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:


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

Complex Competing Failure Modes
In addition to being viewed as a series system, the relationship between the different competing failures modes can be more complex. After performing separate analysis for each failure mode, a diagram that describes how each failure mode can result in a product failure can be used to perform analysis for the item in question. Such diagrams are usually referred to as Reliability Block Diagrams (RBD) (for more on RBDs see ReliaSoft's System Analysis Reference and ReliaSoft's BlockSim software).

A reliability block diagram is made of blocks that represent the failure modes and arrows and connects the blocks in different configurations. Note that the blocks can also be used to represent different components or subsystems that make up the product. Weibull ++ provides the capability to use a diagram to model, series, parallel, k-out-of-n configurations in addition to any complex combinations of these configurations.

In this analysis, the failure modes are assumed to be statistically independent. (Note: In the context of this reference, statistically independent implies that failure information for one failure mode provides no information about, i.e. does not affect, other failure mode). Analysis of dependent modes is more complex. Advanced RBD software such as ReliaSoft's BlockSim can handle and analyze such dependencies, as well as provide more advanced constructs and analyses (see http://www.reliasoft.com/BlockSim).