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:

where  is the log-likelihood function of the failure distribution, described in Chapter 5.

Bounds on Reliability
The competing failure modes reliability function is given by:


 * $$\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}$$


 * where:
 * •	 $${{R}_{i}}$$ is the reliability of the  $${{i}^{th}}$$  mode,
 * •	 $$n$$ is the number of failure modes.

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


 * $$\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}$$

where $$\widehat{R}$$  is calculated using Eqn. (CFMReliability). $${{K}_{\alpha }}$$ is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\underset{\overset{\infty }{\mathop \int }}\,{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

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

The variance of $$\widehat{R}$$  is estimated by:


 * $$Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial R}{\partial {{R}_{i}}} \right)}^{2}}Var({{\hat{R}}_{i}})$$


 * $$\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat$$


 * Thus:


 * $$Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})$$


 * $$Var({{\hat{R}}_{i}})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial {{R}_{i}}}{\partial {{a}_{i}}} \right)}^{2}}Var({{\hat{a}}_{i}})$$

where $$\widehat$$  is an element of the model parameter vector.

Therefore, the value of $$Var({{\hat{R}}_{i}})$$  is dependent on the underlying distribution.

For the Weibull distribution:


 * $$Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})$$


 * where:


 * $${{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})$$

and $$Var(\widehat)$$  is given in Chapter 6.

For the exponential distribution:


 * $$Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})$$

where $$Var(\widehat)$$  is given in Chapter 7.

For the normal distribution:


 * $$Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\hat{\sigma } \right)}^{2}}Var({{\hat{z}}_{i}})$$


 * $${{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}$$

where $$Var(\widehat)$$  is given in Chapter 8.


 * For the lognormal distribution:


 * $$Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})$$


 * $${{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}$$

where $$Var(\widehat)$$  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:


 * $$\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})$$


 * $$i=1,...,n$$


 * where:
 * •	 $$\varphi $$ is inverse function for Eqn. (CFMReliability)
 * •	for the Weibull distribution $${{\hat{a}}_{i}}$$  is  $${{\hat{\beta }}_{i}}$$, and  $${{\hat{b}}_{i}}$$  is  $${{\hat{\eta }}_{i}}$$
 * •	for the exponential distribution $${{\hat{a}}_{i}}$$  is  $${{\hat{\lambda }}_{i}}$$, and  $${{\hat{b}}_{i}}$$  =0
 * •	for the normal distribution $${{\hat{a}}_{i}}$$  is  $${{\hat{\mu }}_{i}}$$, and  $${{\hat{b}}_{i}}$$  is  $${{\hat{\sigma }}_{i}}$$ , and
 * •	for the lognormal distribution $${{\hat{a}}_{i}}$$  is  $$\hat{\mu }_{i}^{\prime }$$, and  $${{\hat{b}}_{i}}$$  is  $$\hat{\sigma }_{i}^{\prime }$$


 * Set:


 * $$u=\ln (t)$$

The bounds on $$u$$  are estimated from:


 * $${{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$


 * and:


 * $${{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$

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


 * $${{t}_{U}}={{e}^}$$


 * and:


 * $${{t}_{L}}={{e}^}$$

$${{K}_{\alpha }}$$  is calculated using Eqn. (ka) and $$Var(\widehat{u})$$  is computed as:


 * $$Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat)+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat)+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat,\widehat) \right)$$