Template:Ipl lognormal

IPL-Lognormal

The [math]\displaystyle{ pdf }[/math] for the Inverse Power Law relationship and the lognormal distribution is given next.

The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:

[math]\displaystyle{ f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]

where:

[math]\displaystyle{ T'=ln(T) }[/math].

and:

[math]\displaystyle{ T }[/math] = times-to-failure.

[math]\displaystyle{ \overline{T}' }[/math] = mean of the natural logarithms of the times-to-failure.

[math]\displaystyle{ \sigma_{T'} }[/math] = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

[math]\displaystyle{ \breve{T}=e^{\overline{T}'} }[/math]

The IPL-lognormal model [math]\displaystyle{ pdf }[/math] can be obtained first by setting [math]\displaystyle{ \breve{T}=L(V) }[/math] in the lognormal [math]\displaystyle{ pdf }[/math]. Therefore:

[math]\displaystyle{ \breve{T}=L(V)=\frac{1}{K \cdot V^n} }[/math]

or:

[math]\displaystyle{ e^{\overline{T'}}=\frac{1}{K \cdot V^n} }[/math]

Thus:

[math]\displaystyle{ \overline{T}'=-ln(K)-n ln(V) }[/math]

So the IPL-lognormal model [math]\displaystyle{ pdf }[/math] is:

[math]\displaystyle{ f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+ln(K)+n ln(V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]

IPL-Lognormal Statistical Properties Summary

The Mean

The mean life of the IPL-lognormal model (mean of the times-to-failure), [math]\displaystyle{ \bar{T} }[/math] , is given by:

[math]\displaystyle{ \bar{T}=\ {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}= {{e}^{{-ln(K)-nln(V)}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} }[/math]

The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\bar{T}}^{^{\prime }}} }[/math], in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right) }[/math]

The Standard Deviation

The standard deviation of the IPL-lognormal model (standard deviation of the times-to-failure), [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

[math]\displaystyle{ \begin{align} {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\,\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} = \sqrt{\left( {{e}^{2\left( -\ln (K)-n\ln (V) \right)+\sigma _{{{T}'}}^{2}}} \right)\,\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \end{align} }[/math]

The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)} }[/math]

The Mode

The mode of the IPL-lognormal model is given by:

[math]\displaystyle{ \tilde{T}={{e}^{{\bar{T}}'-\sigma _{{{T}'}}^{2}}}={{e}^{-\ln (K)-n\ln (V)-\sigma _{{{T}'}}^{2}}} }[/math]

IPL-Lognormal Reliability

The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:

[math]\displaystyle{ R(T,\,V)=\int_{T}^{\infty }f(t,\,V)dt }[/math]

or:

[math]\displaystyle{ R(T,\,V)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Reliable Life

The reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}}, }[/math] is estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ T_{R}^{\prime }=-\ln (K)-n\ln (V)+z\cdot {{\sigma }_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },\,V \right) \right] }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',\,V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt }[/math]

Since [math]\displaystyle{ {T}'=\ln (T) }[/math] the reliable life, [math]\displaystyle{ {{t}_{R}}, }[/math] , is given by:

[math]\displaystyle{ {{t}_{R}}={{e}^{T_{R}^{\prime }}} }[/math]

Lognormal Failure Rate

The lognormal failure rate is given by:

[math]\displaystyle{ \lambda (T,\,V)=\frac{f(T,\,V)}{R(T,\,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt} }[/math]

Parameter Estimation

Maximum Likelihood Estimation Method

The complete IPL-lognormal log-likelihood function is:

[math]\displaystyle{ \begin{align} \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\varphi \left( \frac{\ln \left( {{T}_{i}} \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right) \right] +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] \end{align} }[/math]

where:

[math]\displaystyle{ z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }} }[/math]

[math]\displaystyle{ z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }} }[/math]

and:

• Fe is the number of groups of exact times-to-failure data points.

• Ni is the number of times-to-failure data points in the [math]\displaystyle{ {i}^{th} }[/math] time-to-failure data group.

• [math]\displaystyle{ s_{T'} }[/math] is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

• [math]\displaystyle{ K }[/math] is the IPL parameter (unknown, the second of three parameters to be estimated).

• [math]\displaystyle{ n }[/math] is the second IPL parameter (unknown, the third of three parameters to be estimated).

• [math]\displaystyle{ Vi }[/math] is the stress level of the [math]\displaystyle{ {i}^{th} }[/math] group.

• [math]\displaystyle{ Ti }[/math] is the exact failure time of the [math]\displaystyle{ {i}^{th} }[/math] group.

• [math]\displaystyle{ S }[/math] is the number of groups of suspension data points.

• [math]\displaystyle{ N'_i }[/math] is the number of suspensions in the [math]\displaystyle{ {i}^{th} }[/math] group of suspension data points.

• [math]\displaystyle{ T^{'}_{i} }[/math] is the running time of the [math]\displaystyle{ {i}^{th} }[/math] suspension data group.

• [math]\displaystyle{ FI }[/math] is the number of interval data groups.

• is the number of intervals in the [math]\displaystyle{ {i}^{th} }[/math] group of data intervals.

• is the beginning of the [math]\displaystyle{ {i}^{th} }[/math] interval.

• is the ending of the [math]\displaystyle{ {i}^{th} }[/math] interval.

The solution (parameter estimates) will be found by solving for [math]\displaystyle{ {{\hat {\sigma}}_{{{T}'}}} }[/math], [math]\displaystyle{ \hat {K} }[/math], [math]\displaystyle{ \hat {n} }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0, }[/math], [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial K}=0 }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial n}=0\ \ : }[/math]:

[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial K}= & -\frac{1}{K\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}})) \ -\frac{1}{K\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{+\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime })}{K\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\ \frac{\partial \Lambda }{\partial n}= & -\frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln ({{V}_{i}})\left[ \ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}}) \right] -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln ({{V}_{i}})\frac{\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\ln {{V}_{i}}\left( \phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime }) \right)}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\ \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}}) \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \ +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)\,\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\phi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\phi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \end{align} }[/math]

and:

[math]\displaystyle{ \varphi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}} }[/math]

[math]\displaystyle{ \Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt }[/math]