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Modeling truck accidents at highway interchanges prediction models using both conventional and artificial intelligence approaches

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Title:
Modeling truck accidents at highway interchanges prediction models using both conventional and artificial intelligence approaches regression, neural networks, and fuzzy logic
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Awad, Wael Hassan
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English
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xiii, 241 leaves (some folded) : illustrations ; 29 cm

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Truck accidents ( lcsh )
Truck accidents ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 237-241).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, Civil Engineering.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Wael Hassan Awad.

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University of Colorado Denver
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Auraria Library
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ocm37822191
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LD1190.E53 1997d .A83 ( lcc )

Full Text
MODELING TRUCK ACCIDENTS AT HIGHWAY INTERCHANGES
PREDICTION MODELS USING BOTH
CONVENTIONAL AND ARTIFICIAL INTELLIGENCE APPROACHES
REGRESSION, NEURAL NETWORKS, AND FUZZY LOGIC
by
WAEL HASSAN AWAD
BS, University of Jordan, 1986
MS, University of Colorado at Denver, 1991
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Civil Engineering
1997


1997 by Wael Hassan Awad
All rights reserved.
n


This thesis for the Doctor of Philosophy
degree by
Wael Hassan Awad
has been approved
by
Bruce N. Janson
Dan M. Frangopol
Sarosh I. Khan
3/3/hr
a
Date


A wad, Wael Hassan (Ph.D., Civil Engineering)
Modeling Truck Accidents at Highway Interchanges
Prediction Models Using Both Conventional and Artificial Intelligence Approaches:
Regression, Neural Networks, and Fuzzy Logic
Thesis directed by Associate Professor Bruce N. Janson
ABSTRACT
Large trucks represent a significant proportion of overall vehicle volumes on
the nation's highways, and this proportion is increasing at the same time that larger
and longer trucks are being used. Highway geometric design elements, including
interchange configurations and ramp characteristics, contribute significantly to traffic
accidents that involve trucks. However, this contribution is very difficult to quantify,
because of the confounding influence of other factors, such as human behavior, traffic
conditions, and prevailing weather conditions.
Most previous accident studies used regression analysis to develop equations
to explain accident rates. All previous attempts have had mixed results, and no set of
geometric/accident relationships is widely accepted. Deficiencies of such models
were attributed to different factors, such as quality and quantity of accident data and
statistical methods used for prediction.
IV


Accident reporting systems in most states compile information about many
variables that contribute to accident causation in a non consistent way. For example,
for two different accidents, "wet surface" can be a contributing factor to one accident,
but only a neutral factor in another accident. Existing accident reporting systems do
not solve this problem.
In this study, different approaches were applied to explain truck accidents at
interchanges in Washington state during the period from 1/1/1993 to 3/3/1995. Three
models for each ramp type were developed using linear regression, neural networks,
and a hybrid system using fuzzy logic and neural networks. The study showed that
linear regression was only able to predict accident frequencies that fell within one
standard deviation from the overall mean of the dependent variable. However, the
coefficient of determination was very low in all cases.
The other two artificial intelligence (AI) approaches showed a high level of
performance in identifying different patterns of accidents in the training data, and
presented a better fit when compared to the linear regression. However, the ability of
these models to predict test data that was not included in the training process showed
unsatisfactory results. The results suggest that AI approaches are promising tools for
exploring the problem, but that the data have many deficiencies.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed*
Bruce W. Janson


'DecUaztim
*] dedicate tArie t&ecic to mtf, puvieute, cut^e, {pwtily, and fyUenoU,
{jW tAeir inceecant encouna/^ement and patience tAxouyAout tAe
fraet yeane. Specifically, *1 devote tAio tuonA to uncle /tcoad and
Aic wife.


rfdwotij£eclcymettt&
On the name ofa Adah, moot ynacioue, moot mencifaul
'Zt/e taiee to decree ofa wiodom whom TVe pleaee: hut oven all
endowed with hnowledye ie one, the atl-huowiny
OÂ¥oly tZunau
s. 12/1.76
0 ant deeply indebted to the almiyhty Adah faon hie mency
upon me, and to my panente faon thein tove and euppont.
0 ynatefaully achuowledye the aeeiotance ofa my adoieon,
*Dn. "Siucc Ot. fyaneon. and my committee memhenc: *Dn. tyamee
'Diehmann; *Dn. *Dan 'pxanyopol; *Dn, Aynn tyohtneon; and *Dn,
Sanoeh 'Khan faon thein intiyht and oaluaMe comments. Ado. 0
would (the to thanh both *Dn. 'Zl/illiam 'ZVolfae and *D%. tyamee
Koehlen faon thein euyyeotione.
\finally, 0 wieh to expneee my appneciation to OKathwonhe One.
faon pmooidiny me with OZiAHAA^ eofatwane. Spzeeial thanhe yo
to thein technical euppont enyineene faon thein eenoice.
TUael Awad


CONTENTS
Tables ................................................................x
Figures ..............................................................xi
CHAPTER
1. INTRODUCTION .......................................................1
Background ....................................................1
Problem Statement and Research Objectives .....................6
Review of Relevant Research ...................................7
Traffic Accidents .......................................8
Interchange Safety .....................................10
Truck Accidents ........................................10
Truck Accidents at Interchanges ........................12
Truck Exposure Measures ................................13
Neural Network .........................................15
Fuzzy Logic ............................................20
2. DATA ACQUISITION AND PRELIMINARY ANALYSIS ........................23
Data Inventory ...............................................23
Data Manipulation and Preparation ............................26
Quality of Data ..............................................29
Preliminary Observations from Washington State Truck Accidents
Data .........................................................34
Factor Analysis and Variable Selection .......................37
viii


3. MODELS FORMULATION .............................................40
Conventional Multiple Linear Regression Model ...............44
Intrinsically Linear Regression Model .......................45
Neural Network Model ........................................47
ANFIS Model .................................................51
ANFIS Architecture .....................................52
Model Formulation ......................................56
4. MODELS DEVELOPMENT .............................................59
Conventional Multiple Linear Regression Model ...............59
Intrinsically Linear Regression Model .......................60
Neural Network Model ........................................72
ANFIS Models ................................................84
5. RESULTS AND FINDINGS ...........................................97
Multiple Linear and Intrinsically Linear Regression Models ..97
Neural Network Model .......................................100
ANFIS Models ...............................................102
Sensitivity Analysis for Neural Networks ...................103
Sensitivity Analysis for ANFIS Model .......................106
Evaluation of Models .......................................106
6. CONCLUSIONS, RECOMMENDATIONS, AND FUTURE WORK ................Ill
General Findings and Observations ..........................112
Recommendations for Future Work ............................115
APPENDIX ..........................................................118
A. Traffic and Truck Growth Statistics .....................119
B. Washington State Data ...................................122
C. Possibility Theory and Fuzzy Logic ......................129
IX


D. Linear Regression Models ..........................144
E. Neural Network Models .............................158
F. ANFIS Models ......................................171
G. All Models Comparison .............................232
BIBLIOGRAPHY ...............................................237
x


TABLES
Table
2.1: Comparison of Truck Accidents per Year in Three States by Ramp Type 25
2.2: Washington Truck Accidents on All Ramps ............................. 31
2.3: Washington Truck Accidents By Ramp Type ............................. 32
2.4: Washington Truck Accidents by Ramp Type, Conflict Area .............. 34
2.5: Washington Truck Accidents per RTADT by Ramp Type, Conflict Area .... 35
2.6: Washington Truck Accidents per RTVMT by Ramp Type, Conflict Area .... 36
3.1: Training and Checking Data Size ..................................... 43
4.1: Multi-Linear Regression Models ...................................... 60
4.2: Intrinsically Multi-Linear Regression Model ......................... 61
4.3: Linear Regression RMSE per Ramp Type ................................ 62
4.4: Mean and Standard Deviation per Ramp Type ........................... 63
4.5: Neural Network RSME per Ramp Type With Four Inputs .................. 73
4.6: ANFIS Minimum RMSE .................................................. 86
4.7: ANFIS RSME per Ramp Type ............................................ 87
5.1: Sensitivity Analysis for Neural Network Model....................... 105
5.2: Sensitivity Analysis for ANFIS Model ............................... 107
5.3: Training Data RMSE by Ramp Type by Modeling Method ................. 108
5.4: Checking Data RMSE by Ramp Type by Modeling Method ................. 110
XI


CO
FIGURES
Figure
1.1 : Multiple Layer Neural Network........................................ 16
1.2 : Multiple-Input Neuron ............................................... 18
2.1 : Ramp Conflict Areas ................................................. 26
2.2. a: Truck Accident Distribution Upstream And Downstream of On-Ramps .... 27
2.2. b: Truck Accident Distribution Upstream And Downstream of Off-Ramps .... 28
2.3 : Distribution of Ramps in Washington State by TAF During 27 Months ... 33
2.4 : The Variables Considered in The Modeling From Washington State
Database ............................................................. 39
3.1 : The Methodology for Prediction Models ............................... 41
3.2 : Training and Checking Data by Ramp Type in Washington State ......... 43
3.3 : Two-layer Network (Input-Hidden-Output) ............................. 47
3.4 : Log-sigmoid Transfer Function ........................................48
3.5 : ANFIS Output Function for Sugeno Type 1 Model ........................51
3.6 : Sugeno-type Systems ..................................................52
.7 : Fuzzy Reasoning for Sugeno Type 1 model ...............................53
3.8 : ANFIS Architecture for Sugeno Type 1 model ...........................54
4.1 .a: Diamond Ramps Linear Regression Model (Training Data)............... 64
4.1 .b: Loop Ramps Linear Regression Model (Training Data).................. 65
4.1 .c: Outer Connector Ramps Linear Regression Model (Training Data)....... 66
4.1 .d: Directional Ramps Linear Regression Model (Training Data)........... 67
4.2. a: Diamond Ramps Linear Regression Model (Checking Data)............... 68
4.2. b: Loop Ramps Linear Regression Model (Checking Data).................. 69
4.2. c: Outer Connector Ramps Linear Regression Model (Checking Data)....... 70
4.2. d: Directional Ramps Linear Regression Model (Checking Data)........... 71
xii


Figures Cont.
4.3 : Directional Ramps Neural Network Model (Training and SSE)
With Six Inputs...................................................... 73
4.4. a: Diamond Ramps Neural Network Model With Four Inputs ................75
4.4. b: Loop Ramps Neural Network Model With Four Inputs .................. 76
4.4. c: Outer Connector Ramps Neural Network Model With Four Inputs ....... 77
4.4. d: Directional Ramps Neural Network Model With Four Inputs ........... 78
4.5. a: Diamond Ramps Neural Network Model (Checking data) ................ 80
4.5. b: Loop Ramps Neural Network Model (Checking data) ................... 81
4.5. c: Outer Connector Ramps Neural Network Model (Checking data) ........ 82
4.5. d: Directional Ramps Neural Network Model (Checking data) ............ 83
4.6 : ANFIS Model ....................................................... 85
4.7. a: Diamond Ramps ANFIS (Training Set 1) .............................. 88
4.7. b: Loop Ramps ANFIS Model (Training Set 1) ........................... 89
4.7. c: Outer Connector Ramps ANFIS Model (Training Set 1) ................ 90
4.7. d: Directional Ramps ANFIS Model (Training Set 1) .................... 91
4.8. a: Diamond Ramps ANFIS (Checking Set 1) .............................. 93
4.8. b: Loop Ramps ANFIS Model (Checking Set 1) ........................... 94
4.8. c: Outer Connector Ramps ANFIS Model (Checking Set 1) ................ 95
4.8. d: Directional Ramps ANFIS Model (Checking Set 1) .................... 96
5.1 : Multi-Linear Regression Model for Directional Ramps ................. 99
xiii


CHAPTER 1
INTRODUCTION
Background
Large trucks represent a significant proportion of the overall vehicle flow on
the nation's highways. National statistics show that this proportion is increasing at
the same time that larger and longer trucks are being used. This has been noted
particularly since the Surface Transportation Assistance Act (STAA) was passed in
1982. The STAA minimized restrictions on the size and weight of trucks allowed on
federal highways. The Tandem Truck Safety Act (TTSA), which followed in 1984,
allowed yet greater access for large vehicles to the highway system.
Trucks provide customers with more flexible service than other freight
transportation modes by offering door-to-door service. Thus, the demand for this
service is expected to keep increasing. The average annual increase in truck
registrations is three percent, or approximately four million new trucks entering the
fleet each year.
According to the Transportation Statistics Annual Report 1994:
Trucks and buses account for over one-forth of all vehicle miles of travel in
the U.S. Trucks with six or more tires, ranging from local delivery vehicles to
combination trucks with three trailers, account for less than 10 percent of total
vehicle miles of travel.
1


In terms of freight movement, the same report says:
Trucking has shown slow but continuous gains in modal share since 1950.
Overall shares were about 16 percent in 1950; they reached 20 percent by
1960 and 25 percent in the mid-1980s. Present shares are in the range of 26-
27 percent of ton-miles, and 79 percent of the overall freight movement
revenue.(Appendix A, Figures 1, 2)
An interchange is a system of interconnecting roadways that provides for
movements between two or more grade-separated highways. The connection for
traffic flow between one roadway and the other is called a ramp. Many ramp types
and different interchange configurations could be selected during the design process,
depending on a variety of considerations, including safety, capacity, and roadway-
functions.
The ramp dimensions and design configurations of many existing interchanges
were designed to accommodate small vehicles (passenger cars, vans, and pickups),
rather than large ones. In general, the interstate systems that exist today have been
built with the ultimate geometric standards. However, the federally-aided primary
and secondary systems in many instances include below-standard geometric designs,
which are critical to the safety of large-truck operations.
Many highway safety engineers and researchers believe that the combination
of increased use of these ramps by large trucks and the fact that the ramps were not
designed to meet large-vehicle requirements, are the major reasons for the increasing
number of accidents involving large vehicles on ramps. Nationally, 20 percent of
truck accidents occur at interchanges. In Colorado, this percentage increased from
about 23.6 percent in 1991 to 30.9 percent in 1993. During the period from 1/1/1993
to 3/31/1995, the truck accident record for Washington state revealed that about 41
2


percent of the truck accidents on the Washington state highway system occurred
within the interchange influence areas.
The 0.5 % fatality rate for accidents that involve trucks is higher than the
comparable 0.3 % rate for accidents involving small vehicles. The injury rates were
39 % and 54 %, respectively. The economic consequences of truck accidents,
according to a study by Bowman, B. and Lum, H. ,1990 could be summarized as
follows:
" An estimate of the total annual cost of urban freeway truck accidents was
determined to be $634,000 per freeway mile. This cost consisted of accident
costs of $182,000, delay costs of $440,000, clean-up costs of $3,000, and
operating costs of $9,000 per freeway mile. Expanding this estimate to the
1,937 interstate and 560 freeway miles in the United States with average daily
traffic volumes of over 100,000 vehicles results in a nationwide annual cost of
$1.6 billion."
highway geometric design elements, including interchange configurations and
ramp characteristics, contribute significantly to traffic accidents, especially where
large vehicles such as trucks are concerned. However, this effect is very difficult to
quantify because of the confounding influences of other factors. Those factors
include human behavior, the prevailing environmental conditions, the amount of
traffic, and the operational characteristics of the large vehicles themselves.
To establish a relationship between traffic accidents and the factors involved,
high quality data is required, as well as the use of good statistical methods that
overcome the problems and deficiencies inherent in previous models. The data
compiled for most accident reporting systems are typically produced by police
accident investigations. Police officers usually have neither the time nor the
experience to conduct in-depth accident investigations or collect the necessary data.
3


Also, it is not always practical to investigate fully each accident. Doing so might
delay traffic and require a fully trained crew for investigations.
Many state departments of transportation are already in the process of
improving the quality of their accident reporting systems, and of attempting to include
in them other factors related to the vehicle, driver, road, traffic, and weather
conditions prior to and after an accident takes place.
Washington state was one of the leading states in this area. The Washington
traffic accident file has about 90 fields of information for each recorded accident,
ranging from time and date of accident to pedestrian information. It also includes
other data related to location, driver, vehicle, and the road (Appendix B). Four other
sets of data in a hard copy format were supplemented with the above Washington
accident file (Appendix B).
The first data set contains drawings (not to scale) for all the interchanges in
the Washington highway system (about 500 locations connecting some 7000 miles of
road). Each drawing shows the milepost of the gore and beginning of the deceleration
lane, or the end of the acceleration lane for each ramp (taper). The second data set is
the annual traffic report for years 1992 to 1994. The information included is the
Annual Average Daily Traffic (AADT) volume for the main lane of state routes at
different major crossroads (interchanges), and some truck percentages at selected
locations. The third data set includes traffic counts for most of the ramps (about 75
% of them) and less than 10 % of the truck percentages at such ramps. The fourth
data set has ramp information, such as ramp length, number of lanes, and shoulder
information.
4


Despite the fact that Washington data is the most comprehensive recorded
data in the nation, it remains uncertain and ambiguous in regard to human error (in its
reporting and coding), and in regard to the fact that a period of time often elapses
between the accident and the reporting time. Therefore, the uncertainty of data has to
be considered in the modeling process, with this uncertainty related to the criterion
variable as well as the explanatory variables used in the prediction models.
However, in this study, we still acknowledge that the main deficiency in the
existing traffic accident databases-including Washington state database-was not
treated yet. This deficiency is related to the nature of the recorded variables into the
accident report, and it is partially behind the unsatisfactory results of previous
prediction models.
The existing accident reporting systems nationwide have partial information
about each recorded accident, the reported variables are related to the prevailing
conditions at the time of the accident regardless whether these variables are
contributing to the causation of the accident or not. A complete accident information
database should distinguish among variables in terms of their level of contribution to
the accident causation, and should lead to better prediction results.
Statistically, it is not difficult to find a relationship among randomly generated
variables. In a simulation study, the R2 for a regression model with 50 variables
randomly generated from a standard normal distribution was 0.59. Therefore, using
meaningful variables is very significant in any prediction model.
5


In our study, different approaches will be examined in an attempt to study
truck accidents at interchanges. A traditional regression model will be built and
compared to other models that will employ some Artificial Intelligence techniques
such as neural network and fuzzy logic. The combination of the neural networks and
fuzzy logic is called a hybrid system. The proposed hybrid model will use an
Adaptive-Network-Based Fuzzy Inference System (ANFIS).
The goal behind using new techniques, such as neural network and fuzzy
logic, is to evaluate their performance, and compare them to the traditional regression
methods. The developed models will be used as pilots for future work in order to
improve the prediction models. However, sample size, and software and hardware
limitations will be obstacles affecting the quality of the initial models.
Problem Statement and Research Objectives
Traffic accidents are generally caused by factors and conditions to which the
driver and the vehicle are subjected, and which reflect negatively on driving behavior.
Despite the fact that many researchers have tried to resolve this problem, they agree
on few common factors, and the relationship among them has not yet been well
determined.
As part of the problem, truck accidents within the highway interchange areas
are considered unique. This specific problem is even more complicated, because of
the integration in effect of the interchange geometric characteristics and the
turbulence of the traffic flow that enters and exits the main road. In this regard, one
6


must keep in mind the different operating needs of drivers and the dimensional
characteristics of small vehicles and of large vehicles on ramps.
Thus, the main goal of this study is to evaluate artificial intelligence
techniques as alternative tools in modeling traffic accidents, and to assess the
limitations and capabilities of such tools, so as to improve similar models in the
future. These goals will be achieved by developing prediction models for truck
accidents on different ramp types. They will predict the number of truck accidents
that occurred at each ramp in Washington state during the 27 month period.
These prediction models will establish a relationship between the frequency of
truck accidents at each ramp and selected variables. The selected variables represent
different characteristics related to the ramp geometry, traffic conditions and other
environmental variables. Exploring such relationships should enable us to devise
counter measures for the broad problem, and these counter measures can be the
keystone for a better and safer highway system.
As part of the procedure, we will construct different models and outline the
merits and deficiencies of each one of them. This will lead to final observations,
recommendations and suggestions for further future work.
Review of Relevant Research
This section includes a review of previous work and theory related to truck
accidents and artificial intelligence techniques. This review is presented in the
following order: traffic accidents in general; truck accidents on segments of highway;
7


truck accidents at interchanges; traffic exposure measures; the neural networks; and
fuzzy logic.
Traffic Accidents
The relationships between traffic accidents and highway geometric variables,
such as horizontal curvature, vertical curvature and grade, lane width and shoulder
width, have been the subject of many studies (e.g., Roy Jorgensen Associates, Inc.,
1978; Zegeer et al., 1987; Okamoto and Koshi, 1989; Miaou et ah, 1991; Miaou and
Lum, 1993). These studies employed different statistical models to investigate the
safety issue at a given section of highway with specified attributes and geometric
variables.
Most of these statistical models were developed using either conventional
linear regression (Jovanis and Chang, 1986; Saccomanno and Buyco, 1988; Miao,
1995), or Poisson and Negative Binomial regression {Joshua and Garber, 1990;
Miaou et al., 1991; Shankar et al., 1994; Hadi et al, 1995; Miao, 1995; andPoch
and Mannering, 1996). The results of the conventional linear regression models
show unsatisfactory statistical properties for explaining the traffic-accident
phenomenon. In addition, the application of the Poisson and negative binomial
models had limitations. All previous attempts to predict and explain the nature of
traffic accidents and their connections to geometric characteristics have had mixed
results. Thus no set of geometric-accident relationships is widely accepted.
The deficiencies of such models were attributed to such factors as:
8


1. The quality of accident data used in the model: The data compiled for
most accident-reporting systems are typically produced by police accident
investigations. Police officers usually have neither the time nor the experience to
conduct in-depth accident investigations, nor to collect the necessary data. In
addition, it is not practical for them to investigate fully each accident.
2. The quality of the accident report: The type of data compiled for most
accident-reporting systems does not explain why the accident happened. Moreover,
many of these reports have conflicting statements. A number of states are currently
working to modify and enhance the structures of their accident-reporting systems.
3. Statistical tools: Traffic accidents are random discrete events. The cause of
these events is related to the different factors mentioned above, and assumes that each
variable (factor) is independent which is far from the reality. Most previous linear
regression models assumed that accident frequency could be represented by a
continuous distribution function, others used Poisson or Negative Binomial
regression, which use a discrete distribution function. In any case, priori knowledge
of the distribution function is only an assumption based on approximation. It appears
that this assumption is behind the failure of most of these models. Another reason
could be the assumption of independence among the contributing accident variables.
Also, these statistical models suffered from some inherited problems. Among
them were uncertainty of independent variables (regression models assume no
measurement errors in these variables); ignoring the variation in characteristics
between different locations within the same category, and even for the same location
during different periods of time; and, finally, the omitted variables that might have
9


been excluded from the model. These could have been a significant reason for poor
fit.
Interchange Safety
Different studies indicate that the design of certain types of ramps such as
cloverleaf ramps, scissor ramps, and left-side ramps should be avoided where
possible. In addition, according to Twomey et al. 1993, the potential for accidents at
interchanges has been related to ramp-traffic volume, main-road traffic volume, and
spacing between interchanges. This study also concluded that rehabilitation of ramps
is effective in reducing accident experience.
Truck Accidents
Since 1970, several reports and studies have been published related to truck
accidents and safety. Some of these reports were limited to studying truck accidents
on highways with different functional classes; accident rates of different types and
degrees of severity; relationships between vehicle configuration and accidents; truck
accidents at interchanges, ramps, and work zones; and other truck-safety aspects.
Recent studies attempt to relate truck accident rates to various traffic and
geometric variables. According to the model produced by Joshua and Garber, 1990,
the significant geometric design variables were slope change rate for primary
highways and curvature change rate for freeway type facilities. The major limitations
10


of this study were the small sample size, failure to consider other factors (driver,
environment), and not distinguishing among various truck types.
A study by Chatterjee et al., 1994 revealed that the human factor, especially
driver failure due to fatigue and inadequate training, contributes heavily to truck
accidents. These findings were discovered during many focus-group discussions with
truck drivers and could not be drawn from accident reports alone.
The impact on the design criteria of substantial increases in truck weights and
dimensions in the past few decades was studied by Hutchinson, B., 1990.
Hutchinson's study concluded that many infrastructure design procedures should be
revised to incorporate the new operational needs for trucks based on current truck
dimensions and weights.
The operational effects of larger trucks on rural roadways was also the subject
a study by Zegeer et al, 1990. The results showed that driver behavior and site
differences have more of an effect on vehicle operations than the effect of different
truck types.
Yet another study, by Miaou et al., 1993, employed different regression
models to establish empirical relationships between truck accidents and highway
geometric design. This study suggested areas in which the quality and quantity of
data could be enhanced to improve the developed model, such as by including
detailed truck exposure data. So, despite the limitations of the data used, some
encouraging relationships were developed.
11


Truck Accidents at Interchanges
None of the above work focused specifically on truck accidents at
interchanges. Further, the literature reveals few studies besides that of Ervin et al.,
1986, directed at investigating this problem. The Ervin study focused on the problem
of truck loss-of-control accidents on interchange ramps, from the viewpoint of the
suitability of highway geometric design. The results show that various aspects of the
geometric standards (unchanged in more than 30 years) provided by the American
Association of State Highway and Transportation Officials (AASHTO) offer a slim
margin of safety for trucks, especially on exit ramps. Some AASHTO
recommendations were written to educate truck drivers about locations with potential
risk.
A study conducted by Garber et al., 1992, attempted to identify the
characteristics of large-truck accidents on highway ramps in Virginia. As Garber
claimed, few studies were conducted after 1983 related to this topic, and none of
them explored in detail the relationship between truck accidents and highway
interchanges.
The Garber study was limited to performing proportionality tests that
compared the percentage of truck-accident involvement by ramp type, collision type,
highway type, and severity. An important finding was that the truck-accident
involvement ratio (number of truck accidents on a ramp per total number of accidents
on the road section in which the ramp is located) increases as the difference between
the average speed of the approaching truck and the posted speed limit on the ramp
increases. Also, as the ramp radius increases, that ratio decreases.
12


Another study, conducted by Leonard, J., 1992, focused on large-truck
crashes on freeway-to-freeway connectors, and attempted to simulate the dynamic
response of heavy vehicles at ramps. Basically, it was a single-vehicle-roadway-
driver model to determine the rollover threshold. A linear regression model to predict
the failure speed of large trucks was used, and the results showed that the failure
speed ranged from 12-to-21 miles per hour above design speed limit.
Truck Exposure Measures
In accident analysis, the exposure is a technique based on the opportunity for
interaction among vehicles and used to compute accident rates. So far, researchers
have developed different concepts and used different methods to measure traffic
exposure. The simplest form of exposure can be measured by Vehicle Miles of Travel
(VMT) generated during a specific period over a certain road section. According to
Khasnabis and Al-Assar, 1989, the purpose of measuring exposure is to enable the
analyst a reasonable assessment of "accident risk." Because of the complexity of this
topic, it is still subject to a continuous debate and future research. It has been
discussed in detail by Thorpe, 1964, Haight, 1970, and Hauer, 1982. The most
common measure used to determine accident rate is:
Accident rate = Number of accidents / VMT
This measure is appropriate if used for all accidents in general, but it is hard to justify
if used for subgroups of the traffic stream. However, different approaches have been
adopted in such cases. In truck accident cases, Khasnabis and Al-Assar, 1989
13


explained three approaches to calculate the truck accident rate, and proposed the
following approach:
Truck accident rate = Number of accidents involving trucks / [Truck VMT Factor]
The Factor in the above equation is the ratio of number of trucks involved in truck
accidents to the number of all vehicles involved in the same truck accidents. This
approach recognizes the rule of the traffic volumes of all vehicles during truck
accidents, not only the truck volume in the traffic stream.
However, for considerations of simplicity, it has been decided to model truck
accident frequencies rather than truck accident rates. In our case, considering truck
accidents at interchanges, we see that many accidents occurred at different segments
of the ramp, and some of them were exposed to main road traffic during merging and
diverging, while other accidents were exposed to the ramp traffic only.
Also, modeling accident frequency rather than accident rate has some
advantages, such as eliminating the redundancy and correlation between the
dependent variable and the independent variables. The frequency is not derived from
any other variable, while the accident rate is a function of other variables.
The calculation of VMT is normally related to a distance, which is the length
of the highway section under consideration. In our case, this term in not applicable
and requires special treatment when used.
14


Neural Network
The application of artificial intelligence, specifically neural networks and
fuzzy set theory, in transportation is considered new. Much of the related work was
included in two recent Transportation Research Record publications, TRB#1399 and
#1497, in 1993 and 1995, respectively. According to a paper by Mark Dougherty,
1995, the interest in neural networks by transportation researchers grew dramatically
in the early 90s. The attention of 52 studies was directed to the following subjects:
driver behavior; parameter estimation; pavement maintenance; vehicle detection and
classification; traffic pattern analysis; freight operations; traffic forecasting;
transportation policy and economics; air transportation; maritime transportation;
submarine transportation; metro operations; and traffic control. However, literature
reviews revealed no studies applying Artificial Intelligence to understanding truck
accidents.
Neural networks are designed to develop a mathematical model that connects
input parameters with solutions, without the need to define the model. Solutions are
based on calculating the error, which is the sum of the square of the differences
between the actual and the desired output data. Several iterations are required to reach
the minimum error that is allowed, depending on the learning rate (weight
adjustment). A basic neural network (Figure 1.1) consists of three layers of
interconnected nodes called neurons. Input-layer neurons receive data from the user.
Output-layer neurons send information to the user. Middle (hidden) layer neurons
receive signals from all the neurons in the input layer, and have the option of sending
signals to all the neurons in the output layer. Neural networks do not "learn" by
15


adding representations to their knowledge base; they learn by modifying their overall
structure.
The applications of neural networks may present certain difficulties. For
example, if there is no initial knowledge, learning derives entirely from experience,
the quality level of knowledge and the bias related to the design of the network, the
Inputs Hidden
Layer
Output
VI
Y
Figure 1.1: Multiple Layer Neural Network
number of nodes at each layer, the patterns of connections, and so on. Nonetheless,
neural networks outperformed current methods of analysis because they successfully:
1. Handle noisy or irregular data from the real world
2. Deal with the nonlinearity of real-world events
3. Quickly provide answers to complex issues
4. Are easily and quickly updated
5. Readily provide generalized solutions
6. Interpret information for large numbers of variables or parameters
16


Next, we will summarize the conceptual and theoretical math of neural
networks, beginning with a description of the neuron models that form the basis of the
neural networks. The following two general equations represent the basics of the
neuron behavior and the relationship between the input and the output for each
neuron:
*=ZWX/ (L1)
7-1
T* -0*) (1.2)
where xx,x2,---,Xj are the input signals; wkl,wk2,...,M'kj are the synaptic weights
of neuron k; uk is the linear combiner output; 6k is the threshold; activation function; yk is the output signal of the neuron.
Since we will be using a software designed by MATLAB in this research,
MATLAB notations will be used to present the structure of the designed neural
network in a simple way. As seen in Figure 1.2, the neuron output is calculated as:
a = f(wp + b)
17


Inputs Multiple-Input Neuron
r's f '\
v______j \________l------------------J
a =/(Wp + i>)
. T XT (Ref. 16)
Fig 1.2: Multiple-Input Neuron
where p is the scalar input, w is the scalar weight; another input is 1, which is
multiplied by a bias b and passed to the summer; n is the summer X output, that
goes into a transfer function /, which produces the scalar neuron output a .
Different types of transfer (activation) functions could be used, such as threshold
function (step function), piecewise-linear function (ramp function), linear function,
and sigmoid function.
The architecture of a network is determined by the way the neurons are
structured and connected to each other, and by whether the flow of signals goes in
both directions. The four main different classes of network architecture are: single-
layer feed forward; multilayer feed forward; recurrent; and lattice-structured network.
Adjustment and modification of the weights and biases of the network are
called the learning rules. The purpose of a learning rule is to train the network to
perform certain tasks. All types of learning fall under one of the following three
categories: supervised learning; unsupervised learning; and reinforcement learning.
18


There are different types of learning rules to train the network, Hebbs rule and
the Delta rule being the most commonly used learning rules. The Hebbian learning
rule assumes the change in weight ( Awtj ) between two neurons (/ and j) during
two successive iterations is /j a,a7, where p is a constant representing the
learning rate, and a is the activation of that neuron. This rule could be modified by
replacing the quantity of a for each neuron with its deviation from the average of all
a's .
The Delta learning rule is known as the Least Mean Squared error rule (LMS).
The objective of this rule is to minimize the mean of the square of the difference
between the targeted value and the computed value. This rule belongs to the
backpropagation algorithm, where it employs the steepest-descent method in the
optimization.
An improved backpropagation neural network will be used to build the neural
network model. The performance of this modified network will be enhanced by using
two techniques called momentum and adaptive learning rate, in order to increase the
speed and reliability of backpropagation. Backpropagation was created by
generalizing the Widrow-Hoff learning rule to multiple-layer networks and to
nonlinear differentiable transfer functions. Generally, a three layer backpropagation
networks (input-hidden-output) with biases and a sigmoid transfer function with a
linear output layer are capable of approximating any function with a finite number of
discontinuities.
The backpropagation training may lead to a local rather than a global error
minimum. If this local error minimum is not satisfactory, more neurons in the hidden
19


layers, or more layers in the network could overcome this problem. Alternatively,
using different initial conditions in several runs, in order to see whether or not the
network will converge to the same solution, will help in finding the optimal solution.
Fuzzy Logic
Fuzzy theory has been successfully used to represent, manipulate, and analyze
data that has high levels of uncertainty and ambiguity. Most fuzzy-theory
applications have dealt with control systems and industrial applications, including
building computer chips, stabilizing helicopters, and subway control systems. Many
other applications related to human behavior and decision making in environments of
uncertainty and ambiguity, have successfully employed fuzzy logic to mimic the
behavior of the system, based on expert knowledge.
Fuzzy theory is a relatively new mathematical device that can be used for
dealing with the uncertainty, complexity, and ambiguity of traffic, and in solving
transportation problems. The first attempt to solve a transportation problem using
fuzzy theory was published by Pappis and Mamdani, 1977. Recently, several papers
have used fuzzy theory to solve traffic and transportation problems related to vehicle
routing, scheduling and dispatching (e.g., Teodorovic and Kikuchi, 1990; Kikuchi and
Donelly, 1992), traffic-assignment models {e.g., Teodorovic and Kikuchi, 1990; Lotan
and Koutsopoulos, 1992), freeway incident detection {Chang and Wang, 1994),
rehabilitation projects {Prechaverakul and Hadipriono, 1995), and for identifying
accident-prone locations {Sayed et al., 1995).
20


According to Juang et ai, 1993, the principle of fuzzy sets may be summed up
as the transformation of uncertainty and ambiguity of data into numerical data in a
systematic way, so that subjective information, such as expert opinions, rules of
thumb, and other nonquantifiable but significant information, can be directly
utilized in the solution process. By using fuzzy logic, different analytical approaches
are available to deal with civil engineering problems. The best approach is
determined by the type of input data, model type, and type of output. Ambiguous or
fuzzy relations, such as the interaction between variables, can be expressed first in
linguistic terms. These terms are then transformed into numerical data, which are
expressed as fuzzy numbers to account for their uncertainty.
A probabilistic model is suitable for the expression of precise but dispersed
information. Once precision is lacking, one tends to exit the domain of validity of the
model. The limitations of traditional models of imprecision and uncertainty
(probability theory and interval analysis) are intended to justify the search for a wider
framework that embraces the two concepts. Possibility theory is a natural tool for
summarizing a set of results that are imprecise but coherent. This general framework
emerged from the notion of fuzzy sets developed by Zadeh in the 1960s.
It has been noticed that combining fuzzy logic and a neural network in one
system could develop a more efficient and effective model that will take advantage of
the properties and strengths of both techniques. An example of this combination is a
hybrid neural system called the Adaptive Network Based Fuzzy Inference System
(ANFIS) created by Jang, 1993.
A practical application of a hybrid system is a Fuzzy Logic Incident Patrol
System (FLIPS), proposed by Hsiao et al., 1994 to solve many of the problems
21


inherent in traditional incident-detection algorithms. The results have proved the
effectiveness of this approach for identifying incidents. The author has suggested
future work to improve his model.
Appendix C summarizes related mathematical terms and expressions that
explain the basics and theory of possibility theory and fuzzy logic.
22


CHAPTER 2
DATA ACQUISITION AND PRELIMINARY ANALYSIS
Initially, different sources of truck accident data were considered as potential
candidates for this study. Data from different states, such as California, Colorado,
Utah, and Washington were examined and evaluated carefully in order to assess the
limitations and possible deficiencies in each data set. Considering the scope of this
study, data from California, Colorado, and Utah were dropped. Washington state data
have been selected because of quality and sample-size considerations. (See Table
2.1.)
Data Inventory
Washington truck accident data as received consist of the following five sets:
1. Washington highway geometric. The highway system in Washington state
consists of exactly 473 interchanges constructed to serve about 45 highways. In a
complete set of drawings for all interchanges, each drawing shows the following
information: interchange name; ramp code; state route number; milepost of the gore
and beginning of the deceleration lane, or the end of the acceleration lane of each
ramp; and direction of travel (Appendix B).
23


2. Accident information. The electronic files contain 90 fields of information
for each recorded accident on the highway system during the study period form
1/1/1993 to 3/31/1995 (Appendix B).
There were about 7,438 truck accidents that occurred on the highway system,
of which 4,320 truck accidents occurred outside the interchange influence zone.
The interchange influence zone is a term that represents the boundaries of each
interchange. It is a certain distance beyond the gore or the end of the taper for each
ramp, either upstream for diverging ramps, or downstream of merging ramps. The
procedure and the values will be discussed later.
3. Annual traffic reports for years from 1992 to 1994. The annual traffic
report contains Annual Average Daily Traffic (AADT) volumes at main lanes. The
AADT volumes are included only at locations where actual traffic counts have been
conducted in one or more of the last four years. In most cases, the AADT is reported
at three locations (beginning, center, end) for each interchange (Appendix B).
The percentage of trucks in the traffic stream is recorded at selected locations
throughout the state. The data reflect only the truck percentages during specific
periods and should not be considered as annual truck percentages for these locations.
4. Traffic-count history reports for years from 1992 to 1994. This file
contains traffic volumes (AADT) for about 75 percent of the ramps, although only
about 10 percent of these ramps have truck percentages (Appendix B).
5. State highway Log. This file contains ramp information such as ramp
length, number of lanes, ramp width, and shoulder information (Appendix B).
24


RAMP TYPE # of Ramps Percent # of Accidents per year Percent Average Accident Frequency
Diamond I 27 30.3 16 25.9 0.60
Loop 12 13.5 9 14.8 0.78
OuterConn 11 12.4 6 9.0 0.52
Directional | 39 43.8 32 50.3 0.81
Other! 0 0.0 0 0.0 0.00
Total! 89 100.0 63 100.0 0.71
Colorado Accidents
RAMP TYPE # of Ramps Percent # of Accidents per year Percent Average Accident Frequency
Diamond 19 3.9 20 5.6 1.04
Loop 25 5.1 19 5.4 0.76
OuterConn | 23 4.7 11 3.1 0.48
Directional I 324 65.9 266 75.8 0.82
Other 101 20.5 35 10.1 0.35
Total 492 100.0 351 100.0 0.71
California Accidents
RAMP TYPE # of Ramps Percent # of Accidents per year Percent Average Accident Frequency
Diamond 310 48.1 218 47.5 0.70
Loop 81 12.6 60 13.0 0.74
OuterConn 59 9.2 44 9.7 0.75
Directional 152 23.6 120 26.1 0.79
Other 42 6.5 16 3.6 0.39
Total 644 100.0 458 100.0 0.71
Washington Accidents
Table 2.1: Comparison of Truck Accidents per Year in Three States by Ramp Type
25


Data Manipulation and Preparation
By interpreting available data from the hard copy files, we were able to add
important variables to the truck accident database. Among them were interchange
type, ramp type, conflict type, connection type, ramp length, main road AADT, truck
percentage at main road, ramp AADT, and ramp truck percentage. A FORTRAN
program also was created to associate each accident with an interchange. The second
step was to associate each accident within the interchange influence zone to the ramp
to which it belonged. In addition, each merge or diverge ramp was divided into four
areas: upstream; ramp connection; downstream; and ramp (Fig 2.1).
26


The distribution of total truck accidents occurring on the main road upstream
and downstream of both merge and diverge ramps, based on 0.05 mile section
increments, indicates the following (Figures 2.2.a, b):
1. Truck accident frequency beyond 0.25 miles upstream of the diverge taper
or merge gore did not change significantly, so the border of the interchange from the
upstream direction is 0.25 miles from the merge gore or the end of the diverge taper.
2. In the downstream direction, and based on the same process, the border of
the interchange from the downstream direction is 0.15 miles from the end of the
merge taper and 0.20 miles from the diverge gore.
3. The average length of the merge connection area was 0.219 miles, and the
average length of the diverge connection area was 0.108 miles.
Figure 2.2.a: Truck Accident Distribution Upstream and Downstream of On-Ramps
27


After determining the boundaries of each individual interchange, we found
there were 3,118 truck accidents occurring within the influence zone of 391
interchanges. Some 82 interchanges experienced no accidents during the study
period.
Figure 2.2.b: Truck Accident Distribution Upstream And Downstream of Off-Ramps
28


Quality of Data
Even though Washington state traffic-accident database is considered to be
one of the best in the nation, and has the most comprehensive list of variables and
data elements, it still suffers from a shortage of information in the following areas:
1. Highway geometric, such as grade, curvature, length of taper, and lengths
of acceleration or deceleration lanes.
2. Some ramps have no traffic volume data, so all accidents occurring at these
ramps were dropped from the final analysis.
3. Most of the ramps were missing the truck percentages. A nonlinear
regression model was formulated using a supplementary list of truck percentages that
were obtained from Washington state, and the estimated truck percentages were used
wherever needed.
4. Some locations at the highway were missing AADT. Assuming linear
relationship between stations around that missing value, by interpolation, an
estimated value was used.
Finally, all the ramps included in the final data set used in the modeling were
ramps that experienced at least one truck accident during the study period. No data
base exists yet that includes complete information (such as weather and surface
conditions) about no-accident locations.
The final truck-accidents set that was ready for analysis was reduced to 1,030
truck accidents associated with 581 ramps at 248 interchanges. The major drop in
sample size was related to missing ramp volume; about 350 accidents were dropped
29


because they occurred at intersections with a secondary road, where the vehicle
movement was interrupted by either a stop sign or yield sign.
To this point the data set was tabulated in terms of individual accident, not by
ramp location, so the last step in preparing the data for modeling was to group
accidents by ramps. Thus, each observation represents a different ramp. The
common geometric and traffic variables for every accident were the same for each
ramp, while the variables that present unique information about each individual
accident were averaged for each location. Some of these variables are light, weather,
and surface conditions.
Table 2.2 shows the distribution of the 1,030 accidents among the 581 ramps
that experienced accidents in Washington state, and Table 2.3 divides this distribution
based on the ramp type.
The distribution of ramps based on the number of truck accidents at each ramp
can be seen in Figure 2.3, where 59% of the ramps in Washington state data
experienced only one accident during the study period, while 32% of the ramps
experienced two or three accidents, and only 9% of the ramps had more than three
accidents during the 27 months.
30


Number of Accidents per Ramp Number of Ramps Total Number of Accidents
1 347 347
2 144 288
3 40 120
4 21 84
5 13 65
6 8 48
7 2 14
8 0 0
9 2 18
1C 1 10
11 1 11
12 1 12
13 1 13
Total 581 1030
Table 2.2: Washington Truck Accidents on All Types of Ramps
31


Accidents per Ramp Diamond Loop Outer Connector Directional Other
1 199 45 34 52 17
2 78 17 11 33 5
3 14 6 3 16 1
4 7 3 2 9
5 5 2 1 5
6 1 1 1 4 1
7 1 1
8
9 1 1
10 1
11 1
12 1
13 1
Total Accidents 489 132 101 272 36
Total Ramps 307 75 54 121 24
Table 2.3: Washington Truck-Accident by Ramp Type
32


Figure 2.3: Distribution of Ramps in Washington State by TAF During 27 Months
33


Preliminary Observations from Washington State Truck Accident Data
In this section, we will explain the observations that we gathered from simple
two-tables. Three tables were assembled to show how the accident rate changed when
we considered a new variable. Each table shows different combinations of the two
variables considered: ramp type and conflict type.
CONFLICT AREA RAMP TYPE Tctal Accidents Tctal Conflict Areas Accidents per Conflict Area
Diamond Loop OuterConn Directional Other
Merge # Accidents 91 7 15 31 7 151
0 Upstream # Conflict areas 167 50 25 69 20 331
n Acc / Conf area 0.54 0.14 0.60 0.45 0.35 0 46
Merge # Accidents 116 50 27 63 11 267
R Area # Conflict areas 167 50 25 69 20 331
a Acc / Conf area 0.69 1 00 1.08 0 91 0 55 0 81
m On # Accidents 21 17 8 28 1 75
p Ramp # Conflict areas 168 53 28 69 21 339
s Acc / Conf area 0 13 0.32 0.29 0 41 0.05 0 22
Merge # Accidents 44 15 2 10 3 74
Downstream # Conflict areas 167 50 25 69 20 331
Acc / Conf area 0.26 0 30 0.08 0 14 0.15 0.22
On # Accidents 272 89 52 132 22 567
Ramps # Conflict areas 669 203 103 276 81 1332
Totals Acc / Conf area 041 0 44 0.50 0.48 0.27 0.43

Diverge # Accidents 67 4 12 32 4 119
0 Upstream # Conflict areas 142 24 28 80 20 294
f Acc / Conf area 0 47 0.17 043 0.40 0.20 0 40
f Diverge # Accidents 54 16 13 42 6 131
Area # Conflict areas 142 24 28 80 20 294
R Acc / Conf area 0.38 0.67 0.46 0 53 0.30 0.45
a Off # Accidents 17 23 10 38 3 91
m Ramp # Conflict areas 142 28 31 83 21 305
p Acc / Conf area 0.12 0.82 0 32 0 46 0.14 0 30
s Diverge # Accidents 80 2 13 25 2 122
Downstream # Conflict areas 142 24 28 80 20 294
Acc / Conf area 0.56 0.08 0.46 0.31 0.10 0.41
Off # Accidents 218 45 48 137 15 463
Ramps # Conflict areas 568 100 115 323 81 1187
Totals Acc / Conf area 0 38 045 0 42 0.42 0 19 0 39

All # Accidents 490 134 100 269 37 1030
Ramps. # Conflict areas 1237 303 218 599 162 2519
Totals Acc / Conf area 0 40 0.44 0 46 045 0.23 0 41
Table 2.4: Washington Truck Accidents by Ramp Type and Conflict Area
34


The third value in each cell in Tables 2.4, 2.5, and 2.6 shows accident
frequency per conflict area, accident frequency per ramp truck trips (RTT), and
accident frequency by ramp truck VMT (RTVMT), respectively.
CONFLICT AREA RAMP TYPE Total Accidents Total RTT Total Accident Rate
Diamond Loop OuterConn Directional Other
Merge # Accidents 91 7 15 31 7 151
0 Upstream RTT(millions) 55 18 12 44 16 144
n Accident Rate 1.66 0.38 1.30 0.71 045 1.1
Merge # Accidents 116 50 27 63 11 267
R Area RTT(millions) 55 18 12 44 16 144
a Accident Rate 2.12 2.75 2.34 1.45 0.70 1 9
m On # Accidents 21 17 8 28 1 75
p Ramp RTT(millions) 55 19 13 44 16 147
s Accident Rate 0.38 0.87 0.61 0.64 0.06 0 5
Merge # Accidents 44 15 2 10 3 74
Downstream RTT(millions) 55 18 12 44 16 144
Accident Rate 0.80 0.82 0.17 0.23 0.19 05
On # Accidents 272 89 52 132 22 567
Ramps RTT(millions) 219 74 48 174 63 578
Totals Accident Rate 1 24 1.20 1.09 076 0.35 10

Diverge # Accidents 67 4 12 32 4 119
0 Upstream RTT(millions) 43 8 10 42 15 119
f Accident Rate 1.57 0.50 1.16 0.75 0.26 1.0
f Diverge # Accidents 54 16 13 42 6 131
Area RTT(millions) 43 8 10 42 15 119
R Accident Rate 1.27 1 99 1.25 0.99 0 39 1.1
a Off # Accidents 17 23 10 38 3 91
m Ramp RTT(millions) 43 10 11 43 16 123
p Accident Rate 0.40 2 36 0.89 0 88 0.19 0 7
s Diverge # Accidents 80 2 13 25 2 122
Downstream RTT(millions) 43 8 10 42 15 119
Accident Rate 1 87 0.25 1.25 0 59 0.13 10
Off * Accidents 218 45 48 137 15 463
Ramps RTT(millions) 171 34 42 171 62 480
Totals Accident Rate 1.28 1.33 1.13 0.80 0.24 1.0

All # Accidents 490 134 100 269 37 1030
Ramps RTT(millions) 390 108 90 345 125 1057
Totals Accident Rate 1.26 1.24 1.11 0.78 0.30 10
i
Table 2.5:Washington Truck Accidents per RTT by Ramp Type and Conflict Area
35


CONFLICT AREA RAMP TYPE Total Accidents Total RTVMT (millions) Accidents per RTVMT
Diamond Loop OuterConn Directional Other
Merge # Accidents 91 7 15 31 7 151
0 Upstream RTVMT (millions) 13.66 4.55 2.89 10.88 3.91 36
n Acc / RTVMT 6.66 1.54 5.19 2.85 1.79 4 2
Merge # Accidents 116 50 27 63 11 267
Area RTVMT (millions) 11.05 3.17 2.84 6.19 3.79 27
R Acc / RTVMT 10.50 15.76 9.49 10.18 2.91 9.9
a On # Accidents 21 17 8 28 1 75
m Ramp RTVMT (millions) 19.81 6.52 4.37 16.02 8.36 55
p Acc/RTVMT 1.06 2.61 1.83 1.75 0 12 1.4
s Merge # Accidents 44 15 2 10 3 74
Downstream RTVMT (millions) 8.20 2.73 1.73 6.57 2.35 22
Acc / RTVMT 5.37 5.49 1.15 1 52 1 28 3.4
On # Accidents 272 89 52 132 22 567
Ramps RTVMT (millions) 52.72 16.97 11.84 39 66 18 41 140
Totals Acc / RTVMT 5.16 5.24 4.39 3.33 1 20 4.1

Diverge # Accidents 67 4 12 32 4 119
0 Upstream RTVMT (millions) 10.67 2.01 2.59 10.60 3.87 30
f Acc/RTVMT 6.28 1.99 4 63 3.02 1.03 4 0
f Diverge # Accidents 54 16 13 42 6 131
Area RTVMT (millions) 4.67 0.79 1.24 5.49 2.41 15
Acc / RTVMT 11.57 20.16 10.49 7.65 249 9.0
R Off # Accidents 17 23 10 38 3 91
a Ramp RTVMT (millions) 22.89 2.13 2.92 15.80 8.40 52
m Acc / RTVMT 0.74 10.79 3.43 2.40 0.36 1.7
p Diverge # Accidents 80 2 13 25 2 122
s Downstream RTVMT (millions) 8.54 1.61 2.07 8 48 3.09 24
Acc / RTVMT 9.37 1.24 6.27 2.95 0.65 5.1
Off # Accidents 218 45 48 137 15 463
Ramps RTVMT 46.76 6.55 8.82 40.38 17.77 120
Totals Acc / RTVMT 4 66 6.87 5.44 3 39 0 84 3.8

All # Accidents 490 134 100 269 37 1030
Ramps RTVMT (millions) 99.49 23.52 20.66 80.04 36.17 260
Totals Acc / RTVMT 4 93 5.70 4.84 3 36 1 02 40
Table 2.6: Washington Truck Accidents per RTVMT by Ramp Type and Conflict
Area
36


Conducting a two-way analysis of variance on each table individually shows
that the accident rates in each one of these tables are significantly different by conflict
area at the 95% confidence level, but not by ramp type. However, the rates do not
change in the same way in all tables, which suggests that these rates are influenced
differently by RTT and/or RTVMT.
To conduct an in-depth statistical analysis using the above approach, we
should incorporate more variables in the analysis, the two-way table becomes a multi-
way table. For example, if we have 10 variables, and if each variable has four groups,
the number of cells will be 1,048,576 (410 ), and, to model three-dimensional cross
tabulation, there will be no less than nineteen models to choose from. This statistical
technique, which is dealing mostly with data measured at the categorical level, rather
than the metric level (although it could be used with ordinal data), is used in
sociology research and known as loglinear analysis (Ref. 13 ).
As seen from the comparisons of Washington state truck accident data, the
cross-tabulation procedure revealed no significant explanation of the truck-accident
dilemma at interchanges. However, a comprehensive conclusions require a higher
level of analysis beyond this simple statistical process.
Factor Analysis and Variable Selection
Factor analysis is a data-reduction technique, which is needed in cases where
the number of variables under consideration is large, and the coefficients in the
correlation matrix are impossible to evaluate. This higher-order data reduction
technique is a procedure used to remove the redundancy from a set of correlated
37


variables. Basically, it groups the similar variables into homogeneous sets, and
creates a new variable (factor) that represents each of these sets. In fact this is the
process of clustering a large number of variables into a smaller number of factors.
In our case, the use of factor analysis will be limited to defining the most
important variables to be included in the model, and will exclude any highly
correlated variables. This is done by choosing the variable with the highest score in
each factor, and excluding any other variables that have the next highest score under
that factor.
Practically speaking, it is less convenient to use the factors obtained by the
factor analysis in the prediction models, because the values of these factors are not
accessible in the same way as the regular variables. Also, using these factors in a
neural network model for example, did not seem different than using the variables
directly.
Washington data were divided into five groups b:ised on the ramp type: ten
variables were initially considered to be used in predicting TAF (Figure 2.4). These
variables fall within the following categories:
1. Ramp geometric: distance from gore-to-taper; ramp length.
2. Traffic conditions: traffic volumes for both main road and ramp; truck
percentages on both main road and ramp.
3. Environmental conditions: weather; surface conditions; visibility (light);
and location (urban/rural).
Insights derived from the factor analysis were partially behind the variable
selection. Other considerations were based on previous research.
38


Gore to taper distance
Ramp length
Urban/Rural
Ramp ADT
Main road ADT
Ramp truck ADT
Main road truck ADT
Weather conditions
Light (visibility)
Surface (dry,wet,ice)
Figure 2.4: The Variables Considered in The Modeling From Washington State
Database
39


CHAPTER 3
MODELS FORMULATION
Different models will be developed to predict traffic-accident frequencies at
highway interchanges. Basically, the process used to build a mathematical expression
that describes the relationship between the criterion (dependent) variable and the
explanatory (independent) variables is called mathematical modeling. Different
techniques will be used in the modeling process, such as regression analysis, neural
networks, and ANFIS system (Figure 3.1).
The purpose of all models is to predict the criterion variable, which, in this
case, is truck accident frequency (TAF), using different sets of independent variables.
These independent variables could be quantitative or qualitative. However, the
qualitative variables can not be used in the different models without being
transformed into dummy variables.
The dependent variable (TAF) is represented by the vector Y. The size of
vector Y is nxl, where n is the number of ramps in Washington state that
experienced truck accidents. That number varies according to the model considered
(see Table 2.2) based on ramp type. In the case of an all-ramps model, n is 581
ramps, while the value of Y is in the range between 1 and 13 accidents.
The independent variables are represented by the matrix X with a size nxk,
where k is the number of independent variables. For the regression models, 10
40


Figure 3.1: The Methodology for Prediction Models
41


variables were selected to enter the model. The selection process was discussed in the
previous chapter, in another attempt to compare the different models, the four
variables used in the ANFIS models were forced to a regression model. Also, a
square root transformation of the criterion variable (TAF) was applied as a remedy to
unequal variance conditions.
For a neural networks, three sets of explanatory variables were used. The first
one included all 10 variables used in the regression, the second set used 6 explanatory
variables to predict TAF, and the third set included only four variables, these four
variables are the ones used in the ANFIS model. The different models with different
number of variables was used to evaluate the impact of the omitted variables on the
predictions.
The ANFIS models were limited to 4 input explanatory variables because of
the physical limitation of the hardware and the soft were that were used in the
modeling. Different sets of variables were considered, with a different combinations
of variables. The variables considered in the final models were gore-to-taper
distance, ramp-traffic volume, main-road traffic volume, light, weather, and truck
percentages for both ramps and main road.
The data used in the modeling process using linear regression, neural
networks, and ANFIS was 90% of the total data available, and the other 10% of the
data was allocated for checking and evaluating the designed models (Figure 3.2).
Table 3.1 shows the size of each data set, excluding non classified ramps (other),
where each ramp location was considered as a distinct observation.
42


All classified
ramps
557
Figure 3.2: Training and Checking Data by Ramp Type in Washington State
Ramp Type Training Data Checking Data
Diamond 277 30
Loop 68 7
Outer Con. 49 5
Directional 109 12
All Ramps 503 54
Table 3.1: Training and Checking Data Size
43


In the following sections, the mathematical forms of the different types of
models will be presented.
Conventional Multiple Linear Regression Model
This model involves several independent variables to predict the dependent
variable as follows:
T, ~ Po+P\Xi\ + PlXi2 +---+fikX,k +£i
where z = 1,2, ...,n
e = error term with a zero mean
y truck accident frequency
x = independent var tables such as traffic volume, ramp length, and others
P\,P2....Pk = regression parameters
The probability distribution of the dependent variable Y is assumed to be
normal with a mean = £(_y,) and a variance aj = V(y,)- Also the variance is
assumed to be constant for all settings of independent variables.
The least squares estimates (LSE) method will be used to estimate the
regression parameters /?, ,/?2,... ,pk by minimizing the sum of the square of the
difference between the predicted and actual values, or in a mathematical expression
44


I/- ~y'j
i
A
where y, is the actual value, and y, is the predicted value.
To determine the utility of the model and whether the results have a good fit
or not, a diagnostic procedure could be used to evaluate the regression model as
follows:
1. The value of the coefficient of determination R2, which is the square of the
correlation coefficient between independent variables X and dependent variable Y.
If all observations fall on the regression line, then R2 is 1, and that is a perfect fit.
While, if R2 is 0, then in that model there is no linear relationship between the
dependent variable and independent variables. However, that does not necessarily
mean that there is no association between the dependent variable and independent
variables.
2. Violation of Assumptions: It is necessary to conduct a search focused on
residuals to look for evidence that supports any violations to the regression
assumptions. For example, if the assumption of linearity and homogeneity of
variance were violated, then you would observe a certain type of relationship between
the predicted and residual values. Other violation tests, such as equality of variance,
independence of error, and normality of residual, are needed to evaluate the
regression model.
Intrinsically Linear Regression Model
45


If the R2 were low or if there were evidence of a violation of assumptions,
one could pursue another strategy. Transforming the variables could improve the
model. For example, by taking the logs, square roots, or reciprocals of the original
variables the model could achieve normality, stabilize the variance, or linearize the
relationship between the dependent variable and independent variables.
Since most traffic-accident models are nonlinear, a transformation technique
could be applied to the dependent or independent variables in the nonlinear equation
so as to be linearized. The most common intrinsically nonlinear models are log-linear
and exponential. For example, the following non-linear equation:
T, =e
£,+Pi
vPl V-P'S
A,2 A/3
.. X
Pk
ik
It can be transformed to the following linear equation by taking the natural logarithm
of both sides of the equation:
Ln(yi) = fix + P2 Ln(xn) + /?3 Ln(x:i) + .. .+/?* (xik) + s,
Also, the following exponential model:
P\*n+Pi*ii+~+PhXlk+£i
y, =e
Using a natural logarithm transformation as in the log-linear model, the exponential
model will be transformed to the following linear model:
46


Ln(y,) = j3 ,x +/J2xp_ + ...+/3kxik +ei
After the transformation process, a traditional multiple linear regression could
be applied to the data, using the transformed variables Ln[yj), and Ln(xn), Ln(xik),
if needed.
Neural Network Model
A multilayer network will be developed and trained using a backpropagation
algorithm, which is an approximate steepest-descent algorithm in which the
performance index is mean-square error. The multilayer network, trained by a
backpropagation algorithm, is considered the most widely used method to solve
nonlinear functions using a neural network (Figure 3.3).
Input Log-Sigmoid Layer
Log-Sigmoid Layer


r


a' =/' (w'p+b')
J

a1 =f2(w1-ai +b2)
J
Figure 3.3: Two-layer Network (Input-Hidden-Output)
47


Two and three layer networks will be developed according to the following:
1. The structure of the two-layer network is 6-15-1, where 6 represents the
number of inputs, 15 are the number of the hidden neurons, and finally the output is
only one, which is the TAF.
2. The structure of the three layer network is 10-15-5-1, where 10 represents
the number of inputs, and 15 and 5 are the numbers of the hidden neurons in the first
and second hidden layers respectively, and, finally, the output is only one, which is
the TAF.
3. The transfer functions (/' f2,...) in all layers (hidden, and output) is a
nonlinear function, which takes values between zero and one. For example, log-
sigmoid function is frequently used (Figure 3.4). This transfer function takes the
input with any value and returns an output value between 0 and 1, according the
following formula:
Output =-----
1 + e~mpul
a
a = logsig(n)
Figure 3.4: Log-sigmoid Transfer Function
48


4. The data was normalized to improve the performance of the network. This
was done by dividing each variable by the maximum value of that variable. When we
need to evaluate any variable or output, we can convert the value of that variable back
to its original value by multiplying it by the maximum.
5. The first step is to initialize the weights and biases for a two layer
backpropagation network. The function initff in MATLAB takes the matrix of input
vectors P and the size S, transfer functions of each layer, and returns the initial
weights W and biases b for each layer, using the following command:
[Wl,bl,W2,b2] = initff ( P, S, 'logsig',T,'logsig');
where Wl, bl, W2, b2 are the weights and biases for both layers, S is the number of
the neurons in the hidden layer, logsig is the transfer function, and T is the output.
6. The next step is to define the training parameters tp, which are: the display
frequency of the training df; maximum number of epochs me, which is the number of
runs between the input and output; and the error goal eg. For example:
df =100;
me 10,000;
eg = 0.01;
tp = [df me eg ];
The network normally could be trained without noise for a maximum of 5000
epochs, or until the network sum-squared error SSE falls beneath 0.1. However, in
49


our case, a lot of noise was expected, and the training required more than 50,000
epochs with SSE between 0.02 and .05, depending on the data.
The SSE is calculated using the following formula:
SSE = E{Kf =X('(K)-")!
A'=l K=\
where:
K is a single observation, n is the total number of observations, t is the target (actual)
output, and a is the predicted output.
7. Now, to train the network using backpropagation with adaptive learning,
one uses the following command:
[Wl,bl,W2,b2] = trainbpx ( Wl,bl, Togsig, W2,b2,'logsig', P,T, tp).
If the network fails to converge, you may have to add more neurons to the
hidden layer (S), add another layer, and/or use more training data.
8. Finally, to test the network, which is calculating the expected output (TAF)
from the trained network, one can use the simuff function, as follows:
Expected = simuff ( P,W1, bl, 'logsig', W2, b2, 'logsig')
50


ANFIS Model
The advantage behind using ANFIS, which is a MATLAB built-in algorithm,
is the elimination of the manual effort in tuning the rules and calibrating the
membership functions, where the adjusting process will be carried by this algorithm,
which uses the neural networks to obtain the optimal solution by training the data.
Manual tuning is subjected to human error, can vary from one person to the other, and
may not be optimal, especially when the data size is large.
This algorithm is based on a paper presented by Jang (1993), which is about
taking a fuzzy inference system (FIS) and tuning it with a backpropagation algorithm,
which will allow the fuzzy system to learn. ANFIS architecture could be employed to
model nonlinear functions. However, MATLAB only supports first-order Sugeno-
type systems. These define the output as a monotonic function, such as the one seen
in Figure 3.5, which is approximated to a piecewise linear function The three
different Sugeno-type systems are shown in Figure 3.6.

p q output
Figure 3.5: ANFIS Output Function for Sugeno Type 1 Model Ref (18)
51


pmntm part
oonaaquaat part
-1 {-------TT
DTP* 1
JXBLL
_ fypt3
Figure 3.6: Sugeno-Type Systems Ref (18)
ANFIS Architecture. Assuming a two input one output system, the two
input variables are x, (MADT), x2 (RADT) and the output variable is Y (TAF).
Suppose we have two zones of membership functions, then two fuzzy rules of Sugeno
Type 1 would be used, as follows:
Rule 1: If x, is Ax and x2 is Bx, then Y is f
Rule 2: If x, is A2 and x2 is B2, then Y is f2
Figure 3.7 illustrates the fuzzy reasoning of the Sugeno Type 1 Model, and
the equivalent ANFIS architecture for the same Sugeno Type is shown in Figure 3.8.
Usually, the membership function for the input variables is considered to be a bell-
shaped function in the range of 0 and 1, such that:


***{*) =
L
b,
( \2
X-C,'
V aj
where {a,. ,bj c,.} is the parameter set. The shape of the function varies according to
the changes in the parameter set. Other continuous and differentiable functions such
as trapezoidal or triangular could be used. The output function was shown in figure
3.5, and was used to obtain the output values /, and f2.
Figure 3.7: Fuzzy Reasoning for Sugeno Type 1 Model Ref (l8)
53


f= a MADT + b RADT + c
W Wn
Product Normalized Product Sum
Figure 3.8: ANFIS Architecture for Sugeno Type 1 Model
The ANFIS architecture consists of the five following layers:
Layer 1: The first layer transfers the input data to its corresponding
membership function value O) ju A (x), where x is the input to the node i, and ()]
is the membership function of Ai in layer 1.
Layer 2: In the second layer, all incoming signals are multiplied, and the
product is sent out to the output layer, according to the following formula:
54


=Va,{x\)x
where w, represents the firing strength of rule /' .
Layer 3: In each node in this layer, the ratio of the i'h rule's firing strength to
the sum of all rules' firing strengths is:
W-
W,=-----i ,/ = l,2
Wj + w2
where the output of this layer wj represents the normalized firing strength.
Layer 4- The output of each node in this layer is calculated using the
following equation:
O,4 =",/,
Layer 5: The overall output is calculated in this node by summing all the fired
outputs in the previous layer:

/ = ]
2>.
The hybrid learning algorithm that is used in ANFIS employs the least squares
estimate in the forward pass. In the backward pass, it updates the parameters by
propagating backward the error rates using the gradient descent.
Checking data is used to test the generalization capability of the ANFIS model
at each epoch. The ANFIS routine has the ability to be trained in a way that does not
lose its generality and that responds to the checking data in a satisfactory way.
55


The ANFIS model will be trained for the designated number of epochs. For
each epoch, it will calculate the RMSE (root mean squared error) for both the training
data and the checking data. The RMSE can be calculated using the following
formula:
where a is the ANFIS output, t is the targeted (actual) value, and n is the number of
observations used in the model.
The final model will be the one that holds the minimum RMSE value during
the training epoch, and will keep the matrix that satisfies this minimum value as the
optimum model and the final solution.
Model Formulation. For each data set, the following steps will be used to develop the
ANFIS models:
1. First, identify the data set in a matrix form, where the last column has the
targeted data (TAF).
2. Next, generate a Fuzzy Inference System (FIS) matrix, assuming numbers
and types of membership functions, or use the default values. Different types of
membership functions do influence the training. Some membership functions can be
defined by 3 parameters, while others require more. Since the job of ANFIS is to
adjust the parameters of the system, the more parameters there are, the longer it will
56


take to train the system, and the more computer memory that is required. The
following command will be used:
numMFs = 3; % number of membership functions.
mfType = 'gbellmf; % membership functions type is generalized bell for all
input variables.
Fismat = genfisl (tmData, numMFs, mfTypes); % generate a FIS called
fismat.
The fismat matrix comprises the initial membership functions, and the genfisl
algorithm places these initial membership functions so that they are equally spaced
with about 25% overlap within the input range.
3. Now, the training process is started by invoking the ANFIS training routine
for the Sugeno-type FIS. ANFIS uses a hybrid learning algorithm to identify
parameters of the Sugeno-type FIS. It applies the least squares method and the
backpropagation gradient descent for linear and nonlinear parameters, respectively.
The following command will be used:
[Fismatl, tmErr, ss, Fismat2, chkErr] = anfis (tmData, Fismat, Epochs
Number, NaN, chkData); % Fismatl is the trained data matrix, and Fismat2 is the
checking data matrix.
4. To test Fismatl and Fismat2, we evaluate the trained FIS by applying
Fismatl to all of the training input data, and by applying Fismat2 to all of the input
checking data. Then, we compare the result with the original TAF using the
following command:
57


Outl = evalfis (Input trainData, Fismatl); % Outl is the expected TAF for the
training data.
Out2 = evalfis (Input chkData, Fismat2); % Out2 is the expected TAF for the
checking data.
58


CHAPTER 4
MODELS DEVELOPMENT
This chapter will explain the development procedure of the different models,
and the results of each model will be shown. The results will be discussed in the
following chapter.
Conventional Multiple Linear Regression Model
The data was prepared to conduct six different regression models. The first
model included all ramp types, where the ramp type variable was included as a
dummy variable along with the other 10 variables. Then, for each ramp type, a
separate model was designed individually. A multiple linear regression procedure
was applied to predict the criterion variable, which is the Truck-Accident Frequency
(TAF).
Originally, all of the 10 variables were considered as the explanatory
variables, and entered into the model. Different methods were used to select the most
significant explanatory variables, such as Forward selection, Backward elimination,
and stepwise selection procedures. The coefficients of the final variables which
remained in each model are listed in Table 4.1. The results of these models will be
discussed in the following chapter.
59


Model Diamond Loop Outer Directional MADT RADT Ramp Length Taper to Gore Constant Error R2
All 0.655 0.825 0.777 0.899 6.05E-6 4.47E-5 0.446 1.37 0.101
Diamond 1.22E-5 1.072 1.19 0.076
Loop 8.43E-5 1.167 1.13 0 123
Outer 8.75E-5 -3.85 6.27 1.394 1.54 0.233
Directional 5.14E-5 1.600 1.70 0.062
Other 1.10 0.00
variables listed passed t-test at 0.05 level of significance
Table 4.1: Multi-Linear Regression Models
Intrinsically Linear Regression Model
In cases where the conventional linear regression model has a bad fit, or when
there is evidence of a violation of assumptions, a different strategy could be pursued
to improve the model. One method is to take the log of either the dependent variable
or the independent variables, or both. This type of transformation of variables may
improve the fit of the model and linearize the relationship.
The choice of transformation depends on several considerations. When the
true model governing the relationship is known, the choice will be relatively simple.
However, if the true model is not known, the choice of transformation method should
be determined by individually examining the plotted data of the dependent variable
(or residuals) and each one of the independent variables.
60


In our case, one model for all ramps was developed to see if a better fit could
be achieved by transforming the dependent variable only. Even though no particular
relationship (linear or nonlinear) could be traced between the dependent variable and
independent variables, an exponential relationship was examined. By taking the log
of the dependent variable, an exponential model could be transformed to a linear
expression. The final variables in this intrinsically linear model are the same as the
final variables in the conventional linear model, and the coefficients and other outputs
are listed in Table 4.2. The square root transformation of the criterion variable
(TAF) was tested, the RSME of this model will be shown in the next chapter.
Model Diamond Loop Outer Directional MADT RADT Ramp Length Taper to Gore Constant Error R2
All 0.246 0.330 0.269 0.388 2.55E-6 1.51E-5 -0.120 0.51 0.111
variables listec
passed t-test at 0.05 level of significance
Table 4.2: Intrinsically Multi-Linear Regression Model
Another term was used to measure the prediction power of all of the
regression models. This term is the Root Mean Square Error (RMSE), which is used
in the ANFIS models to unite the comparison tool among models and compare
results. The RMSE values for each ramp type are listed in Table 4.3. Also, for each
model, the mean and standard deviation for the actual and linear regression estimate
of TAF are shown in Table 4.4.
61


The quality of the linear regression prediction models as well as the other
prediction models can be seen graphically by plotting the prediction ratio (ratio of
predicted TAF to actual TAF) verses the actual TAF for each model. Figure 4.1 (a, b,
c, and d), and Figure 4.2 (a, b, c, and d), show the prediction capability of the
different regression models with 4 inputs using the training data and the checking
data, respectively Notice that, for a perfect prediction, the ratio should be unity.
And each point represent one ramp site.
Ramp Type Training Data Checking Data
Diamond 1.18 1.68
Loop 1.10 1.76
Outer Connector 1.52 1.02
Directional 1.65 1.42
Table 4.3: Linear Regression RMSE per Ramp Type
62


Ramp Actual TAF Linear Regression Estimated TAF
Type Training Checking Training Checking
Data Data Data Data
Mean SD Mean SD Mean SD Mean SD
Diamond 1.56 1.23 1.87 1.68 1.56 0.34 1.65 0.32
Loop 1.72 1.19 2.14 1.86 1.72 0.42 1.55 0.13
Outer Connector 1.84 1.70 2.20 2.17 1.89 0.82 1.88 1.21
Directional 2.25 1.74 2.25 1.55 2.25 0.44 2.76 0.39
Table 4.4: Mean and Standard Deviation of TAF per Ramp Type
63


Prediction ratio
4
CO
2 s 1 X 1
6
1 I |
X i X * * 5 S s
0 1 1 1 |
0 2 4
Actual TAF
10
Figure 4.1.a: Diamond Ramps Linear Regression Model (Training Data)


Prediction ratio
4
3 I i I I - -
2 X X X X XX*
s
X H
1 X a X X
X X X s X X
0 I i I I I i
0 2 4 6
Actual TAF
10
Figure 4. Lb: Loop Ramps Linear Regression Model (Training Data)


prediction ratio
B
5
3
2
1
0
X
X
X
£
0
X
*
2
X
X
i
4 6 8 10
Actual TAF
Figure 4.1 .c: Outer Connector Ramps Linear Regression Model (Training Data)


Prediction ratio
4
I
3
2
1
x
0
0
2
4 6 8 10 12 14
Actual TAF
Figure 4.1.d: Directional Ramps Linear Regression Model (Training Data)


Prediction ratio
4
3
2 :

t
a
1; *.
a !
0
0
2 4 6 8 10 12 14
Actual TAF
Figure 4.2.a: Diamond Ramps Linear Regression Model (Checking Data)


Prediction ratio
3
i
l
I
j
I
2
1
I
0
0
s
2 4 6
Actual TAF
Figure 4.2.b: Loop Ramps Linear Regression Model (Checking Data)


3
2
0
0
2 4 6
Actual TAF
Figure 4.2.c: Outer Connector Ramps Linear Regression Model (Checking Data)


Prediction ratio
1
0
0
I I i i i i i I i
l 4 6 8 10
Actual TAF
figure 4.2.d: Directional Ramps I,inear Regression Model (Checking Data)
12 14


Neural Network Model
we tested different neural models, and used 10, 6, and 4 input variables to
evaluate the impact of using more variables on the training process. The first attempt
was to develop one model for all-ramps, without considering the ramp-type effect.
However, the training process did not indicate that the neural network would
converge, even though different network structures were considered, such as
including more hidden neuron and adding another layer.
Four models were developed, one for each known ramp type (diamond, loop,
outer connector, and directional). The training time and the number of epochs needed
to reach the targeted SSE for each model were different. After more than 50,000
epochs, the SSE for the directional ramps network dropped to less than 0.02. This
was considered satisfactory and the training was stopped. Figure 4.3 shows the
change in learning rate and SSE for 30,000 epochs during the training of the
directional ramps network.
After the network converged (reached the targeted error), the expected TAF
for each model was obtained and compared to the actual TAF for both training data
and checking data. Table 4.5 shows the RSME for each ramp type.
72


Sum-Squared Error
Training for 30000 Epochs
Figure 4.3: Directional Ramps Neural Network Model (Training and SSE)
With Six Inputs.
Ramps Training Checking
Type Data Data
Diamond 0.67 2.28
Loop 0.23 2.07
Outer Connector 0.31 2.78
Directional 0.76 4.20
Table 4.5: Neural Network RMSE per Ramp Type
With Four Inputs
73


Figure 4.4 (a, b, c, and d) shows the prediction quality of the different neural
networks models by plotting the prediction ratio verses the actual TAF for the
different ramps in the training data. Notice that the number of prediction points in
each graph represent the number of ramps used in that prediction model.
74


4
o
MBM
03

a 2
o
TD
CD
Ql 1
X
i
X X X
X
0 : I : i t -t I ( I t
0 2 4 6 8 10 12
Actual TAF
Figure 4.4.a; Diamond Ramps Neural Network Model With Four Inputs
14


Prediction ratio
1
0
0
6
Actual TAF
Figure 4.4.b: Loop Ramps Neural Network Model With Four Inputs


Prediction ratio
2
1.5
X
X
1
X
I
0
0
2 4 6 8
Actual TAF
Figure 4.4.c: Outer Connector Ramps Neural Network Model With Four Inputs


Prediction ratio
1
0
0
x
X X X X
X
X
X
I I
2 4 6 8 10 12 14
Actual TAF
Figure 4.4.d: Directional Ramps Neural Network Model With Four Inputs


i
|
i
Figure 4.5 (a, b, c, and d) shows the prediction quality of the different neural
networks models developed (figure 4.4) when applied to the checking data. The
prediction quality can be shown by plotting the prediction ratio verses the actual TAF.
Notice that the number of prediction points in each graph represent the number of
ramps used in that prediction model.
The results of the neural networks models will be discussed in more details in
the following chapter. Also, the results of the neural networks models will be
compared to the results of other models, such as linear regression and ANFIS method.
79


Prediction ratio
14
12
10
8
6
4
2
0
0
! i -
2
4
6 8 10
Actual TAF
Figure 4.5.a: Diamond Ramps Neural Network Model (Checking Data)


4
OO
2
CL
1

0 I I i i i T I
0 2 4 6 8
Actual TAF
Figure 4.5.b: Loop Ramps Neural Network Model (Checking Data)


Prediction ratio
5
3
2 i
j
1
I
|
0
0
2 3 4 5
Actual TAF
Figure 4.5.e: Outer Connector Rumps Neural Network Model (Checking Data)


Actual TAF
Figure 4.5.d: Directional Ramps Neural Network Model (Checking Data )


ANFIS Models
The execution of the ANFIS model consumes a lot of computer memory, and
the number of inputs to the system determines the number of fuzzy rules. Further, the
number of rules created is equal to fN where / is the number of membership
functions used (linguistic regions), and N is the number of inputs. The ANFIS
structure is a much more complex network structure than neural networks, where the
number of nodes of the ANFIS network can be calculated using the following
formula:
number of nodes = number of inputs + 2 x number of input membership functions
+ 2 x {number of rules + 2) +1
The number of parameters depend on the type of membership function, input
variables, number of rules, and number of nodes. In our case, for only four inputs, the
total number of nodes was 193 node with 441 parameters, which required about 16 M
of RAM memory. So, for 13 inputs ANFIS model, the RAM memory required is
about 200 M based on an estimate by MATLAB technical engineers. According to
these estimates, MATLAB recommend the use of ANFIS network only when N is
less than 7.
Based on these limitations, the initial models were designed to include only 4
inputs, which are gore-to-taper distance, truck percentage on main lane, and total
traffic volumes on both ramp and main lane.
84


First, a model was developed to include all ramps, then 4 different models
were developed for each classified ramp type using different combinations of input
variables as seen in Figure 4.6.
4 inputs
2 inputs
membership
function
Figure 4.6: ANFIS Model
Two sets of input variables were selected to evaluate the impact of including
different variables in the ANFIS model. The first set included the variables gore-to-
85


taper distance, main road truck percentage, and traffic volume on both main road and
ramp. The second set included the variables gore-to-taper distance, truck percentage on
both main road and ramps, and main-road traffic volume. Table 4.6 shows the minimum
RMSE for all the ANFIS models.
Ramp Type Data Set 1 Training Checking Data Set 2 Training Checking
All Ramps 0.055468 0.434693 0.0777780 0.154142
Diamond 0.026087 0.792343 0.0419986 0.330881
Loop 0.001808 0.378998 0.0003830 0.412940
Outer Connector 0.000032 0.041015 0.0000500 0.208608
Directional 0.001106 0.559130 0.0023230 0.287107
Table 4.6: ANFIS Minimum RMSE
Notice that RMSE in Table 4.6 reflects the magnitude of the normalized TAF,
while the RMSE values in Table 4.7 reflect the actual magnitude of the error for the
original TAF using the first data set.
Table 4.7 shows the real RSME for each model. These values reflect the variation
in performance of the ANFIS models between the training data and the checking data.
The results and interpretation of the different tables and figures will be discussed in
Chapter 5.
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Ramps Training Checking
Type Data Data
Diamond 0.34 10.30
Loop 0.01 2.65
Outer Connector 0.00 0.41
Directional 0.01 6.71
Table 4.7: ANFIS RSME per Ramp Type
Figures 4.7 (a. b, c, and d), show the prediction quality of the different ANFIS
models for the first training data set by plotting the prediction ratio verses the actual TAF.
Notice that the number of prediction points in each graph represent the number of ramps
used in that prediction model. And a unity prediction ratio indicates a perfect prediction.
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