Markov process applied to degradation modelling

.................................................................................................................... 2 Preface ....................................................................................................................... 3 Acknowledgements................................................................................................... 4 Chapter 1: Introduction .......................................................................................... 7 Chapter 2: Characteristics of the data ................................................................... 9 Chapter 3: Model description ...............................................................................11 3.1 General introduction of Markov processes ...................................................................................... 11 3.1.1 Markov Chain ............................................................................................................................. 11 3.1.2 Markov process .......................................................................................................................... 12 3.1.3 Semi-Markov process ................................................................................................................. 13 3.2 Nonhomogeneous Poisson Process ................................................................................................. 15 3.3 Model description ............................................................................................................................. 18 3.3.1 Model A and Model B ................................................................................................................ 18 3.3.2 Model C and Model D ................................................................................................................ 20 3.4 Semi-Markov process with sojourn time belonging to Weibull distribution .................................... 21 Chapter 4: Model application ...............................................................................23 4.1 Description of information ................................................................................................................ 23 4.2 Parameter estimation ....................................................................................................................... 26 4.2.1 Maximum likelihood method for Model A, B, C and D .............................................................. 26 4.2.2 Maximum likelihood method for Model E ................................................................................. 27 4.3 Results of parameter estimation ...................................................................................................... 28 4.3.1 Results of Model A and Model B ................................................................................................ 28 4.3.2 Result of Model C and Model D ................................................................................................. 32 4.3.3 Result of semi-Markov process with sojourn time belonging to Weibull distribution .............. 35 4.4 Summary ........................................................................................................................................... 36 Chapter 5: Evaluation of Markov models ...........................................................38 5.1 Comparison of models ...................................................................................................................... 38 5.2 Simulation ......................................................................................................................................... 39 5.3 Verification of Model E ..................................................................................................................... 44