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 The Weibull Distribution.

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 The Exponential Distribution.

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 The Normal Distribution.

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 The Lognormal Distribution.

Bounds on Time

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:

[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 the reliabilty equation for competing faiure modes.

• 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 the inverse standard normal distribution 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]