Deterministic models are usually applied to phenomena where relationships between components of the rail structure are identified. Linear and exponential forms of deterministic models were the first attempt in rail degradation modelling, due to their simplicity in mathematical expressions and ability to show a direct relationship between the input and output variables [26]. Using a mathematical or statistical expression, deterministic models can identify the relationship between the factors affecting rail degradation and the condition of the track. Hence, these models predict the condition of the rail and its degradation deterministically by ignoring random errors in prediction.

Deterministic models require different data parameters, including train speed, geometry and the operations of the rail (i.e. axle weight, line speed and traffic volume) [27] and accumulated tonnage (MGT) [2, 15, 28]. By using data on rail conditions, the deterministic approach provides a general pattern of the statistical distribution of the rail and the affecting parameters.

Deterministic models following a linear relationship are commonly used [2, 15, 29], even though some studies have found a non-linear relationship based on other forms of the model such as polynomial [30], exponential [31] and multi-stage linear [14].

The Office for Research and Experiments (ORE) of the International Union of Railways (UIC2) studied the fundamentals of the degradation mechanism of rail tracks in the 1980s [32]. A deterministic ORE model was proposed to estimate rail degradation according to various studies [33-36]. Accordingly, traffic volume, dynamic axle and speed are the important variables/parameters influencing rail track degradation. The structure of the ORE deterministic model is as follows:

(3)

Where, e₀ is the degradation directly after tamping, h is a constant, T is the traffic volume, Q is the dynamic axle, v is the speed, and α, β, and γ are parameters estimated from experimental data.

Basically, the deterministic approach uses railway track geometry data to contribute to the understanding and management of the long-lasting deterministic model. However, although this model may work well for large datasets, the rate of degradation may vary, which may affect the asset management of the track. Moreover, this type of model does not account for uncertainty. In other words, the parameters governing the equations are assumed to be known and the solutions are hence unique. Therefore, even if the methodology of the deterministic models is understandable and simple, it is difficult to trust the results of these models because the input parameters and the model geometry are not well defined. In addition, these studies potentially miss an important degradation factor during application. Therefore, more research is needed for further analysis (a detailed summary is shown in Table 1).

Various probabilistic models have been proposed, although it is difficult to analyse the probability of track degradation due to a number of factors, such as the environment, the structural materials and the construction quality. Probabilistic models can be grouped into three classifications (Fig. 3).

• Continuous Probability Distributions

Continuous probability distribution models are usually applied in certain states and within a known elapsed time since the last maintenance activity. A study by Jernbaneverket (JBV), the Norwegian National Rail Administration, was carried out using a probability distribution model [37]. This study calculates the risks and costs of rail defects and outlines the issues of rail failure by using the inspection and maintenance strategies followed by the railway company [37]. It was found that time, speed and rail route location are the variables, which most affect rail degradation using the continuous probability distribution model.

Following equation presents the model, which evaluates the probability of rail breakage of detecting a crack.

(4)

Where, E[D(τ, τ^’, t_w)] is the expected number of derailments per year, f_I is the yearly frequency of crack initiations, Q (τ, τ′, t_w) is the number of cracks missed by USI per year and τ, τ^’, t_w are different variables directly related to rail derailment, including time, speed and route of the railway.

Another continuous probability distribution model was proposed by Zio et al. [39], defining the progression of defects according to the Norwegian National Rail Administration (Jernbaneverket, JBV). A probability distribution model was applied within a multi-state perspective. In other words, each rail track section is analysed in different discrete states, depending on the rail track degradation and its conditions. Fig. (3) shows a state diagram of the defects using this degradation model. There are 6 sections in this diagram; the rail track condition hj in each section j (j = 1, 2, …, n) is discretised in δ + 1 = 6 levels. Level 5 corresponds to a section with zero defects (i.e. perfect condition). However, the degradation levels hj (hj= 1, 2, 3, 4) correspond to gradual critical rail conditions. Level hj = 0 corresponds to rail breakage (i.e. complete failure). The downward and upward arrows in the diagram correspond to the stochastic transitions of defect growth and repair, respectively, as shown in Fig. (3) [39].

Overall, the purpose of the probability distribution model is to inform realistic and reasonable decision-making on inspection and maintenance optimisation practices in the railway industry. The study of this model found that the growth of defects depends on the expected times of failure, which sort the defect as ‘high risk’ or ‘low risk’. Failure also depends on the speed at which trains pass over the track. Although there have been a number of recent continuous probability distribution studies, this model is recommended for use in a certain state and within a known time frame.

Fig. (3). Model of defect growth [39].

• Hierarchical Bayesian Models

Hierarchical Bayesian Models (HBMs) are flexible statistical models that provide a general prediction of railway track geometry degradation. HBMs allow the assessment of the relationship between different components of consecutive rail track sections, including the deterioration rates and the initial qualities parameters. For example, a study of an HBM was developed for the main Portuguese railway line Lisbon-Oporto, assessing two main quality parameters in relation to the degradation of rail track geometry, which are the Standard Deviation of Longitudinal Level defects (SDLL) and the Standard Deviation of Horizontal Alignment defects (SDHA) [40]. This model adopts the parameters as random variables that can be uncertainly calculated by a prior distribution [40]. This prior distribution P(θ) is then combined with the traditional likelihood p(y|θ) to obtain the posterior distribution of the parameters of interest [40]. The posterior distribution P(θ|y) of the parameters θ given the observed data y can be calculated according to Bayes' rule as presented in the following equation:

(5)

Where, θ is the initial quality parameter, y is a random variable, the value or probability distribution of which is known, P(θ | y) is posterior distribution of θ given y which relates to θ via a model, P(y | θ) is the likelihood to observe y given unknown θ or the sampling distribution of D given known θ, and P(θ) is the prior probability of θ.

It was found that the calculation of the prior distribution is a vital step in every Bayesian model study. Nevertheless, every case using this model shows that the joint posterior distribution p(θ|y) has a rational high dimension and combination through numerical methods must rely on Markov Chain Monte Carlo (MCMC) methods, which are built such that their stationary distribution is the desired posterior distribution [40-42].

Using HBMs on a sample of operation and maintenance data indicated that they present a worse fit of the quality indicator SD_HA compared to the quality indicator SD_LL. Horizontal alignment defects also appear to be less predictable [40]. This literature review found that Bayesian model case studies are limited in the published research. They also rely on Markov models, especially when high numerical dimensions occur. Other studies have also developed Bayesian models, including those of [41-45].

• Markov Models

Markov models are statistical models that assess the infrastructure of rail tracks at various condition levels over time. They also analyse the hazard degradation rate while assessing the uncertainty of rail track degradation. The main task of developing a Markov model is to calculate the transition probability from the sampling data. Some calibration techniques are usually used to calibrate the data and calculate the transition probability matrix. Hence, these techniques can be divided into state-based Markov models and time-based Markov models. State-based models are the main focus in this literature as they are commonly used in previous rail degradation prediction studies. For instance, Shafahi and Hakhamaneshi [36] developed a state-based Markov model computing the Track Quality Index (TQI) in a range between 0 and 100 (mapped onto 5 states) based on different parameters including track unevenness, twist, alignment and gauge measurements. The structure of the model is based on a transition matrix showing the probabilities of various states of rail tracks at any time, n, as follows:

(6)

Where, X(n) is the track state at time n, p{X(n) = j} is the probability that a track is in state j at time n.

Shafahi and Hakhamaneshi [36] found that the Markov model appears to be superior to conventional regression models, such as the ORE model (refer to Section 2.3.2.1). Although further studies and enhancement of the model are desired, the use of the Markov model on Iranian railways has been shown to be a reasonable method for the allocation of maintenance funds [36].

Lyngby et al. [3] also analysed a 50-state Markov model on Norwegian railway tracks. This study considered the twist on each section of the track up to 50mm, with the intention of assessing the failure rates over the railway line. Moreover, this model analysed the deterioration rates depending on the geometry of the track section, whether it was straight, curved or transition. It also analysed the frequency optimisation between track geometrical sections [3].

Another stated-based Markov model was proposed by Prescott and Andrews [46], who studied the degradation, inspection and maintenance of a single one-eighth of a mile section of UK railway track. The model analyses the changing deterioration rate and maintenance of the rail track section. It also studies the effects of changing the level of rail geometry degradation, starting from the good condition of the rail until it reaches the critical value at which maintenance is required.

Overall, probabilistic models are not common, due to the lack of historical data and research related to the geometrical quality of rail tracks. This might be due to the difficulty of predicting the probability of rail degradation. There are also limitations of Markov models, as they are effectively applied to small rail track sections and based on certain states. Moreover, the transitions between states using Markov modelling might face a problem as they must occur at constant rates [3]. This, in turn, restricts the details of the model analysis (as shown in Table 1).

Stochastic models are mainly statistical models based on historical records and data [47]. They aim to understand the distribution of time to degradation events and predict their performance. These models are based on what has actually happened and account for variability through the use of probability, although they do not deliver insight into the underlying physics [47].

Various research studies have developed stochastic models for rail degradation prediction. Rail characteristics (i.e. rail type, sleeper type), time and rail geometry (including tamping activities) are the main variables used in stochastic modelling. Yousefikia et al. [24] proposed a review of stochastic models based on their data analysis of rail tracks along tram routes in Melbourne, Australia. From this study’s viewpoint, the rail track is considered to be regular when it carries out its function under operating conditions for a certain period of time; if this is not the case, the rail track fails [24]. Hence, the failure progress can be identified in several ways. The gamma process is the most common model used for failure progression and continuous time stochastic processes. Refer to [48] for further details on systems with gamma deterioration activity.

Lyngby et al. [3] also analysed Markov and stochastic degradation models. They stated that rail track geometry could be displayed in a better way as a stochastic model due to the observed variability [3]. Other rail degradation prediction and maintenance planning studies have used stochastic modelling in their research, including [49-54].

A study carried out by Guler et al. [15] analysed a rail geometry deterioration model. This study focuses on the effects of track characteristics, environmental conditions, and maintenance and renewal policies on the deterioration of track parameters measured by recording vehicles. The study determined that natural disasters such as flooding and falling rocks increase the rate of geometry deterioration, whereas snow and landslides have no effect [15]. This study also concluded that the expansion of curvature or gradient or line speed increase the rate of geometry deterioration. This indicates that sleeper type and rail type (Continuously Welded Rail (CWR) or jointed rail) have an influence on geometry deterioration, with CWRs deteriorating at a slower rate [15]. However, the study found that an increase in the annual tonnage reduced the deterioration rates. Therefore, this invalidates the model, as it is known that the increase of the annual tonnage increases deterioration rates.

Quiroga and Schnieder [43] proposed a stochastic degradation model following a heuristic-based method for scheduling tamping intervention, and then developed it into a Monte Carlo simulation for track ageing and restoration. The model is based on 20 years of rail track measurement data collected from the French railway operator SNCF [43]. This model does not take into account data taken in the first three months after an intervention because it assumes that all rail tracks face bedding-in [43]. Hence, only data sets where the time period between tamps was at least one full year were used to raise the precision of the model [55]. However, this assumption reduces the application of this model for the UK network and a number of geometry deterioration practices. The model hypothesis of this study has two assumptions:

• The degradation value NLinit_n is achieved after the nth tamping intervention. It is defined as a log-normally distributed stochastic variable:

(7)

Where, μ is the mean value, and σ² is the variance.

• The evolution of the degradation value between two tamping activities. It is defined by an exponential function of the form:

(8)

Where, t is the time, t_n is the time at which the last tamping activity took place, b_n is a log normally distributed stochastic variable of equation bn ~ LN (μ_b (n), σ_b² (n)), and ε_n(t) is a normally distributed variable with mean value 0 of equation: ε_n(t) ~ N(0,σ_ε².

Although it is assumed in this model that the track undergoes exponential deterioration, there is no evidence to validate the claim of an exponential deterioration pattern. Therefore, the plots of the SNCF rail network sections shown in the results do not indicate that the track geometry undergoes an exponential deterioration pattern [43].

In addition, Andrews [47] developed a Petri net stochastic model to analyse the degradation, inspection, maintenance and renewal of rail track sections. This model studies the efficiency of the asset management process employed and predicts the state of the track geometry. Statistical distributions of times to given levels of geometry deterioration are applied, taking into account the effects of maintenance on the rate of deterioration. However, a better understanding of the degradation process needs to be established. Subsequently, this could support the development of accurate models based on historical data such as the effects of maintenance data [27].

Vale and Lurdes [56] developed a stochastic model of geometrical track degradation using the Portuguese railway Northern Line as a case study. Statistical and probabilistic analyses were achieved for different vehicle speed groups, showing that the rate of degradation of the standard deviation of the longitudinal level is similar for both rails. The degradation rate of longitudinal level has an asymmetric distribution with heavy tailedness shown as:

(9)

Where, γ is the skewness of a random variable X, μ is the third moment about the mean, and σ is the standard deviation of the variable.

In this case, the Dagum distribution, usually adopted for representing income distribution, fitted very well the geometrical track degradation activity of the Portuguese railway Northern Line in terms of the longitudinal level [56]. The Dagum distribution represents the model in the analysis of three parameters of function F(x), defined as:

(10)

Where, α, β and k are positive parameters, parameter β is a scale parameter while α and k are shape parameters.

In general, stochastic models are widely used in deterioration prediction studies. However, these models require more understanding of the application used and more explanation of it (see Table 1 for a summary of details). In turn, this can increase the accuracy of these models and their application.

Shafahi et al. [55] assumed that the rail track started its life at a time where it was in perfect condition and was then subjected to a sequence of duty cycles that caused its condition to deteriorate. It was also assumed that the duty cycle of the rail track was one year. CTR was also used to define the rail track condition. The following factors were used as effective parameters of rail track degradation in the study:

• CTR index of a year, the previous year, and two and three years before, which was classified from 1 to 5 [55].
• Traffic volume was divided into two groups: light and heavy.
• Maximum allowable speed was classified into 5 classes as follows: maximum speed less than 60 km/h class 1; 60-80 km/h class 2; 80-100 km/h class 3; 100-120 km/h class 4; and more than 120km/h class 5.
• Geographical location was classified into three classes: plain, hilly, and mountainous.
• The maximum gradient of the block was classified into five classes as follows: maximum gradient from 0% - 0.5% gradient class 1; 0.5% - 1% gradient class 2; 1% - 1.5% gradient class 3; 1.5% - 2% gradient class 4; and 2% -3% gradient class 5.
• Minimum radius of curves in the blockwas classified into 7 classes as follows: maximum radius of curves less than 250 m radius class 1; 250 - 400m class 2; 400 – 750m class 3; 750-1000m class 4; 1000 – 2000m class 5; 2000 – 4000m class 6; and radii larger than 4000m class 7.

The following steps were undertaken to build the structure of the ANN model:

Step 1: The topology of the network was created with the inclusion of parameters such as number of layers and nodes of the network, type of network, initial and activation functions.

Step 2: On the basis of the training process in the network, the weighted parameters were corrected and the data on every situation as training data were shown to the network many times.

Step 3: The neural network was examined using known data so that probable errors could be corrected.

A network with 3 layers and 5 neurons in the internal layer was selected as the optimal network. Then the data were randomly directed into two sets: the training set which included 82% of the data, and the test set with 18% of the data. Shafahi et al. [55] showed a comparison of the model predictions and the observed data for one of the sample data sets used. The predictions of the ANN indicated that the following year CTR indices were at the level of the CTR of the previous year or one level lower. Therefore, it was concluded that there was 67% correctness of neural network results using correct sample sets, while there was 33% correctness within one level wrong [55].

Another study carried out in Turkey by Guler [59] indicated excellent results by using an ANN model to predict rail track degradation. This study was a case study for Turkish state railways. The researcher conducted a thorough field investigation over a period of 2 years. The observed track length for this study was approximately 180 km. The dataset was collected based on different variables, including track structure, traffic characteristics, track layout and environmental factors. The author developed separate ANN models for the main track geometry parameters, and performed sensitivity analysis to calculate the importance of each predictor in determining the neural networks.

The neural network was developed using SPSS software and used 70% of the total data from 2009 to 2011 was used as the training data and the remaining 30% for testing. ANN models produced strong relationships between the deterioration rate and the variables. The R² given as twist=0.727, gauge=0.795, alignment=0.765, cross level= 0.831 and longitudinal levelling =0.742

These results again indicate that ANN could be used as a better alternative for rail track degradation prediction than other models.

Shafahi et al. [55], developed the ANN model in an effort to compare the accuracy of it by comparing the results produced through that model to few other different models. This study showed that there was 73% correctness of neural network results using correct data sets, while there was 27% correctness with one level wrong [55]. When comparing the results of these two AI models, it was clear that the neuro-fuzzy model showed results 6% better than the ANN model [55].

Neuro-fuzzy models are becoming as popular as ANN models among researchers and look promising for degradation modelling in the future for a number of reasons. The main reason is that they show high accuracy compared to other approaches. The major drawbacks of these models are the lack of literature since they are relatively new to degradation prediction. In addition, it is hard to understand the structure of the models and how they work. They lack the transparency of other models, such as statistical or mechanical models. This discourages researchers from using them in their studies (Table 1).

Based on the literature review, it is possible to conclude that all existing degradation models have a number of strengths and weaknesses. The selection of the most suitable degradation model depends on the research area and the data sets provided to researchers. Table 1 below shows a brief outline of the strengths and weaknesses of each degradation model discussed in the review. It also shows the variables used in each model to help with decision-making on the most appropriate degradation model for any particular research.