1. Application to bridge condition data in the Netherlands.
Statistical inference and hypothesis testing for Markov chains with interval censoring
Application to bridge condition data in the Netherlands.
Monika Skuriat-Olechnowska
Delft University of Technology

2. Presentation outline Importance of bridge management
Data and Work plane
Markov chains
Markov deterioration model
Conclusions
Recommendations
Questions
July 28, 2005
TU Delft

3. Importance of bridge management
Bridge management is essential for today’s transportation infrastructure system
Bridge management systems (BMSs) help engineers and inspectors organize and analyze data collected about specific bridges
BMSs are used to predict future bridge deterioration patterns and corresponding maintenance needs
July 28, 2005
TU Delft

4. Data Statistical analysis of deterioration data (DISK)
Condition rating scheme
Data analysis
Include a total of 5986 registered inspection events for 2473 individual superstructures
Ignoring the time between inspections there are 3513 registered transitions between condition states
July 28, 2005
TU Delft

5. Work plane Check Markov property
Create a deterioration model using Markov chains
Determine and compare the expected deterioration over time and annual probability of failure
Separate bridges into sensible groups and determine the effect of this grouping on the parameters of the Markov model
Take into account the inspector’s subjectivity into condition rating
July 28, 2005
TU Delft

6. Markov chains The Markov Chain is a discrete-time stochastic process
that takes on a finite or countable number of possible values.
July 28, 2005
TU Delft

7. Markov chains (cont’d)
The main assumption in modelling deterioration using a Markov chain is that the probability of a bridge moving to a future state only depends on the present state and not on its past states.
This is called Markov property and is given by:
July 28, 2005
TU Delft

8. Markov chains (cont’d)
Verification of the Markov property
In order to test the Markov property we need to verify if the transition probabilities from the present to future state don’t depend on past states.
Test based on the contingency tables
Test to verify if a chain is of given order
Test to verify if the transition probabilities are constant in time
July 28, 2005
TU Delft

9. Markov chains (cont’d)
Test based on the contingency tables
Generated frequency are from the same distributions;
Result: can not reject
Test to verify if a chain is of given order
The chain is of first-order;
Test to verify if the transition probabilities are constant in time
Transition probabilities do not depend on time; stationarity
July 28, 2005
TU Delft

10. Markov deterioration model
Transition probability matrix
July 28, 2005
TU Delft

11. Markov deterioration models
Modeling transition probability matrix
July 28, 2005
TU Delft

12. Markov deterioration models
Modeling transition probability matrix
case
State-independent
State-dependent
std
lower
upper
state
All
data
0.154
0.151
0.158
0.410
0.394
0.427 1
0.310
0.230
0.321 2
0.096
0.092
0.100 3
0.033
0.029
0.036 4
0.114
0.086
0.142
July 28, 2005
TU Delft

13. Markov deterioration models
5% lower
Expected Lifetime
5% upper
State-independent 15
33,26 57
State-dependent 19
53,83
119
July 28, 2005
TU Delft

14. Markov deterioration model
Data analysis
Year of construction of the structure (we consider two groups of age; all bridges constructed before 1976 and all bridges constructed starting 1976)
Location of structure: bridges “in the road”- heavy traffic, against bridges “ over the road”- light traffic;
Type of bridge: separated into “concrete viaducts” and “concrete bridges” ;
Use, which means the type of traffic which uses the bridge: traffic only with trucks and cars, and mixture of traffic (also bikes, pedestrians, etc.);
Province in which bridge is located:
Groningen
Friesland
Drenthe
Overijssel
Gelderland
Utrecht
Noord-Holland
Zuid-Holland
Zeeland
Noord-Brabant
Limburg
Flevoland
West-Duitsland
Population density
Higher
Proximity to the sea
Close to the sea
Lower
Inland
July 28, 2005
TU Delft

15. Markov deterioration model
Model parameters
Type of data
Estimated parameters
All data
[0.4104, , , , ]
Province in
which bridge
is located
Groningen
[0.2912, , , , ]
Friesland
[0.2285, , , , ]
Drenthe
[0.4425, , , , ]
Overijssel
[0.2843, , , , ]
Gelderland
[0.2321, , , , ]
Utrecht
[0.8176, , , , ]
Noord-Holland
[0.5625, , , , ]
Zuid-Holland
[0.5324, , , , ]
Zeeland
[0.5575, , , , ]
Noord-Brabant
[0.5622, , , , ]
Limburg
[0.5059, , , , ]
Flevoland
[1.0000, , , , ]
West-Duitsland
[0.9785, , , , ]
Type of data
Estimated parameters
All data
[0.4104, , , , ]
Construction year
Built before 1976
[0.8026, , , , ]
Built after 1976
[0.3988, , , , ]
Location
“in the road”- heavy traffic
[0.4490, , , , ]
“ over the road”- light traffic
[0.3180, , , , ]
Type
“concrete viaducts”
[0.3933, , , , ]
“concrete bridges”
[0.5883, , , , ]
Use
Only with trucks and cars
[0.4193, , , , ]
Mixture of traffic
[0.3491, , , , ]
Type of data
Estimated parameters
All data
[0.4104, , , , ]
Population density
Higher
[0.5398, , , , ]
Lower
[0.2677, , , , ]
Proximity to the sea
Close to the sea
[0.4739, , , , ]
Inland
[0.3465, , , , ]
July 28, 2005
TU Delft

16. Markov deterioration model
July 28, 2005
TU Delft

17. Markov deterioration model
Logistic regression
The goal of logistic regression is to correctly predict the influence of the independent (predictor) variables (covariates) on the dependent variable (transition probability) for individual cases.
July 28, 2005
TU Delft

18. Markov deterioration model
Logistic regression
Location
(Over the road)
No influence
Type of bridge
(Concrete bridges)
Influence
Type of traffic
(Mixture of traffic)
Population density (High)
Construction year (built before 1976)
July 28, 2005
TU Delft

19. Markov deterioration model
Inspector’s subjectivity
State assessed by inspector is uncertain given an actual state.
This part of analysis was an extra subject and due to lack of time is not finish yet. We derived formula from which we can estimate transition probabilities, but it is very long and difficult so it will not be presented.
July 28, 2005
TU Delft

20. Conclusions
We may assume that Markov property holds for DISK deterioration database
State-dependent model fits better to the data
Grouping bridges with respect to construction year, type of traffic, location, etc. has influence on the model parameters:
The most statistically significant are covariates “Population density (high)” and “Construction year (built before 1976)”
No influence for covariate “Location (over the road)”
Problem of taking into account subjectivity of inspectors into model parameters is not finish due to time constraints.
July 28, 2005
TU Delft

21. Recommendations
Choose another combination of covariates in logistic regression and look on the changes (how this influence on transition probabilities).
Repeat tests for Markov property with a new deterioration data. We didn’t do this due to time constraints.
Model used in this analysis does not include maintenance. It would be interesting to incorporate this into Markov model.
July 28, 2005
TU Delft

22. Questions ?
July 28, 2005
TU Delft

23. Resulting models (cont’d)
Inspectors subjectivity
July 28, 2005
TU Delft