Stochastic models are proposed for sleep and for the sleep related electroencephalogram (EEG), electrooculogram (EOG), and electromyogram (EMG). The evolution of sleep through its various stages is described as a Markov chain. The EEG is modelled using Wiener processes. The EOG and EMG are modelled as combinations of Poisson point processes and Gaussian processes, respectively. The EEG models contain a feedback structure that is based on physiological data. The maximum likelihood sleep stage monitor, that uses the sleep-related observations, has been derived and implemented. The agreement between automatic and human stage classifications of six sleep recordings was 70.6%, which was 4.5% worse than the average agreement between six human classifiers. Monitoring of simulated sleep suggests that the difficulty in separating wakefulness from stage 1 is due to poor modelling. If one ignores this difference, which, from a diagnostic point of view is fairly unimportant, the above mentioned agreement reaches 81.8%, which is 0.5% better than the corresponding average human vs human agreement.
A model has been proposed for a Markov-jumping sleep depth that modulates a white-noise driven structure generating the sigma rhythm in the electroencephalogram. The corresponding maximum likelihood monitor, that continuously detects the current sleep stage from the observed electroencephalogram, has been derived and implemented. Simulations show high detection performances.
In the railway sector, many redundant subsystem structures are applied to increase the safety and availability of the overall railway system. Failures to single paths of these structures occur and are found during routine inspection. Routine inspections are, depending on their type and the equipment location, quite costly and limit the vehicle availablity.
The present paper analyses occurrences based on simulated data resembling field data of a fleet of rail vehicles. The system is analysed statistically to identify the wear mechanisms leading to the failures. Failure data is then used to identify wear models which are consequently used in a Markov Chain (MC) to simulate the probability of multiple path failure.
The failure rate of the overall system is typically expected to be in the range $\left(10^{-9}\cdots10^{-6}\right)\, \mathrm{h}^{-1}$ due to the safety critical nature of the railway system. For this reason, it is required to apply rare event simulation techniques to the MC simulation in order to limit the number of simulations.
The simulation results are then applied to an optimisation of the inspection routine, which yields an appropriate failure rate for the associated hazards.
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