Stochastic Algorithms

Data-driven prognostics based on stochastic approaches provide a probability distribution of RUL that may be analysed statistically but may not be esti-

mated precisely. These methods are, in general, Bayesian-based approaches, which predict the state of a process by a minimum prediction covariance ob- tained from measurements. They can estimate both current and future states of nonlinear systems, and they can predict RUL by tracking the trends of growing deterioration before the asset hits a prearranged threshold (An et al., 2013).

In Bayes’ theorem, the probability of an event is described by prior knowledge of conditions related with that event. This relation forms a reference point for updating estimations with due consideration of relevant evidence (Bayes et al., 1763).

P(A|B) = P(B|A)P(A)

P(B) (2.18)

whereA and B are two different observable events andP(B)6= 0 . • P(A) and P(B) are the probabilities of A and B respectively.

• P(A|B), conditional probability, is the probability of A given that con- dition B is true.

• P(B|A) is the probability of B given that condition A is true.

When there is useful condition monitoring data available, a Bayesian network can model the degradation changes over time and provide results for prognostics which are invariably undertaken using time series forecasting. The most common variants of such Bayesian networks used in prognostics include Particle filters (Orchard and Vachtsevanos, 2009; Orchard et al., 2005; Saha and Goebel, 2009; An et al., 2013; Miao et al., 2013; Wang and Gao, 2014), Kalman filters (Hu et al., 2012a; Julier and Uhlmann, 1997; Swanson, 2001)

and Markov models (Bunks et al., 2000; Camci, 2005; Baruah and Chinnam, 2005; Dong and He, 2007; Ramasso, 2009)

Particle filtering (PF) is known to be a powerful tracking technique based on sequential Monte-Carlo methodology for sequential signal process- ing. Orchard and Vachtsevanos (2009) pioneered this approach in prognostic estimations, and carried it out by the approximation of the conditional state probability distribution using a swarm of points called “particles”. It is stated that the model can provide results in long-term predictions, and it is suitable for on-line implementation.

In PF, the Bayesian theorem is used in a sequential way with samples (particles) which have probability information of unknown parameters. This process is mainly based on a state transition function fs and a measurement

functionfm.

¨

xt=fs(¨xt−1,p¨t,ε˙t) (2.19)

¨

zt=fm(¨xt,ε¨t) (2.20)

where ¨x is the damage state, ¨p is a vector of model parameters, ¨z is measurement data, ˙ε and ¨ε are respectively process and measurement noises, and t is time (Orchard et al., 2005). The filtering problem includes the esti- mation of ¨xt, given all the measurements up to time “t” (¨z(1:t)). Considering the Bayesian theorem, this estimation can be formalised as the calculation of the distribution P(¨xt|z¨(1:t)), which can be done recursively in estimation and update stages where the computations are carried out by Monte Carlo sampling.

In prognostics, the state transition function is referred to a damage model (An et al., 2013). If this model is accurately defined to represent the

governing system dynamics, PF can be applied into nonlinear systems. How- ever, in a linear system, the Kalman filtering (KF) is optimal since they have lower computational requirements than particle filters. In the case of KF, the state and measurement equations reduce to the following forms (Orhan, 2012).

¨

xt=Fsx¨t−1,p¨t,ε˙t (2.21)

¨

zt=Fmx¨t,ε¨t (2.22)

where ˙εand ¨εare Gaussian noises, andFs andFm are respectively the state evolution and measurement matrices which are assumed to be known. With these equations, KF estimates the state of a process and minimises es- timation covariance by including the measurement related to the state. This filtering model can correct the estimations with the latest measurements to minimise state error covariance.

Since both PF and KF methods are based on time series and their rela- tionship with prior conditions, their application is limited in complex systems operating under various conditions. A damage state at any given time instant may not match with the upcoming conditions. Additionally, it is difficult to describe the behaviour of damage model for multiple sensor data.

A simpler Bayesian network, Hidden Markov Model (HMM), is also applied in the field of prognostics by Bunks et al. (2000), Camci (2005), and Baruah and Chinnam (2005). Accordingly, Zhang et al. (2005) investigated the use HMMs in bearing fault prognosis and applied to obtain the degrada- tion index of bearings. Dong and He (2007) and Ramasso (2009) extended the use of HMMs in prognostics as an estimation tool for representing prob- ability distributions over sequences of observations. In these applications, it is assumed that the observation at “t” is generated by some process whose state P(¨x) is hidden from the observer and the state of this hidden process

satisfies the Markov property (in which future behaviour is predicted from the current or present behaviour) (Ghahramani, 2001). These applications can be used to recognise different fault types and states, but they might have prac- ticability issues in physically observing a defect in an operating unit (Heng et al., 2009). Similar to the PF and KF method, HMM also has difficulties with multi-dimensional data and requires accurate degradation modelling.