See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/333149747 Predicting rail defect frequency: An integrated approach using fatigue modeling and data analytics Article in Computer-Aided Civil and Infrastructure Engineering · May 2019 DOI: 10.1111/mice.12453 CITATIONS 0 READS 98 5 authors, including: Some of the authors of this publication are also working on these related projects: Viscoplasticity of polycrystalline solids under high strain rates View project Travel Mode Identification with Smartphone Sensors View project Faeze Ghofrani University at Buffalo, The State University of New York 5 PUBLICATIONS 32 CITATIONS SEE PROFILE Abhishek Pathak University at Buffalo, The State University of New York 5 PUBLICATIONS 7 CITATIONS SEE PROFILE Qing He University at Buffalo, The State University of New York 47 PUBLICATIONS 664 CITATIONS SEE PROFILE All content following this page was uploaded by Faeze Ghofrani on 10 June 2019. The user has requested enhancement of the downloaded file.
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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/333149747
Predicting rail defect frequency: An integrated approach using fatigue
modeling and data analytics
Article in Computer-Aided Civil and Infrastructure Engineering · May 2019
DOI: 10.1111/mice.12453
CITATIONS
0READS
98
5 authors, including:
Some of the authors of this publication are also working on these related projects:
Viscoplasticity of polycrystalline solids under high strain rates View project
Travel Mode Identification with Smartphone Sensors View project
Faeze Ghofrani
University at Buffalo, The State University of New York
5 PUBLICATIONS 32 CITATIONS
SEE PROFILE
Abhishek Pathak
University at Buffalo, The State University of New York
5 PUBLICATIONS 7 CITATIONS
SEE PROFILE
Qing He
University at Buffalo, The State University of New York
47 PUBLICATIONS 664 CITATIONS
SEE PROFILE
All content following this page was uploaded by Faeze Ghofrani on 10 June 2019.
The user has requested enhancement of the downloaded file.
Paris law can be converted to the following equations giving
number of cycles for a particular growth in size:
𝑁 =(𝑎𝑐
1−𝑚∕2 − 𝑎01−𝑚∕2)
𝐶 ⋅ (1 − 𝑚∕2)⋅(Δ𝑆eff
)−𝑚(2)
Here, 𝑎𝑐 is the final crack size and 𝑎0 is the initial crack
size (in meters). Furthermore, this number of cycles for crack
growth can be converted to equivalent accumulated traffic
load (MGT), by multiplying it by the load from each wheel.
This methodology is used to perform crack propagation stud-
ied with several initial crack sizes and a lookup table was gen-
erated. We chose seven different crack sizes starting from 0.5
to 3.5 mm in the increment of 0.5 mm. The data obtained from
one particular crack size are used to propagate crack from its
initial value to the next higher size of the crack for which sim-
ulation has been performed, for example, from 2.5 to 3.0 mm.
The stress intensity factors are updated as soon as the crack
grows to reach a level where new data are available. Follow-
ing this procedure, any initial crack can be propagated to any
other higher crack size by using relevant stress intensity fac-
tors as it grows. These data are consolidated in a lookup table
that acts as input for the further statistical analysis and is pre-
sented in Table 1.
According to existing railroad practice, we assume the min-
imum crack size to be detected by ultrasonic device is 3.5 mm
(Lanza di Scalea et al., 2005) around 5% of the railhead cross-
section. Once detected, a crack shall be labeled as a rail defect.
3.2 Bayesian inferenceTo calculate the posterior distribution of rail defects based
on the recorded data, the statement of Bayes’ Theorem that
describes the conditional probability of a parameter 𝜃 based
on another parameter D could be used as follows:
𝑝 (𝜃|𝐷) = 𝑝 (𝐷|𝜃) 𝑝 (𝜃)𝑝 (𝐷)
(3)
GHOFRANI ET AL. 7
In Equation (3), 𝑝(𝜃|𝐷) refers to the posterior, 𝑝(𝐷|𝜃)denotes the likelihood, p(𝜃) the prior, and p(D) the evidence
(which is also referred to as the prior predictive probability of
the data).
Any characteristics associated with rail defects that we wish
to model (defect size, its occurrence, etc.) could be treated as
𝜃 and the relevant data recorded (defect size, occurrence time,
respectively) as D. In most of the real-life problems, 𝑝(𝐷) acts
only as a normalizing factor and is usually ignored in the anal-
ysis (Sunnåker et al., 2013). If we choose to go by analytically
tractable route, some information about prior distribution 𝑝(𝜃)is needed as well as the calculation of the likelihood 𝑝(𝐷|𝜃).However, barring some very simple problems, it is generally
computationally expensive to evaluate the likelihood. ABC is
a method that gives posterior distribution of any parameter
without having to calculate the likelihood. Using computa-
tional efficiency of modern day simulations, ABC framework
circumvents the need of calculating likelihood by the compar-
ison between simulation data and recorded data. This method
has been successfully used in the field of biology, specifi-
cally population genetics, where the problem of large data sets
and many parameters makes it unfeasible for using analyti-
The final processed data set includes the unique ID, year,
average tonnage, frequency of rail and geometry defects,
inspection frequency, and grinding presence (output in
Figure 5) for 9,780 rail segments with more than 52,000
defects during 6 years. These are all the input for the modeling
procedure.
4.2 Integration of mechanistic and statisticalmodel for a US Class I RailroadTo conduct the integration of the mechanistic and statistical
model, the ABC framework is used. The aim is to find the
optimal parameter values that achieve the minimum error for
predicting the number of defects occurring in each segment
given a 6-year period of relevant data, including the observed
number of defects.
It is worth mentioning that we do not have any informa-
tion about the cracks appearing inside the rail. Therefore, we
take advantage of simulation for crack emergence inside the
rail and then we move forward in our simulation to see which
of those cracks are supposed to become defects according to
FEM output.
Before we jump into the steps of the main algorithm, we
need to define three functions named FE, G, and POSTE-RIOR, each of which is used inside the previous one, respec-
tively, and all are used in main algorithm.
Function FE: This function gets the initial crack size as
input and returns the required MGT for that crack to grow
as large as 3.5 mm (to be detected as a defect) according
to Table 1 (output of FEM). As an example, according to
Table 1, FE (0.5) and FE (1) are equal to 196.5 and 123.4
MGT, respectively.
Function G: This function is used to simulate number of
defects for each segment (shown with index p), given the
information of that segment). The general steps of Function
G are provided in the flowchart in Figure 6. According to this
figure, Function G gets the information of each segment as
an input to the function. For each segment, number of cracks
for each segment (which is unknown) is drawn from a Pois-
son distribution with parameter 𝜆p, considering the Poisson
distribution for the case of arrival rate is very common in the
literature (Tonge & Ramesh, 2016).
As mentioned before, the minimum crack size required
to be detected as a defect by ultrasound is assumed to be
3.5 mm. In this essence, the size of each crack is considered
to be drawn from a uniform distribution with lower bound
0.5 and upper bound 5 to first make sure that the size range
GHOFRANI ET AL. 9
The function gets segment information as
input
It draws number of cracks for each
segment (λp) from a Poisson distribution
It draws size of each crack from a discrete uniform distribution
It checks if the drawn cracks would turn into a defect (using MGTp and
FE function)
It calculates number of cracks that turn into
defects
The function returns simulated number of
defects for each segment
F I G U R E 6 General steps in Function G for calculating the simulated number of defects for each segment
is adequate, and to cover the 3.5 mm in the range, secondly.
Having the MGT of each segment in each year and using
the FE function, Function G checks whether each of the
drawn cracks would turn into a defect (become 3.5 mm or
more in size) after a certain time (year) or not. The function
ultimately returns the simulated number of defects for each
segment. More details on the implementation of the steps of
Function G are given in pseudocode provided in Algorithm
1. In this pseudocode, T is the time in years, n_cracks defines
number of cracks, n_defects denotes number of defects, and
MGTP accounts for the annual MGT for segment p.
Algorithm 1. Simulating number of defects
Function G (p) # given the data related to segment p, simulate the number of defects
define cracks as a list of size Tdefine n_defects as a list of size T initialized with 0 For t in 1 to T do
if t>1 for crack in cracks[t-1] do
if t*MGTp > FE(crack) n_defects[t] = n_defects[t] + 1 remove crack from cracks[t-1]
endend
endn_cracks ~ Poisson (λp) For i in 1 to n_cracks do
cracks[t][i] ~ DiscreteUniform(0.5, 5) if MGTp > FE(cracks[t][i]) or cracks[t][i] >= 3.5
n_defects[t] = n_defects + 1 end
endend
end
Function POSTERIOR: This function applies the logic of
the ABC framework to estimate the posterior distribution of
cracks arrival (𝜆) for each segment given observed number
of defects and cumulative MGT. 𝜆p is the Poisson distribu-
tion parameter for segment p. A schematic of what happens
in POSTERIOR function is shown in Figure 7.
Since we started with no data on past experience of
internal rail cracks, we assume noninformative or uniform
prior distribution for parameter of the model 𝜆 (Chatterjee &
Modarres, 2012). In this function, 𝜆 is assumed to be drawn
from a prior uniform distribution with 0 and 10 lower and
upper bounds, respectively. Considering the fact that the
average rate of defects for each segment in our study is almost
one defect per mile per year, applying a discrete uniform
distribution of crack rate with lower and upper bounds 0–10,
is conservative enough for the purpose of our simulation.
We run a series of M simulations (M = 1,000 in our study)
for each segment by drawing parameter values from the prior
distribution. For each simulation, we use Function G to cal-
culate the simulated number of defects and then the dis-
tance between the simulated and observed number of defects
is computed. According to the ABC framework, the simu-
lation runs with distances over a threshold (є) are rejected,
where є is set as the 90th percentile of all distances. In other
words, only the 10% of the lowest distances of the simula-
tion runs are kept for each segment. The mean of the distribu-
tion of 𝜆 for those kept simulations are proposed as the poste-
rior distribution of the parameters of the model. The selected
𝜆 distribution based on the rejection algorithm for two sam-
ple segments of our study is provided in Figure 8. More
algorithmic details for POSTERIOR function are provided in
Algorithm 2.
Algorithm 2. Calculating posterior distribution of 𝜆
Main algorithm: After defining each of the mentioned
functions, we can now explain the details on the steps of the
main designed algorithm. To do so, we first introduce notation
of the parameters and variables of the model as follows:
10 GHOFRANI ET AL.
λ1 λ2 λ3 … θ
n_defects1
Prior distribution of the model parameter, number of cracks (λ): assumed as discrete uniform distribution
n_defects
Observa�onal data
1. Summary statistic (
5. The posterior distribution of λ isapproximated using the distribution of parameter values λ, of accepted simulations
n_defects) from observational data
2. n
ˆ
simulations are performed by drawing parameter values from the prior distribution for each segment
3. The summary statistic(n_defect) is computed for each
(n_defectsi,n_defectsi ≤ ε))
simulation using Function G
4. Based on the distance and a tolerance, we decide for a simulation whether its summary statistics to be kept or to be rejected (considering the closeness of prediction to observed data) ( Posterior distribution
of model parameter λ
Simulation 1 Simulation 2 Simulation 3 Simulation n
n_defects2 n_defects
3n_defects
n
F I G U R E 7 General steps in Function POSTERIOR for calculating the simulated number of defects for each segment
F I G U R E 8 Posterior distribution of 𝜆 for two sample segments
M: Number of trials
T: Total time in years
K: Number of folds
MGTp: Annual MGT for segment pWeightp: Rail weight for segment p
Speedp: Speed limit for segment pGeo_Defp: Number of geometry defects per year for
segment pInspectionp: Frequency of inspection in each year for
segment p
GHOFRANI ET AL. 11
Grindingp: Frequency of grinding in each year for segment pn_defectsp: Array of length T storing the number of defects
per year for segment pDISTANCE(𝑛defect, 𝑛d𝑒𝑓𝑒𝑐𝑡)∶ Euclidean distance between
observed number of defects (𝑛defect) and simulated number
of defects (𝑛d𝑒𝑓𝑒𝑐𝑡)
The general steps of the main algorithm are given in
Figure 9 and more details are provided in Algorithm 3.
Algorithm 3. ABC framework
To conduct the main algorithm, we first divide the data set
into three folds for the purpose of threefold cross-validation.
In this essence, each fold would serve as training data set twice
and as test data set once. Given the real number of defects
per mile and MGT of each segment in training data set, the
number of cracks per mile (𝜆p) for each segment is estimated
using POSTERIOR function. The variables vector is set as
Xp, which includes six variables: (a) the average annual MGT
of the segments (MGTp), (b) weight of the rail in each segment
(Weightp), (c) freight speed limit in the segment (Speedp),
(d) number of geometry defects per year for each segment
(Geo_Defp), (e) frequency of inspection in each year for each
segment (Inspectionp), and (f) presence of grinding in each
year for each segment (Grindingp).
These variables are required to set a log-regression model
on the training data set so that we would achieve the coef-
ficient estimates for each of these variables. By fitting the
log-regression model and finding its coefficients, 𝜆p could be
predicted for the test data set. The predicted 𝜆ps are used in
Function G to predict number of defects in each segment of
the data set. The mentioned functions and the simulation runs
have been all undertaken using Python 3. Later, we compare
the predicted values with the observed data to check the vali-
dation of the model.
5 RESULTS AND FINDINGS
As mentioned before, we have conducted our proposed
approach for data collected from a Class I US Railroad
between 2011 and 2016.
The prediction accuracy of the model is evaluated by
comparing the predicted (𝑛_𝑑𝑒𝑓𝑒𝑐𝑡𝑠) and actual values
(n_defects) for number of defects in the test data sets (by three-
fold cross-validation as explained before), using two mea-
surements: the Mean Absolute Error (MAE) and Root Mean
Square Error (RMSE) which are formulated as:
MAE =
∑𝑛
𝑖=1|||𝑛𝑑𝑒𝑓𝑒𝑐𝑡𝑠 − 𝑛defects
|||𝑁
(5)
RMSE =
√√√√√∑𝑛
𝑖=1
(𝑛𝑑𝑒𝑓𝑒𝑐𝑡𝑠𝑖
− 𝑛_defect𝑠𝑖)2
𝑁(6)
where N is the number of segments in test data set,
𝑛𝑑𝑒𝑓𝑒𝑐𝑡𝑠𝑖
and 𝑛defects refer to the predicted and observed num-
ber of defects for ith segment in test data set, respec-
tively. These two measurements are among the most popular
and most common validation measurements when predicting
numeric variables (Willmott, 1982).
A discussion on the superiority of MAE over RMSE is
presented in Willmott and Matsuura (2005). Since RMSE is
based on the sum of the squared error, it does not describe
average error alone, tends to become increasingly larger than
MAE and the interpretation tends to be more difficult. In this
essence, MAE is usually a far better measure of error (MOE)
compared to RMSE and even other MOEs.
We have compared the results of our model with Negative
Binomial (NB) model which is a well-known traditional
statistical model for frequency prediction (Washington,
Karlaftis, & Mannering, 2010). The results of both models
are provided in Table 2.
The average predicted number of defects and real num-
ber of defects are shown in Table 2. As understood from this
table, considering MAE and RMSE, the proposed method
is an improvement of 20% and 16%, respectively, over the
NB regression model for predicting the expected number of
defects on each segment.
12 GHOFRANI ET AL.
Partition the data set into train and test sets
For the segments in train set, the posterior
distribution of λ is calculated (using
POSTERIOR Function)
Fitting a log-linear regression model on
train data segments, the coefficients of each
variable is determined
The computed variable coefficients are used to predict λ for test data set
Simulated number of defects are computed
using Function G
The average difference between simulated
defects and observed defects of all segments
are calulated as the error metric
F I G U R E 9 General steps in the main algorithm for predicting number of defects
T A B L E 2 Results of the proposed model compared to the results
of the negative binomial model
ItemProposedmodel
Negative binomialmodel
Average predicted no.
of defects (annual
per mile)
0.85 0.88
Average real no. of
defects (annual per
mile)
0.89 0.89
MAE 0.68 0.85
RMSE 1.11 1.32
Number of segments
in test data set
3,260
0
20
40
60
80
100
120
125
150
175
110
0112
5115
0117
5120
0122
5125
0127
5130
0132
51
Abs
olut
e E
rror
Segment Number
Proposed_Model NB_Model
F I G U R E 1 0 Distribution of the absolute error by the proposed
model as well as NB model on segments of test data set
The distribution of average error of prediction for segments
of the test data set using both NB and our proposed model is
presented in Figure 10.
As seen in Figure 10, for almost all of the samples in test
data set, the MAE of the proposed model is less than that of
the NB model. This is justifiable as the proposed methodol-
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How to cite this article: Ghofrani F, Pathak A,
Mohammadi R, Aref A, He Q. Predicting rail defect fre-
quency: An integrated approach using fatigue modeling
and data analytics. Comput Aided Civ Inf. 2019;1–15.