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| === The GRP Model ===
| | #REDIRECT [[Recurrent Event Data Analysis]] |
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| In this model, the concept of virtual age is introduced. Let <math>{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}</math> represent the successive failure times and let <math>{{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}}</math> represent the time between failures ( <math>{{t}_{i}}=\sum_{j=1}^{i}{{x}_{j}})</math> . Assume that after each event, actions are taken to improve the system performance. Let <span class="texhtml">''q''</span> be the action effectiveness factor. There are two GRP models:
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| Type I:
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| ::<span class="texhtml">''v''<sub>''i''</sub> = ''v''<sub>''i'' − 1</sub> + ''q''''x'''''<b><sub></sub> = </b>''q'''''t''<sub>''i''</sub></span>
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| Type II:
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| ::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}</math>
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| where <span class="texhtml">''v''<sub>''i''</sub></span> is the virtual age of the system right after <span class="texhtml">''i''</span> th repair. The Type I model assumes that the <span class="texhtml">''i''</span> th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age <span class="texhtml">''x''<sub>''i''</sub></span> to <span class="texhtml">''q''''x'''''<b><sub>''i''</sub></b></span> . The Type II model assumes that at the <span class="texhtml">''i''</span> th repair, the virtual age has been accumulated to <span class="texhtml">''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub></span> . The <span class="texhtml">''i''</span> th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <span class="texhtml">''q''(''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub>)</span> .
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| The power law function is used to model the rate of recurrence, which is:
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| ::<span class="texhtml">λ(''t'') = λβ''t''<sup>β − 1</sup></span>
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| The conditional <span class="texhtml">''pdf''</span> is:
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| ::<math>f({{t}_{i}}|{{t}_{i-1}})=\lambda \beta {{({{x}_{i}}+{{v}_{i-1}})}^{\beta -1}}{{e}^{-\lambda \left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i-1}^{\beta } \right]}}</math>
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| MLE method is used to estimate the model parameters. The log likelihood function is [[Appendix: Weibull References|[28]]]:
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| ::<math>\begin{align}
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| & \ln (L)= n(\ln \lambda +\ln \beta )-\lambda \left[ {{\left( T-{{t}_{n}}+{{v}_{n}} \right)}^{\beta }}-v_{n}^{\beta } \right] \\
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| & -\lambda \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i}^{\beta } \right]+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{x}_{i}}+{{v}_{i-1}})
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| \end{align}</math>
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| where <span class="texhtml">''n''</span> is the total number of events during the entire observation period. <span class="texhtml">''T''</span> is the stop time of the observation. <span class="texhtml">''T'' = ''t''<sub>''n''</sub></span> if the observation stops right after the last event.
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