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2019-01-01

Towards Multiple Model Approach to BridgeDeteriorationJin A. CollinsUniversity of Texas at El Paso

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TOWARDS MULTIPLE MODEL APPROACH

TO BRIDGE DETERIORATION

JIN A COLLINS

Master’s Program in Civil Engineering

APPROVED:

Jeffrey Weidner, Ph.D., Chair

Carlos M. Ferregut, Ph.D.

Jose Espiritu Nolasco, Ph.D.

Stephen Crites, Ph.D. Dean of the Graduate School

Copyright ©

by

Jin A Collins

2019

TOWARDS MULTIPLE MODEL APPROACH

TO BRIDGE DETERIORATION

by

JIN A COLLINS, BS

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at El Paso

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE

Department of Civil Engineering

THE UNIVERSITY OF TEXAS AT EL PASO

August 2019

iv

Table of Contents

Table of Contents ....................................................................................................................... iv

List of Tables ............................................................................................................................. vi

List of Figures ...........................................................................................................................vii

Chapter 1 : Introduction .............................................................................................................. 1

Background ........................................................................................................................ 1

Objectives .......................................................................................................................... 1

Organization of Thesis........................................................................................................ 2

Chapter 2 : Literature Review of Bridge Deterioration Modeling................................................. 3

Condition Rating ................................................................................................................ 3

Explanatory Variables ........................................................................................................ 6

Deterministic models .......................................................................................................... 7

Stochastic models ............................................................................................................... 9

Markovian process ..................................................................................................... 9

Weibull distribution ................................................................................................. 11

Artificial Intelligence models ........................................................................................... 14

Mechanistic-based models ................................................................................................ 16

Reliability-based models .................................................................................................. 17

Chapter 3 : Methodology ........................................................................................................... 20

Markov chain models ....................................................................................................... 20

Transition Probability Matrix (TPM) ................................................................................ 21

Regression Nonlinear Optimization (RNO) ...................................................................... 23

Bayesian Maximum Likelihood (BML) ............................................................................ 24

Ordered Probit Model (OPM) ........................................................................................... 25

Poisson Regression (PR)................................................................................................... 28

Negative Binomial Regression (NBR) .............................................................................. 30

Proportional Hazard Model (PHM) ................................................................................... 31

Hazard Ratio ............................................................................................................ 32

Kaplan-Meier Estimator ........................................................................................... 32

Summary .......................................................................................................................... 33

v

Chapter 4 : Case Study .............................................................................................................. 36

Data Review and Filtration ............................................................................................... 36

Results ............................................................................................................................. 39

Transition Probability Matrix (TPM) ................................................................................ 40

Deterioration Rate Curves ................................................................................................ 46

Comparison ...................................................................................................................... 48

Chi-squared goodness-of fit Test .............................................................................. 48

Modal Assurance Criterion (MAC) .......................................................................... 50

Chapter 5 : Conclusions and Future Work ................................................................................. 54

Summary .......................................................................................................................... 54

Conclusions ...................................................................................................................... 54

Future Work ..................................................................................................................... 56

References ................................................................................................................................ 57

Vita 60

vi

List of Tables

Table 2.1: Condition rating codes and descriptions for bridge......................................................5 Table 2.2: Condition status and description of bridge .................................................................5 Table 2.3: Condition state codes and description of bridge elements revised in 2011 ...................5 Table 2.4: Explanatory variables applied on bridge deterioration modeling .................................7 Table 3.1: Number of bridge decks of transition ........................................................................ 23 Table 3.2: Summary of objective functions and random variables of models. ............................ 35 Table 4.1: Mean transition probability matrices for bridge components with 2008-2010 data .... 42 Table 4.2: Transition probability matrices of bridge decks with 2008-2010 data ........................ 43 Table 4.3: Transition probability matrices of bridge superstructures with 2008-2010 data ......... 44 Table 4.4: Transition probability matrices of bridge substructures with 2008-2010 data ............ 45 Table 4.5: Results of Chi-square goodness-of-fit of bridge components ..................................... 49 Table 4.6: Results of Modal Assurance Criterion for bridge deck .............................................. 51 Table 4.7: Results of Modal Assurance Criterion for bridge superstructure ................................ 51 Table 4.8: Results of Modal Assurance Criterion for bridge substructure ................................... 52 Table 4.9: Transition probability matrices of decks using proportional hazard model ................ 52

vii

List of Figures

Figure 2.1: Weibull probability density function with different shape parameters ...................... 14 Figure 2.2: Schematic diagram of ENN process (Winn and Burgueño 2013) ............................. 15 Figure 2.3: Sketch of the BPM process (Huang 2010) ............................................................... 16 Figure 3.1: Schematic illustration of the ordered probit model (H. D. Tran 2007) .................... 28 Figure 4.1: Total number of bridges in Texas from 2000 to 2010............................................... 37 Figure 4.2: Number of bridges according to structure material in 2010 ...................................... 38 Figure 4.3: Number of bridge decks in 2004 by condition rating groups .................................... 38 Figure 4.4: Number of bridges by component rating after filtering ............................................ 39 Figure 4.5: Multiple model approach workflow chart ................................................................ 40 Figure 4.6: Deterioration rate curves for bridge deck ................................................................. 47 Figure 4.7: Deterioration rate curves for bridge superstructure .................................................. 47 Figure 4.8: Deterioration rate curves for substructure ................................................................ 48 Figure 4.9: Deterioration curves estimated using by proportional hazard model......................... 53 Figure 5.1: Actual bridge deck condition ratings in 2010 ........................................................... 55 Figure 5.2: Deterioration curves of multiple model approach and bridge deck condition ratings in 2010 .......................................................................................................................................... 56

1

Chapter 1 : Introduction

Background

The Intermodal Surface Transportation Efficiency Act of 1991 (ISTEA) marked the start

of a new era in transportation in the United States. ISTEA required transportation agencies to take

a proactive approach to planning and management of their assets. This included requirements for

managing pavement, bridges, safety, congestion, public transportation, and intermodal systems.

For horizontal transportation assets (i.e., bridges and pavement), deterioration modeling is an

essential tool (Yanev and Chen 1993). Since ISTEA, Bridge Management Systems (BMS) have

been utilized to inform decision-making regarding bridge projects such as maintenance,

rehabilitation, and replacement (MR&R) under financial limitations (Agrawal, Kawaguchi, and

Chen 2010). The goal of a BMS is to optimize the performance of bridge networks by

implementing planned MR&R events to the selected bridges in the correct manner and at the

correct time. Reliable predictions of future condition states are critical for optimizing MR&R

activities. The condition rating of bridges is the most vital variable to predict the future

performance of bridges (Jiang 2010). Deterioration models are used to estimate a future condition

ratings. There are numerous approaches to develop deterioration models such as deterministic,

stochastic, mechanistic, artificial intelligent, reliability-based approaches. Each of these

approaches is defensible but is based on different assumptions and could potentially provide

different results. A stochastic approach using Markov chains was the focus of this research.

Objectives

The objectives of this research are twofold; 1). to identify common Markovian-based

deterioration modeling approaches and apply them to a population of Texas bridges to explore the

similarities and differences between approaches 2). to generate a deterioration curve combines the

individual models, minimizing the individual basis of any single model.

2

Organization of Thesis

Chapter 2 presents a review of the literature focused on infrastructure deterioration

modeling approaches. Chapter 3 presents the individual methodologies of modeling a transition

probability matrix. The mathematical models of regression nonlinear optimization, ordered probit,

Poisson regression, negative binomial regression, Bayesian maximum likelihood, and proportional

hazard are presented. Chapter 4 presents a case study including data collection and filtering,

transition probability matrices and deterioration curves obtained from each model, comparison

with existing data, results of consistence test of each model, and a deterioration curve generated

using multiple model approach. Chapter 5 presents conclusions and recommendations for future

work.

3

Chapter 2 : Literature Review of Bridge Deterioration Modeling

The goal of a bridge management system is to support decision-making at Departments of

Transportation in order to maximize the performance of a bridge network system under financial

constraints. The condition rating of bridges is one of the most important factors considered in of

the allotment of capital bridge projects. Deterioration models can be used to estimate future

condition ratings of a bridge. If the estimated future condition rating is reliable, the selection of

bridge MR&R projects can be optimized (Agrawal, Kawaguchi, and Chen 2010; Jiang 2010).

However, it must be noted that the bridge rating system has significant constraints based on two

assumptions (Yanev and Chen 1993):

• The selection of structural components and corresponding weights is based on engineering

experience and reliability estimates.

• The lowest component condition rating is used to decide for the entire structure.

Condition Rating

Bridges are an important element in the highway transportation system. These structures

are expected to be safe. The issue of bridge safety was made prominent by the collapse of the

Silver Bridge located at the Ohio River in 1967. The Secretary of Transportation was required by

Congress to develop and implement the National Bridge Inspection Standards (NBIS) for

estimating the deficiencies of existing bridges. The NBIS requires visual inspection biennially.

The National Bridge Inventory (NBI) database contains information on every bridge in the nation.

The database is primarily populated with bridge inspection data. Bridge owners are responsible for

the inspections and for reporting the information to FHWA for inclusion in the NBI database.

(FHWA 2004).

NBI contains 116 items including the following (Ryan et al. 2012):

• Identification – bridges are identified by coordinates and qualitative descriptions.

4

• Structure material and type – bridges are classified by structural material (i.e., concrete,

steel, etc.), number of spans, and design type (i.e., truss, multi-girder, slab).

• Age and service – built year and functionality of a bridge.

• Geometric data – overall dimensions (i.e., structure length and width, skew) but not section

level geometric data

• Condition – inspection date and the condition ratings of bridge components.

Condition ratings are subjectively assigned by bridge inspectors using a 0 to 9 rating scale. The

inspectors, who must be trained and certified by FHWA, determine the ratings for each bridge

component (deck, superstructure, substructure) based on engineering expertise and experience

(Ryan et al. 2012). The general guideline of condition rating for bridge components are described

in the 1995 edition of the FHWA Coding Guide in Table 2.1 (Ryan et al. 2012). In 1995, FHWA

revised the standards to focus on bridge elements, a more refined discretization of bridges than

components. The Commonly Recognized (CoRe) Structural Elements manual was accepted as an

official American Association of State Highway and Transportation Officials (AASHTO) manual

in 1995.

Table 2.2 describes an initial guideline used in evaluation of element condition rating (Congress

2012). In 2011, the AASHTO Guide Manual for Bridge Element Inspection was published. It

included four standardized condition states utilizing a 1-4 rating scale. The AASHTO element-

level data can be aggregated to determine the condition ratings of bridge components in NBI

(Yanev and Chen 1993).

Table 2.3 describes the final condition ratings of bridge elements that were adopted. These

condition states provide severity and extent (i.e. total element quantity) of deterioration of bridge

elements (Congress 2012). The National Bridge Investment Analysis System (NBIAS) introduced

in 1999 models the investment needs for bridge maintenance, repair, and rehabilitation. This

system incorporates analytical approaches such as a Markovian modeling, optimization, and

simulation. Also, the NBIAS model can perform an analysis of bridge conditions using element

level data (Ryan et al. 2012). States were required to report element level data of all bridges to

5

FHWA by the Moving Ahead for Progress in the 2lst Century legislation (MAP-21) signed into

law in 2012 (Congress 2012).

Table 2.1: Condition rating codes and descriptions for bridge

Codes Descriptions N Not applicable 9 Excellent condition 8 Very good condition – no problems noted 7 Good condition – some minor problems 6 Satisfactory condition – structural elements show some minor deterioration 5 Fair condition – all primary structural elements are sound but may have minor

section loss, cracking, spalling or scour 4 Poor condition – advanced section loss, deterioration, spalling, or scour 3 Serious condition – loss of section, deterioration, spalling, or scour have

seriously affected primary structural components. Local failures are possible. Fatigue cracks in steel or shear cracks in concrete may be present.

2 Critical condition – advanced deterioration of primary structural elements. Fatigue cracks in steel or shear cracks in concrete may be present or scour may have removed substructure support. Unless closely monitored it may be necessary to close the bridge until corrective action is taken.

1 “Imminent” Failure condition – major deterioration or section loss present in critical structural components, or obvious vertical or horizontal movement affecting structure stability. Bridge is closed to traffic, but corrective action may put bridge back in light service.

0 Failed condition – out of service; beyond corrective action.

Table 2.2: Condition status and description of bridge

Status Descriptions Good Element has only minor problems. Fair Structural capacity of element is not affected by deficiencies Poor Structural capacity of element is affected or jeopardized by deficiencies.

Table 2.3: Condition state codes and description of bridge elements revised in 2011

States Descriptions 1 Good – No deterioration to minor deterioration 2 Fair – Minor to Moderate deterioration 3 Poor – Moderate to Severe deterioration 4 Severe – Beyond the limits of 3

6

Explanatory Variables

Explanatory variables are defined here as external factors that affect bridge deterioration.

Bridges are generally classified with explanatory variables prior to deterioration modeling

analysis. Morcous et al. (2003) classified concrete bridge decks in Quebec, Canada with

explanatory variables including functionality, location, average daily traffic, percentage of truck

traffic, and environments. Wellalage et al. (2015) grouped railway bridges in Australia by

explanatory variables including structure material, number of tacks, average ton passed per week,

element type, environments, and span length. Agrawal et al. (2010) classified bridge elements in

New York State, U.S.A. based on explanatory variables including design type, location, structure

material, ownership, annual average daily truck traffic, deicing salt usage, snow accumulation,

environments, and functionality. Huang (2010) identified 11 explanatory variables statistically

relevant to concrete bridge deck deterioration in Wisconsin by using the artificial neural network

approach. The explanatory variables that were identified included maintenance history, age,

previous condition, district, design load, structure length, deck dimension, average daily traffic,

skew, number of spans, and environments. Table 2.4 summarizes explanatory variables based on

application. Note that in nearly all cases, the selection of explanatory variables was based on

heuristics and engineering judgment alone. Analysis was rarely conducted to determine which

explanatory variables to consider.

7

Table 2.4: Explanatory variables applied on bridge deterioration modeling

Application Explanatory variables

Bridge elements

• Design type • Location • Structure material • Ownership • Annual average daily truck traffic • Deicing salt usage • Snow accumulation • Environments • Functionality

Bridge components

• Functionality • Location • Average daily traffic • Percentage of truck traffic • Maintenance history • Age • Previous condition • District • Design load • Structure length • Deck dimension • Skew • Number of spans • Environments

Railway bridge components

• Structure material • Number of tacks • Average ton passed per week • Element type • Environments • Span length

Deterministic approaches to deterioration modeling

Deterministic approaches are the simplest method to obtain bridge condition predictions.

In deterministic models the most possible condition rating can be estimated as a function of age

and other explanatory variables via regression. The basic assumption of deterministic approaches

is that the relationship between the future condition of bridges over time is certain. The models

8

neglect the uncertainty and randomness of deterioration processes (Ranjith et al. 2013). Since the

probabilistic nature of models is not considered, the same outcome from a deterministic model will

be obtained if the input variables are the same (Kotze, Ngo, and Seskis 2015). When an analysis

of each bridge does not influence on an bridge network analysis , a statistical method is suitable to

develop an equation in bridge conditions (Hyman and Hughes 1983). Deterioration rates can be

developed by a regression analysis of data (Agrawal, Kawaguchi, and Chen 2010). In deterministic

models the relationship between explanatory variables influencing bridge deterioration and the

condition rating are illustrated by this regression analysis. The average duration at each condition

state, the average condition rating by age, and the minimum rating of an element are the common

deterministic methods to estimate deterioration rates (Agrawal, Kawaguchi, and Chen 2010).

Yanev and Chen (1993) estimated the future condition state of bridge decks in New York City by

linear regression analysis. The rate of change for each condition rating state was obtained and

averaged. Also, the average condition rating at various ages was calculated. The non-linear

regression method was applied to formulate a third order polynomial model to describe the

relationship between the condition rating of bridge components and the bridge age in Indiana

through statistical and regression analysis by Jiang and Sinha (1989). Limitations to deterministic

models include (Agrawal, Kawaguchi, and Chen 2010):

• The uncertainty due to intrinsic probabilistic nature of infrastructure deterioration and

unobserved explanatory variables is not considered.

• The average condition of a group of bridges is calculated without consideration of the

current and historical condition of each individual bridge.

• Maintenance actions are not considered because it is difficult to calculate the impact of

MR&R activities.

• The influence of interaction between components is not considered.

• New deterioration rates must be calculated when new data are obtained.

9

Stochastic approaches to deterioration modeling

The condition states of bridge components/elements or the duration at each condition state

are treated as random variables in stochastic models. Probability distributions are used to develop

deterioration models (Kotze, Ngo, and Seskis 2015). Probabilistic deterioration models can be

grouped into two categories: state-based and time-based models. In stated-based models, such as

a Markov chain approach, the probability of transition from one condition state to another in a

discrete time is estimated, conditional on a set of explanatory variables. In time-based models,

such as Weibull distribution function, the probability of duration of time at a condition state is

estimated, conditional on a set of explanatory variables (Mauch and Madanat 2002).

Markovian process

A Markovian process, named after Russian mathematician Andrey Markov, is a stochastic

process that satisfies the Markov property which states that a prediction for the future of the process

only depends on the present state of a system. Markov used “random walk” and “gambler’s ruin”

as the examples of Markovian processes in his first paper published in 1906. These examples are

the Brownian motion and Poisson processes, which are the main components of the theory of

stochastic processes. A Markov chain has a discrete state space (Leonard 2011). Markovian

deterioration models have been utilized in modeling the deterioration of infrastructure facilities

such as pavement (A. Butt et al. 1987; Carnahan et al. 1987; DeLisle, Sullo, and Grivas 2003;

Golabi, Kulkarni, and Way 1982), storm water pipe (Micevski, Kuczera, and Coombes 2002),

sewer pipe (Baik, Jeong, and Abraham 2006), bridge components (Hatami and Morcous 2015;

Jiang 2010; Jiang, Saito, and Sinha 1988), bridge elements (Ranjith et al. 2013; J. O. Sobanjo

2011), and culverts and traffic signs (Thompson et al. 2012). Markovian models are based on the

following assumptions (Ranjith et al. 2013):

• The deterioration process is homogenous with a constant transition probability in an

inspection period.

10

• A condition state can transition to across multiple states. For example, the condition

rating can change to any lower condition rating in consecutive inspection period.

• The deterioration process is assumed a continuous process within a discrete time

interval and constant bridge population.

• The future condition of a bridge relies only on the present condition (Markov property)

A Markov chain is a chain of random variables 𝑋𝑋𝑘𝑘 , 𝑋𝑋𝑘𝑘−1, 𝑋𝑋𝑘𝑘−2, … which are finite and which

satisfy the Markov property. The probability of transition from the current state to the next state is

𝑃𝑃𝑃𝑃{𝑋𝑋𝑘𝑘+1 = 𝑥𝑥𝑘𝑘+1|𝑋𝑋𝑘𝑘 = 𝑥𝑥𝑘𝑘,⋯ ,𝑋𝑋0 = 𝑥𝑥0} = 𝑃𝑃𝑃𝑃{𝑋𝑋𝑘𝑘+1 = 𝑥𝑥𝑘𝑘+1|𝑋𝑋𝑘𝑘 = 𝑥𝑥𝑘𝑘} Eq. 1

The Markov chain can be described as a matrix called the transition probability matrix (TPM). The

Markov chain as applied to bridge performance prediction models is used by defining discrete

condition ratings and obtaining the probability of transition from one condition rating to another

in discrete time. This method is commonly used when prior maintenance records are not available

in BMS databases (Jiang, Saito, and Sinha 1988; Morcous 2006).

The example format of a transition probability matrix as follows:

𝑃𝑃 =

⎣⎢⎢⎢⎡𝑝𝑝9900⋮0

𝑝𝑝98𝑝𝑝880⋮0

𝑝𝑝97𝑝𝑝87𝑝𝑝77⋮0

𝑝𝑝96𝑝𝑝86𝑝𝑝76⋮0

⋯⋯⋯⋯⋯

𝑝𝑝90𝑝𝑝80𝑝𝑝70⋮𝑝𝑝00⎦

⎥⎥⎥⎤ Eq. 2

The bridge condition ratings are from 9 to 0, and 9 is the maximum rating number for excellent

condition. The prediction can be estimated by:

𝑃𝑃(𝑡𝑡) = 𝑃𝑃(0) × 𝑃𝑃 × 𝑃𝑃⋯𝑃𝑃 = 𝑃𝑃(0) × 𝑃𝑃𝑇𝑇 Eq. 3

11

where 𝑃𝑃(𝑡𝑡) = the probability distribution at time, t; 𝑃𝑃(0) = the initial-state vector; P = the transition

probability matrix; and T = the power that is the same value with the time, t (Jiang, Saito, and

Sinha 1988). The accuracy of Markovian models is dependent on obtaining an accurate transition

probability matrix with a reliable estimation method (Wellalage, Zhang, and Dwight 2014). Jiang

et al. (1988) estimated a transition probability matrix for bridges in Indiana, USA by using

regression nonlinear optimization method with zone technique developed by Butt et al. (1987) to

improve the accuracy of prediction of future condition state of pavements. In the zone technique,

pavements are grouped into zones based on their age. The deterioration rate was assumed to be

different between zones, but it was constant in each zone. Therefore, a transition matrix was

developed for each zone. Wellalage et al. (2015) generated transition probability matrices for

railway bridges in Australia using regression nonlinear optimization (RNO), Bayesian maximum

likelihood (BML), and Metropolis-Hasting algorithm (MHA)-based Markov chain Monte Carlo

(MCMC) simulation technique methods. He concluded that transition probabilities obtained using

MHA-based MCMC and BML are similar, and they were more accurate than ones estimated using

RNO. The ordered probit model was used to estimate a transition probability matrix for storm

water pipes in the City of Greater Dandenong, Victoria, Australia by Tran (2007) and for bridges

in Indiana, USA by Madanat et al. (2002). Madanat and Ibrahim (2002) estimated a transition

probability matrix for bridges in Indiana, USA using Poisson regression and negative binomial

regression models. Cavalline et al. (2015) developed a transition probability matrix for bridges in

North Carolina, USA by the proportional hazard model combining the Kaplan-Meier estimation

and hazard ratio.

Weibull distribution approach to deterioration models

The Weibull distribution is commonly used in reliability analysis due to its flexibility. It

can take on the features of other distributions depending on the values of the shape (𝛽𝛽) and scale

12

(𝛼𝛼) parameters. For example, the curve which has 𝛽𝛽 = 1 and 𝛼𝛼 = 1 is an exponential distribution

as shown in Figure 2.1. The probability density function is

𝑓𝑓(𝑥𝑥) =𝛽𝛽𝛼𝛼 �

𝑥𝑥𝛼𝛼�

𝛽𝛽−1𝑒𝑒−(𝑥𝑥 𝛼𝛼⁄ )𝛽𝛽 𝑓𝑓𝑓𝑓𝑃𝑃 𝑥𝑥 ≥ 0. Eq. 4

Weibull survival models have been use for deterioration modeling of infrastructure facilities such

as pipe culverts, roadway lighting fixtures, pavement markings (Thompson et al. 2012), reinforced

concrete bridge decks (Mishalani and Madanat 2002), bridge elements (Agrawal, Kawaguchi, and

Chen 2010), bridge decks (J. O. Sobanjo 2011), and bridge components (J. Sobanjo, Mtenga, and

Rambo-Roddenberry 2010). Weibull models capture the effects of age and uncertainty more than

Markov chain models (Agrawal, Kawaguchi, and Chen 2010). Time series data and information

of historical MR&R activities are needed (Agrawal, Kawaguchi, and Chen 2010). The Weibull

survival function is:

𝑦𝑦𝑔𝑔 = 𝑒𝑒𝑥𝑥𝑝𝑝�−1.0 × (𝑔𝑔 𝛼𝛼⁄ )𝛽𝛽� Eq. 5

where 𝑦𝑦𝑔𝑔 is survival probability at age 𝑔𝑔; 𝛽𝛽 is the shape parameter and 𝛼𝛼 is the scale parameter:

𝛼𝛼 =𝑇𝑇

(𝑙𝑙𝑙𝑙2)1/𝛽𝛽 Eq. 6

where T is the median life expectancy from the Markov model (Thompson et al. 2012). When 𝛽𝛽

is less than 1, the failure rate (also known as a hazard rate) is decreasing. When 𝛽𝛽 is equal to 1, the

failure rate is constant. When 𝛽𝛽 is greater than 1, the failure rate is increasing. Increasing the failure

rate indicates that a component/element has been at a condition rating for a long time, so the

component/element will transition to a lower condition rating in the next inspection period

(Agrawal, Kawaguchi, and Chen 2010). The shape parameter (𝛽𝛽) of 1 is equivalent to a Markov

13

deterioration model, meaning the transition probability does not change with time. A bigger shape

parameter means that the initial deterioration rate is slow, and then the deterioration rate increases

faster as the age of a facility increases. Figure 2.1 shows the effects of the shape parameter on the

resulting distribution. The average duration that a component/element stays at a condition rating,

𝐸𝐸(𝑇𝑇𝑖𝑖) is:

𝐸𝐸(𝑇𝑇𝑖𝑖) = 𝛼𝛼𝑖𝑖𝛤𝛤 �1 +1𝛽𝛽𝑖𝑖� Eq. 7

where Γ is the Gamma function defined as Γ(𝑙𝑙) = (𝑙𝑙 − 1)!; 𝛼𝛼𝑖𝑖 and 𝛽𝛽𝑖𝑖 are scale and shape

parameters at condition rating i respectively. The average durations for different condition ratings

are calculated cumulatively (Agrawal, Kawaguchi, and Chen 2010). Some or all of the components

or elements in a given condition rating can transition to the next lower condition rating within a

discrete interval. The duration or time at which p% of the components/elements will transition to

lower condition ratings, 𝑡𝑡𝑝𝑝 is:

𝑡𝑡𝑝𝑝 = 𝛼𝛼[−𝑙𝑙𝑙𝑙(1 − 𝑝𝑝)]1/𝛽𝛽 Eq. 8

where 𝛼𝛼 and 𝛽𝛽 are the scale and shape parameters (Agrawal, Kawaguchi, and Chen 2010).

14

Figure 2.1: Weibull probability density function with different shape parameters

Artificial Intelligence approaches to deterioration modeling

Artificial intelligence (AI) techniques have been used to model the deterioration of

infrastructure facilities such as neural network for storm water pipes (D. H. Tran, Perera, and Ng

2009), artificial neural network (ANN) for water mains, ensemble of neural network (ENN) for

bridge elements and components (Bu et al. 2014; Lee et al. 2014; Li and Burgueño 2010),

backward prediction model (BPM) for steel bridge structures and bridge elements (Lee et al. 2008;

Pandey and Barai 1995), multilayer perceptron (MLP) for abutment walls (Li and Burgueño 2010),

and combination of ANN and Case-Based Reasoning (CBR) for the Dickson Bridge (Morcous and

Lounis 2005). An ANN is a computational model derived from the study of nerve cells or neurons

in physiology. An artificial neuron contains receiving sites, receiving connections, a processing

element and transmitting connections. Neural network models are stipulated by the network

topology, node features, and training or learning processes. A multilayer perceptron has an input

x

Prob

abili

ty d

ensit

y fu

nctio

n, f(

x)

15

layer, an output layer, and a number of hidden layers (Pandey and Barai 1995). An ANN can

generate accurate response even though there is noise or uncertainty in training data and evaluate

the results of complex matters with a higher order of nonlinear behavior. An ANN learns to map

the accurate input-output by an iterative procedure. In the learning process, the weighted

connections between neurons are modified, and the error of the model reduced to produce more

accurate results. If data are trained only to focus to reduce the errors, the ANN model might not

generalize the connection between inputs and outputs. In the process, neurons receive inputs from

the previous layer, compute an output through a pre-defined function, and send the output to

neurons on the next layer. Figure 2.2 shows an ENN process diagram. An ENN consists of many

individual MLP models and predict the outcomes by combining these models (Winn and Burgueño

2013). Lee et al. (2008) used an ANN-based BPM to generate unavailable historical bridge

condition ratings for using limited bridge inspection data. Figure 2.3 illustrate the process of BPM.

The missing information such as condition ratings can be estimated by using the correlation

between explanatory variables and condition states. The correlation can be obtained from the

existing historical data by using ANN process (Huang 2010).

Figure 2.2: Schematic diagram of ENN process (Winn and Burgueño 2013)

16

Figure 2.3: Sketch of the BPM process (Huang 2010)

Mechanistic-based approaches to deterioration modeling

The methods mentioned above are based on observation of bridge condition states using

visual inspection. Mechanistic-based approaches rely on modeling the physical processes of bridge

deterioration; most commonly corrosion of rebar in reinforced concrete. A passive film (oxide

film) is formed on the surface of the steel reinforcement during concrete hydration and protects

the reinforcement from corrosion. The film can be destroyed by chloride ions that penetrate to the

reinforcement from the surface of the concrete (Roelfstra et al. 2004). Corrosion models have been

applied to predict reinforced concrete bridge deck conditions. Morcous and Lounis (2007) applied

Fick’s second law to estimate the corrosion initiation time and used the Monte Carlo simulation

technique to generate the probability density function and cumulative distribution function of the

corrosion initiation time:

𝐶𝐶(𝑥𝑥, 𝑡𝑡) = 𝐶𝐶𝑠𝑠 �1− 𝑒𝑒𝑃𝑃𝑓𝑓 �𝑥𝑥

2√𝐷𝐷𝑡𝑡�� Eq. 9

17

where C(x,t) is the chloride concentration at depth (x) and time (t) 𝐶𝐶𝑠𝑠 is surface chloride

concentration, D is diffusion coefficient of chlorides, and erf is error function. Roelfstra et al.

(2004) divided the deterioration of reinforced concrete due to corrosion into two phases, the

initiation phase which chlorides penetrates through the concrete to the reinforcement from the

surface and the propagation phase which the reinforcement actively corrodes. The deterioration

curve generated from the proposed mechanistic model was compared with the curve developed

using the standard Markov chain method with the difference between two curves stemming from

explicit inclusion of the corrosion initiation time. Hu et al. (2013) and Nickless and Atadero (2017)

divided the corrosion process into three phases; corrosion initiation at the rebar surface, initiation

of cracking at the interface between concrete and rebar, and propagation of cracking to the concrete

surface. A numerical model was selected for each phase. The Monte Carlo simulation technique

was implemented to estimate cumulative deterioration of an entire deck. The damage over the deck

was mapped to condition rating scale described by the NBI (Morcous and Lounis 2007).

Reliability-based approaches to deterioration modeling

Bridge management systems are often reliability-based to obtain the spread of uncertainties

in the lifetime process. Commonly used approaches to predict bridge condition (e.g., state and

time-based approaches) do not consider the reliability of performance of elements, components,

or systems of a bridge. Bridge design, however, is based on achieving a uniform probability of

failure using the reliability index, 𝛽𝛽. The same concept can be applied to bridge condition. A bridge

is in excellent condition where 𝛽𝛽 ≥ 9.0, in very good condition where 9.0 > 𝛽𝛽 ≥ 8.0, in good

condition where 8.0 > 𝛽𝛽 ≥ 6.0, in fair condition where 6.0 > 𝛽𝛽 ≥ 4.6, and in unacceptable

condition where 𝛽𝛽 < 4.6 (Frangopol et al. 2001). The performance function, g(t) for a given failure

mode, the probability of failure of a system with several failure modes, Psys(t), and the reliability

index, 𝛽𝛽(𝑡𝑡) associated with the failure state of the system are

18

𝑔𝑔(𝑡𝑡) = 𝑃𝑃(𝑡𝑡)− 𝑞𝑞(𝑡𝑡) Eq. 10

𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡) = 𝑃𝑃[𝑎𝑎𝑙𝑙𝑦𝑦 𝑔𝑔𝑖𝑖(𝑡𝑡) < 0],𝑓𝑓𝑓𝑓𝑃𝑃 𝑎𝑎𝑙𝑙𝑙𝑙 𝑡𝑡 > 0 Eq. 11

𝛽𝛽(𝑡𝑡) = 𝛷𝛷−1 �1− 𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡)� Eq. 12

where 𝑃𝑃(𝑡𝑡) and 𝑞𝑞(𝑡𝑡) are the instantaneous resistance and load effect at time t, respectively, 𝑔𝑔𝑖𝑖(𝑡𝑡)is

the performance function associated with the ith system failure mode, and Φ is the standard normal

cumulative distribution function (Barone and Frangopol 2014). There are three methods to

approximate the failure probability, the mean value first-order second-moment method, the first

order reliability method, and the Monte Carlo simulation method (Frangopol, Kallen, and

Noortwijk 2004). The time-dependent bi-linear and nonlinear reliability index models, 𝛽𝛽(𝑡𝑡)

without maintenance are as follows:

𝛽𝛽(𝑡𝑡) = � 𝛽𝛽0,𝑓𝑓𝑓𝑓𝑃𝑃 0 ≤ 𝑡𝑡 ≤ 𝑡𝑡𝐼𝐼𝛽𝛽0 − 𝛼𝛼1(𝑡𝑡 − 𝑡𝑡𝐼𝐼),𝑓𝑓𝑓𝑓𝑃𝑃 𝑡𝑡 > 𝑡𝑡𝐼𝐼

Eq. 13

𝛽𝛽(𝑡𝑡) = � 𝛽𝛽0, 𝑓𝑓𝑓𝑓𝑃𝑃 0 ≤ 𝑡𝑡 ≤ 𝑡𝑡𝐼𝐼 ,𝛽𝛽0 − 𝛼𝛼2(𝑡𝑡 − 𝑡𝑡𝐼𝐼) − 𝛼𝛼3(𝑡𝑡 − 𝑡𝑡𝐼𝐼)𝑝𝑝,𝑓𝑓𝑓𝑓𝑃𝑃 𝑡𝑡 > 𝑡𝑡𝐼𝐼 ,

Eq. 14

where 𝛼𝛼1,𝛼𝛼2,𝛼𝛼3 are reliability index deterioration rates, 𝑡𝑡𝐼𝐼 is the deterioration initiation time, and

p is a parameter related to the nonlinear effect in terms of a power law in time. An increase in p

results in an increase in the rate of reliability index deterioration. The time-dependent condition

index model, 𝐶𝐶(𝑡𝑡) at time 𝑡𝑡 ≥ 0 is following:

𝐶𝐶(𝑡𝑡) = � 𝐶𝐶0,𝑓𝑓𝑓𝑓𝑃𝑃 0 ≤ 𝑡𝑡 ≤ 𝑡𝑡𝐼𝐼𝐼𝐼𝐶𝐶0 − 𝛼𝛼𝐼𝐼(𝑡𝑡 − 𝑡𝑡𝐼𝐼𝐼𝐼),𝑓𝑓𝑓𝑓𝑃𝑃 𝑡𝑡 > 𝑡𝑡𝐼𝐼𝐼𝐼

Eq. 15

where 𝐶𝐶0 is the initial condition, 𝛼𝛼𝐼𝐼 is the condition deterioration rate, 𝑡𝑡𝐼𝐼𝐼𝐼 is the condition index

which is considered constant for a period equal to the time of damage initiation, and 𝐶𝐶(𝑡𝑡) is the

19

condition at time t which is assumed to decrease with time. The reliability-based model with

maintenance is as follows:

𝛽𝛽𝑗𝑗(𝑡𝑡) = 𝛽𝛽𝑗𝑗 ,0(𝑡𝑡) + �∆𝛽𝛽𝑗𝑗,𝑖𝑖(𝑡𝑡)

𝑛𝑛𝑗𝑗

𝑖𝑖=1

, Eq. 16

where 𝑙𝑙𝑗𝑗 is the number of maintenance actions associated with reliability index profile j, 𝛽𝛽𝑗𝑗,0(𝑡𝑡) is

the reliability index profile without maintenance, and ∆𝛽𝛽𝑗𝑗,𝑖𝑖(𝑡𝑡) is the additional reliability index

profile associated with the ith maintenance action. The reliability index profile of the system is

obtained by combining the reliability index profiles of all individual elements and limit states

considered (Frangopol, Kallen, and Noortwijk 2004).

20

Chapter 3 : Methodology

Six different deterioration methods are presented in this chapter. The selected methods are

• Regression Nonlinear Optimization

• Bayesian Maximum Likelihood

• Ordered Probit

• Poisson Regression

• Negative Binomial Regression

• Proportional Hazard

Though different, each of these approaches has some common attributes. Each model optimizes

an objective function to obtain input parameters (Morcous, Lounis, and Mirza 2003) and then

applies a mathematical model to estimate transition probabilities. Each approach results in a unique

TPM.

Markov chain models

In the Markov chain models, finite number of states and a set of initial probabilities for all

states are needed to estimate transition probabilities. These probabilities are time-independent and

satisfy the Markov property. The Markovian property states that the transition of a condition state

to any future condition state only depends on the present condition state (Eq. 1) (I and Glagola

1994). The probability of transition from state i to state j in an interval, s is following:

𝑝𝑝𝑖𝑖𝑗𝑗(𝑠𝑠) = 𝑃𝑃𝑃𝑃{𝑋𝑋(𝑡𝑡 + 𝑠𝑠) = 𝑗𝑗|𝑋𝑋(𝑡𝑡) = 𝑖𝑖} 𝑤𝑤𝑖𝑖𝑡𝑡ℎ 𝑠𝑠, 𝑡𝑡 ≥ 0 Eq. 17

The probability of being state i at time t is following:

𝑝𝑝𝑖𝑖(𝑡𝑡) = 𝑃𝑃𝑃𝑃{𝑋𝑋(𝑡𝑡) = 𝑖𝑖} Eq. 18

21

It can be a vector form representing the probability distribution of state at time t.

The expected value at time t is following:

𝐸𝐸[𝑋𝑋(𝑡𝑡)] = �𝑗𝑗 ∙ 𝑝𝑝𝑘𝑘(𝑡𝑡)

∀𝑘𝑘

Eq. 19

where j is the condition rating, j = 9, 8, 7, …, 0 and 𝑝𝑝𝑘𝑘(𝑡𝑡) is the transition probability at condition

rating, k at time t.

Transition Probability Matrix (TPM)

TPMs can be categorized into two types, one-step transition probability matrices and multi-

step transition probability matrices. In a one-step TPM, a condition state can transition only the

next lower condition state. However, in a multi-step TPM, a condition state can transition to

multiple lower states (Wellalage, Zhang, and Dwight 2014). A one-step TPM is a subset of a multi-

step TPM where all probabilities outside of the primary pair of the current state and next lower

state are equal to zero.

An example of a one-step TPM is following:

𝑃𝑃

=

⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡𝑝𝑝99000000000

1− 𝑝𝑝99𝑝𝑝8800000000

01 − 𝑝𝑝88𝑝𝑝770000000

00

1 − 𝑝𝑝77𝑝𝑝66000000

000

1 − 𝑝𝑝66𝑝𝑝5500000

0000

1 − 𝑝𝑝55𝑝𝑝440000

00000

1− 𝑝𝑝44𝑝𝑝33000

000000

1 − 𝑝𝑝33𝑝𝑝2200

0000000

1− 𝑝𝑝22𝑝𝑝110

00000000

1 − 𝑝𝑝111

⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤

Eq. 20

An example of a multi-step TPM is following:

22

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎢⎢⎡𝑝𝑝99000000000

𝑝𝑝98𝑝𝑝8800000000

𝑝𝑝97𝑝𝑝87𝑝𝑝770000000

𝑝𝑝96𝑝𝑝86𝑝𝑝76𝑝𝑝66000000

𝑝𝑝95𝑝𝑝85𝑝𝑝75𝑝𝑝65𝑝𝑝5500000

00𝑝𝑝74𝑝𝑝64𝑝𝑝54𝑝𝑝440000

000𝑝𝑝63𝑝𝑝53𝑝𝑝43𝑝𝑝33000

0000𝑝𝑝52𝑝𝑝42𝑝𝑝32𝑝𝑝2200

00000𝑝𝑝41𝑝𝑝31𝑝𝑝21𝑝𝑝110

000000𝑝𝑝30𝑝𝑝20𝑝𝑝101

⎦⎥⎥⎥⎥⎥⎥⎥⎤

Eq. 21

It is assumed that the bridge condition rating could stay in its current state or transit to the next

lower state in one year without any repair or rehabilitation. While it is technically possible for a

condition rating to transition two states in one period, this occurs very infrequently. For example,

Table 3.1 shows the number of bridge decks in 2010 classified into condition rating groups. This

population of bridges was selected for the comparison conducted in Chapter 4 and is presented in

more depth there. The specifics of the bridge population are immaterial for the discussion of

adoption of one or multi-step TPM. The first column indicates condition rating, 𝑖𝑖. The first row

represents the number of bridge components in condition rating, 𝑁𝑁𝑖𝑖𝑗𝑗, the condition rating transits

from a state, 𝑖𝑖 to a state, 𝑗𝑗 (𝑗𝑗 = 9, 8, … , 3). For example, in 2008, 631 bridges were in condition

rating 8. 381 of 631 bridges stayed the same condition rating, 242 bridges transited to condition

rating 7, and 8 bridges transited to condition rating 6 in the next cycle in 2010. This equates to

about 1 % of the bridges which transitioned to condition rating 6. As such a low percentage

transitioned more than one state, a one-step TPM was adopted. Also, since the lowest condition

rating in the data was 3, a truncated TPM was adopted. Based on these characteristics of the data,

the following TPM proposed by Jiang et al. (1988) was used in this research.

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡𝑝𝑝99000000

1 − 𝑝𝑝99𝑝𝑝8800000

01− 𝑝𝑝88𝑝𝑝770000

00

1 − 𝑝𝑝77𝑝𝑝66000

000

1 − 𝑝𝑝66𝑝𝑝5500

0000

1− 𝑝𝑝55𝑝𝑝440

00000

1 − 𝑝𝑝441 ⎦

⎥⎥⎥⎥⎥⎤

Eq. 22

23

Table 3.1: Number of bridge decks of transition

CR 𝑁𝑁𝑖𝑖9 𝑁𝑁𝑖𝑖8 𝑁𝑁𝑖𝑖7 𝑁𝑁𝑖𝑖6 𝑁𝑁𝑖𝑖5 𝑁𝑁𝑖𝑖4 𝑁𝑁𝑖𝑖3 Sum 𝑖𝑖 = 9 0 3 2 0 0 0 0 5 𝑖𝑖 = 8 0 381 242 8 0 0 0 631 𝑖𝑖 = 7 0 0 2,672 136 6 0 0 2,814 𝑖𝑖 = 6 0 0 0 413 22 0 0 435 𝑖𝑖 = 5 0 0 0 0 42 0 0 42 𝑖𝑖 = 4 0 0 0 0 0 2 0 2 𝑖𝑖 = 3 0 0 0 0 0 0 0 0

Regression Nonlinear Optimization (RNO)

In RNO method, a transition probability matrix is estimated by minimizing the absolute

difference between the regression curve, 𝑆𝑆(𝑡𝑡), that best fits the actual condition rating at age, t,

and the estimated condition rating for the corresponding age, 𝐸𝐸(𝑡𝑡,𝑃𝑃) obtained by the Markovian

process. The formula of the objective function is as follows (Jiang et al. 1988):

𝑚𝑚𝑖𝑖𝑙𝑙�|𝑆𝑆(𝑡𝑡)− 𝐸𝐸(𝑡𝑡,𝑃𝑃)|𝑇𝑇

𝑡𝑡=1

Eq. 23

𝑆𝑆(𝑡𝑡) = 𝐴𝐴 + 𝐵𝐵𝑡𝑡 + 𝐶𝐶𝑡𝑡2 + 𝐷𝐷𝑡𝑡3 Eq. 24

𝐸𝐸(𝑡𝑡,𝑃𝑃) = 𝑄𝑄0 × 𝑃𝑃𝑡𝑡 × 𝑅𝑅 Eq. 25

where:

𝑇𝑇 = the largest age of the bridge in a data set;

𝑆𝑆(𝑡𝑡) = the average condition rating at time, t;

A, B, C, and D = coefficients;

𝐸𝐸(𝑡𝑡,𝑃𝑃) = the expected value;

𝑃𝑃 = transition probabilities;

𝑄𝑄0 = the initial condition state, 𝑄𝑄0 = [1 0 0 0 0 0 0] when t = 0;

24

𝑃𝑃𝑡𝑡 = TPM at any time, t;

𝑅𝑅 = the condition rating, [9 8 7 6 5 4 3].

Bayesian Maximum Likelihood (BML)

The Bayes’ theorem states that the conditional distribution of 𝜃𝜃 (an unknown variable set)

given by 𝑌𝑌 (a known data set of bridge condition ratings) is the following:

𝑃𝑃(𝜃𝜃|𝑌𝑌) ∝ 𝑃𝑃(𝜃𝜃)𝐿𝐿(𝑌𝑌|𝜃𝜃) Eq. 26

where 𝑃𝑃(𝜃𝜃|𝑌𝑌), 𝑃𝑃(𝜃𝜃), and 𝐿𝐿(𝑌𝑌|𝜃𝜃) are called a target distribution, prior distribution, and likelihood

distribution respectively. From Bayes-Laplace “principle of insufficient reason,” the prior

distribution, 𝑃𝑃(𝜃𝜃) can be assumed to be a uniform distribution. Therefore, the target distribution

can be proportional to the likelihood distribution (Wellalage, Zhang, and Dwight 2014). From the

joint probability theory, the likelihood distribution can be expressed (H. D. Tran 2007).

𝐿𝐿(𝑌𝑌|𝜃𝜃) = ��(𝐶𝐶𝑖𝑖𝑡𝑡)𝑁𝑁𝑖𝑖𝑡𝑡

9

𝑖𝑖=1

𝑇𝑇

𝑡𝑡=1

Eq. 27

To simply computations, the distribution function can be transformed into a logarithm likelihood

function and applied to bridges as follows:

𝑙𝑙𝑓𝑓𝑔𝑔[𝐿𝐿(𝑌𝑌|𝜃𝜃)] = ��𝑁𝑁𝑖𝑖𝑡𝑡𝑙𝑙𝑓𝑓𝑔𝑔(𝐶𝐶𝑖𝑖𝑡𝑡)9

𝑖𝑖=1

𝑇𝑇

𝑡𝑡=1

Eq. 28

where T is the largest age in the data set, 𝑁𝑁𝑖𝑖𝑡𝑡 is the number of bridge components in condition state

i at year t, and 𝐶𝐶𝑖𝑖𝑡𝑡 is the probability of condition state 𝑖𝑖 at year 𝑡𝑡. Since only transition probabilities,

25

𝐶𝐶𝑖𝑖𝑡𝑡 are random variables, a vector of transition probabilities, 𝐶𝐶𝑖𝑖𝑡𝑡 can be obtained by maximizing

the log likelihood function, 𝑙𝑙𝑓𝑓𝑔𝑔[𝐿𝐿(𝑌𝑌|𝜃𝜃)].

𝐶𝐶𝑖𝑖𝑡𝑡 = [𝐶𝐶9𝑡𝑡 𝐶𝐶8𝑡𝑡 𝐶𝐶7𝑡𝑡 𝐶𝐶6𝑡𝑡 𝐶𝐶5𝑡𝑡 𝐶𝐶4𝑡𝑡] Eq. 29

where 9, 8, …, and 4 are condition rating.

Ordered Probit Model (OPM)

The ordered probit model is most commonly used to model unobservable features of a

population in social sciences. Applied to infrastructure, facility deterioration can be considered

unobservable, while condition rating is observed. However, the model can still be used because of

the assumption that condition rating is a constant unobservable variable. An incremental

deterioration model can be generated from the observed condition rating as an “indicator” of the

unobserved deterioration. The variable, 𝑍𝑍𝑛𝑛 is the number of transitions of condition state of bridge

n between two successive inspection periods. It is assumed that the deterioration is constant within

the same condition rating group, and each group has different deterioration mechanism. Therefore,

a different deterioration model is needed for each condition rating group. Madanat et al. (2002)

categorized the deterioration process of bridge decks into two steps. The transitions from condition

rating 9 to 7 were determined by the change of the electrical potential intensity and the chloride

content amount. The transition from condition rating 6 to 5 is determined by the amount of spall

of concrete. Based on the deterioration process of reinforced concrete defined by Hu et al. (2013)

and Nickless and Atadero (2017), the first stage (condition rating 9 to 7) is related to the phase

from corrosion initiation to cracking initiation. The second stage (condition rating 6 to 5) is related

to the phase of cracking propagation to the surface. The unobserved deterioration of a bridge, n in

given condition state, i, 𝑈𝑈𝑖𝑖𝑛𝑛 can be expressed as a function of explanatory variable, 𝑋𝑋𝑛𝑛.

𝑙𝑙𝑓𝑓𝑔𝑔(𝑈𝑈𝑖𝑖𝑛𝑛) = 𝛽𝛽𝑖𝑖′𝑋𝑋𝑛𝑛 + 𝜀𝜀𝑖𝑖𝑛𝑛 Eq. 30

26

where 𝛽𝛽𝑖𝑖′ is a set of parameters in condition state i, 𝑋𝑋𝑛𝑛 is a set of explanatory variables for bridge

n; 𝜀𝜀𝑖𝑖𝑛𝑛 is an error term (Madanat, Mishalani, and Ibrahim 2002). Figure 3.1 schematically illustrates

a ordered probit model (H. D. Tran 2007). There are two y-axes showing pipe deterioration in

different variables. One is threshold, 𝜃𝜃 and the other is state and ranges from 1 to 3. The graph has

one curve line which is actual deterioration curve of a pipe and a straight line, 𝑍𝑍𝑖𝑖 which consists

of two parts, deterministic (𝛽𝛽𝑖𝑖′𝑋𝑋𝑛𝑛) and random (𝜀𝜀𝑖𝑖𝑛𝑛) parts. 𝜃𝜃1 and 𝜃𝜃2 are thresholds and 1, 2, and 3

indicate condition states (1 is corresponding to condition rating 9 in NBI). Since 𝑈𝑈𝑖𝑖𝑛𝑛 is

unobservable, the relationship can be expressed by using 𝑍𝑍𝑖𝑖𝑛𝑛 and between two thresholds, 𝛾𝛾𝑖𝑖𝑗𝑗 and

𝛾𝛾𝑖𝑖(𝑗𝑗+1) as follows (Madanat, Mishalani, and Ibrahim 2002):

𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗 𝑖𝑖𝑓𝑓 𝛾𝛾𝑖𝑖𝑗𝑗 ≤ 𝑈𝑈𝑖𝑖𝑛𝑛 < 𝛾𝛾𝑖𝑖(𝑗𝑗+1); Eq. 31

𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗 𝑖𝑖𝑓𝑓 𝑙𝑙𝑓𝑓𝑔𝑔 𝛾𝛾𝑖𝑖𝑗𝑗 − 𝛽𝛽𝑖𝑖′𝑋𝑋𝑛𝑛 ≤ 𝜀𝜀𝑖𝑖𝑛𝑛 < 𝑙𝑙𝑓𝑓𝑔𝑔 𝛾𝛾𝑖𝑖(𝑗𝑗+1) − 𝛽𝛽𝑖𝑖′𝑋𝑋𝑛𝑛; Eq. 32

for j = 0, …, i. Based on the assumption that 𝜀𝜀𝑖𝑖𝑛𝑛 is a normal cumulative distribution, 𝐹𝐹(𝜀𝜀𝑖𝑖𝑛𝑛), the

transition probability from condition state i to condition state i-j for a bridge in an inspection period

can be written as follows:

𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗) = 𝐹𝐹�𝛿𝛿𝑖𝑖(𝑗𝑗+1) − 𝛽𝛽𝑖𝑖𝑋𝑋𝑛𝑛� − 𝐹𝐹(𝛿𝛿𝑖𝑖𝑗𝑗 − 𝛽𝛽𝑖𝑖𝑋𝑋𝑛𝑛) Eq. 33

where 𝛿𝛿𝑖𝑖𝑗𝑗 = 𝑙𝑙𝑓𝑓𝑔𝑔𝛾𝛾𝑖𝑖𝑗𝑗 . The parameter 𝛽𝛽𝑖𝑖 and thresholds 𝛾𝛾𝑖𝑖1 ,𝛾𝛾𝑖𝑖2 , … , 𝛾𝛾𝑖𝑖𝑖𝑖 can be obtained by optimizing

the logarithm objective function as follows:

𝐿𝐿𝑖𝑖∗ = ��𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗)𝑑𝑑𝑛𝑛𝑗𝑗𝑖𝑖−1

𝑗𝑗=0

𝑁𝑁𝑖𝑖

𝑛𝑛=1

Eq. 34

27

𝑙𝑙𝑓𝑓𝑔𝑔(𝐿𝐿𝑖𝑖∗) = ��𝑑𝑑𝑛𝑛𝑗𝑗𝑙𝑙𝑓𝑓𝑔𝑔[𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗)]𝑖𝑖−1

𝑗𝑗=0

𝑁𝑁𝑖𝑖

𝑛𝑛=1

Eq. 35

where:

𝐿𝐿𝑖𝑖∗ is the likelihood function of the ordered probit model for condition state 𝑖𝑖

𝑁𝑁𝑖𝑖 is total number of bridges in condition state 𝑖𝑖 in the data set

𝑑𝑑𝑛𝑛𝑗𝑗 is equal to 1 if 𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗, and 0 otherwise.

After the parameter and thresholds are obtained, the transition probabilities, �̂�𝑝(𝑗𝑗|𝑋𝑋𝑛𝑛 , 𝑖𝑖) for each

bridge can be computed as follows:

�̂�𝑝(𝑗𝑗 = 0|𝑋𝑋𝑛𝑛 , 𝑖𝑖) = 𝐹𝐹�𝛿𝛿𝑖𝑖1 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛�

�̂�𝑝(𝑗𝑗 = 1|𝑋𝑋𝑛𝑛 , 𝑖𝑖) = 𝐹𝐹�𝛿𝛿𝑖𝑖2 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛� − 𝐹𝐹�𝛿𝛿𝑖𝑖1 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛�

�̂�𝑝(𝑗𝑗 = 2|𝑋𝑋𝑛𝑛 , 𝑖𝑖) = 𝐹𝐹�𝛿𝛿𝑖𝑖3 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛� − 𝐹𝐹�𝛿𝛿𝑖𝑖2 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛�

⋮

�̂�𝑝(𝑗𝑗 = 𝑖𝑖|𝑋𝑋𝑛𝑛 , 𝑖𝑖) = 1 − 𝐹𝐹�𝛿𝛿𝑖𝑖𝑖𝑖 − �̂�𝛽𝑖𝑖′𝑋𝑋𝑛𝑛� Eq. 36

for j = 0, 1, 2, …, i. Then the probabilities of each bridge are grouped to estimate the mean value

of transition probabilities, �̂�𝑝𝑖𝑖(𝑖𝑖−𝑗𝑗)𝑔𝑔 as follows:

�̂�𝑝𝑖𝑖(𝑖𝑖−𝑗𝑗)𝑔𝑔 =

1𝑁𝑁𝑔𝑔

� �̂�𝑝(𝑗𝑗|𝑋𝑋𝑛𝑛 , 𝑖𝑖)

𝑁𝑁𝑔𝑔

𝑛𝑛=1

; 𝑗𝑗 = 0, … , 𝑖𝑖;𝑔𝑔 = 1, … ,𝐺𝐺 Eq. 37

where 𝑁𝑁𝑔𝑔 is the total number of bridges in group 𝑔𝑔 and G is the total number of groups (Madanat,

Mishalani, and Ibrahim 2002).

28

Figure 3.1: Schematic illustration of the ordered probit model (H. D. Tran 2007)

Poisson Regression (PR)

The Poisson regression model describes events that occur randomly and independently

over time. Incremental bridge deterioration can be modeled as a function of the number of

transitions in an inspection period by using the Poisson Regression method. From the model,

transition probabilities of a data set of bridge components/elements can be obtained. Different

deterioration models are needed for each condition state because mechanistic deterioration

procedures are different. The model estimates the deterioration of a bridge in an inspection period

using the change in the condition states between two successive inspections. In a continuous-time

Markov process, a deterioration rate is a negative exponential distribution, so the Poisson

distribution is applicable for the deterioration model. The Poisson mass function is as follows

(Madanat, Mishalani, and Ibrahim 2002):

𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗) =𝑒𝑒𝜆𝜆𝑖𝑖𝑛𝑛𝜆𝜆𝑖𝑖𝑛𝑛

𝑗𝑗

𝑗𝑗! , 𝑗𝑗 = 0,1,2, … , 𝑖𝑖; 𝑖𝑖 = 1,2, … , 𝑘𝑘 − 1 Eq. 38

29

where 𝜆𝜆𝑖𝑖𝑛𝑛 is a deterioration rate in condition state 𝑖𝑖; 𝑗𝑗 is the number of transitions in condition state

in an inspection period; 𝑘𝑘 is the highest condition state. The deterioration rate, 𝜆𝜆𝑖𝑖𝑛𝑛 as a function of

explanatory variables is as follows.

𝜆𝜆𝑖𝑖𝑛𝑛 = 𝑒𝑒(𝛽𝛽𝑋𝑋𝑛𝑛) Eq. 39

where 𝛽𝛽 is a set of parameters and 𝑋𝑋𝑛𝑛 is a set of explanatory variables for a bridge, 𝑙𝑙. By optimizing

the objective function (log of the likelihood distribution), the 𝛽𝛽𝑖𝑖 can be obtained. The likelihood

distribution and log of the likelihood are as follows.

𝐿𝐿𝑖𝑖∗ = �𝑒𝑒−𝜆𝜆𝑖𝑖𝑛𝑛𝜆𝜆𝑖𝑖𝑛𝑛

𝑍𝑍𝑖𝑖𝑛𝑛

𝑍𝑍𝑖𝑖𝑛𝑛!

𝑁𝑁𝑖𝑖

𝑛𝑛=1

Eq. 40

𝑙𝑙𝑓𝑓𝑔𝑔(𝐿𝐿𝑖𝑖∗) = �−𝜆𝜆𝑖𝑖𝑛𝑛 + 𝑍𝑍𝑖𝑖𝑛𝑛 𝑙𝑙𝑓𝑓𝑔𝑔(𝜆𝜆𝑖𝑖𝑛𝑛)𝑁𝑁𝑖𝑖

𝑛𝑛=1

Eq. 41

Substitute Eq. 39, then

𝑙𝑙𝑓𝑓𝑔𝑔(𝐿𝐿𝑖𝑖∗) = �𝑍𝑍𝑖𝑖𝑛𝑛(𝛽𝛽𝑋𝑋𝑛𝑛) − 𝑒𝑒(𝛽𝛽𝑋𝑋𝑛𝑛)

𝑁𝑁𝑖𝑖

𝑛𝑛=1

Eq. 42

𝑍𝑍𝑖𝑖𝑛𝑛 is the number of transitions of condition states in an inspection period. It is assumed the

maximum value of 𝑍𝑍𝑖𝑖𝑛𝑛 is equal to 𝑖𝑖. The transition probabilities for each bridge, 𝑙𝑙 is as follows

(Madanat, Mishalani, and Ibrahim 2002):

𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗|𝑋𝑋𝑛𝑛 , 𝑖𝑖) =𝑒𝑒−𝜆𝜆𝑖𝑖𝑛𝑛𝜆𝜆𝑖𝑖𝑛𝑛

𝑗𝑗

𝑗𝑗! , 𝑗𝑗 = 0,1,2, … , 𝑖𝑖 Eq. 43

30

For network-level, bridges in a data set are grouped into condition states. The average transition

probability for each group is computed as follows:

𝑝𝑝𝑖𝑖(𝑖𝑖−𝑗𝑗)𝑔𝑔 =

1𝑁𝑁𝑔𝑔

�𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗|𝑋𝑋𝑛𝑛 , 𝑖𝑖)

𝑁𝑁𝑔𝑔

𝑛𝑛=1

, 𝑗𝑗 = 0,1, … , 𝑖𝑖;𝑔𝑔 = 1, … ,𝐺𝐺 Eq. 44

where 𝑁𝑁𝑔𝑔 is the total number of bridges in group 𝑔𝑔 and G is the total number of facility groups

(Madanat, Mishalani, and Ibrahim 2002).

Negative Binomial Regression (NBR)

In Negative Binomial Regression models, a disturbance term in the parameters of the

Poisson distribution is introduced leading to an “overdispersion” situation where the variance of

data is larger than the mean (Madanat and Ibrahim 2002). 𝜆𝜆𝑖𝑖𝑛𝑛∗ is a random variable and a function

of the selected explanatory variables:

𝜆𝜆𝑖𝑖𝑛𝑛∗ = 𝑒𝑒(𝛽𝛽𝑋𝑋𝑛𝑛+𝜀𝜀𝑛𝑛) Eq. 45

where 𝜀𝜀𝑛𝑛 is a random error term. A negative binomial probability distribution is as follows:

𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗) =𝛤𝛤 � 1

𝛼𝛼𝑖𝑖+ 𝑗𝑗�

𝛤𝛤 � 1𝛼𝛼𝑖𝑖� 𝑗𝑗!

�1

1 + 𝛼𝛼1𝜆𝜆𝑖𝑖𝑛𝑛∗�1 𝛼𝛼𝑖𝑖⁄

�1−1

1 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗�𝑗𝑗

Eq. 46

where Γ( ) is gamma function, 𝛼𝛼𝑖𝑖 is rate of “overdispersion,” and 𝑍𝑍𝑖𝑖𝑛𝑛 is the number of transitions

of condition states in an inspection period. The likelihood distribution function of the negative

binomial for condition state is as follows:

31

𝐿𝐿𝑖𝑖∗ = �𝛤𝛤� 1

𝛼𝛼𝑖𝑖+ 𝑍𝑍𝑖𝑖𝑛𝑛�

𝛤𝛤 � 1𝛼𝛼𝑖𝑖�𝑍𝑍𝑖𝑖𝑛𝑛!

𝑁𝑁𝑖𝑖

𝑛𝑛=1

�1

1 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗�1 𝛼𝛼𝑖𝑖⁄

�1 −1

1 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗�𝑍𝑍𝑖𝑖𝑛𝑛

Eq. 47

By optimizing the log of the objective function, 𝜆𝜆𝑖𝑖𝑛𝑛∗ and 𝛼𝛼𝑖𝑖 can be estimated.

𝑙𝑙𝑓𝑓𝑔𝑔(𝐿𝐿𝑖𝑖𝑛𝑛∗ ) = �𝑙𝑙𝑓𝑓𝑔𝑔�𝛤𝛤 �1𝛼𝛼𝑖𝑖

+ 𝑍𝑍𝑖𝑖𝑛𝑛��− 𝑙𝑙𝑓𝑓𝑔𝑔�𝛤𝛤 �1𝛼𝛼𝑖𝑖��− 𝑙𝑙𝑓𝑓𝑔𝑔(𝑍𝑍𝑖𝑖𝑛𝑛!)

𝑁𝑁𝑖𝑖

𝑛𝑛=1

+1𝛼𝛼𝑖𝑖𝑙𝑙𝑓𝑓𝑔𝑔 �

11 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗

� + 𝑍𝑍𝑖𝑖𝑛𝑛𝑙𝑙𝑓𝑓𝑔𝑔 �1−1

1 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗�

Eq. 48

Transition probabilities of condition states for each bridge can be estimated by applying the

obtained parameter to the negative binomial probability distribution function, and a transition

probability matrix can be obtained by the same process in the Poisson regression method (Madanat

and Ibrahim 2002).

Proportional Hazard Model (PHM)

In the proportional hazard model, the effects of explanatory variables can be explicitly

expressed as hazard ratio in a transition probability matrix. This is the only approach that explicitly

addresses the effects of the explanatory variables in the development of the TPM. The example

format of a transition probability matrix is following (Cavalline et al. 2015).

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎢⎡𝑝𝑝99

𝐻𝐻𝐻𝐻9

000000

1 − 𝑝𝑝99𝐻𝐻𝐻𝐻9

𝑝𝑝88𝐻𝐻𝐻𝐻8

00000

01 − 𝑝𝑝88

𝐻𝐻𝐻𝐻8

𝑝𝑝77𝐻𝐻𝐻𝐻7

0000

00

1− 𝑝𝑝77𝐻𝐻𝐻𝐻7

𝑝𝑝66𝐻𝐻𝐻𝐻6

000

000

1 − 𝑝𝑝66𝐻𝐻𝐻𝐻6

𝑝𝑝55𝐻𝐻𝐻𝐻5

00

0000

1 − 𝑝𝑝55𝐻𝐻𝐻𝐻5

𝑝𝑝44𝐻𝐻𝐻𝐻4

0

00000

1 − 𝑝𝑝44𝐻𝐻𝐻𝐻4

1 ⎦⎥⎥⎥⎥⎥⎥⎤

Eq. 49

where 𝑝𝑝99, 𝑝𝑝88, … ,𝑝𝑝44 are transition probabilities and 𝐻𝐻𝑅𝑅9,𝐻𝐻𝑅𝑅8, … ,𝐻𝐻𝑅𝑅4 are hazard ratios.

32

In the model, the transition probability matrix consists of two parts:

(1) Transition probabilities that are obtained by the simplified Kaplan-Meier method

(2) Hazard ratios associated with the selected explanatory variables

Transition probabilities and hazard ratios are estimated for each condition state. If a hazard ratio

is less than 1, the deterioration process is slower. If a hazard ratio is greater than 1, the deterioration

process accelerates.

Hazard Ratio

Hazard rate is the instantaneous rate of change from a condition state to another condition

state. It can be expressed as a function of explanatory variables as follows:

ℎ(𝑡𝑡, 𝑧𝑧) = ℎ0(𝑡𝑡)𝑒𝑒𝑧𝑧𝛽𝛽��⃗ = ℎ0(𝑡𝑡)𝑒𝑒(𝑧𝑧1𝛽𝛽1+𝑧𝑧2𝛽𝛽2+⋯+𝑧𝑧𝑛𝑛𝛽𝛽𝑛𝑛) Eq. 50

ℎ(𝑡𝑡, 𝑧𝑧) = ℎ0(𝑡𝑡)𝑒𝑒𝑧𝑧1𝛽𝛽 Eq. 51

where 𝛽𝛽 is the regression coefficient associated with the hazard rate z and ℎ0(𝑡𝑡) is the baseline

hazard function. Hazard ratio is the ratio of the hazard rates, the relative risk of failure �ℎ(𝑡𝑡, 1)�

to the value of baseline hazard function �ℎ(𝑡𝑡, 0)�.

𝐻𝐻𝑅𝑅 = ℎ(𝑡𝑡, 1)ℎ(𝑡𝑡, 0) = 𝑒𝑒𝛽𝛽(1−0) = 𝑒𝑒𝛽𝛽 Eq. 52

Kaplan-Meier Estimator

Kaplan and Meier (KM) method is used to estimate non-parametric cumulative transition

probability corresponding to transition times and events, 𝑇𝑇𝑃𝑃(𝑡𝑡𝑥𝑥). This is assumed as a one-step

transition as follows (Archilla, DeStefano, and Grivas 2002):

33

𝑇𝑇𝑃𝑃(𝑡𝑡𝑥𝑥) = 1− 𝑅𝑅�(𝑡𝑡𝑥𝑥) Eq. 53

𝑅𝑅�(𝑡𝑡𝑥𝑥) = [(𝑃𝑃𝑥𝑥 − 1)/𝑃𝑃𝑥𝑥] × 𝑅𝑅𝑥𝑥−1 Eq. 54

where 𝑅𝑅�(𝑡𝑡𝑥𝑥) is the reliability at 𝑡𝑡𝑥𝑥 equal to 𝑇𝑇𝑇𝑇𝑖𝑖𝑗𝑗𝑘𝑘 , 𝑃𝑃𝑥𝑥 is order of times observed in a dataset. 𝑇𝑇𝑇𝑇𝑖𝑖𝑗𝑗𝑘𝑘

is the time associated with one-step transition of bridge 𝑖𝑖 and component 𝑗𝑗 to condition rating 𝑘𝑘.

𝑇𝑇𝑇𝑇𝑖𝑖𝑗𝑗𝑘𝑘 =��𝐹𝐹𝐶𝐶𝑅𝑅𝑘𝑘 − 𝐿𝐿𝐶𝐶𝑅𝑅𝑘𝑘"� + (𝐿𝐿𝐶𝐶𝑅𝑅𝑘𝑘′ − 𝐹𝐹𝐶𝐶𝑅𝑅𝑘𝑘′)�𝑖𝑖𝑗𝑗

2 Eq. 55

where 𝐿𝐿𝐶𝐶𝑅𝑅𝑘𝑘" is last date a component was observed in a prior condition rating 𝑘𝑘". 𝐹𝐹𝐶𝐶𝑅𝑅𝑘𝑘/𝐿𝐿𝐶𝐶𝑅𝑅𝑘𝑘′ is

first/last date a component was observed in initial condition rating 𝑘𝑘′ (Archilla, DeStefano, and

Grivas 2002).

Summary

From the literature review, six models related to the Markovian process were collected and

studied. These included regression nonlinear optimization (RNO), Bayesian maximum likelihood

(BML), ordered probit model (OPM), Poisson regression (PR), negative binomial regression

(NBR), and proportional hazard model (PHM). Some observations about the individual methods

are included herein:

• In the RNO method the third order polynomial was used to describe the deterioration rate of a

bridge. However, the polynomial does not always fit well to the data. The 𝑅𝑅2 value for the

polynomial was less than 0.1 for all data sets considered in Chapter 4.

• In OPM, PR, and NBR methods, the explanatory variable of age is explicitly applied to

estimate the TPM for each condition rating (CR) group. Some CR groups show a wide range

of age. For example, in the group of CR 8 or 7 the age ranges from about 10 to 60 years.

34

• The assumption that deterioration rates are constant in each group is not realistic. For example,

the condition rating can be transition to multiple lower condition ratings according to the data

sets.

Table 3.2 summarizes the objective functions and parameters for each model with some additional

comments.

35

Table 3.2: Summary of objective functions and random variables of models

Models Objective Functions Parameters Comments

Regression Nonlinear

Optimization (RNO)

𝑚𝑚𝑖𝑖𝑙𝑙�|𝑆𝑆(𝑡𝑡) − 𝐸𝐸(𝑡𝑡,𝑃𝑃)|𝑇𝑇

𝑡𝑡=1

𝑃𝑃 Explanatory variables are not explicitly expressed.

Bayesian Maximum Likelihood

(BML)

𝑙𝑙𝑓𝑓𝑔𝑔[𝐿𝐿(𝑌𝑌|𝜃𝜃)] = ��𝑁𝑁𝑖𝑖𝑡𝑡𝑙𝑙𝑓𝑓𝑔𝑔(𝐶𝐶𝑖𝑖𝑡𝑡)9

𝑖𝑖=1

𝑇𝑇

𝑡𝑡=1

𝐶𝐶𝑖𝑖𝑡𝑡 Explanatory variables are not explicitly expressed.

Ordered Probit Model

(OPM)

𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗) = 𝐹𝐹�𝛿𝛿𝑖𝑖(𝑗𝑗+1) − 𝛽𝛽𝑖𝑖𝑋𝑋𝑛𝑛� − 𝐹𝐹(𝛿𝛿𝑖𝑖𝑗𝑗− 𝛽𝛽𝑖𝑖𝑋𝑋𝑛𝑛)

log(𝐿𝐿𝑖𝑖∗) = ��𝑑𝑑𝑛𝑛𝑗𝑗𝑙𝑙𝑓𝑓𝑔𝑔[𝑝𝑝(𝑍𝑍𝑖𝑖𝑛𝑛 = 𝑗𝑗)]𝑖𝑖−1

𝑗𝑗=0

𝑁𝑁𝑖𝑖

𝑛𝑛=1

𝛿𝛿𝑖𝑖𝑗𝑗, 𝛽𝛽𝑖𝑖

Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.

Poisson Regression

(PR) log(𝐿𝐿𝑖𝑖∗) = �𝑍𝑍𝑖𝑖𝑛𝑛(𝛽𝛽𝑋𝑋𝑛𝑛)− 𝑒𝑒(𝛽𝛽𝑋𝑋𝑛𝑛)

𝑁𝑁𝑖𝑖

𝑛𝑛=1

𝛽𝛽

Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.

Negative Binomial

Regression (NBR)

log(𝐿𝐿𝑖𝑖𝑛𝑛∗ ) = �𝑙𝑙𝑓𝑓𝑔𝑔�Γ �1𝛼𝛼𝑖𝑖

+ 𝑍𝑍𝑖𝑖𝑛𝑛��𝑁𝑁𝑖𝑖

𝑛𝑛=1

− 𝑙𝑙𝑓𝑓𝑔𝑔�Γ �1𝛼𝛼𝑖𝑖��

− 𝑙𝑙𝑓𝑓𝑔𝑔(𝑍𝑍𝑖𝑖𝑛𝑛!)

+1𝛼𝛼𝑖𝑖𝑙𝑙𝑓𝑓𝑔𝑔 �

11 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗

�

+ 𝑍𝑍𝑖𝑖𝑛𝑛𝑙𝑙𝑓𝑓𝑔𝑔 �1

−1

1 + 𝛼𝛼𝑖𝑖𝜆𝜆𝑖𝑖𝑛𝑛∗�

𝛼𝛼𝑖𝑖 ,𝜆𝜆𝑖𝑖𝑛𝑛∗

Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.

Proportional Hazard Model

(PHM) Baseline probability function Hazard

ratio

Explanatory variables are explicitly expressed in hazard ratio. Baseline probability function can be any continuous probability function.

36

Chapter 4 : Case Study

Data Review and Filtration

Data describing Texas bridges spanning 2000 to 2010 year were collected from the

National Bridge Inventory (NBI) database and reviewed. Figure 4.1 shows the total number of

bridges in Texas. The number of bridges increased from 2000 to 2008 and it decreased in 2010.

Though these fluctuations are substantial, they are likely do issues with data management at

TxDOT. Most of the bridges that “disappear” between 2008 and 2010 are likely duplicates in the

earlier dataset. The bridges in Texas were grouped according to structure material and design type

in this research.

• Structure material – concrete (non-prestressed)

• Design type – stringer/multi beam or girder; simple span

Figure 4.2 shows the number of bridges categorized by structure material in 2010. The pairs or

years that define the three data sets were randomly selected, and the environment was assumed to

be similar throughout the state. This assumption falls into the general category of “explanatory

variable” and is not the focus of this study. The data were paired as two consecutive inspection

periods for the following reasons:

• To check whether a bridge was subject to the MR&R activities. If there is an increase of

condition rating, the component may have been repaired. All models applied to estimate a

TPM are based on “do nothing” condition, meaning there can be no MR&R activities on

the bridges.

• Some models required bridges to be divided by condition rating groups. For example, in

OPM, PR, and NBR the data in past year such as 2000, 2004, or 2008 was grouped

according to condition ratings and transition probabilities within the groups in present

inspection year such as 2002, 2006, or 2010 were calculated.

Set A consisted of 2000 and 2002 data, set B included 2004 and 2006 data, and set C contained

2008 and 2010. The 2002, 2006, and 2010 data were considered as the present and the 2000, 2004,

and 2008 data were considered as the past in this research. Figure 4.3 shows the number of bridge

37

decks in 2004 grouped by condition ratings. The total number of sample bridge decks in 2004 were

3,870. The condition rating 8 group has about 1,000 decks and the condition rating group 7 has

near 2,500 decks. The selected three data sets were utilized for the consistence test. The

reconstructed bridges (defined by a designation as reconstructed in NBI) were filtered if their

condition ratings were not 9. These bridges were considered to be subject to the MR&R activities,

not considered as new bridges. The bridges more than 60 years were eliminated because the

probability of major repair, retrofit or rehabilitation is much greater. After bridge level filtering,

the data were split by components and filtered again. The bridge components were eliminated if

they showed any increases in condition rating within two consecutive inspection periods. Figure

4.4 shows the total number of bridges before filtering and total number of bridge components after

filtering. The bars hatched with dot indicate decks, the bars filled indicate superstructures, and the

bars hatched with angled straight line indicate substructures. Overall, less than 10 % of the total

number of bridges in the state of Texas were utilized in this research. This is the direct effect of a

explanatory variable selection (i.e., simple span, reinforced concrete multi-girder bridges).

Figure 4.1: Total number of bridges in Texas from 2000 to 2010

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

2000 2002 2004 2006 2008 2010

Num

ber o

f brid

ges

Year

38

Figure 4.2: Number of bridges according to structure material in 2010

Figure 4.3: Number of bridge decks in 2004 by condition rating groups

0

5,000

10,000

15,000

20,000

25,000

30,000N

umbe

r of b

ridge

s

Structure material

0

500

1,000

1,500

2,000

2,500

3,000

9 8 7 6 5 4 3

Num

ber o

f brid

ges

Condition rating

39

Figure 4.4: Number of bridges by component rating after filtering

Results

A TPM was calculated for every method described in Chapter 3 using a combination of

Microsoft Excel 2018 and Matlab 2017b. Figure 4.5 describes the proposed framework for a

multiple model approach. This research focuses on one branch of analysis - Markovian models

within the umbrella of stochastic approaches. The other arms, and the combination of these

approaches with the Markovian models is considered future work. Texas bridges collected from

the NBI database were classified into groups and filtered. Parameters were obtained by regression

analysis and optimization of objective functions. These parameters were applied to estimate

transition probabilities as described in Chapter 3. Each deterioration rate curve was obtained using

the transition probabilities by the Markovian process. The mean transition probability matrix was

computed by averaging the transition probabilities estimated from each model. The deterioration

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

53,520 53,904 54,898 57,278 58,709 51,454

2000 2002 2004 2006 2008 2010

Num

ber o

f brid

ges

Total number of bridgesYear

deck superstructure substructure

40

rate curve was obtained using the mean transition probability matrix by the Markovian process.

Then the upper and lower boundaries were estimated by combining the 95% confidence intervals

of each individual model.

Figure 4.5: Multiple model approach workflow chart

Transition Probability Matrices (TPM)

Table 4.1 shows the combined transition probability matrices for the three bridge

components estimated by averaging each transition probability matrix developed using the six

identified methods. The probabilities in each component are similar values. For example, the

probabilities of a bridge deck with a CR of 9 staying at the same condition rating (P99) are 0.22,

0.24, and 0.26 for bridge deck, superstructure, and substructure respectively. These TPMs

41

represent a basic approach to multiple model deterioration models wherein the TPM from various

methods, a common point in all analyses, are combined prior to developing a deterioration model.

Table 4.2, Table 4.3, and Table 4.4 show transition probability matrices for bridge components

estimated by each model with 2008-2010 data. These values are used to build the TPMs in Table

4.1. In regression nonlinear optimization (RNO), Bayesian maximum likelihood (BML), and

proportional hazard model (PHM), most condition states have appreciable probabilities to either

transition or to stay at the current condition. However, in ordered probit model (OPM), Poisson

regression (PR), and negative binomial regression (NBR), many of the probabilities are 1, except

the probabilities staying at a condition rating 9 or 8. This is true across components.

These findings indicate that three of the approaches to deterioration modeling selected

(OPM, PR, NBR), would not predict any changes from condition states 7 or 6 ever for a bridge

that falls into the family of bridges as defined by the explanatory variables identified. This runs

counter to our understanding of deterioration but appears to be consistent with the data.

42

Table 4.1: Mean transition probability matrices for bridge components with 2008-2010 data

Bridge Component Transition Probability Matrix

Deck 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.22

000000

0.780.82

00000

00.180.99

0000

00

0.010.99

000

000

0.010.91

00

0000

0.090.79

0

00000

0.241 ⎦⎥⎥⎥⎥⎥⎤

Superstructure 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.24

000000

0.760.69

00000

00.310.99

0000

00

0.010.98

000

000

0.020.82

00

0000

0.180.95

0

00000

0.051 ⎦⎥⎥⎥⎥⎥⎤

Substructure 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.26

000000

0.740.87

00000

00.130.98

0000

00

0.020.99

000

000

0.010.82

00

0000

0.180.97

0

00000

0.031 ⎦⎥⎥⎥⎥⎥⎤

43

Table 4.2: Transition probability matrices of bridge decks with 2008-2010 data

Model Transition Probability Matrix

Regression Nonlinear

Optimization 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.84

000000

0.160.78

00000

00.220.99

0000

00

0.011000

0000

0.4900

0000

0.510.56

0

00000

0.441 ⎦⎥⎥⎥⎥⎥⎤

Bayesian Maximum Likelihood

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.93

00000

00.070.99

0000

00

0.010.99

000

000

0.01100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Ordered Probit Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.71

00000

00.29

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Poisson Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.76

00000

00.24

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Negative Binomial

Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.76

00000

00.24

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Proportional Hazard Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.5000000

0.50.97

00000

00.030.97

0000

00

0.030.93

000

000

0.07100

0000000

0000011⎦⎥⎥⎥⎥⎥⎤

44

Table 4.3: Transition probability matrices of bridge superstructures with 2008-2010 data

Models Transition Probability Matrix

Regression Nonlinear

Optimization 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.82

000000

0.180.75

00000

00.250.99

0000

00

0.010.99

000

000

0.010.95

00

0000

0.050.90

00000

0.11 ⎦⎥⎥⎥⎥⎥⎤

Bayesian Maximum Likelihood

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.800000

00.2

0.990000

00

0.010.99

000

000

0.01100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Ordered Probit Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.53

00000

00.47

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Poisson Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.56

00000

00.44

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Negative Binomial

Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.56

00000

00.44

10000

0001000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Proportional Hazard Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.63

000000

0.37091

00000

00.090.96

0000

00

0.040.9000

000

0.1000

00001

0.80

00000

0.21 ⎦⎥⎥⎥⎥⎥⎤

45

Table 4.4: Transition probability matrices of bridge substructures with 2008-2010 data

Models Transition Probability Matrix

Regression Nonlinear

Optimization 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.83

000000

0.170.77

00000

00.230.99

0000

00

0.010.97

000

000

0.030.94

00

0000

0.060.88

0

00000

0.121 ⎦⎥⎥⎥⎥⎥⎤

Bayesian Maximum Likelihood

𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.900000

00.1

0.980000

00

0.020.99

000

000

0.010.99

00

0000

0.010.99

0

00000

0.011 ⎦⎥⎥⎥⎥⎥⎤

Ordered Probit Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.85

00000

00.050.99

0000

00

0.011000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Poisson Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.84

00000

00.160.99

0000

00

0.011000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Negative Binomial

Regression 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0000000

10.87

00000

00.130.99

0000

00

0.011000

0000100

0000010

0000001⎦⎥⎥⎥⎥⎥⎤

Proportional Hazard Model 𝑃𝑃 =

⎣⎢⎢⎢⎢⎢⎡0.71

000000

0.290.96

00000

00.040.97

0000

00

0.030.96

000

000

0.04000

00001

0.950

000000.1⎦⎥⎥⎥⎥⎥⎤

46

Deterioration Rate Curves

Deterioration rate curves were plotted for each component in Figure 4.6, Figure 4.7 and

Figure 4.8. The graphs show nine individual curves. Grey lines depict deterioration curves of each

of the six models and black solid line is the deterioration curve estimated with the mean transition

probability matrix. The black dash line is the upper boundary (UB) and black dash-dot line is the

lower boundary (LB). To generate a deterioration rate curve applying a multiple model approach,

the individual models were combined. Using Eq. 25 with the mean transition probability matrix

(Table 4.1), a deterioration rate curve was generated.

The upper and lower boundary envelopes were estimated by:

1. checking the goodness-of-fit for each curve

2. selecting the third order polynomial for all curves based on the value of 𝑅𝑅2, closer to 1

3. applying the polynomial to each curve and obtaining the lower and upper boundary with

95% confidential interval

4. comparing the boundary values of each curve and

5. determining the lowest and highest values at any time.

If the value showed lower than zero, the lower limit value was zero. If the value increased, the

upper limit value was the value before increasing. Therefore, the estimated future condition rating

is within the boundaries in 95% certainty. For example, the CR of a bridge deck at 50 years old

can be 4, 5, 6, 7 or 8.

The bridge components deteriorate fast up to about 20 years and then the deterioration slows down

after 20 years shown in most of the models, except in PHM. In the lower boundary curve, the

condition rating reaches to zero at 58 years and 54 years in deck and superstructure, respectively.

47

Figure 4.6: Deterioration rate curves for bridge deck

Figure 4.7: Deterioration rate curves for bridge superstructure

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

Cond

ition

ratin

g

Age

PR PHMOPM BMLRNO NBRMean LB

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

Cond

ition

ratin

g

Age

PR PHMOPM BMLRNO NBRMean LB

48

Figure 4.8: Deterioration rate curves for substructure

Comparison

The estimated probability distribution obtained from each individual model was compared

with the actual distribution by using the chi-squared goodness-of-fit test to find which model best

estimated the probability distribution of the actual value. The consistency between TPMs obtained

using three different data sets in each model was measured using the modal assurance criterion test

as well as the deterioration curves obtained from the TPMs.

Chi-squared goodness-of fit Test

This test measures the closeness between the predicted probability distribution and actual

probability distribution. The formulation of chi-square goodness-of-fit is

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

Cond

ition

ratin

g

Age

PR PHMOPM BMLRNO NBRMean LB

49

𝜒𝜒2 = �(𝑅𝑅𝑖𝑖 − 𝐸𝐸𝑖𝑖)2

𝐸𝐸𝑖𝑖

𝑘𝑘

𝑖𝑖=1

Eq. 56

where 𝑘𝑘 is number of observations; 𝑅𝑅𝑖𝑖 is actual condition state of the ith observation; and 𝐸𝐸𝑖𝑖 is

expected value of the ith observation and computed by using Eq. 25. The smaller chi squared value

means the estimated value is closer to actual value. For a simple computation, set the initial

condition probability distribution, 𝑄𝑄0 (Eq. 57) using 2008 data. The probability distribution of

time, t equal to 2 was estimated (Eq. 58). The results are present in Table 4.5. The most values

are less than 0.02, except the value of RNO substructure, which is 0.10. Overall, the models predict

future condition state distributions of bridge components well.

𝑄𝑄0 = [0.001 0.161 0.715 0.112 0.011 0.001 0] Eq. 57

The expected value of condition state distribution, 𝑃𝑃(2) at time, t=2 is computed as follows:

𝑃𝑃(2) = 𝑄𝑄0 × 𝑃𝑃2 Eq. 58

Table 4.5: Results of Chi-square goodness-of-fit of bridge components

Model Deck Superstructure Substructure

RNO 0.02 0.01 0.10

BML 0.01 0.01 0.02

OPM 0.02 0.01 0.01

PR 0.01 0.01 0.01

NBR 0.02 0.01 0.01

Mean 0.02 0.02 0.01

50

In this test, the proportional hazard model was not included because a probability distribution of

bridge components cannot be estimated with this model. However, the future condition rating of

each bridge component can be predicted by using this model. An example is following.

If a bridge deck is 3 years old and we want to know what condition rating of the deck after 5 years.

𝐸𝐸 = 𝑃𝑃8 × 𝑅𝑅 Eq. 59

where E is the estimated condition rating; P is the transition probability matrix by using PHM; and

R is the condition rating vector, [9 8 7 6 5 4 3].

Modal Assurance Criterion (MAC)

MAC is used as a statistical indicator to compare modes quantitatively in modal analysis.

The MAC value represents the consistency of modes of two structures. This concept was applied

to compute the consistency of TPMs and deterioration curves between the three data sets used in

this research. The value of the MAC is between 0 and 1. A value larger than 0.9 represents that the

modes are consistent and a value closer to 0 represents that the modes are less consistent (Pastor

et al. 2012). The calculation of the MAC is as follows:

𝑀𝑀𝐴𝐴𝐶𝐶(𝐴𝐴,𝑋𝑋) =�∑ {𝜑𝜑𝐴𝐴}𝑗𝑗{𝜑𝜑𝑋𝑋}𝑗𝑗𝑛𝑛

𝑗𝑗=1 �2

�∑ {𝜑𝜑𝐴𝐴}𝑗𝑗2𝑛𝑛𝑗𝑗=1 ��∑ {𝜑𝜑𝑋𝑋}𝑗𝑗2𝑛𝑛

𝑗𝑗=1 �

Eq. 60

where {𝜑𝜑𝐴𝐴} and {𝜑𝜑𝑋𝑋} are the two sets of vectors. The results are shown in Table 4.6 and Table 4.7.

The first observation of note is that all of the MAC values for deterioration curves are 1. Second,

the values in RNO, BML, OPM, PR, and NBR are 0.9 or greater. This means estimated transition

probabilities using these models with different data sets are similar. Some MAC values in PHM

are lower than 0.9. In the deck, the values of A&B and A&C are 0.73 and 0.67 respectively. In

other words, TPMs estimated with different data set using PHM are not consistent. In Table 4.9,

51

the transition probability matrices of bridge decks using PHM with set A, set B, and set C are

shown. However, the deterioration rate curves using these TPMs are consistent. Therefore, a

deterioration rate does not change by different data sets.

Table 4.6: Results of Modal Assurance Criterion for bridge deck

Models TPM Deterioration curve

A&B A&C B&C A&B A&C B&C RNO 0.97 0.96 1.00 1.00 1.00 1.00 BML 1.00 0.98 0.98 1.00 1.00 1.00 OPM 0.97 0.97 0.91 1.00 1.00 1.00 PR 0.99 0.97 0.94 1.00 1.00 1.00

NBR 0.99 0.97 0.92 1.00 1.00 1.00 PHM 0.73 0.67 0.95 1.00 1.00 1.00 Mean 0.99 0.97 0.98 1.00 1.00 1.00

Table 4.7: Results of Modal Assurance Criterion for bridge superstructure

Models TPM Deterioration curve

A&B A&C B&C A&B A&C B&C

RNO 0.99 1.00 0.98 1.00 1.00 1.00

BML 1.00 0.98 0.98 1.00 1.00 1.00

OPM 0.95 0.98 0.89 1.00 1.00 1.00

PR 0.97 0.97 0.92 1.00 1.00 1.00

NBR 0.96 0.97 0.89 1.00 1.00 1.00

PHM 0.97 0.88 0.91 1.00 0.98 0.98

Mean 0.99 0.99 0.98 1.00 1.00 1.00

52

Table 4.8: Results of Modal Assurance Criterion for bridge substructure

Table 4.9: Transition probability matrices of decks using proportional hazard model

Set A 𝑃𝑃 =

⎣⎢⎢⎢⎢⎡0.77

000000

0.230.98

00000

00.020.96

0000

00

0.040.81

000

000

0.190.98

00

0000

0.020.98

0

00000

0.021 ⎦⎥⎥⎥⎥⎤

Set B 𝑃𝑃 =

⎣⎢⎢⎢⎢⎡0.24

000000

0.760.98

00000

00.020.97

0000

00

0.030.75

000

000

0.25000

00001

0.330

00000

0.671 ⎦⎥⎥⎥⎥⎤

Set C 𝑃𝑃 =

⎣⎢⎢⎢⎢⎡0.50

000000

0.500.97

00000

00.030.97

0000

00

0.030000

000

0.93000

000

0.07100

0000011⎦⎥⎥⎥⎥⎤

Models TPM Deterioration curve

A&B A&C B&C A&B A&C B&C

RNO 1.00 1.00 1.00 1.00 1.00 1.00

BML 1.00 0.99 0.98 1.00 1.00 1.00

OPM 0.94 0.99 0.92 1.00 1.00 1.00

PR 0.98 0.99 0.95 1.00 1.00 1.00

NBR 0.98 0.99 0.95 1.00 1.00 1.00

PHM 0.97 0.84 0.83 1.00 1.00 1.00

Mean 0.99 0.99 0.98 1.00 1.00 1.00

53

Figure 4.9: Deterioration curves estimated using by proportional hazard model

0123456789

10

0 10 20 30 40 50 60 70

Cond

ition

Rat

ing

Age

Bridge Deck

set Aset Bset C

54

Chapter 5 : Conclusions and Future Work

Summary

This research presented an effort to produce a multiple model approach to deterioration

modeling in an effort to combat the inherent biases of selecting a single model for a task. Transition

probability matrices and deterioration curves obtained from six different models, including

regression nonlinear optimization, Bayesian maximum likelihood, ordered probit model, Poisson

regression, negative binomial regression, and proportional hazard model, were combined. For data

analysis and computation of parameters for each model, Microsoft Excel 2018 and Matlab 2017b

were utilized. The lower and upper boundaries estimated with 95% confidence for each

deterioration model by regression analysis were combined to generate bounds. By including all

models, and establishing reasonable bounds of confidence, the decision-maker has more

confidence in their deterioration modeling results. Thus, the multiple model approach is more

robust and flexible than a single approach to deterioration modeling. Finally, chi-square goodness-

of-fit and modal assurance criterion tests were presented to measure the closeness and consistence

of models. Conclusions and future work recommendations are presented herein.

Conclusions

Models are a useful tool to understand future behavior of systems. In many fields, like weather

forecasting, multiple models are used to develop predictions. In civil engineering, typically a single

model is used, and deterioration modeling is no exception. Any single deterioration modeling

approach has idiosyncrasies and differences that can cause uncertainty and false confidence in the

results in certain circumstances. The proposed multiple model approach provides an average future

condition state of bridge components at each age. However, all bridges at the same age in actual

data are not the same condition rating. Figure 5.1 shows the actual data of bridge decks in 2010.

The plot presents three sets of data. The black dots (Mean) are the mean values of condition ratings

at any age. The dark grey dots (UB) are the upper limit values at any age and the light grey dots

55

(LB) are the lower limit values at any age. From single model approach, for example, the condition

rating of bridge deck at 50 years might be 7. However, the condition rating of some decks shows

8 or 4. Figure 5.2 presents actual bridge deck data in 2010 and a deterioration curve (black solid

line) with upper (black dash line) and lower (black dash-dot line) boundaries obtained by using

multiple model approach. This approach provides an average condition rating and other possible

condition ratings of decks at each age. For example, the average condition rating of decks at 50

years is about 6.5 and the range of possible condition ratings is from about 8 to 4. What is evident

from this analysis is that the selected data set, inclusive of reinforced concrete multi-girder bridges

in Texas, does not provide enough variation to highlight the differences in any individual model.

As such, it is difficult to demonstrate the potential benefits of the multiple model approach. For

this population of bridges, there is little difference between any single model nor between any of

the single models and the multiple model approach. This is partially an effect of the data itself,

which is subjective and is also inexorably tied to financial, logistical, and political forces that use

condition data to make administrative decisions. If the data does not demonstrate enough

variability, then model choice is irrelevant.

Figure 5.1: Actual bridge deck condition ratings in 2010

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70

Cond

ition

Rat

ing

Age

MeanLBUB

56

Figure 5.2: Deterioration curves of multiple model approach and bridge deck condition ratings in 2010

Future Work

Some future works were identified during this research to improve reliability of predicting

future condition states of bridges. The recommendations for future work are followings:

• Identification of a different research dataset which includes more inherent variability in

deterioration.

• Establish the importance and influence of explanatory variables in development of

deterioration models.

• Integration of more models such as mechanistic or artificial intelligence approaches into the

multiple model approach.

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

Cond

ition

Rat

ing

Age

MeanLBUBestimated Meanestimated LB

57

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Vita

Jin A Collins earned a Bachelor of Science in Civil Engineering from California State

University Long Beach in 2007. She worked for the County of Los Angeles before attending the

University of Texas El Paso for a Bachelor of Science in Physics which she obtained in 2017.

She then began pursuing her Master of Science in Civil Engineering at The University of Texas

El Paso, and plans to continue working the field of infrastructure deterioration analysis with a

focus on bridges while she pursues her Ph.D.