Download Modelling of road deterioration including lifetime distribution and the Cox model

MODELING OF ROAD DETERIORATION EVOLUTION WITH
LIFETIME DISTRIBUTION AND COX MODEL
E. Courilleau - Laboratoire Central des Ponts et Chaussées (France)
Abstract
Maintenance program is necessary for optimizing national road network surveys. The
maintenance cost could be minimized by the knowledge of the road future state. For
this purpose, evolution of different road deterioration characteristics should be
determined. Surveys of some test roads had been made during periods of five to
thirteen years. Only transverse cracks on semi-rigid pavements were considered in
this paper. Lifetime distributions were used to determine deterioration evolution laws.
Data correspond to a random censoring with left, right and interval censoring.
LIFEREG SAS procedure was used to select the best probability law among those
available. Some explanatory variables were used to improve the model. The non
parametric Cox model was also used with the PHREG SAS procedure. A new
approach was developped to handle left and right censoring with Cox model. To
choose the best model, a comparison of the results was made. Confidence intervals
of the retained model were determined from law parameters.
1.
INTRODUCTION
Roads deteriorate under traffic and environment actions. Without an adequate
maintenance, user's security will no longer be insured. In France, road maintenance
is planned yearly. During a network survey, several road state indexes like transverse
cracks, longitudinal cracks, etc, are recorded. Maintenance works are then planned
with priority allowed to roads subjected to heavy traffic or to deteriorated roads.
Optimization should be done due to limited funds. Part of planned works will be
realized, the remainder being deferred to one or several years later. In order to
evaluate the consequence of deferred works on road state, one needs to predict the
evolution of road deterioration characteristics, called indexes, by means of models.
Optimum planning will be then possible as shown in figure 1.
survey year 0
indexes
roads evaluation
yearly programming
program
indexes evolution
laws
laws of works effects
on indexes
nothing
works
program
year1
year2
year3
...
Figure 1 : Maintenance planning for several years
Indexes are recorded at year 0 for the overall road network. Evolution models
allow then to determine values of the indexes at year 1, on sections which were not
subjected to maintenance. This process will be applied several times to simulate
network aging. Comparison of different maintenance strategies by means of
simulations allows to select the optimum strategy with minimum cost.
Several statistical models were used to construct evolution models. This paper
presents two types of models : the survival laws method and the Cox model. Steps to
obtain evolution models will be first described.
2.
STEPS TO CONSTRUCT EVOLUTION MODELS
To illustrate our purpose, transverse cracks on semi-rigid pavements are
studied. Transverse cracking rate is defined as :
τ=
in which
N * 50
* 100
L * 21
(Eq. 1)
τ = transverse cracking rate;
N : number of cracks;
L : sub-section length, expressed in meters.
Before transverse cracking rate evolution could be modelled, a study of
cracking mecanism should be done to identify factors having an influence on
cracking apparition and its subsequent development. Three factors, which will be
named explanatory variables, were selected to model cracking evolution :
- material nature, noted mat,
- traffic, noted traf,
- equivalent thickness, noted theq, calculated with Odemark [1] formula. This
variable depends on layers thickness and their nature.
Database SISERD (Saisie In Situ et Exploitation des Relevés de Dégradation)
was used to test different statistical models. More than two hundred roads test
sections of the national network have been surveyed over periods ranging from five
to thirteen years. Section lengths vary from five hundred to one thousand meters.
Each section is cut into sub-sections of fifty meters long from which the following
information could be obtained :
- thickness and nature of different layers road,
- construction date,
- traffic in 1975, 1980, 1985 and 1990,
- deteriorations (cracking, crazing, etc.).
Study has been restricted to semi-rigid sub-sections without maintenance.
More than 3900 records were done, several could be done on the same sub-section.
In order to construct evolution models, robustness notion has been introduced.
Firstly, group is defined as a set of sub-sections whose values of explanatory
variables are the same. Within a group, a sub-section is considered as weaker than
another if its transverse cracking rate is higher. Sub-section belonging to the r%
weakest of a group is supposed to stay in this r% weakest during its life. This
assumption has been verified for 80% of database sub-sections. The parameter r,
referred to by robustness, translates the notion of more or less well constructed subsections. An example of evolution model is shown in figure 2.
30
r=0.4
Cracking rate
25
20
15
10
5
0
0
5
10
15
20
Age (years)
Figure 2 : Evolution curve
On a given evolution curve, robustness r is constant. To obtain evolution
curve, the probability of exceeding different transverse cracking rates (0%, 5%, etc.)
has been calculated in function of age. Network curves are then built (Fig. 3).
Si ( ti , wi )
1
Probability
0.8
0.6
0.4
0.2
0
0
5
10
15
20
Age (years)
Cracking rate :
0%
5%
10%
15%
20%
25%
Figure 3 : Probability of exceeding different cracking rates for sub-sections
caracterised by explanatory variables (mat, traf, theq)
Let a sub-section Si, characterized by its cracking rate wi at age ti. Robustness
(in this example r = 0.4) is determined by placing the point representing Si on figure 3
or, if necessary, by interpolating evolution curves. In our example, simply read on
diagrams at which age, 40% of sub-sections exceeded different cracking rates (0%,
5%, etc.). In the following, the age at which cracking rate is exceeded will be referred
to by exceeding age.
3.
SURVIVAL LAWS
Cracking rate exceeding is not observed on some sub-sections for which only
boundaries are available. Survival laws allow to treat these data. In practice, it is not
possible to wait until all sub-sections have exceeded a given cracking rate to stop
experience for security of users for instance. For some sub-sections, as exceeding
age is not observed, three cases are possible. In the first case, cracking rate has
already been exceeded at the first observation, i.d., age t 0 , hence, exceeding age will
belong to the interval [0;t 0 [ , one talks about left censoring. In the second case,
cracking rate was exceeded between two observations at times t1 and t2 , exceeding
age is then comprised in the interval [t1 ; t 2 ] , one talks about interval censoring. In the
third case, cracking rate is not exceeded yet at the last observation at age t 2 , so
exceeding age will be comprised in the interval [t 2 ;+∞[ , one talks about right
censoring.
Censoring is random in SISERD database with data left, interval and right
censoring. For a given cracking rate, censoring type and associated boundaries are
determined for all sub-sections.
In fact, curves on figure 3 are cumulative distribution functions. Cumulative
distribution function Fw ( t ) represents probability that age " Tw " of transverse cracking
rate w is less than t :
Fw ( t ) = Pr( Tw ≤ t ) =
t
∫f
w ( x ) dx
= 1 − S w (t )
(Eq. 2)
0
in which
Fw ( t ) : cumulative distribution function. Fw(t) is monotonous increasing
with Fw ( 0 ) = 0 ;
f w ( t ) : density function;
S w ( t ) : survival distribution function.
LIFEREG[2] PROCEDURE of SAS/STAT was used to determine cumulative
distribution function. Usually, log of exceeding age is modelled. For instance,
consider transverse cracking rate wi . Let Ti j be the exceeding age of this rate on subsection j, denoted by S j . That is :
( )
ln Ti j = X j β + σε j
in which
(Eq. 3)
[
]
X j : vector of explanatory variables on S j , that is mat j , traf j , é paij ;
β : vector of coefficients associated to explanatory variables
a mat , a traf , a é pai ;
[
]
σ : a scale parameter;
ε j : error value on S j .
The errors ε j follow probability laws like exponential, Weibull, lognormal, Gamma,
loglogistic, etc.
From (Eq. 3) :
εj=
( )
ln Ti j − X j β
σ
(Eq. 4)
Maximum likelihood method was used to estimate probabilty laws parameters
of ε j and vector β. To determine initial values of vector β parameters, LIFEREG
PROCEDURE use the least squares method with noncensored data.
With initial values of β, log of maximum likelihood function L is calculated by the
following formula :
L=
∑ log( f (ε )) + ∑ log(S (ε )) + ∑ log( F (ε )) + ∑ log( F (ε ) − F (ε ))
wi
j
wi
j
wi
j
wi
j
wi
'
j
(Eq. 5)
The first sum term represent observed data, the second right censoring data,
the third left censoring data and the last interval censoring data with ε =
'
j
( )
ln Zi j − X j β
σ
,
and Z ij lower interval bound. Newton-Raphson [4] method allows to estimate these
parameters.
Using probability law of ε j , one can determine that of Twji by the following
relationship :

ln ( t ) − X j β 
Fwi ( t ) = Pr Twji ≤ t = Pr exp X j β + σε j ≤ t = Pr ε j ≤
σ


(
) ( (
) )
(Eq. 6)
When applied to different transverse cracking rates, equation Eq. 6 allows to
obtain cumulative distribution functions of figure 3, and then, the transverse cracking
rate evolution. Several probability laws of ε j (exponentielle, Weibull, etc.) were
tested. Those for which maximum likelihood function is maximized for all studied
cracking rate was selected for evolution model. The following code allowed to test
several probability laws for a given cracking rate with its explanatory variables (mat,
traf, theq) :
proc lifereg data=data1 outest=outset1 covout noprint;
a : model(lower, upper) = mat traf theq/distribution = exponential corrb;
b : model(lower, upper) = mat traf theq/distribution = weibull corrb;
c : model(lower, upper) = mat traf theq/distribution = lnormal corrb;
d : model(lower, upper) = mat traf theq/distribution = llogistic corrb;
output out=out1 quantiles = .1 .2 .3 .4 .5 .6 .7 .8 .9;
run;
Lower and upper are intervals boundary for censored sub-section.
4.
THE COX MODEL
The Cox model [4] was used to determine in non parametric way, survival
distribution function, and hence, cumulative distribution function (cf. Eq. 3). As for
survival laws, explanatory variables could be introduced in model.
To calculate survival distribution function S w ( t ) , proportionnal hazard model is
used (see later in the paper). Hazard function hw ( t ) represents the probability that
transverse cracking rate exceeded a value w in the interval [t , t + ∆t ] , cracking rate
being less than w between 0 and t. That is :
 Pr( t ≤ T ≤ t + ∆t / T ≥ t )  f ( t )
w
w
 = w
hw ( t ) = lim 
∆t →0
t
∆

 Sw (t )
(Eq. 7)
Proportionnal hazard model for sub-section S j is :
(
)
(
hw t , X j = h0 ( t ) *exp X j β
in which
)
(Eq. 8)
X j : explanatory variables vector;
(
)
hw t , X j : hazard function at time t knowing X j ;
h ( t ) : an arbitrary hazard function;
0
β : vector of coefficients associated to explanatory variables.
Survival distribution function is deduced :
(
) (
Sw t, X j = S 0 ( t )
)
(
exp X j β
)
(Eq. 9)
where
 t

S ( t ) = exp − h 0 ( u ) du

 0
0
∫
(Eq. 10)
Survival distribution function S 0 ( t ) was estimated with PHREG PROCEDURE
[3] of SAS/STAT. This procedure did not allow to treat simultaneously data with left
and right censoring [5]. To overtake this problem, envelope curves method have
been developped. Initial sample is cut into two sub-samples. The first one contained
all sub-sections whose cracking rate exceeding is observed and data are right
censoring. The second one contained all sub-sections whose cracking rate
exceeding is observed and data are left censoring. PHREG PROCEDURE is applied
on each sub-sample. For a given cracking rate with explanatory variables (mat, traf,
theq), the program is the following :
Proc PHREG data=data1;
model (lower, upper) = mat traf theq;
run;
Figure 4 shows non parametric survival function obtained for each subsample. Survival function is calculated by weighting envelope curves by the two subsamples sizes.
1
0.9
Survival function
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
Age
Right censoring
Left censoring
weighting survival curve
Figure 4 : Envelope curves
For each cracking rate, a weighted nonparametric survival function is
calculated, as shown in figure 4. Weighted survival functions were fitted by different
probability laws (exponential, Weibull, etc.). Once the "best fit" probability law is
selected, cumulative distribution functions were calculated (cf. Eq. 3).
5.
COMPARISON OF THE TWO METHODS
In SISERD database, transverse cracking rate, age and explanatory variables
are known for each sub-section. Transverse cracking evolution model is adjusted
such that the evolution curve crosses the point representing the sub-section in the
plan defined by the factors age and transverse cracking rate. For instance, let us
consider a sub-section which has 10% of transverse cracking at five years, with its
explanatory variables, the evolution curve is shown in figure 5 :
25
Cracking rate
20
15
10
5
0
0
5
10
15
20
Age (years)
Figure 5 : Evolution curve
Transverse cracking rate estimations are on this evolution curve.
To compare the survival laws method, referred to by M1, and the Cox model,
referred to by M2, predictive performance of each of these methods was analysed.
The criteria used for comparisons is the quadratic mean of differences between
observations and estimations. It is defined as :
EM =
∑ ( observation − estimation )
2
(Eq. 13)
sample size
For a given sub-section, observed at age n, transverse cracking evolution
curve was calculated. Observed transverse cracking rates at ages (n+i), for i = 1 to 5,
and (n+10) on this sub-section were then compared with their estimations made at
the same ages by means of evolution curves. Results are given in table 1:
Table 1 : Predictions at 1, 2, 3, 4, 5 et 10 years
Prediction age
Sample size
M1 EM
M2 EM
1 year 2 years 3 years 4 years 5 years 10 years
1487 1596 1134
954
598
96
6.18
7.2
7.97
7.27
8.92
14.82
8.13
8.71
9.08
9.78 11.58 14.98
For all predictions, EM obtained with the survival method are less than those
obtained with the Cox model. Values of the root mean squares remain low for all
predictions. Prediction performance might be improved by refining the calculus of
survival laws using the Cox model. New statistical developments could be done to
improve this calculus.
6.
CONFIDENCE INTERVALS
Once the survival laws are chosen, confidence intervals should be determined
for evolution curve. Several parameters are involved in the calculus of confidence
intervals such as : survival laws parameters and explanatory variables parameters.
LIFEREG PROCEDURE [2] estimates parameters with their standard deviations.
Results for exceeding cracking rate of 0% with loglogistic probability law are given in
table 2.
Table 2 : LIFEREG procedure results
Variable
Intercept
mat
traf
theq
scale
df
1
1
1
1
1
Estimate
0.80
0.38
0.58
0.001
0.50
Std Err
0.32
0.09
0.06
0.006
0.03
ChiSquare
6.10
18
96.82
0.04
Pr>Chi
0.01
0.0001
0.0001
0.8335
Note : theq parameter is not significantly different of 0.
Using standard deviations of the parameters, confidence intervals of evolution curves
could be determined. Figure 6 shows evolution curve obtained for a sub-section
which, at 10 years, have reached a cracking rate of 22%, and associated confidence
intervals :
30
Cracking rate
25
20
15
10
5
0
0
5
10
15
20
25
Age(years)
estimation
limit sup
limit inf
Figure 6 : Confidence intervals example
One can see that confidence intervals limits did not start at point (10 years,
cracking rate 22%), due to the fact that observation (10 years, cracking rate 22%)
was considered as a random variable. This reason explains the pattern of the
confidence intervals shown in figure 6.
7.
CONCLUSION
In road field, surveying of network evolution between construction to first
maintenance is difficult. Oftently, only an observation window is available. On some
sub-sections, transverse cracking rate exceeding is not observed. The survival laws
(LIFEREG PROCEDURE) and the Cox model (PHREG PROCEDURE) allow to treat
these "incomplete" data. However, for the Cox model, one should work with data with
right and left censoring to improve the construction of the envelope curve method.
The method described in this paper could be applied to another deterioration
index like scaling, and also on other types of roads like bituminous concretes for
instance. This method also allows to take another deterioration as explanatory
variable.
8.
ACKNOWLEDGMENT
The author would like to thank the SETRA (Service d'Etudes Techniques des Routes
et Autoroutes) who enabled the access to database SISERD.
9.
REFERENCES
[1] Per Ullidtz, (1987), "Pavement analysis",Elsevier.
[2] SAS/STAT User's Guide, Version 6, Fourth Edition, Volume 2 . The LIFEREG
procedure pp 997-1026
[3] SAS Technical Report P-229, SAS/STAT Software : Changes and enhancements,
Release 6.07, The PHREG procedure pp 433-479
[4] Lawless, JF (1982), " Statistical Models and Methods for lifetime Data", Edition
Wiley
[5] Andersen P.K. and Gill R.D. (1982), "Cox's regression model counting process : a
large sample study", Annals of statistics 10, pp 1100-1120
*
*
*
Contact :
Emmanuel COURILLEAU
PhD student, Department of "Pavement Surveys and Management",
Laboratoire Central des Ponts et Chaussées,
BP n° 19,
44340 Bouguenais, FRANCE
Tel : (33) 02-40-84-56-24
Fax : (33) 02-40-84-59-92
Email : [email protected]