Transcript Slide 1

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
Presentation outline
Importance of bridge management
 Data and Work plane
 Markov chains
 Markov deterioration model
 Conclusions
 Recommendations
 Questions

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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
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Data

Statistical analysis of deterioration data (DISK)
 Condition
rating scheme
 Data analysis


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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
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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
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Markov chains

The Markov Chain is a discrete-time
stochastic process  X t , t  0,1,2,... 
that takes on a finite or countable number
of possible values.
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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:
Pij  Pr  X t 1  j | X t  i, X t 1  it 1,..., X 1  i1, X 0  i0 
 Pr  X t 1  j | X t  i  Pij (t )
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t  0,1,2,....
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Markov chains (cont’d)
Verification of the Markov property

In order to test the Markov property we need to verify if
the transition probabilities Pr  X t 1  m | X t  j, X t 1  i 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
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Markov chains (cont’d)



Test based on the contingency tables

H 0 : Generated frequency are from the same distributions;

Result: can not reject
Test to verify if a chain is of given order

H 0 : The chain is of first-order;

Result: can not reject
Test to verify if the transition probabilities are constant in
time

H 0 : Transition probabilities do not depend on time; stationarity

Result: can not reject
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Markov deterioration model
Transition probability matrix
 p00
p
P   10

p
 k0
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p01
p11
pk 1
p0 k 
p1k 


pkk 









0.4581
0.3553
0.2782
0.2222
0.1096
0.2857
0.0062 
0.0092 

0.0137 

0.0780 0.2813 0.3097 0.0993 0.0095 
0.0959 0.2877 0.3425 0.1164 0.0479 

0.1429 0.1429 0.2500 0.1429 0.0357 
0.1181
0.1684
0.0989
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0.2881
0.2921
0.3604
0.0978 0.0317
0.1276 0.0474
0.1890 0.0597
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Markov deterioration models
Modeling transition probability matrix
0
1  p01 p01
 0 1 p
p12
12

 0
0 1  p23

0
0
 0
 0
0
0

0
0
 0
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 p00
 0

 00
 0
 0
 0p23
1  p
 0 34
0
0
p01
p11
00
00
00
0p34
1  p45
0
p02
p12
p022 

00 
00 

0
0
p45 

1 
p03
p13
p23
p33
0
0
p04
p14
p124 p
 0
p34
p 0
 44
 00
 0

 0
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p05 
p15 
0
0
0
0
p25 p
1  p
p
0
0
0 
p35

p
0
0
p45 0 1  p

0
0
1

p
p
0

p55 
0
0
0
1  p p

0
0
0
0
1 
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Markov deterioration models
Modeling transition probability matrix
case
State-independent
ˆ
All
data
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0.154
std
0.00171
lower
0.151
State-dependent
upper
0.158
state
ˆ
std
lower
upper
0
0.410
0.00823
0.394
0.427
1
0.310
0.00532
0.230
0.321
2
0.096
0.00198
0.092
0.100
3
0.033
0.00187
0.029
0.036
4
0.114
0.01435
0.086
0.142
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Markov deterioration models
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5% lower
Expected Lifetime
5% upper
State-independent
15
33,26
57
State-dependent
19
53,83
119
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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.);

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Province in which bridge is
located:
Groningen
Friesland
Drenthe
Overijssel
Gelderland
Utrecht
Noord-Holland
Zuid-Holland
Zeeland
Noord-Brabant
Limburg
Flevoland
West-Duitsland

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

Population density
Higher
Utrecht
NoordHolland
ZuidHolland
NoordBrabant
Limburg
Proximity to the sea
Close to the sea
Groningen
Friesland
NoordHolland
ZuidHolland
Zeeland
Flevoland
Lower
Groningen
Friesland
Drenthe
Overijssel
Gelderland
Zeeland
Flevoland
West-Duitsland
Inland
Drenthe
Overijssel
Gelderland
Utrecht
Noord-Brabant
Limburg
West-Duitsland
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Markov deterioration model

Model parameters
Type of data
Estimated parameters
Type
of data
All data
Estimated
parameters
[0.4104, 0.3102,
0.0963,
0.0328, 0.1143]
All data
Groningen
[0.4104,
0.3102,
0.0963,
0.0328,
[0.2912, 0.9476,
0.2608,
0.0050,
0.0000]0.1143]
Friesland
[0.2285,
0.3324, 0.0952,
0.0546, 0.0000]
Type of data
Estimated
parameters
Built before 1976
[0.8026, 0.4135, 0.1027, 0.0335, 0.0816]
Construction
[0.4425, 0.3712, 0.1739, 0.0805, 0.0000]
All dataDrenthe
[0.4104,
0.3102, 0.0963, 0.0328, 0.1143]
year
Built
after
1976
[0.3988,
0.2542, 0.0842, 0.0305, 0.2073]
Overijssel
[0.2843, 0.2350, 0.2201, 0.0282, 0.0637]
Higher
[0.5398, 0.2790, 0.0789, 0.0255, 0.1422]
Population
Location
“in the road”heavy traffic
[0.4490,
0.3115,
0.0966,
0.0328,
0.1140]
Gelderland
[0.2321, 0.3743,
0.1964,
0.0599,
0.0957]
density
Lower
[0.2677,
0.3856,
0.1817,
0.0477,
0.0847]
“ over the road”[0.3180,
0.3017,
0.0949,
0.0329,
0.1150]
Utrechtlight traffic
[0.8176, 0.4492,
0.0624,
0.0000,
0.0001]
Province in
Proximity
Type
which
bridge to
the sea
is located
Use
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Close
to the
sea
“concrete
viaducts”
Noord-Holland
[0.4739,
0.2501,
0.0787,
0.0233,
0.1176]
[0.3933,
0.2976,
0.0946,
0.0327,
0.1332]
[0.5625,
0.2183,
0.0932,
0.0075,
0.1509]
“concrete
bridges”
Inland
Zuid-Holland
[0.5883,
0.3867,
0.1025,
0.0332,
0.0493]
[0.3465,
0.3725,
0.1223,
0.0416,
0.1111]
[0.5324,
0.2211,
0.0629,
0.0351,
0.1231]
Only withZeeland
trucks and cars
[0.4193,
0.3029,
0.0951,
0.0326,
0.1152]
[0.5575, 0.6394,
0.0512,
0.0000,
0.8807]
Noord-Brabant
Mixture
of traffic
[0.5622, 0.3578,
0.0926,
0.0356,
0.3216]
[0.3491,
0.3675,
0.1020,
0.0340,
0.1096]
Limburg
[0.5059, 0.4049, 0.1290, 0.0263, 0.0000]
Flevoland
[1.0000, 0.3461, 0.2899, 0.0688, 0.0000]
West-Duitsland
[0.9785, 0.8198, 0.0000, 0.0183, 1.0000]
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Markov deterioration model
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Markov deterioration model
Logistic regression

p
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.
exp(  1 x1   2 x2  ...   n xn )
1  exp(  1 x1   2 x2  ...   n xn )
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 p 
log 
    1 x1  2 x2  ...  n xn
1

p


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Markov deterioration model

Logistic regression
p01
p12
p23
No influence
No influence
No influence
No influence
No influence
Influence
Influence
No influence
No influence
No influence
No influence
Influence
No influence
No influence
No influence
Population density
(High)
Influence
Influence
Influence
Influence
Influence
Construction year
(built before 1976)
Influence
Influence
Influence
No influence
Influence
Location
(Over the road)
Type of bridge
(Concrete bridges)
Type of traffic
(Mixture of traffic)
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p34
p45
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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.
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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.
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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.
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Questions ?
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Resulting models (cont’d)

Inspectors subjectivity
Pr  X t | X t 1 , X t 2 ,..., X1 , X 0   Pr  X t | X t 1
X t  Yt   t
m
L(Y ,..., Y | p)   (...
1
m
n
i 1
m

1
m
(Y ,..., Y | p)   (
pu
i 1
j 1
i
j
n
...  P



0
j 1
k j 1  j 1 , k j  j

 k j   j | X i j 1  k j 1   j 1  Pr  j )
1
n
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0
 Pr  X
n
(t j  t j 1 )  Pr  j 
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...



n
0
 n
( Pk  , k  (t j  t j 1 )  Pr  j ))
pu j 1 j1 j1 j j
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