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Hierarchical Bayesian modelling of rail track geometry degradation António Ramos Andrade and Paulo Fonseca Teixeira Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 2013 227: 364 originally published online 24 May 2013 DOI: 10.1177/0954409713486619 The online version of this article can be found at: http://pif.sagepub.com/content/227/4/364

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Special Issue Original Article

Hierarchical Bayesian modelling of rail track geometry degradation

Proc IMechE Part F: J Rail and Rapid Transit 227(4) 364–375 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954409713486619 pif.sagepub.com

Antońio Ramos Andrade and Paulo Fonseca Teixeira

Abstract This paper explores hierarchical Bayesian models that can be used to predict rail track geometry degradation and thus guide planning maintenance and renewal actions. Hierarchical Bayesian models allow great flexibility in their specification, especially if they are combined with conditional autoregressive terms that can take into account spatial dependencies between model parameters. For rail track geometry degradation, conditional autoregressive terms are specified to tackle spatial interactions between consecutive rail track sections in rail track lines. An analysis of inspection, operation and maintenance data from the main Portuguese line (Lisbon–Oporto) motivates and illustrates the proposed predictive models. Inference is then conducted based on Markov Chain Monte Carlo (MCMC) simulation, which is proposed for fitting different model specifications. Finally, model comparison and a sensitivity analysis on prior distribution parameters are assessed. Keywords Track degradation, track geometric quality, uncertainty, life cycle, Bayesian model Date received: 1 June 2012; accepted: 25 March 2013

Introduction Planning maintenance and renewal actions in transport infrastructures is a complex problem. Infrastructure managers (IMs) should plan their maintenance and renewal actions, bearing in mind several criteria, for example: maintenance and renewal costs, delays caused to different operators, risks, logistic and resource constraints, environmental impacts, etc. These plans mainly rely on predictive models for the degradation of different infrastructure components, given their present condition and the past maintenance interventions. In this sense, it is believed that the development of more reliable predictive models may enhance the decision-making processes in maintenance and renewal planning. Railway infrastructure as one of the major transport infrastructures is not an exception. In addition, in our opinion, Bayesian approaches are still overlooked by railway engineers. Since Bayesian approaches combine prior information from past samples or from expert judgment with new data, they constitute a learning mechanism that can be used to update information over time. Different areas have witnessed the development of many applications supported by the Bayesian learning mechanism, from image processing in engineering applications, to public policy in healthcare

applications, for example, up to socio-economic predictive models. Therefore, they seem to be a promising technique for infrastructure management and thus, the present paper intends to explore a hierarchical Bayesian approach to rail track geometry degradation. The structure of the remainder of this paper is as follows: the next section reviews rail track geometry degradation concepts whereas the section ‘Statistical modelling of rail track geometry’ explores the most important contributions to the statistical modelling of rail track geometry degradation. Afterwards, the section ‘Hierarchical Bayesian modelling’ explores the proposed modelling approach and the section ‘Analysis of track geometry degradation’ discusses model results and estimates. In the section ‘Sensitivity analysis’ we conduct several sensitivity tests on the hierarchical Bayesian models, and finally

Instituto Superior Tećnico, Universidade Tećnica de Lisboa, Portugal Corresponding author: Paulo Fonseca Teixeira, CESUR, DECIVIL, Instituto Superior Tećnico, Universidade Tećnica de Lisboa, Av. Rovisco Pais 1049-001 Lisboa, Portugal. Email: [email protected]


Andrade and Teixeira

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Figure 1. Schematic representation of rail track geometry defects: LLL, RLL, LHA, RHA, C, GD and T.

conclusions are drawn and new directions for future research are discussed.

Rail track geometry degradation The degradation of rail track geometry is usually quantified in terms of seven track geometry defects: the left and right longitudinal level defects (LLL and RLL), the left and right horizontal alignment defects (LHA and RHA), cant defects (C), gauge deviations (GD) and track twist (T). These defects are measured using automated measuring systems, typically integrated into inspection vehicles, and saved as signal data. Signal digital processing techniques are then used to derive indicators (for each type of defect) that can support maintenance and renewal decisions as part of planned maintenance or eventually as unplanned maintenance actions. Figure 1 provides a schematic representation of these defects. Many, IMs tend to combine all these defects into a track quality index, which is typically a function of the standard deviations of each defect and/or a train’s permissible speed (as reported in Zhao et al.1 and ElSibaie and Zhang2). Nevertheless, according to the UIC guide on best practices for optimum track geometry durability3, the standard deviation of the short wavelength (3–25 m) longitudinal level defects is still perceived as the crucial indicator for planned maintenance decisions for many European IMs. This can be attributed in our opinion to two reasons: first, the simplicity of the empirical expressions that describe its evolution (which depends on the accumulated

tonnage, usually in million gross tons (MGT)), and second, the fact that it correlates well with the vertical force4,5, which is a proxy for the vertical acceleration felt by the passenger and thus, of ride quality. Rail track geometry defects should be within certain limits according to the European Standard EN 13848-5 which provides limits for each type of defect depending on maximum permissible speed and for the following three main levels.6 1. The immediate action limit (IAL): refers to the value which, if exceeded, requires imposing speed restrictions or immediate correction of track geometry. 2. Intervention limit (IL): refers to the value which, if exceeded, requires corrective maintenance before the immediate action limit is reached. 3. Alert limit (AL): refers to the value which, if exceeded, requires that the track geometry condition is analysed and considered in the regularly planned maintenance operations. Although IAL limits are considered normative, providing the highest admissible limits to ensure safety and ride comfort; IL and AL limits are purely indicative, reflecting common practice among most European IMs. They are even expressed as a range rather than a single value. In fact, the EN 13848-5 also directs each IM to select their own IL and AL limits according to their inspection and maintenance systems, which in turn relate to different targets for safety, ride quality, lower life-cycle costs and track


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Proc IMechE Part F: J Rail and Rapid Transit 227(4)

Table 1. Standard deviation limits of the longitudinal level defects for different train speeds (V) and for the AL quality level according to the EN 13848-5. Standard deviation limits of the longitudinal level defects

AL (mm)

V 4 80 km/h 80 < V 4 120 km/h 120 < V 4 160 km/h 160 < V 4 230 km/h 230 < V 4 300 km/h

2.3–3.0 1.8–2.7 1.4–2.4 1.2–1.9 1.0–1.5

access availability. AL range limits for the standard deviation of longitudinal level defects are listed in Table 1.

Statistical modelling of rail track geometry Statistical modelling of rail track geometry has come a long way in the last 30 years. First published approaches date back to the 1980s. Back then, researchers focused on obtaining quantitative measures, which could be used to describe rail-vehicle performance, mainly through two components: ride comfort and safety (probability of derailment). Corbin and Fazio7 called them performance-based track-quality measures, and used simple linear models to obtain response modes for a given vehicle to given track geometry perturbations. They computed envelopes of allowed profile (longitudinal level defect) deviations for different speeds depending on the spatial frequency (cycles per metre) or on its inverse (i.e. the wavelength in metres). For that purpose, they used the power spectral density of the rail profile (longitudinal level defect) as a statistical representation of rail track geometry, for ride comfort and safety purposes. In parallel, Hamid and Gross8 also discussed the need to objectively quantify track quality for maintenance planning and ride comfort purposes, analysing rail track geometry data collected over a period of a year on approximately 290 miles (467 km) of a mainline track, investigating statistical dependencies on track geometry defects (measured at 1 ft (0.3048 m) intervals using a T-6 vehicle) and some indicators (e.g. mean, 99th percentile, standard deviation and higher-order moments). Moreover, they developed empirical degradation models through linear autoregressive techniques that could describe the relationship between track quality indices (defined as the standard deviation of the profile) and physical parameters. For instance, simple linear regressions were proposed to relate the track quality index with the root mean square of the vertical acceleration. Nevertheless, as the resulting empirical equations contained autoregressive terms (i.e. they included previous values as

explaining variables), these expressions proved unreliable when used to predict track quality for mediumor long-term operation. The development of rail track degradation models to predict future track quality indices for maintenance planning purposes benefited from a major contribution from Bing and Gross9 who used a multiplicative form for their model including as explaining variables: traffic information (equivalent train speed), track structure, maintenance (e.g. time since surfacing) or even ballast index in order to explain the rate of degradation at two consecutive time periods. Another important contribution was made by Hamid and Yang10 who followed an approach based on analytically describing typical variations of track geometry, distinguishing random waviness, periodic behaviours at joints, and isolated variations, which occur occasionally but with regular patterns. In fact, they represented some track geometry defects as stationary random processes (modelled through the power spectral density) and others as periodic processes. In addition to these published works, investigations were carried out in the 1970s by the former Office for Research and Experiments (ORE) in an attempt to understand the fundamentals of the deterioration mechanism and to control this phenomenon. The ORE examined data available from a number of administrations and showed that the factors governing the rate of deterioration were not obvious, and unknown factors in the track were critical in determining both the average quality and the rate of deterioration. Original tests on the rate of deterioration of track geometry were carried out by ORE committee D 117. Although the results were not very conclusive, track quality on being relayed (initial quality) was identified as the most important factor. More measurements on track geometry were recorded and the following qualitative conclusions were drawn.11 After the first initial settlement, both vertical quality and alignment deteriorate linearly with tonnage (or time) between maintenance operations. 1. The rate of deterioration varies drastically from section to section even for apparently identical sections carrying the same traffic. 2. There is no proved effect on the quality and on the rate of deterioration by the type of traffic or track construction. 3. The rate of deterioration appears to be a constant parameter for a section regardless of the quality achieved by the maintenance machine. 4. Tamping machines improve the quality of a section of track to a more or less constant value. The second conclusion emphasizes the importance of modelling degradation at the track section level, whereas the fourth conclusion (together with the second conclusion) makes it clear that the inherent uncertainty of the degradation model parameters



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should be assessed with an associated learning mechanism. In the 1990s, Iyengar and Jaiswal12 started to model rail track irregularities using a non-Gaussian model approach and concluded that in terms of level crossing and peak statistics, the proposed nonGaussian model was consistently better than a Gaussian model in predicting the number of upward level crossings and peaks. Their non-Gaussian model for statistically representing railway track irregularities used a finite series of uncorrelated terms: the first term was a Gaussian process and the remaining terms were derived using a Gram–Schmidt procedure. Iyengar and Jaiswal13 eventually modelled track irregularities (both absolute vertical profile (longitudinal level defects) and unevenness (standard deviations)) as a stationary Gaussian random field so that the classic level-crossing and peak-statistics theory could be exploited to relate the sample deviation to the highest peak value in a simpler way than in their first nonGaussian model. In a way, Iyengar and Jaiswal opted to simplify their initial approach so that they could make probabilistic approaches that would appeal to practicing engineers. Nevertheless, as discussed by Kumar and Stathopoulos14, unevenness data (standard deviations) indicated a non-Gaussian character with skew and kurtoses values significantly different from the typical values for Gaussian processes. More recently, several approaches have been proposed to capture the non-linear characteristics of track quality deterioration15–17, though they typically do not consider all track geometry defects, and instead model a quality index. An important contribution was given in Lopez-Pita et al.18 for the case of a Spanish high-speed rail line; they identified the embankment height as being a dependent variable on the density of maintenance works, though the study analysed the past tamping actions and vertical accelerations rather than the track geometry records themselves. In that sense, that work analyses the ‘outputs’ of maintenance decisions, rather than their inputs (i.e. degradation) so that one may assess whether or not that decision was a good one. Another important issue in statistical modelling of rail track geometry is how to model the tamping/ maintenance recuperation, i.e. the impact of levelling and tamping operations on the rail track geometry defects and their evolution. In fact, specialized literature usually does not cover this improvement in great detail and only a few references have been found, such as Veit17 and Lichtberger.5 Quiroga and Schnieder19 used accumulated tamping interventions as an explaining variable, and showed higher variances for higher numbers of accumulated tamping interventions. However, to the best of our knowledge, current statistical modelling of rail track geometry has overlooked the spatial correlations between degradation rates and initial quality for the standard deviation of longitudinal level defects (3–25 m) and the learning

mechanism from a Bayesian statistical perspective as more inspection data becomes available over time. In fact, this idea follows from an initial exploratory work previously conducted by the two current authors that showed that spatial correlation between deterioration rates and initial quality were statistically significant from a classic statistics perspective.20

Hierarchical Bayesian modelling Having discussed in the previous section the current state of the art of statistical modelling of rail track geometry degradation, this section will explore the proposed hierarchical Bayesian models for rail track geometry degradation. However, let us first discuss Bayesian modelling approaches in general and then focus on hierarchical Bayesian modelling in particular and on the proposed modelling approach for this application on rail track geometry degradation. Bayesian approaches diverge from classic statistics theory in that that they consider parameters as random variables, whose uncertainty can be expressed by a prior distribution. This prior distribution pðÞ is then combined with the traditional likelihood pð yjÞ to obtain the posterior distribution of the parameters of interest. The calculation of the posterior distribution pðj yÞ of the parameters given the observed data y can be computed according to Bayes’ rule as pðjyÞ ¼ R

pðyjÞ pðÞ / pðyjÞ pðÞ pðyj0 Þ pð0 Þd0

A very important step is specifying the prior distribution using a non-informative (or vague prior) or incorporating preceding known information, using old samples hopefully under the same boundary conditions or from expert judgment techniques. For further details on Bayesian statistics, please see, for example, Bernardo21 and Paulino et al.22, or for a more practical approach, Ntzoufras.23 In many realistic applications, the joint posterior distribution pðj yÞ has a reasonably high dimension, and integration through numerical methods must rely on MCMC-based methods; they are built so that their stationary distribution is the desired posterior distribution. Hierarchical Bayesian models, in particular, benefit from a key property: they can be factorized using a directed acyclic graph (DAG) in a convenient way, so that it not only enables arbitrarily complex full probability models to be specified based on the simple local components, but also it makes the identification of full conditional distributions straightforward. Once full conditional distributions are available, we can sequentially sample from them using the Gibbs sampler to sample from the posterior distribution of interest, or any more general sampling method (e.g. adaptive rejection sampling, Metropolis–Hastings algorithm). Note that a DAG is a graph that is


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directed because each node is linked through an arrow (i.e. with a defined direction) and it is acyclic because no cycles are formed with the arrows (i.e. it is not possible to return to a given node once you have left it). Parent nodes from a given node (i.e. parents[v]) are all nodes that are connected through arrows from them to the node v; whereas children nodes from a given node (i.e. children[v]) are all nodes that are connected through arrows from the node v to them. In fact, according to Lunn et al.24 on the WinBUGS use of DAG factorization, let V denote the set of all nodes ðvÞ in a DAG for a given hierarchical Bayesian model, it can be shown that pðj yÞ / pð, yÞ ¼ pðVÞ ¼

Y

pðvjparents½vÞ

v2V

Let V \v denote ‘all elements of V except v’. The full conditional pðvjVnvÞ is then proportional to the product of terms in pðVÞ which contain v pðvjVnvÞ / pðvjparents½vÞ

Y

pðwjparents½wÞ

w2children½v

Note that OpenBUGS – the most recent development of WinBUGS, allows user-friendly inference for these Bayesian hierarchical models, especially if the adaptive rejection sampling algorithm from Gilks and Wild25 is needed. Having introduced the main concept of hierarchical Bayesian models, we are able to start modelling the rail track geometry degradation. At the end of this section, we provide the associated DAG for the more complicated models explored in this paper. However, let us first revise the main assumptions on statistical modelling of rail track geometry degradation that our models will rely on, having in mind the research findings discussed in the previous section. A typical assumption made in the statistical modelling of rail track geometry degradation is to consider that the standard deviation of longitudinal level defects ðysvkl Þ at inspection l for track section k from track segment v from area s is normally distributed with mean msvkl and variance s2 , i.e. ysvkl Nðmsvkl , s2 Þ. Figure 2 should allow the reader to better understand the meaning of indices s, v and k for a typical doubletrack line.

The following assumptions on the mean value result from a combination of factors. 1. A constant linear evolution with accumulated tonnage (Tsvkl – accumulated tonnage since last tamping or renewal operation), given by the deterioration rate - svk , assuming different values for each track section k in track segment v for area s. 2. An initial standard deviation of longitudinal level defects, given by the initial quality - svk , assuming different values for each track section k in track segment v for area s. 3. A disturbance effect ðsv Þ of the initial standard deviation of longitudinal defects after each tamping operation, i.e. it does not recover to its initial value svk , but rather it is affected by a rate 1 þ sv , given by sv , assuming different values for each track segment v for area s. Therefore, note that at each new tamping cycle the initial quality would be svk ð1 þ sv ÞNsvkl , where Nsvkl is the number of tamping operations conducted since the last renewal. 4. Distinction between renewed track sections ðRsvkl ¼ 1Þ and non-renewed track sections ðRsvkl ¼ 0Þ is ensured through the separation of the initial quality and the deterioration rates for a non-renewed track section k from segment v for area s – 0svk and 0svk respectively; whereas the disturbance effect ðsv Þ is considered the same for renewed and non-renewed track sections. Therefore, bearing these assumptions in mind, we may write the mean msvkl as msvkl ¼ svk ð1 þ sv ÞNsvkl þsvk Tsvkl Rsvkl þ 0svk ð1 þ sv ÞNsvkl þ0svk Tsvkl ð1 Rsvkl Þ In this expression, we should regard , , 0 , 0 and as parameters, to which a hierarchical probability structure should be assigned, whereas N, T and R should be regarded as known variables. Figure 3 provides a graphical representation of the intended behaviour of the standard deviation of longitudinal level defects expressed by the presented equation. Of course, we may want to predict the future state of the track to plan a future intervention ðN, RÞ based

Figure 2. Schematic representation of a typical double-track line and indices from area s, segment v and track section k.



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Figure 3. Graphical representation of the track geometry degradation model for a given track section k in segment v in area s.

on observed data and a future usage ðTÞ, and thus a typical transportation demand model can be hierarchically assigned to T and different values for ðN, RÞ to find a reasonable maintenance and renewal plan. Nevertheless, this extension of the present degradation model with a demand model for future usage is beyond the scope of the present paper and it is left for future research. In order to keep the number of parameters to a minimum, i.e. in an attempt to be parsimonious in modelling, a good strategy is considering (Gaussian) conditional autoregressive (CAR) terms. Besag26 showed in 1974 that conditional probability structures can deal with spatial interactions in complex structures. The CAR model is used to model the spatial dependencies in degradation rates and initial qualities for consecutive track sections. For example, take the initial quality svk for a given track section k in segment v in area s, we will consider it as a combination of two additive components: an average value sv and a spatially correlated term "svk so that svk ¼ sv þ "svk . For "svk , we can assign a conditional probability structure such as "svk "svðkÞ , 2 N "svk , 2 =nsvk , P in which "svk ¼ j2N svk "svj =nsvk , N svk denotes the set of track sections which are considered neighbours to track section k (in segment v in area s), and nsvk is the number of neighbours of track section k (in segment v in area s) and finally "svðkÞ is the vector with all components "svk from segment v in area s except the component related to track section k. Two well-known CAR structures were tested as hierarchical structures for , , 0 , 0 : the first-order random walk ðRWð1ÞÞ and the second-order random walk ðRWð2ÞÞ. The first-order random walk ðRWð1ÞÞ is defined by considering as neighbours structure nsvk ¼ 1 for k ¼ 1

and k ¼ Ks , and nsvk ¼ 2 for k ¼2, . . . , Ks 1, and "svk ¼ "svðkþ1Þ for k ¼ 1, "svk ¼ "svðk1Þ þ "svðkþ1Þ =2 for k ¼ 2, . . . , Ks 1, and "svk ¼ "svðk1Þ for k ¼ Ks . For the second-order random walk ðRWð2ÞÞ these expressions become a little more complicated but they can be derived from the following equivalence, where the symbol * denotes the different possible parameters , , 0 , 0 "svk RWð2Þ ,"svk "sv k , 2 8 N 2"svðkþ1Þ "svðkþ2Þ , 2 , > > > > > > k¼1 > > > > > N 2"svðk1Þ þ 4"svðkþ1Þ "svðkþ2Þ =5, 2 =5 , > > > > > > < k ¼ 2 N "svðk2Þ þ 4"svðk1Þ þ 4"svðkþ1Þ > > > "svðkþ2Þ =6, 2 =6 , k ¼ 3, . . . , Ks 2 > > > > > > N "svðk2Þ þ 4"svðk1Þ þ "svðkþ1Þ =5, 2 =5 , > > > > k¼K 1 > > s > > : N " þ 2" , 2 , k ¼ K svðk2Þ

svðk1Þ

s

Moreover, for the disturbance effect of the initial quality after each tamping operation ðsv Þ, we define a typical probability structure expressing vague information on that parameter, i.e. sv N 0, 2 . Finally, for each variance component in each hierarchical structure, we end by assigning inverse gamma distributions to each component, i.e. s2 IGðc0 , d0 Þ, 2 IGðc1 , d1 Þ, 2 IGðc2 , d2 Þ, 2 IGðc3 , d3 Þ, 20 IGðc4 , d4 Þ and 20 IGðc5 , d5 Þ, where IGðc, dÞ denotes an inverse gamma distribution with shape parameter c and scale parameter d, whose density is proportional to xðcþ1Þ expðd=xÞ, x 4 0. As mentioned by one reviewer, the use of inverse gamma prior distributions for the variance terms has


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Figure 4. DAG representation of the proposed hierarchical Bayesian models M2, M3 and M4.

been criticized by Gelman.27 He recommends starting with a non-informative uniform prior density, or when more prior information is desired, working within the half-t family of prior distributions, which are more flexible and have a better behaviour near zero, compared with the inverse-gamma family, which are very sensitive in datasets where the variance terms may assume low values of 2 . However, the choice of assigning inverse gamma distributions ‘is an attempt at non-informativeness within the conditional conjugate family’27, which mainly translates into full conditional posterior distributions for each variance component within the same distributional family, i.e. also inverse gamma distributions, as later seen in equations (1), (3), (5), (7), (9) and (11) in Appendix 1. This choice is not only attractive for pedagogical purposes, but it is also a common choice in many BUGS software applications. Figure 4 provides a DAG representation of the proposed hierarchical Bayesian models, focusing on the three models explored later on in this paper: models M2, M3 and M4. Model M1 has a simple representation without the eight upper nodes (2 and "svk for ¼ , , 0 , 0 ). Stochastic nodes are represented in ovals with a solid line, whereas constants are in rectangular boxes. Hyperparameters (c, d) are not represented for clarity and ovals with dashed contours are logically dependent on their parent nodes and are not stochastic. To derive the joint posterior distribution, prior independence is usually assumed among the model

parameters so that the joint posterior density is then proportional to where msvkl ¼ svk ð1 þ sv ÞNsvkl þsvk Tsvkl Rsvkl þ 0svk ð1 þ sv ÞNsvkl þ0svk Tsvkl ð1 Rsvkl Þ svk ¼ sv þ "svk ,0svk ¼ 0sv þ "0svk svk ¼ sv þ "svk and 0svk ¼ 0sv þ "0svk : In order to ensure that the CAR model structures are identifiable, we follow the typical constraint suggested 28 by P Besag and Kooperberg that is to impose that the k "svk ¼ 0 and use a flat prior for the constant sv on the whole real line. This is also adopted for , 0 and 0 CAR structures. As the joint posterior is rather complex, we derive the full conditional posterior distribution in Appendix 1, so that a Gibbs sampling strategy can iteratively draw for each parameter and use them as current values for each conditional posterior distribution. We explored four models (M1, M2, M3 and M4), whose differences are related to the hierarchical structures for , , 0 and 0 . Model 1 assigns simple flat priors to each parameter sv , sv , 0sv , 0sv , model 2 assigns a flat prior to each parameter sv , sv , 0sv , 0sv and a normal prior with mean equal to zero and variance 2 , 20 , 2 , 20 for each parameter "svk , "svk , "0svk , "0svk , respectively. Model 3 assigns a flat prior to each parameter sv , sv , 0sv , 0sv and a first-order random walk for each parameter



371

"svk , "svk , "0svk , "0svk and model 4 assigns a flat prior to each parameter sv , sv , 0sv , 0sv and a second-order random walk to each parameter "svk , "svk , "0svk , "0svk . For all models M1, M2, M3 and M4, inverse gamma priors IGð0:5, 0:0005Þ were assigned for each variance parameter (s2 , 2 , 20 , 2 , 20 and 2 ). Note that flat priors are improper distributions (i.e. do not integrate to one, but assume a constant value everywhere, attempting to describe vague or no prior information on that parameter). In terms of simulation details, MCMC samples were of size 20,000, taking every tenth iteration (thin ¼ 10) of the simulated sequence, after 10,000 iterations of burn-in period, for the four models. Initial values were set as: the variance terms s2 , 2 , 2 , 20 , 20 and 2 equal to 10 (which is equivalent to a precision (1= 2 ) equal to 0.1), the spatially correlated terms "svk , "svk , "0svk and "0svk equal to zero, and finally, each parameter sv , sv , 0sv , 0sv and sv equal to zero. To have a reasonable confidence on MCMC convergence, we ran an alternative MCMC simulation for every model using different initial values, and found very similar results for the posterior estimates. We also used the diagnostic convergence tests from the BOA program29, which did not reveal any evidence against convergence for all MCMC outputs. In terms of model comparison, hierarchical Bayesian models tend to rely on a standard measure proposed by Spiegelhalter et al.30 for comparing models of any degree of complexity. In fact, the deviance information criterion (DIC), whose assessment is straightforward using MCMC inference methods, balances two components: the expected posterior

deviance (i.e. the goodness of fit) and the effective number of parameters (i.e. the complexity of the model). It is defined as: DIC ¼ 2DðÞ D , where DðÞ is the posterior mean of the deviance and is the posterior mean of the model parameters. In practical terms, a lower DIC value means a better model in terms of relative comparison. From Table 2, providing a comparison of the four models based on the DIC, we can observe that model M3 with first-order random walk CAR structure has the lowest DIC value among the four models explored. However, note that the DIC for model M4, which has secondorder random walk structures, is not so distant from the DIC value for model M3.

Analysis of track geometry degradation This section will analyse some results obtained from the application of the proposed hierarchical Bayesian models explored in the previous section to a sample of historical data obtained for the main Portuguese line (Lisbon–Oporto). This historical data mainly refers to: . the inspection records from the EM 120 vehicle to get the standard deviation of longitudinal level defects with respect to 200 m long track sections ðysvkl Þ; . the operation records to get the accumulated tonnage ðTsvkl Þ; . the maintenance records to get the past maintenance and renewal actions ðNsvkl , Rsvkl Þ. Since the motivation of this paper is the hierarchical Bayesian model itself rather than a comprehensive description of Lisbon–Oporto line inspection, operation and maintenance past experience, we decided to focus on a particular exemplifying area. It mainly involves a double-track ðVs ¼ 2Þ area of about 15 km ðKs ¼ 74Þ, from a total of 36 inspections ðLs ¼ 36Þ from February 2001 up to October 2009. Table 3 provides estimates of the posterior parameters for this example track segment. Regarding Table 3, note that both the parameters related to renewed track sections, i.e. initial quality sv and deterioration rate sv , present very close values,

Table 2. Comparison of the DIC for different models. Models defined from svk ¼ svk , svk , 0svk , 0svk M1: M2: M3: M4:

svk svk svk svk

DIC 7192 2794 3274 3234

¼ sv ¼ sv þ "svk ; "svk Nð0, 2 Þ ¼ sv þ "svk ; "svk RWð1Þ ¼ sv þ "svk ; "svk RWð2Þ

Note: The symbol represents the parameters , , 0 , 0 .

Table 3. Estimates of the posterior parameters for an example track segment (s ¼ 1, v ¼ 1). sv (mm)

sv (mm/100 MGT)

0sv (mm)

0sv (mm/100MGT)

sv

Model

Mean

s.d.

Mean

s.d.

Mean

s.d.

Mean

s.d.

Mean

s.d.

M1 M2 M3 M4

0.3407 0.3065 0.3102 0.3034

0.020 0.024 0.012 0.013

1.203 1.469 1.460 1.499

0.143 0.133 0.081 0.090

1.787 1.379 1.381 1.375

0.038 0.067 0.015 0.016

1.680 6.406 6.247 6.169

0.329 0.488 0.179 0.170

0.1016 0.0035 0.0015 0.0055

0.019 0.008 0.009 0.009

Note: s.d. ¼ standard deviation.


372


especially for models M2, M3 and M4. This is also true for the parameters related to non-renewed track sections (0sv and 0sv Þ, though for model M1 values differ considerably, particularly for the deterioration rate. Regarding the disturbance effect ðsv Þ, the standard deviations associated with its estimates are high compared with its mean values for models M2, M3 and M4. Moreover, note that all values for the disturbance effect seem to be very close to zero. Also, note the deterioration rates for non-renewed track sections are (on average) at least four times higher than the deterioration rates for renewed track sections for models M2, M3 and M4, and only 40% higher for model M1; whereas the initial quality for nonrenewed track sections seem to be at least four times higher than for the renewed track sections (on average) for models M2, M3 and M4, and five times higher for model M1.

gamma priors; whereas 2 and 20 seem to be very sensitive for priors D and G and 2 seems to vary considerably for all priors with associated standard deviation higher than its mean values, which indicates that priors are conveying a lot of information to the variance of the disturbance effect - 2 . We also conducted a sensitivity analysis on model selection, i.e. we analysed if different priors would affect the selection of model 3 as the best model based on the DIC. Table 5 explores the sensitivity of DIC for different models using different priors. For this analysis, we decided to leave out model M1 because its DIC value for prior A was high compared with the others. The last column of the table presents

Table 5. Sensitivity analysis of the model selection based on DIC for different inverse gamma priors.

Sensitivity analysis To analyse the influence of the priors’ specifications, a sensitivity analysis is conducted in this section, assuming different inverse gamma priors IGðc, d Þ. For this purpose, we followed the same experimental design from a similar investigation contained in Silva et al.31 on Bayesian hierarchical models to analyse revascularization odds, using only inverse gamma distributions but different combinations: ðc, dÞ ¼ ð0:5, 0:0005Þ, ð0:001, 0:001Þ, ð0:01, 0:01Þ, ð0:1, 0:1Þ, ð2, 0:001Þ, ð0:2, 0:0004Þ andð10, 0:25Þ, which are denoted as A, B, C, D, E, F and G, respectively. As noted in Silva et al.31, priors C and D (with c ¼ d) are variants of prior B with a larger associated dispersion (compared with B) in increasing order, whereas prior E and F correspond to distributions with the same mode ðd=ðc þ 1ÞÞ as prior A, but with lower (E) and larger (F) dispersion. Prior G is a less vague prior from this experimental design set. Table 4 provides some estimates for model M3 of the variances s2 , 2 , 20 , 2 , 20 and 2 for different parameters and the sensitivity of these estimates using different priors. Note that the mean values for s2 , 2 and 20 are not very sensitive to different inverse

Prior

Model

DIC

A

M1 M2 M3 M4 M2 M3 M4 M2 M3 M4 M2 M3 M4 M2 M3 M4 M2 M3 M4 M2

7192 2794 3274 3234 2796 3337 3230 2790 3106 3107 2775 3275 3154 2794 3083 3217 2795 3232 3189 2779

B

C

D

E

F

G

Selected model M3

M3

M4

M3

M4

M3

M3

Table 4. Estimates of the spatial variance components based on model M3 with different inverse gamma priors. s2 (mm2)

2 (mm2)

20 (mm2)

2 ((mm/100 MGT)2)

20 ((mm/100 MGT)2)

2

Mean

s.d.

Mean

s.d.

Mean

s.d.

0.2379 0.2489 0.4299 1.9770 0.2432 0.2358 3.6110

21.950 23.100 24.610 42.070 21.610 22.380 57.510

3.609 3.758 4.025 5.544 3.565 3.592 6.801

0.0304 0.1732 0.2596 0.7139 0.0121 0.0947 0.0274

0.105 0.787 4.513 6.716 0.013 0.725 0.010

Prior Mean

s.d.

Mean

s.d.

Mean

s.d.

A B C D E F G

0.00141 0.00069 0.00199 0.00067 0.00241 0.00073 0.00067

0.0531 0.0522 0.0454 0.0429 0.0506 0.0535 0.0360

0.00816 0.00818 0.00729 0.00723 0.00833 0.00856 0.00569

0.3478 0.3679 0.3551 0.3618 0.3522 0.3535 0.3048

0.04848 1.4520 0.04834 1.6220 0.04917 3.3140 0.04591 16.380 0.04694 1.5470 0.04529 1.4480 0.03832 32.700

0.0323 0.0318 0.0327 0.0312 0.0326 0.0318 0.0316



373

the selected model according to the lowest DIC for each prior. Regarding Table 5, note that model M3 seems to be the best model for priors A, B, D, F and G; whereas model M4 seems to present the lowest DIC value for priors C and E. Therefore, the selection between models M3 and M4 is sensitive to the prior used for the variance components. Apart from prior C, for which the DIC values are almost the same, it seems that model M4 has the lowest DIC value for more vague priors in the experimental set used, and model M3 has the lowest DIC value for less vague (more precise or more informative) priors.

Conclusions and further research This paper has put forward a statistical modelling approach for rail track geometry degradation based on Bayesian hierarchical models. Its main purpose was to alert railway researchers and practitioners to the need to start modelling spatial dependencies in rail track geometry degradation. From the models explored, the model where the deterioration rates and the initial quality for consecutive track sections follow random walk priors, capturing the spatial correlation between them, proved to be the best model based on the DIC value. This model is parsimonious in its complexity, while preserving reliability of predicted results. In addition, we have provided an inference method through the derivation of the full conditional posterior distribution in Appendix 1. For further research, we would like to emphasize that we opt to use non-informative (or vague) priors to test the sensitivity of the model, though it can be enhanced by introducing more informative priors from past samples or expert judgments using elicitation techniques. Moreover, another unexplored direction is the use of more complicated correlation structures than the first- and second-order random walks. In fact, other spatial formulations, using, for example, power exponential functions to model the decline of correlation in terms of the distance between track sections, may enhance the description of the spatial correlation between degradation models. Furthermore, we believe that for practical purposes (similar to what happens in other application areas) to support maintenance decision processes at the tactical decision level, mapping deterioration rates at the network level for each pair (s,v) may provide a good understanding of degradation at the railway network level. Finally, another future direction worth pursuing would be to try to let the model comparison be conducted for each area s, and then select as an overall predictive model the combination of all models selected at a particular s, rather than running model selection for all the areas. In that sense, we would be letting the overall prediction model select which model (i.e. M3 or M4 for example), it would use for each area s.

Funding This work was financially supported by the Portuguese Foundation for Science and Technology (FCT) (project reference PTDC/SEN-TRA/112975/2009) and MIT Portugal (FCT PhD grant SFRH/BD/33785/2009).

Acknowledgements The authors would like to thank the support and collaboration of the Portuguese Railway Infrastructure Manager, REFER, E.P.E. We are also grateful to Professor Carlos Paulino for letting the first author attend his classes on Bayesian statistics and to Professor Giovani Silva for his suggestions and comments on a previous version of the manuscript. Needless to say, that any errors or lapses are totally our responsibility. Finally, we thank the anonymous reviewers for their comments and suggestions which improved the readability of the final version.

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we can derive the full conditional posterior distributions (denoted below by jhj ), which are given by ! 1 1X s2 hs2 IG c0 þ Vs Ks Ls , d0 þ ðysvkl msvkl Þ2 , 2 2 v,k,l s ¼ 1, . . . , S

ð1Þ sv jhsv

! 1 2 1 X 2 / exp 2 sv 2 ðysvkl msvkl Þ , 2s k,l 2

s ¼ 1, . . . , S,

v ¼ 1, . . . , Vs ð2Þ

2 h2

1X 1X 2 IG c1 þ V s , d1 þ 2 s 2 s,v sv

! ð3Þ

2 1 nsvk "svk h"svk / exp 2 "svk "svk 2 2s 2 ! X ðysvkl msvkl Þ2 , s ¼ 1, . . . , S, v ¼ 1, . . . , Vs , s,v,k,l

v ¼ 1, . . . , Vs ð4Þ

2 h2

2 1X 1X IG c2 þ Vs Ks ,d2 þ nsvk "svk "svk 2 s 2 s,v,k

!

ð5Þ 2 1 nsvk "svk h"svk / exp 2 "svk "svk 2 2s 2 ! X ðysvkl msvkl Þ2 , s ¼ 1, . . . , S, v ¼ 1, . . . , Vs s,v,k,l

v ¼ 1, . . . , Vs ð6Þ

2 h2

2 1X 1X IG c2 þ Vs Ks , d2 þ nsvk "svk "svk 2 s 2 s,v,k

!

ð7Þ 2 nsvk

"0svk h"0 / exp 2 "0svk "0svk svk 20 ! X 1 2 ðysvkl msvkl Þ , s ¼ 1, . . . , S, 2 2s s,v,k,l

ð8Þ

v ¼ 1, . . . , Vs v ¼ 1, . . . , Vs

Appendix 1 Let h be the vector of the model parameters, with elements s2 , sv , 2 , "svk , 2 , "svk ,2 , "0svk , 20 , "0svk , 20 , sv , sv , 0sv and 0sv , with s ¼ 1, . . . , S, v ¼ 1, . . . , Vs , k ¼ 1, . . . , Ks . From the joint posterior,

1X 20 h20 IG c2 þ Vs Ks , d2 2 s

2 1X þ nsvk "0svk "0svk 2 s,v,k


ð9Þ


375

2 nsvk

"0svk h"0 / exp 2 "0svk "0svk svk 20 ! 1 X 2 2 ðysvkl msvkl Þ , s ¼ 1, . . . , S, 2s s,v,k,l

! X ðysvkl msvkl Þ2 P½sv sv hsv / exp k,l

ð10Þ

ð13Þ

s ¼ 1, . . . , S, v ¼ 1, . . . , Vs ! X 2 0sv h0sv / exp ðysvkl msvkl Þ P 0sv

v ¼ 1, . . . , Vs , v ¼ 1, . . . , Vs

k,l

1X 20 h20 IG c2 þ Vs Ks , d2 2 s !

2 1X þ nsvk "0svk "0svk 2 s,v,k

s ¼ 1, . . . , S, v ¼ 1, . . . , Vs ð14Þ ð11Þ

0sv h0sv

k,l

!

sv jhsv / exp

! X 2 / exp ðysvkl msvkl Þ P 0sv ,

s ¼ 1, . . . , S, v ¼ 1, . . . , Vs

X ðysvkl msvkl Þ2 P½sv ,

ð15Þ

k,l

s ¼ 1, . . . , S, v ¼ 1, . . . , Vs ð12Þ
