaystyle{ {{\hat{a}}_{i}} }[/math] is [math]\displaystyle{ \hat{\mu }_{i}^{\prime } }[/math] , and [math]\displaystyle{ {{\hat{b}}_{i}} }[/math] is [math]\displaystyle{ \hat{\sigma }_{i}^{\prime } }[/math]

Set:

[math]\displaystyle{ u=\ln (t) }[/math]

The bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

and:

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

Then the upper and lower bounds on time are found by using the equations

[math]\displaystyle{ {{t}_{U}}={{e}^{{{u}_{U}}}} }[/math]

and:

[math]\displaystyle{ {{t}_{L}}={{e}^{{{u}_{L}}}} }[/math]

[math]\displaystyle{ {{K}_{\alpha }} }[/math] is calculated using Eqn. (ka) and [math]\displaystyle{ Var(\widehat{u}) }[/math] is computed as:

[math]\displaystyle{ 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]