383671

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

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

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.


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


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 .


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.


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].


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


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


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


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


(c)

0.948

0.950

0.952

0.954

0.956

0.958

0.960

0.962

0.964

0.966

CR


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


(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


(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.


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














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.


0.60

0.65

0.70

0.75

0.80D

R


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


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


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


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


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


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.


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
















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.


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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