Estimation of Deterioration Rate

A range of techniques has been proposed to estimate the rate of asset deterioration, in this section, we distribute them into two types: a) point estimation techniques to produce the best estimate of an unknown population parameter to predict deterioration and b) techniques that describe deterioration with a statistical distribution.

Point Estimation Technique

A time-dependent stochastic process {X (t),t ≥ 0}, where X (t) is a collection of random variables for all t ≥ 0, is one primary form when modelling deterioration [50]. The variable X(t) gives the state of the system – for example, working or broken – at time t. Among the stochastic processes, most studies assume deterioration to be a Markov process [12]. Markov process is a stochastic process where the future state only depends on the present state (this Markov property is very useful for modelling of multi-state asset deterioration, it is further emphasised in the next section). In deterioration modelling, the transition from one state to another state is represented by the transition probability. Followed the Markov property, the conditional probability of moving into future state St+1 at time t + 1 given the present state

S_t at time t is:

P(Xt+1= St+1|Xt= St, . . . , X1= S1, X0= S0) = P(Xt+1= St+1|Xt = St) (2.1)

Extensions of the Markov process, such as Brownian motion with independent normal- distributed increments and decrements, and the gamma process with independent gamma- distributed increments are also widely researched in deterioration modelling. These exten- sions are further discussed and compared in Van Noortwijk and Pandey [172], Frangopol et al. [50] and Gorjian et al. [58].

Point estimation is often used in these studies to estimate the transition probabilities of stochastic processes. Point estimation produces a single value that can best represent the population parameter. A range of methods can be used to fit the transition probability, such as maximum likelihood estimation for Markov process in Kallen and Van Noortwijk [75] and method of moments for gamma process in Ohadi and Micic [130].

Meanwhile, with the development of machine learning techniques in recent years, several approaches have been employed as alternatives to the stochastic process to model and predict asset deterioration with point estimators. Instead of estimating the transition probabilities like what stochastic processes do, these techniques describe the deterioration rate as an arbitrary linear or nonlinear function. For example, Winn and Burgueño [184] built an Artificial Neural Networks model for bridge deck condition prediction, which was further improved

by an Ensemble Neural Networks. Similarly, Fink et al. [49] adopted a Multilayer Feedfor- ward Neural Networks based on Multi-Valued Neurons to provide time-series aggregated predictions for railway turnout system’s deterioration. These two examples describe the deterioration process in the form of a linear regression and a nonlinear regression respectively, and output the best-estimated values of deterioration given the age of assets.

Though these studies have shown remarkable performance for aggregated prediction of the global population - for example, how many assets will fail in the next 10 years; they also agreed that current approaches from point estimators suffer difficulty to provide an accurate prediction for individuals - for example, for two assets with the same age, which asset is more likely to fail. Also, in these studies, deterioration time was assumed to be precise (complete data, describing exactly how long does it takes for an asset to deteriorate) and the data size was big, whereas in practice, as discussed in Section 2.2.1, data are often censored, and for some specific type of assets, the amount of data is relatively small. Some studies had tackled the limitation of the current approaches applied in this field when dealing with data uncertainty. For example, Ferguson et al. [47] modelled the uncertain data as censored survival time within a Markov model. To evaluate the transition probabilities, they employed a non-parametric estimator called Datta-Satten estimator for hazard rate function estimation. This methodology has been applied many fields, for example, in predicting the deterioration of patient’s clinical states [31], but not in reliability. Later in Section 5.4.3, we compare the performance of this method with our proposed methods. However, most other methods are still data-driven approaches and did not take deterioration uncertainty between different individuals into consideration.

Statistical Distribution Technique

Unlike point estimation technique, which describes the deterioration probability from one state to another state with a single value, a parametric statistical distribution expresses its prediction with a stochastic function that takes the uncertainty during the deterioration process into consideration [157]. With a statistical distribution, we can describe the deterioration with, for example, an interval estimation. An interval estimator consists of a range of plausible values; we can represent them in the form of confidence or credible intervals. These statistical techniques are also often applied to deterioration prediction [50]. In a lifetime distribution, the expected lifetime of an asset is derived from its likely time to failure. A set of failure time data is gathered and fitted to a statistical distribution, describing the failure probability of an asset at a given time. The lifetime distribution function (denoted as F) is:

where t is the time point, Pr denotes the probability and T is a random variable stands for the time to failure. This means the lifetime distribution function is the probability that the time to failure T is greater or equals to a specified time point t.

Therefore, the failure rate (denoted as λ ), representing the probability of the asset has been working for a time t, but will fail with an additional time dt is:

λ (t) = lim

dt→0

Pr_{(t ⩽ T < t + dt)}

dt· Pr(T > t) (2.3)

Various statistical distributions have been used to fit asset deterioration behaviour. For example, He et al. [67] and Guler et al. [63] use exponential distributions to estimate the deterioration of the railway track. Studies of bridges in Agrawal et al. [3] and railways track in Andrews [5] provide two examples showing the use of Weibull distributions to model a range of asset deterioration behaviours.

In Le [94], lifetime data of bridge components of the same type and material are grouped to fit with a series of distributions. The fit of different distributions including normal, exponential, lognormal and Weibull distributions are compared using Anderson-Darling tests. Weibull distributions have the closest fit in most cases. Given this result and its versatility in describing a range of deterioration behaviours, the Weibull distribution is adopted in this thesis.

Apart from the advantage of allowing us to represent prediction in the form of an interval estimator, another advantage of using a statistical distribution is the interpretability over its parameters. For example, in a Weibull distribution, its probability density function (pdf) over time t is: f(t) = β η( t η) β −1_e−(_ηt)β_{, f (t) ≥ 0, β ≥ 0, η ≥ 0} (2.4)

and is characterised by parameter shape β and parameter scale η. For example, with a shape value that is lower than 1, we can model a failure rate that decreases with time while a shape greater than 1 describes wear-out, giving an increasing failure rate. For a given shape, increasing the scale increases the mean failure time [73]. The interpretability of the parameters gives a natural way to extract knowledge from engineers, which may reduce the need for data. A more detailed explaining about how to elicit this knowledge will be disused in Section 4.1.3.