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Research ArticleMultiple Na\ve Bayes Classifiers Ensemble for
TrafficIncident Detection
Qingchao Liu,1,2 Jian Lu,1,2 Shuyan Chen,1,2 and Kangjia
Zhao3
1 Jiangsu Key Laboratory of Urban ITS, Southeast University,
Nanjing 210096, China2 Jiangsu Province Collaborative Innovation
Center of Modern Urban Traffic Technologies, Nanjing 210096,
China3Department of Civil & Environment Engineering, National
University of Singapore, Singapore 119078
Correspondence should be addressed to Jian Lu; lujian
[email protected]
Received 16 January 2014; Revised 26 March 2014; Accepted 27
March 2014; Published 28 April 2014
Academic Editor: Erik Cuevas
Copyright 2014 Qingchao Liu et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
This study presents the applicability of the Nave Bayes
classifier ensemble for traffic incident detection. The standard
NaiveBayes (NB) has been applied to traffic incident detection and
has achieved good results. However, the detection result of
thepractically implementedNBdepends on the choice of the optimal
threshold, which is determinedmathematically by using
Bayesianconcepts in the incident-detection process. To avoid the
burden of choosing the optimal threshold and tuning the parameters
and,furthermore, to improve the limited classification performance
of the NB and to enhance the detection performance, we propose anNB
classifier ensemble for incident detection. In addition, we also
propose to combine the Nave Bayes and decision tree (NBTree)to
detect incidents. In this paper, we discuss extensive experiments
that were performed to evaluate the performances of
threealgorithms: standard NB, NB ensemble, and NBTree. The
experimental results indicate that the performances of five rules
of theNB classifier ensemble are significantly better than those of
standard NB and slightly better than those of NBTree in terms of
someindicators. More importantly, the performances of the NB
classifier ensemble are very stable.
1. Introduction
The functionality of automatically detecting incidents
onfreeways is a primary objective of advanced traffic manage-ment
systems (ATMS), an integral component of the NationsIntelligent
Transportation Systems (ITS) [1]. Traffic incidentsare defined as
nonrecurring events such as accidents, disabledvehicles, spilled
loads, temporarymaintenance and construc-tion activities, signal
and detector malfunctions, and otherspecial and unusual events that
disrupt the normal flow oftraffic and cause motorist delay [2, 3].
If the incident cannotbe handled timely, it will increase traffic
delay, reduce roadcapacity, and often cause second traffic
accidents. Timelydetection of incidents is critical to the
successful implemen-tation of an incident management system on
freeways [4].
Incident detection is essentially a pattern
classificationproblem, where the incident and nonincident traffic
patternsare to be recognized or classified [5]. That is to say,
incidentdetection can be viewed as a pattern recognition problem
that
classifies traffic patterns into one of the two classes:
noninci-dent and incident classes.The classification is normally
basedon spatial and temporal traffic pattern changes during an
inci-dent. The spatial pattern changes refer to the traffic
patternalterations over a stretch of a freeway. The temporal
trafficpattern changes refer to the traffic pattern alterations
overconsecutive time intervals. Typically, traffic flow maintainsa
consistent pattern at upstream and downstream detectorstations.When
an incident occurs, however, traffic flow at theupstream of
incident scene tends to be congested while that atthe downstream
station tends to be light due to the blockageat incident site.
These changes in the traffic flow are reflectedin the detector data
obtained from both the upstream anddownstream stations [5].
Therefore, an AID problem isessentially a classification problem.
Any good classifier is apotential tool for the incident detection
problem. Based onthis idea, in our approach to incident detection
in general, wetreat the problem as one of pattern classification
problems.Under normal traffic operation, traffic parameters
(speeds,
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2014, Article ID 383671, 16
pageshttp://dx.doi.org/10.1155/2014/383671
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2 Mathematical Problems in Engineering
occupancies, and volumes) both upstream and downstreamfrom a
given freeway site are expected to have more or lesssimilar
patterns in time (except under bottleneck situations).In the case
of an incident, this normal pattern is disrupted.Patterns of
incident develop increased occupancies and dropin speeds for
instance.
Automated incident detection (AID) systems, whichemploy an
incident detection algorithm to detect incidentsfrom traffic data,
aim to improve the accuracy and efficiencyof incident detection
over a large traffic network. Early AIDalgorithmdevelopment focused
on simple comparisonmeth-ods using raw traffic data [6, 7]. To
enhance algorithm perfor-mance and to achieve real-time incident
detection, advancedmethods have been suggested, which include image
process-ing [8], artificial neural networks [9], support
vectormachine[4], and data fusion [10]. Although these new
publishedmethods represent significant improvements,
performancestability and transferability are still major issues
concerningthe existing AID algorithms. To enhance freeway
incidentdetection and to fulfill the universality expectations for
AIDalgorithms, the classic Bayesian theory has attracted
manyscientists attention. Due to the Nave Bayes ensemble andNave
Bayes, and decision tree (NBTree) algorithm simplicityand easy
interpretation, the applications of this approach canbe found in an
abundant literature. However, the report ofits application to
traffic engineering is rare. It provides amplemotivation to
investigate thismodel performance on incidentdetection.
There are drawbacks which limit its applications; theoptimal
threshold and parameter of Nave Bayes (NB) havea great effect on
the generalization performance, and settingthe parameters of the NB
classifier is a challenging task.At present, there is no structured
method to choose them.Typically, the optimal threshold and
parameters have to beenchosen and tuned by trial and error. Some
studies haveapplied search techniques for this problem; however, a
largeamount of computation time will still be involved in
suchsearch techniques, which are themselves
computationallydemanding.
A natural and reasonable question is whether we canincrease or
at least maintain NB performance, while, at thesame time, avoiding
the burden of choosing the optimalthreshold and tuning the
parameters. Some researchers haveproposed classifier ensembles to
address this problem. Theperformance of classifier ensemble has
been investigatedexperimentally, and it appears to consistently
give betterresults [1113]. Kittler et al. [12] used different
combiningschemes based on multiple classifier ensembles, and six
ruleswere introduced into ensemble learning to search for
accurateand diverse classifiers to construct a good ensemble.
Thepresented method works on a higher level and is more directthan
other search based methods of ensemble learning. Theprevious
research illustrated that ensemble techniques areable to increase
the classification accuracy by combining theirindividual outputs
[14, 15]. While these general methods arepreexisting, their
application into the specific problem andtheir integration into the
proposed model for the detectionof traffic incident are new. It is
expected that the NB classifierensemble has more generalization
ability, but the method has
not been seen discussed in the context of incident
detection.Coupling this expectation of reasonably accurate
detectionwith the attractive implementation characteristics of the
NBclassifier ensemble, we propose to apply the NB
classifierensemble achieved by the combination of different rules
todetect incidents. The NB classifier ensemble algorithm trainsmany
individual NB classifiers to construct the classifierensemble and
then uses this classifier ensemble to detectthe traffic incidents.
It needs to train many times. Takingthis into account, we also
propose NBTree [16] for incidentdetection. The NBTree splits the
dataset by applying anentropy-based algorithm and uses standard NB
classifiers atthe leaf node to handle attributes.TheNBTree applies
strategyto construct a decision tree and replaces leaf nodes with
NBclassifiers. We have performed some experiments to evaluatethe
performances of the standard NB, NB ensemble, andNBTree
algorithms.The experimental results indicate that theperformances
of five rules of the NB classifier ensemble aresignificantly better
than those of standard NB and slightlybetter than those ofNBTree in
terms of some indicators.Moreimportantly, the performances of the
NB classifier ensembleare very stable.
The remaining part of the paper is structured as follows.Section
2 introduces the combination schemes of the NBclassifier ensemble.
The general performance criteria of AIDare presented in Section 3.
Section 4 is devoted to empiricalresults. In this section, we
evaluate the performances of thethree algorithms: standard NB, NB
ensemble, and NBTree.Finally, the conclusions are drawn in Section
5.
2. Na\ve Bayes Classifier andCombining Schemes
2.1. Classification and Classifier. A dataset generally
consistsof feature vectors, where each feature vector is a
descriptionof an object by using a set of features. For example,
take alook at the synthetic dataset as shown in Figure 1. Here,
eachobject is a data point described by the features
-coordinate,-coordinate, and color, and a feature vector looks like
(0.8,0.9, yellow) or (0.9, 0.1, red). Features are also called
attributes,a feature vector is also called an instance, and
sometimes adataset is called a sample.
Nave Bayes classifier ensemble is a predictive model thatwewant
to construct or discover from the dataset.Theprocessof generating
models from data is called learning or training,which is
accomplished by a learning algorithm. In supervisedlearning, the
goal is to predict the value of a target featureon unseen
instances, and the learned model is also called apredictor. For
example, if we want to predict the color of thesynthetic data
points, we call yellow and red labels, andthe predictor should be
able to predict the label of an instanceforwhich the label
information is unknown, for example, (0.7,0.7). If the label is
categorical, such as color, the task is calledclassification and
the learner is also called classifier.
2.2. Nave Bayes Classifier. To classify a test instance ,
oneapproach is to formulate a probabilistic model to estimatethe
posterior probability ( | ) of different s and predict
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Mathematical Problems in Engineering 3
10
1
X
Y
Instance: (X, Y; color)(1) If (X < 0.25 and Y > 0.75)
or
(X > 0.75 and Y < 0.25) then
(2) If (X > 0.75 and Y > 0.75) then
Y [0.25, 0.50]) then
(3) If (X < 0.25 and Y < 0.25) then
(4) If (X [0.25, 0.50] and
Figure 1: The synthetic dataset.
the one with the largest posterior probability; this is
themaximum a posteriori (MAP) rule. By Bayes Theorem, wehave
( | ) =
( | ) ()
()
, (1)
where () can be estimated by counting the proportion ofclass in
the training set and() can be ignored sincewe arecomparing
different s on the same . Thus we only need toconsider ( | ). If we
can get an accurate estimate of ( |), we will get the best
classifier in theory from the giventraining data, that is, the
Bayes optimal classifier with theBayes error rate, the smallest
error rate in theory. However,estimating ( | ) is not
straightforward, since it involvesthe estimation of exponential
numbers of joint-probabilitiesof the features. To make the
estimation tractable, someassumptions are needed. The naive Bayes
classifier assumesthat, given the class label, the features are
independent ofeach other within each class. Thus, we have
( | ) =
=1
(| ) (2)
which implies that we only need to estimate each feature valuein
each class in order to estimate the conditional probability,and
therefore the calculation of joint-probabilities is avoided.In the
training stage, the naive Bayes classifier estimates
the probabilities () for all classes and (| ) for
all features = 1, 2, . . . , and all feature values from the
training set. In the test stage, a test instance will be
predictedwith label if leads to the largest value of all the class
labels
( | ) ()
=1
(| ) . (3)
As demonstrated in paper [17], for a threshold level of0.0006,
65.4% of incidents were identified. However, if thethreshold cannot
be chosen appropriately, fewer incidentswill be identified
correctly. As known frommachine learning,the naive Bayesian
classifier provides a simple approach,with clear semantics, for
representing, using, and learningprobabilistic knowledge. The
method is designed for use insupervised induction tasks, in which
the performance goalis to accurately predict the class of test
instances and inwhich the training instances include class
information. In thisway, it avoids choosing the optimal threshold
and tuning theparameters manually.
2.3. Combining Schemes
2.3.1. Five Rules. Themost widely used probability combina-tion
rules [12] are the product rule, the sum rule, themin rule,and the
max rule. Given classifiers and
1, . . . ,
, classes
1, . . . ,
, these are defined as follows.
Product rule:
(| x1, . . . , x
) =
1
()
=1
(| x) , (4)
where xis the input to the th classifier and (
) is the a
priori probability for class .
Sum rule:
(| x1, . . . , x
) =
1
=1
(| x) . (5)
Min rule:
(| x1, . . . , x
) =
min (| x)
=1min (| x)
. (6)
Max rule:
(| x1, . . . , x
) =
max (| x)
=1max (| x)
. (7)
Majority vote rule:
=
{
{
{
1 if (| x) =
max=1
(| x)
0 otherwise,
=1
=
max=1
=1
.
(8)
Note that for each class , the sum on the right hand side of
(8) simply counts the votes received for this hypothesis fromthe
individual classifiers. The class that receives the largestnumber
of votes is then selected as the majority decision.
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4 Mathematical Problems in Engineering
2.3.2. Combine Decision Tree and Nave Bayes. The NaveBayesian
tree (NBTree) algorithm is similar to the classicalrecursive
partitioning schemes, except that the leaf nodes cre-ated are nave
Bayesian classifiers instead of nodes predictinga single class
[16]. First, define a measure called entropy thatcharacterizes the
purity of an arbitrary collection of instances.Given a collection ,
if the target attribute can take on different values, then the
entropy of relative to this -wiseclassification is defined as
Entropy () =
=1
log2() , (9)
where is the proportion of belonging to class . The
information gain, Gain(, ), of an attribute , the
expectedreduction in entropy caused by partitioning the
examplesaccording to this attribute relative to , is defined as
Gain (, ) = Entropy () Vvalue()
V
||
Entropy (V) ,
(10)
where value () is the set of all possible values for attributes
and V is the subset of for which attribute has valueV. NBTree is a
hybrid approach that attempts to utilizethe advantage of both
decision trees and nave Bayesianclassifiers. It splits the dataset
by applying an entropy-basedalgorithm and uses standard nave
Bayesian classifiers at theleaf node to handle attributes. NBTree
applies strategy toconstruct a decision tree and replaces leaf
nodes with NBclassifiers.
3. Performance Criteria of Aid
3.1. Definition of DR, FAR, MTTD, and CR. Four primarymeasures
of performance, namely, detection rate (DR), falsealarm rate (FAR),
mean time to detection (MTTD), andclassification rate (CR), are
used to evaluate traffic incidentdetection algorithms. We will
quote the definitions from[18, 19].
DR is defined as the number of incidents correctlydetected by
the traffic incident detection algorithmdivided bythe total number
of incidents known to have occurred duringthe observation
period:
DR = number of incident cases detectedtotal number of incident
cases
100%. (11)
FAR is defined as the proportion of instances that
wereincorrectly classified as incident instances based on the
totalinstances in the testing set. Out of the total number
ofapplications of the model to the dataset, FAR is calculatedto
determine how many incident alarms were falsely set.In order to
decrease FAR, persistent test is often used. Anincident alarm is
triggered whenever consecutive outputsof the model exceed the
threshold, which is called persistentcheck of
FAR = number of false alarm casestotal number of non-incident
cases
100%.(12)
MTTD is computed as the average length of time between thestart
of the incident and the time the alarm is initiated.Whenmultiple
alarms are declared for a single incident, only thefirst correct
alarm is used for computing the detection rateand the mean time to
detect
MTTD =1+ 2+ +
+ +
. (13)
Besides these three measures, we also use classificationrate
(CR) as an index to test traffic incident detectionalgorithms. Of
the total number of applications of cyclelength data or input
instances, the percentage of correctlyclassified instances
(including both incident and nonincidentinstances) by the model is
computed as CR
CR =number of instances correctly classified
total number of instances 100%.
(14)
3.2. Area under the ROC Curve (AUC). Receiver opera-tor
characteristic (ROC) curves illustrate the relationshipbetween the
DR and the FAR from 0 to 1. Often thecomparison of two or more ROC
curves consists of eitherlooking at the area under the ROC curve
(AUC) or focusingon a particular part of the curves and identifying
which curvedominates the other in order to select the
best-performingalgorithm. AUC, when using normalized units, is
equal tothe probability that a classifier will rank a randomly
chosenpositive instance higher than a randomly chosen negative
one(assuming positive ranks higher than negative) [20]. Itcan be
shown that the area under the ROC curve is closelyrelated to
theMann-Whitney, which testswhether positivesare ranked higher than
negatives. It is also equivalent to theWilcoxon test of ranks [21].
The AUC is related to the Ginicoefficient (
1) by the formula [22]. In this way, it is possible
to calculate the AUC by using an average of a number
oftrapezoidal approximations:
AUC = 1 + 12
, (15)
where 1= 1
=1( 1)(+ 1).
3.3. Statistics Indicators. In statistics, the mean absolute
error(MAE) is a quantity used to measure how close forecasts
orpredictions are to the eventual outcomes. The mean absoluteerror
is given by
MAE = 1
=1
. (16)
The root-mean-square error (RMSE) is a frequently usedmeasure of
the differences between values predicted by amodel or an estimator
and the values actually observed.These individual differences are
called residuals when thecalculations are performed over the data
sample that wasused for estimation and are called prediction errors
whencomputed out of sample. The RMSE serves to aggregate
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Mathematical Problems in Engineering 5
the magnitudes of the errors in predictions for various
timesinto a single measure of predictive power:
RMSE = 1
=1
( )
2
. (17)
The equality coefficient (EC) is useful for comparing
differentforecast methods; for example, whether a fancy forecast
isin fact any better than a nave forecast repeating the
lastobserved value. The closer the value of EC is to 1, the
betterthe forecast method. A value of zero means the forecast is
nobetter than a nave guess:
EC = 1
=1( )
2
=12
+
=12
.(18)
Kappa measures the agreement between two raters who eachclassify
items into mutually exclusive categories. Kappais computed as
formula (19), () is the observed agreementamong the raters, and ()
is the expected agreement; thatis, () represents the probability
that the raters agree bychance. The values of Kappa are constrained
to the interval[1, 1]. Kappa = 1means perfect agreement, Kappa =
0meansthat agreement is equal to chance, and Kappa = 1 meansperfect
disagreement:
Kappa = () ()1 ()
. (19)
4. Experiments on Traffic Incident Detection
4.1. Parameters and Procedures of Experiments. To describethe
experiments clearly, we first present the definitions forall
parameters and symbols used in experiments. Then, wedescribe the
experiment procedures in detail.
4.1.1. Parameters of Experiments. Some parameters areadopted to
make the procedures of the experiments moreautomatic and optimized.
In addition, some symbols areused to denote specified conceptions.
For clarity, we havepresented the definitions of each parameter and
symbol inTable 1.
4.1.2. Construction of Datasets for Training and Testing.
Thetraffic datamentioned refers to three basic traffic flow
param-eters, namely, volume, speed, and occupancy. The incident
isdetected based on section, which means that the traffic
datacollected from the upstream and the downstream
detectionstations are usually used as model inputs in AID systems.
InFigure 2, A1, namely, the first attribute, so the number of
-variables (predictor variables) is 6.Thismeans that thematrixused
in training themodel has the size
6.The test data
form amatrix of size 6. The formal description of matrix
and can be written as shown in Figure 2.One instance consists of
at least the following items:
(i) speed, volume, and occupancy of the upstream detec-tor,
(ii) speed, volume, and density of the downstream detec-tor,
(iii) traffic state (incident or nonincident),
where the item traffic state is a label. The value of the
labelis 1 or 1, referring to nonincident or incident,
respectively,which is determined by the incident dataset.
Typically, themodel is fit for part of the data (the training set),
and thequality of the fit is judged by how well it predicts the
otherpart of the data (the test set). The entire dataset was
dividedinto two parts: a training set that was used to build the
modeland a test set that was used to test the models
detectionability. Where each row is composed of one observation,
isthe number of instances and
{1, 1}. The data analysis
problem is to relate the matrix as some function of thematrix to
predict (e.g., traffic state) using the data of , = (). The
training set was used to develop a Nave Bayesclassifier ensemble
that was, in turn, used to detect incidentsfor the test set
samples. The output values of the detectionmodels were then
compared with the actual ones for eachof the calibration samples,
and the performance criteria werecalculated and compared.
4.1.3. Experiments Procedures. The experiments were per-formed
according to the procedures as shown in Figure 3.
Step 1. Divide the whole dataset into training set and testset .
We take part of the whole dataset as the training set, the other as
test set and use the parameter
to control
the size of the training set , and is obtained by taking
outsamples from the front to the back of.
Step 2. Perform sampling with replacement from training set
times and then obtain the training subsets
1, 2, . . . ,
.
We use the parameter to control the ratio of the number
ofsamples in the training subset to the number of samples inthe
training set; that is, the number of samples in the trainingsubset
is sub = . The range of is [0, 1].
Step 3. While = 1, 2, 3, . . . , , perform the following:
(1) use the training subset to train the th individual
NB classifier INBC;
(2) use the training subset to train the th NBTree
classifier INBTreeC;
(3) use the all the existing individual NB classifiers
toconstruct the th ensemble NB classifier ENBC
.
Step 4. While = 1, 2, 3, . . . , , perform the following:
(1) test the performances of the th individual NB classi-fier
INBC
on test set ;
(2) test the performances of the th NBTree
classifierINBTreeC
on test set ;
(3) test the performances of the th ensemble NB classi-fier
ENBC
on test set .
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6 Mathematical Problems in Engineering
A1 A2 A3 A4 A5 A6 Class
Upstream data Downstream data
Matrix X Matrix Y
First instance
Last instance
......
...
Traffic state1
Traffic staten
Speedup 1 Volumeup 1 Occupancyup 1 Speeddn 1 Volumedn 1
Occupancydn 1
Speedup n Volumeup n Occupancyup n Speeddn n Volumedn n
Occupancydn n
Figure 2: Construction of traffic datasets.
Detector 1 Detector 2 Detector 3 Detector n
Signal processing unit
Speed1
Volume1Occupancy
1
Speed2
Volume2Occupancy
2
Speed3
Volume3Occupancy
3
Speedn
VolumenOccupancy
n
Buildtrafficow
Flow Flow Flow
database
Historical dataReal-time data
New instance
Training setensemble model
Real-timetraffic state
Alarm
IncidentIf state = 1 If state = 1
State Incident-free
Nave Bayes classifier
Figure 3: A freeway incident detection model based on Nave Bayes
classifier ensemble.
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Mathematical Problems in Engineering 7
Table 1: Definition of Symbols and Parameters.
Parameter/Symbol Definition
Symbol
The whole dataset Training set Test set
The th subset of training setINBC
The th individual Nave Bayes classifier
ENBC
The th ensemble Nave Bayes classifierINBTreeC
The th individual NBTree classifier
Parameter
The total number of the training subsets The ration of sub
to
The number of samples in test set
The number of samples in training set all The total number of
samples in the whole datasetsub The number of samples in the
training subset (each training subset has the same number of
samples)incident The number of incident samples in the whole
datasetnonincident The number of non-incident samples in the whole
dataset
Table 2: Parameter Setting of Experiments.
Parameter Setting of Experiments for I-880 DatasetParameter
sub all incident nonincident
Value 20 0.05 45138 45518 2275 90656 4136 86520Parameter Setting
of Experiments for AYE Dataset
Parameter
sub all incident nonincident
Value 20 0.05 16000 13500 675 29500 6000 23500
4.2. Experiments on I-880 Dataset
4.2.1. Data Description for I-880 Dataset. We proceeded toreal
world data. These data were collected by Petty et al. fromthe I-880
Freeway in the San Francisco Bay area, California,USA. This is the
most recent and probably the most well-known freeway incident
dataset collected, and these data havebeen used inmany studies
related to incident detection. Loopdetector data, with and without
incident, was collected froma 9.2 miles (14.8 km) segment of the
I-880 Freeway betweenthe Marina and Wipple exits, in both
directions. There were18 loop detector stations in the northbound
direction and17 stations in the southbound direction. The data
collectedincluded traffic volume, occupancy, and speed,
averagedacross all lanes in 30 s intervals at the same station.
Insummary, the training dataset has 45,518 training instances,of
which 2100 are incident instances (from 22 incident cases).The
testing dataset has 45,138 instances in all, including 2036incident
instances (from 23 incident cases). Thus, incidentexamples are very
rare in this dataset, as only approximately4.6% and 4.5% incident
examples are contained in thetraining set and the testing set,
respectively. Each instancehas seven features. In addition to the
measurements ofspeed, volume, and occupancy collected at both the
upstreamdetector station and the downstream detector station,
thelast one is the class label, 1 for nonincident state and 1
forincident state.
4.2.2. Parameter Setting of Experiments for I-880 Dataset.
Todivide
into sub properly, there are some parameters that
need to be set, which can be seen in Table 2. We set values
foreach parameter and present the results in Table 2.
In our experiments, we constructed 20 individual NBclassifiers,
20NB ensemble classifiers, and 20 NBTree clas-sifiers. We tested
the performances of the five rules ofeach classifier on the I-880
dataset. Then, we calculatedthe averages and variances of the
performances of the 20individual NB classifiers, 20NB ensemble
classifiers, and 20NBTree classifiers. The summarized results are
in Table 3. InTable 3, the results are presented with the form
average variance, and the best results are highlighted in bold.
Tomakea visual comparison of the performances of all classifiers,
weplotted them in Figures 4 and 5.
4.3. Experiments on AYE Dataset with Noisy Data
4.3.1. Data Description for AYE Dataset. The traffic data usedin
this study for the development of the incident detectionmodels was
produced from a simulated traffic system. A5.8 km section of the
Ayer Rajah Expressway (AYE) inSingapore was selected to simulate
incident and nonincidentconditions.This site was selected for
incident detection studybecause of its diverse geometric
configurations that can covera variety of incident patterns [23,
24].
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8 Mathematical Problems in Engineering
Table3:Ex
perim
entalR
esultsof
NB,
Five
Rulesa
ndNBT
reeB
ased
asAp
pliedto
theI-880
Dataset(Th
eperform
ancesa
representedin
theform
average
varia
nce).
Algorith
mDR
FAR
MTT
DCR
AUC
Kapp
aMAE
RMSE
ECNave
Bayes
classifier
0.82282.5005
0.03986.32E07
1.299
10.01727
0.95406.43E07
0.89156.50E06
0.594792.63E05
0.016193.11E07
0.179959.4
7E06
0.991848.03E08
Prod
uctrule
ensemble
0.84042.91E04
0.03803.03E06
1.76151.0
4744
0.95572.22E06
0.90127.78E05
0.613832.28E04
0.016025.33E07
0.178981.6
0E05
0.991921.3
8E07
Sum
rule
ensemble
0.89
635.47
E06
0.02975.87E07
1.69060.89473
0.96363.66
E07
0.93331.0
6E06
0.69321.9
1E05
0.0159
92.04
E07
0.1788
16.34
E06
0.99
1945.26
E08
Max
rule
ensemble
0.89605.58E06
0.02971.6
2E07
1.62360.84537
0.96361.6
3E07
0.93311.33E06
0.692686.81E06
0.016068.09E08
0.179242.52E06
0.991902.09E08
Min
Rule
ensemble
0.89622.29E06
0.03042.99E06
1.61930.85452
0.96293.13E06
0.93292.53E
060.688451.2
8E04
0.016031.0
6E07
0.179023.28E06
0.991922.73E08
MVrule
ensemble
0.81936.25E05
0.04102.72E06
1.78021.15158
0.95272.02E06
0.88921.2
9E05
0.586395.16E05
0.016305.09E07
0.180521.52E05
0.991781.32E07
NBT
reec
lassifier
0.81430.00112
0.00
891.4
63E5
1.36220.00798
0.98
311.4
74E5
0.90272.7953E4
0.80
520.00
147
0.026266.3487E6
0.228941.2
02E4
0.986691.6
7E6
-
Mathematical Problems in Engineering 9
The simulation system generated volume, occupancy, andspeed data
at upstream and downstream sites for both inci-dent and nonincident
traffic conditions. The traffic datasetconsisted of 300 incident
cases that had been simulated basedon AYE traffic.The simulation of
each incident case consistedof three parts. The first part was the
nonincident periodthat lasted for 5min. This was after a simulation
of a 5minwarm-up time. During the warm-up time, the data
containsnoise. The second part was the 10min incident period.
Thiswas followed by a 30min postincident period. Each inputpattern
included traffic volume, speed, and lane occupancyaccumulated at 30
s intervals, averaged across all the lanes, aswell as the traffic
state. The value of the traffic state label is 1or 1, referring to
nonincident or incident states, respectively.
4.3.2. Parameter Setting of Experiments for the AYE Dataset.As
we used a new dataset to perform the experiments, theparameter
values of the experiments needed to be updated.The updated
parameter values can be seen in Table 2.
4.3.3. Experimental Results with the AYE Dataset. As men-tioned
in Section 4.3.1, the AYE dataset includes noisy data,which
seriously reduces the quality of the detection. There-fore, the
experimental results obtained with the AYE datasetare much worse
than the experimental results from the I-880 dataset overall. The
AYE dataset experimental results aresummarized in Figures 6 and 7
and Table 4. In Table 4, foreach algorithm of standard NB, NB
ensemble, and NBTree,we calculate the averages and variances of the
performancesof the total 20 individual classifiers or ensemble
classifiers.The results are presented as average variance, and the
bestresults are highlighted in bold.
4.4. Performance Evaluation. In this subsection, we
haveevaluated the performances of all three algorithms, standardNB,
NB ensemble, and NBTree, using the I-880 dataset andthe AYE dataset
with noisy data. In Figures 4 and 6, thesubfigures (a)(f) evaluate
the performances of five rules onthe indicators DR, FAR, MTTD, CR,
AUC, and Kappa. InFigures 5 and 7, the subfigures evaluate the
performances ofMAE, RMSE, and EC.
4.4.1. Performance Evaluation for the I-880 Dataset. FromFigures
4 and 5, we can see that the performances of thesum rule, the max
rule, and the min rule are stable andsignificantly better than the
performances of the productrule and the majority vote rule. The
performances of theproduct rule and the majority vote rule
fluctuate violentlydynamically, which demonstrates that the
performances areunstable. The reason for this is that the results
of the NaveBayes algorithm depend on the appropriate threshold
andparameters, and the algorithm is not resilient to
estimationerrors. For example, in Figure 4(a), the DR of the
productrule and the majority vote rule reaches 88% and 82% oreven
higher. In contrast, DR can also reach as low as 78%or even lower.
In Figure 5 concerning the product rule,when the number of
classifiers is less than 10, the RMSEreaches as low as 0.17 or even
lower. In contrast, the RMSE
can reach 0.19 or even higher. These results indicate thatwhen
the threshold and parameters are chosen appropriately,the detection
performance can improve. We need to selectan appropriate threshold
and parameters to make the NBensemble achieve optimal performances,
But the proceduresfor choosing an appropriate threshold and
parameters needto use the trial and error method. Until now, there
has notbeen a structuredway to choose these values.There is
anothersignificant phenomenon that can be found from the data
inFigures 4 and 5. When the number of classifiers increases,the
performance of the five rules mitigates their fluctuation,and they
tend to achieve a stable value. The reason for thisis that the
multiple classifiers ensemble can compensate forthe defect of a
single classifier to some extent. From Figure 4,we can also see
that the performances of the sum rule, themax rule, and the min
rule on each indicator are very closeto each other. As for the
indicators DR, FAR, AUC, andKappa, the performance of the sum rule
is slightly betterthan that of the max rule and the min rule. As
mentionedabove, AUC and Kappa can evaluate the performances
morecomprehensively than can the other four indicators. To makean
overall evaluation, the performances of the NB ensembleare slightly
better than those of the Nave Bayes.
In Table 3, the average DR value of Nave Bayes is 82.28%,whereas
the average DR value of the five rules ranges from81.93% to 89.63%.
This indicates that the NB ensemble algo-rithm is more sensitive to
the traffic incidents and can detectmore traffic incidents than can
the standard Nave Bayesalgorithm. The average MTTD value of NBTree
is 1.36min,which indicates that the NBTree algorithm can detect
theincidents more quickly than NB ensemble algorithm. Theaverage
value of CR of NBTree is 98.31%, which indicates thatNBTree can
achieve the higher classification accuracy of theincident instance
than can the Nave Bayes and NB ensemblealgorithm. The average Kappa
value of NBTree is 0.8052,which indicates that the performance of
NBTree classifieris significantly better than those of the other
classifiers. Inaddition, it can detect more incidents than the NB
ensemble,but the experimental results are not the same. The
reasonfor this is that the FAR of NBTree is the highest.
FARimproves the performance of the NBTree classifier.TheMAE,RMSE,
and EC of the product rule are best, which indicatesthat the
predicted value of the product rule is the
closestapproximation.
4.4.2. Performance Evaluation Using AYE Dataset with NoisyData.
From Figures 6 and 7, we can see that when the datasetincludes
noisy data, the performances of the sum rule, themax rule, and the
min rule are still stable and significantlybetter than the
performances of the product rule and themajority vote rule. In
addition, we also find that the perfor-mances of the sum rule, the
max rule, and the min rule aremuch better than those of the product
rule and the majorityvote rule on the indicators of DR, FAR, CR,
and Kappa.Among the five rules, it appears that the min rule yields
thehighest average MAE, whereas the other rules are similar. Asfar
as EC is concerned, the five rules perform at a similarlevel that
is better than that of the NBTree. The opposite case
-
10 Mathematical Problems in Engineering
Table4:Ex
perim
entalR
esultsof
NB,
Five
Rulesa
ndNBT
reeB
ased
asAp
pliedto
theA
YEDataset(Th
eperform
ancesa
representedin
theform
average
varia
nce).
Algorith
mDR
FAR
MTT
DCR
AUC
Kapp
aMAE
RMSE
ECNave
Bayes
classifier
0.71120.00348
0.04998.16E04
1.3940.08869
0.90
940.00
111
0.86634.39E04
0.62479.14E04
0.167693.07E04
0.578955.31E04
0.908541.8
3E04
Prod
uctrule
ensemble
0.64570.00127
0.05572.40
E06
1.86851.2
4139
0.87321.6
4E06
0.86194.85E07
0.74793.58E05
0.166901.0
4E06
0.577753.11E06
0.908953.68E07
Sum
rule
ensemble
0.79235.57E05
0.05006.73E07
1.43840.63868
0.87743.55E07
0.86
673.75E07
0.74391.2
9E04
0.167394.94E07
0.57861
1.47E06
0.908661.7
5E07
Max
rule
ensemble
0.78691.7
9E04
0.05002.21E06
1.4571
0.72272
0.87711.2
5E06
0.86632.02E07
0.75114.12E05
0.168034.96E06
0.579691.4
6E05
0.908281.7
7E06
Min
Rule
ensemble
0.79
601.2
5E04
0.04982.76E06
1.46460.73539
0.87811.10E06
0.86
671.14E06
0.60521.8
6E06
0.1663
83.25E06
0.576849.8
3E06
0.909261.15E06
MVrule
ensemble
0.62461.15E05
0.05722.29E06
1.83581.19124
0.87203.84E07
0.78371.18E06
0.73415.31E05
0.166872.27E06
0.577696.87E06
0.90
8978.03E07
NBT
ree
classifier
0.72753.25E4
0.02
501.8
5E5
1.21030.1038
30.90313.52E5
0.85121.0
4E4
0.70853.26E4
0.17116.42E5
0.49
1892.62
E4
0.905532.06
E5
-
Mathematical Problems in Engineering 11
0 5 10 15 20
0.80
0.82
0.84
0.86
0.88
0.90
0.92
DR-product ruleDR-sum ruleDR-max rule
DR-min ruleDR-majority vote rule
DR
The number of Nave Bayes classifier ensembles
(a)
0.028
0.030
0.032
0.034
0.036
0.038
0.040
0.042
0.044
0.046
FAR-product ruleFAR-sum ruleFAR-max rule
FAR-min ruleFAR-majority vote rule
FAR
0 5 10 15 20The number of Nave Bayes classifier ensembles
(b)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
MTTD-product ruleMTTD-sum ruleMTTD-max rule
MTTD-min ruleMTTD-majority vote rule
MTT
D
0 5 10 15 20The number of Nave Bayes classifier ensembles
(c)
0.948
0.950
0.952
0.954
0.956
0.958
0.960
0.962
0.964
0.966
CR
0 5 10 15 20The number of Nave Bayes classifier ensembles
CR-product ruleCR-sum ruleCR-max rule
CR-min ruleCR-majority vote rule
(d)
0.88
0.89
0.90
0.91
0.92
0.93
0.94
AUC-product ruleAUC-sum ruleAUC-max rule
AUC-min ruleAUC-majority vote rule
AUC
0 5 10 15 20The number of Nave Bayes classifier ensembles
(e)
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
Kappa-product ruleKappa-sum ruleKappa-max Rule
Kappa-min ruleKappa-majority vote rule
Kapp
a
0 5 10 15 20The number of Nave Bayes classifier ensembles
(f)
Figure 4: Experimental results of five rules for the Nave Bayes
ensembles as applied to the I-880 dataset: (a) performance with DR;
(b)performance with FAR; (c) performance with MTTD; (d) performance
with CR; (e) performance with AUC; (f) performance with Kappa.
-
12 Mathematical Problems in Engineering
0 5 10 15
200.01450.01500.01550.01600.01650.01700.01750.01800.01850.0190
MAE-product rule
0.170
0.175
0.180
0.185
0.190
0.195
RMSE-product rule
0.9905
0.9910
0.9915
0.9920
0.9925
The number of NB classifier ensembles
EC-product rule
0.01450.01500.01550.01600.01650.01700.01750.01800.01850.0190
MAE-sum rule
0.170
0.175
0.180
0.185
0.190
0.195
RMSE-sum rule
0.9905
0.9910
0.9915
0.9920
0.9925
EC-sum rule
0.01520.01540.01560.01580.01600.01620.01640.01660.0168
MAE-max rule
0.1750.1760.1770.1780.1790.1800.1810.1820.183
RMSE-max rule
0.9916
0.9918
0.9920
0.9922
EC-max rule
0.01540.01560.01580.01600.01620.01640.01660.0168
MAE-min rule
0.1760.1770.1780.1790.1800.1810.1820.183
RMSE-min rule
0.9916
0.9918
0.9920
0.9922
EC-min rule
0.01500.01550.01600.01650.01700.01750.01800.0185
MAE-majority vote rule
0.175
0.180
0.185
0.190
RMSE-majority vote rule
0.9910
0.9915
0.9920
EC-majority vote rule
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
Figure 5: Bar chart comparison of five rules for Nave Bayes
classifier ensembles as applied to the I-880 dataset. The red bar
is MAE, thegreen bar is RMSE, and the blue bar is EC.
-
Mathematical Problems in Engineering 13
0.60
0.65
0.70
0.75
0.80D
R
0 5 10 15 20The number of Nave Bayes classifier ensembles
DR-product ruleDR-sum ruleDR-max rule
DR-min ruleDR-majority vote rule
(a)
0.0460.0480.0500.0520.0540.0560.0580.0600.0620.064
FAR
0 5 10 15 20The number of Nave Bayes classifier ensembles
FAR-product ruleFAR-sum ruleFAR-max rule
FAR-min ruleFAR-majority vote rule
(b)
0.00.51.01.52.02.53.03.54.04.5
MTT
D
0 5 10 15 20The number of Nave Bayes classifier ensembles
MTTD-product ruleMTTD-sum ruleMTTD-max rule
MTTD-min ruleMTTD-majority vote rule
(c)
0.870
0.872
0.874
0.876
0.878
0.880CR
0 5 10 15 20The number of Nave Bayes classifier ensembles
CR-product ruleCR-sum ruleCR-max rule
CR-min ruleCR-majority vote rule
(d)
0.770.780.790.800.810.820.830.840.850.860.870.88
AUC
0 5 10 15 20The number of Nave Bayes classifier ensembles
AUC-product ruleAUC-sum ruleAUC-max rule
AUC-min ruleAUC-majority vote rule
(e)
0.580.600.620.640.660.680.700.720.740.760.78
Kapp
a
0 5 10 15 20The number of Nave Bayes classifier ensembles
Kappa-product ruleKappa-sum ruleKappa-max Rule
Kappa-min ruleKappa-majority vote rule
(f)
Figure 6: Experimental results of five rules forNave Bayes
ensemble as applied to theAYEdataset: (a) performancewithDR; (b)
performancewith FAR; (c) performance with MTTD; (d) performance
with CR; (e) performance with AUC; (f) performance with Kappa.
-
14 Mathematical Problems in Engineering
MAE-min rule RMSE-min rule EC-min rule
MAE-majority vote rule RMSE-majority vote rule EC-majority vote
rule
0.16500.16550.16600.16650.16700.16750.16800.16850.1690
0.5730.5740.5750.5760.5770.5780.5790.5800.5810.582
0.9040.9050.9060.9070.9080.9090.9100.911
0.16600.16650.16700.16750.16800.16850.1690
0.576
0.577
0.578
0.579
0.580
0.581
0.582
0.9040.9050.9060.9070.9080.9090.9100.911
0.16380.16520.16660.16800.16940.17080.17220.17360.1750
0.5740.5760.5780.5800.5820.5840.5860.5880.5900.5920.594
0.9040.9050.9060.9070.9080.9090.9100.911
0.16290.16380.16470.16560.16650.16740.16830.1692
0.5700.5720.5740.5760.5780.5800.582
0.908
0.909
0.910
0.16200.16320.16440.16560.16680.16800.1692
0.5680.5700.5720.5740.5760.5780.5800.582
0.9070.9080.9090.9100.9110.912
0.911
0.912
0.913
MAE-product rule RMSE-product rule EC-product rule
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
0 5 10 15 20The number of NB classifier ensembles
MAE-sum rule RMSE-sum rule EC-sum rule
MAE-max rule RMSE-max rule EC-max rule
Figure 7: Bar chart comparison of five rules for Nave Bayes
classifier ensembles as applied to the AYE dataset.The red bar is
MAE, the greenbar is RMSE, and the blue bar is EC.
-
Mathematical Problems in Engineering 15
is observed with RMSE. Comparing Figure 4 and Figure 5,we find
that the performances of all five rules are reduced inFigure 5. The
reason for this is that the noisy data involvedin the process of
training a single classifier result in thefinal output of the NB
ensemble being worse. However, theperformances of the sum rule, the
max rule, and the min ruleare reduced, which indicates that the sum
rule, the max rule,and themin rule have a better ability to
tolerate the noisy dataamong the five rules.
In Tables 3 and 4, the dataset we used is extremelyunbalanced. A
very high CR, even exceeding 90%, doesnot indicate good detection
performance. In Table 4, theaverage DR value of the Nave Bayes
algorithm is 71.12% andthe average DR value of the NBTree algorithm
is 72.75%.Both values are less than 75%, which is unsatisfactory
forpractical applications. The DR values of the product ruleand the
majority rule are even lower (64.57% and 62.46%,resp.). These
results indicate that if the average accuracy ofthe individual NB
classifiers is lower, the average accuracyof the ensemble
classifiers, which are constructed usingthese individual
classifiers, will become even lower than theaverage accuracy of the
individual NB classifiers in certaincombination rules.
Therefore, we should avoid drawing noisy data into theNB
ensemble.The standard Nave Bayes and NBtree are bothindividual
classifiers, and they only need to train one time. Incontrast to
standard Nave Bayes and NBtree, NB ensembleneeds to train many
individual NB classifiers to constructthe NB ensemble. The training
time of the NB ensembleis relatively long. From Figures 4(a)4(f),
we can see thatin order to obtain relatively better performance,
the NBensemble needs approximately 15 individual NB to
constructtheNB ensemble; that is, theNB ensemble algorithmneeds
totrain 15 times.Thus, compared withNB ensemble, the
NBtreealgorithm saves a large amount of time cost.
5. Conclusions
The Nave Bayes classifier ensemble is a type of
ensembleclassifier based on Nave Bayes for AID. In contrast toNave
Bayes, the NB classifier ensemble algorithm trainsmany individual
NB classifiers to construct the classifierensemble and then uses
this classifier ensemble to detect thetraffic incidents, and it
avoids the burden of choosing theoptimal threshold and tuning the
parameters. In our research,we take the traffic incident detection
problem as a binaryclassification problem based on the ILD data and
use theNB ensemble to divide the traffic patterns into two
groups:an incident traffic pattern and nonincident traffic pattern.
Inthis paper, we have performed two groups of experiments
toevaluate the performances of the three algorithms: standardNave
Bayes, NB ensemble, and NBTree. In the first group ofexperiments,
we used all the three algorithms as applied to theI-880 dataset
without noisy data. The results indicate that theperformances of
the five rules of the NB ensemble are signif-icantly better than
those of standard Nave Bayes and slightlybetter than those ofNBTree
in terms of some indicators.Moreimportantly, the NB ensemble
performance is very stable. To
further test the stability of the three algorithms, we
appliedthe three algorithms to the AYE dataset with noisy data in
thesecond group of experiments. The experimental results indi-cate
that the NB ensemble has the best ability to tolerate thenoisy data
among the three algorithms. After analyzing theexperimental
results, we found that if the average accuracyof the individual NB
classifiers is lower, the average accuracyof the ensemble
classifiers constructed by these individualclassifiers will become
even lower than the average accuracyof the individual NB
classifiers. To obtain good results for theNB ensemble classifier,
we should avoid drawing the noisydata into the ensemble. NBTree is
an individual classifier thatneeds to train only one time, whereas
the NB ensemble needsto train many individual NB classifiers to
construct the NBensemble. As a result, compared with the NBTree
algorithm,theNBTree algorithm reduces the time cost.The
contributionof this paper is that it presents the development of a
freewayincident detection model based on the Nave Bayes
classifierensemble algorithm.TheNB ensemble not only improves
theperformances of traffic incidents detection but also enhancesthe
stability of the performances with an increased numberof
classifiers. The advantage of NBTree is that the MTTDvalue is
better than that of the NB ensemble algorithm.We believe that the
NB ensemble algorithm and NBTreecan be successfully utilized in
traffic incidents detection andthe other classification problems.
In a future study, we willconcentrate on constructing an NB
ensemble and scaling upthe accuracy of NBTree to detect traffic
incidents.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgments
This work is part of an ongoing research Supported byNational
High Technology Research and Development Pro-gram of China under
Grant no. 2012AA112304 and Researchand Innovation Project for
College Graduates of JiangsuProvince no. CXZZ13 0119. Qingchao Liu
would like to makea grateful acknowledgment to all the members of
TrafficEngineering Lab in SoutheastUniversity, China, for their
helpand many useful suggestions in this study.
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