Outlier Detection In Large-scale Traffic Data By Naïve Bayes Method and Gaussian Mixture Model Method

by   Philip Lam, et al.
Hong Kong Baptist University

It is meaningful to detect outliers in traffic data for traffic management. However, this is a massive task for people from large-scale database to distinguish outliers. In this paper, we present two methods: Kernel Smoothing Naïve Bayes (NB) method and Gaussian Mixture Model (GMM) method to automatically detect any hardware errors as well as abnormal traffic events in traffic data collected at a four-arm junction in Hong Kong. Traffic data was recorded in a video format, and converted to spatial-temporal (ST) traffic signals by statistics. The ST signals are then projected to a two-dimensional (2D) (x,y)-coordinate plane by Principal Component Analysis (PCA) for dimension reduction. We assume that inlier data are normal distributed. As such, the NB and GMM methods are successfully applied in outlier detection (OD) for traffic data. The kernel smooth NB method assumes the existence of kernel distributions in traffic data and uses Bayes' Theorem to perform OD. In contrast, the GMM method believes the traffic data is formed by the mixture of Gaussian distributions and exploits confidence region for OD. This paper would address the modeling of each method and evaluate their respective performances. Experimental results show that the NB algorithm with Triangle kernel and GMM method achieve up to 93.78



page 2


A Sparse PCA Approach to Clustering

We discuss a clustering method for Gaussian mixture model based on the s...

Skewed Distributions or Transformations? Modelling Skewness for a Cluster Analysis

Because of its mathematical tractability, the Gaussian mixture model hol...

PCA Reduced Gaussian Mixture Models with Applications in Superresolution

Despite the rapid development of computational hardware, the treatment o...

Using Subset Log-Likelihoods to Trim Outliers in Gaussian Mixture Models

Mixtures of Gaussian distributions are a popular choice in model-based c...

Online Vehicle Detection For Estimating Traffic Status

We propose a traffic congestion estimation system based on unsupervised ...

Multi-Kernel Correntropy for Robust Learning

As a novel similarity measure that is defined as the expectation of a ke...

Two-way kernel matrix puncturing: towards resource-efficient PCA and spectral clustering

The article introduces an elementary cost and storage reduction method f...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

It is desired that most events to be safe, stable or rather predictable. Identifying an unusual event is a typical and vital topic in different fields, such as intrusion detection in cyber security, fraud detection for credit cards, insurance or health care and fault detection in safety critical systems [9]. We may call this unusual event as outlier or abnormality which is different from a usual event. Outlier is usually minor in a group of events/data while inlier is the majority.

OD refers to detect any abnormal element in data which is not consistent with an expected behavior [5]. A good OD method should be accurate to detect outlier and less erroneous judgment on inlier data. Many OD methods [8] have been developed in recent years. In [5],[9], nearest neighbor approach was proposed to consider the distance or the similarity between two data instances. The assumption behind the approach is that inliers in data should be dense and outlier(s) is far from these dense inliers.

NB classifiers were developed in the 1950s. The NB classifiers are constructed based on Bayes’ theorem, and are widely used for text categorization with superior performance

[15]. Another popular application is spam filtering among numerical emails [16]. The kernel smooth NB method in this paper supposes the traffic data could be modeled by kernel distributions so that the Bayes’ theorem is applied. In contrast, the GMM method presumed a mixture of Gaussian distributions and confidence region is employed . GMM, firstly introduced by Aitkin and Wilson (1980) [2], is  an appropriate way of handle data with multiple outliers [17]. The GMM method is useful in dealing with speech recognition [18].

For traffic data, detecting anomaly traffic event would be better to deal with the traffic problems. This research aims at detecting outliers for a large-scale traffic database from Hong Kong. The original video was taken at a four-arm junction as shown in Fig. 0(a). The 4-arm junction can be expressed as an ideal map like Fig. 0(b). In total, the traffic data for 31 days was recorded with two sessions per day: AM (07:00-10:00) and PM (17:00-20:00). For each session, the original video data are dividing into 19 spatial temporal (ST) signals. Fig. 0(c) demonstrates an sample of 4 normal ST signals. These ST signals have different number of traffic cycles because smoother traffic flow would lead to a shorter traffic cycle and result in more traffic cycles in one session. Also, the ST signals in each direction suffer a high degree of similarities (Fig. 0(d)) among each other or across signals in different traffic direction. Therefore, it requires a truncation process (to standardize the length of a signal’s cycles) and a signal domain transformation (to remedy the signal similarities). Herein, each ST signal was processed by principal component analysis (PCA) [11] to reduce the dimension of signal representation. Then, an OD method was carried out based on the PCA-processed (x,y)-coordinates plane.

(a) The real scene
(b) 4-arm junction
(c) Session 2 Entry: All signals are normal
(d) Session 50: Entries E,W,N are normal, Entry S is abnormal
Figure 1: (a) A generic diagram of the 4-arm junction; (b) sample of the real scene; (c) normal ST signals; (d) abnormal ST signals (with arrow).

In this paper, a detail investigation of NB and GMM classifiers in large-scale traffic data is carried out, for these two methods are newly applied in traffic OD. Experimental results demonstrate that the NB and GMM methods can achieve 93.78% and 94.50% detection success rates (DSRs), respectively. These performance is comparable to previous evaluation on other OD methods [12], [13]

including Gaussian mixture model (80.86%), one-class SVM (59.27%), S-estimator (76.20%) and kernel density estimation (95.20%).

The organization of this paper is as follows: Section II gives a review of related work about OD. Sections III and IV present details of two proposed OD methods and their experimental results, and Section V concludes this paper.

2 Related Work

The popular OD methods include statistical, nearest neighborhood, spectral approaches and learning based approaches. Details about these methods are given as follows.

2.1 Statistical Approach

Statistical approach is one of the earliest approaches in OD. It assumes that all inliers occur in a high probability region of this distribution model when outliers deviate strongly from the inliers. Statistical OD methods can perform parametric techniques like GMM or regression, or non-parametric techniques like histogram based or kernel function based. This approach would perform well if data distribution is assumed well. However, if the data distribution assumption is false, the result would be far away from the correct situation

[3], [8].

2.2 Nearest Neighbor Approach

Nearest neighbor approach considers the distance or the similarity between two data instances. The assumption behind the approach is that inliers in data should be dense and outlier(s) is far from these dense inliers [9], [5]. This approach can be easily realized in an unsupervised way that could require high computational complexity. As a result, it is hard to deal with very complex data [8].

There are two major variant methods related to this approach, density measurement based methods, such as Local Outlier Factor (LOF) [19], Influenced Outlierness (INFLO) [3] and Local Outlier Correlation Integral (LOCI) [3], or distance measurement based methods, like Mahalanobis Distance [6], DB()-Outliers, Outlier Scoring based on k-NN distances [4], Resolution-based Outlier Factor (ROF) [8].

2.3 Spectral Approach

Spectral approach suggests that inliers and outliers could appear significantly different in a spectral domain. The approach detects outliers by embedding data into a lower dimensional subspace. The approach is useless if inliers and outliers in data are not separable in the lower dimensional subspace. In addition, it requires high computational complexity [8].

2.4 Learning Approach

Learning approach is using various training methodologies [10]

(or so-called trained machines) to train the input data. The related OD methods are neural networks, naïve Bayesian network, support vector machines (SVM). The assumption is that inliers and outliers in data can be easily distinguished by the trained machines. Therefore, the machines can test and classify a test instance into either an inlier group or an outlier group in OD

[8]. There are many algorithms that can be used in multi-learning class approach, and they have a faster testing phase than other approach. However, accurate labeling for various normal classes is often impossible [8].

In this paper, an investigation of NB classifier and GMM classifier to model the PCA-processed (x,y)-data and their performance for OD in traffic data would be carried out.

3 Kernel Smoothing NB Method

NB classifier was developed in 1950s for text retrieval [18]. It is also widely used for spam filtering among numerical emails [1].

3.1 NB Classifier

NB classifier [3] is developed based on Bayes’ Theorem which takes the form of


Based on the probability P(E),P(H), and the conditional probability P(E|H), the posteriori probability P(H|E), which denotes the possibility of event H conditioned on an occurred event E, can be obtained. Naive Bayes classifiers are based on the information of the training data, and then determined the highest possible class of testing data from their information.

3.2 Kernel Distribution

Kernel Distribution is a non-parametric distribution to estimate each training point as some independent distributions into the whole distribution [7].


Kernel Distribution is estimated by  sample data points with a kernel density. The commonly used kernels are listed as follows:
Box kernel:


Triangle kernel:


Epanechnikov kernel:


Normal kernel:


where x is replaced by y for y-axis kernel.

3.3 Kernel Smoothing NB classifier for OD

Kernel smoothing NB (KSNB) classifier is a trained Kernel Distribution based classifier. Different kernels with different widths for each predictor or class are available in KSNB classifier. We use the trained KSNB classifiers to detect outliers, where the classifiers would automatically set a bandwidth value for each feature and class which is optimal for a Gaussian model. Then, a corresponding region from contours surrounding various kernel distributions is formed an inlier region E, therefore any data out of that region is classified as an outlier. In this method, box, triangle, Epanechnikov and normal kernels are used for OD. The OD procedures based on KSNB classifier are given as follows:

Fig. 2 illustrates the flowchart of the proposed NB based method, which includes a training stage and a testing stage. The details are as below.

Figure 2: Flowchart of the proposed NB method.

A. Training Stage

Step 1: Feed training data (all inliers) into NB classifiers.
Step 2: Fit training data into the specified kernel distributions defined in Eqns. (3-6).
Step 3: Combine each kernel distribution into a whole distribution.
Step 4: Set a bandwidth (BIB) of the inlier region automatically from the kernel distribution.

B. Testing Stage

Step 1: Input testing data into the trained NB classifiers.
Step 2: Data label as outlier if the data is out of the inlier region.
Step 3: Output result.

In addition, Algorithm 1 is provided below.

1:   The inliers data set and testing data set
2:procedure Employ Kernel Distribution
3:     define outliers set
4:      every data point
5:     form in kernel distribution
8:     set the bandwidth value by
9:     form an inlier region
10:     for all in
11:     if  then
13:     else
15:     output labels
Algorithm 1 NB classifier with different kernel distribution

3.4 Experimental results

Experimental results based on the kernel smoothing NB method are listed in Table 1. Accuracy (Acc.), positive predictive value (PPV), negative predictive value (NPV), sensitivity (Sen.), false positive rate (FPR) are employed as measurement metrics and their definitions can be referred to [12]. From the results, we can see that overall accuracy among all kernels is almost higher than 90%. Overall NPV among all kernels is higher than 95%.

Kernel Gaussian Box Epanechnikov Triangle
AM Acc. 97.13 91.39 95.22 96.17
PPV NA 18.18 40.00 50.00
NPV 97.13 97.93 98.09 98.09
Sen. 0.00 33.33 33.33 33.33
FPR 0.00 6.84 2.87 1.91
PM Acc. 91.87 85.65 89.47 91.39
PPV 100.00 22.71 29.49 36.81
NPV 91.65 96.21 96.49 96.49
Sen. 5.00 55.50 55.50 55.50
FPR 0.00 12.60 8.31 6.30
All Acc. 94.50 88.52 92.345 93.78
PPV 100.00 20.45 34.75 43.41
NPV 94.39 97.07 97.29 97.29
Sen. 2.50 44.42 44.42 44.42
FPR 0.00 9.72 5.59 4.11
Table 1: Performance of the NB Method Among Different Kernels.

The OD scatters corresponding to the four kernels are shown in Fig. 3. From the figure, one observes that the scatter based on Triangle kernel is closest to the perfect classification point. Among the NB method with different kernels, Gaussian kernel usually has better performance. However, in our evaluation, the OD performance based on NB method with Gaussian kernel is very close to the random-guess line ( black line) as shown in ROC space of Fig. 3. It is just a little better than random guessing that all testing data are inliers. Therefore, Gaussian kernel is found to be not a good kernel in this method for OD. For other three kernels listed in Table I, the triangle kernel shows the best performance that is closest to the perfect classification point with the highest accuracy (93.78%). For AM or PM OD, the NB method with the triangle kernel still outperforms other three kernels That may indicate that the NB method with a triangle kernel is significantly good to describe the behaviors of PCA traffic data.

Figure 3: ROC plot for the NB method based on four kernels.

4 GMM Method

The GMM method was firstly introduced in [2], the mixture model as a way of handle data with multiple outliers [4]. The GMM method is good in deal with speech recognition [18].

4.1 Gaussian Distribution

In mathematics, a one-dimensional Gaussian distribution is a distribution function of the possibility density form:


where , denotes the expectation of ,

denotes the variance of

. In term of a two-dimensional expression, the Gaussian distribution is


where and .

4.2 Gmm


is a parametric probability density function represented as a weighted sum of Gaussian component densities. The p.d.f. of the mixture model among

is which takes the form of


where is the mixture weight [14].

4.3 Methodology

For OD in this section, we will use confidence region (CR) with a Bonferroni adjustment as criterion in  significant level. Herein, .

For a CR with Bonferroni adjustment, GMM CR for X is


GMM confidence region for Y is


where is student’s value in significant level. The GMM method models fit the training data by using several Gaussian distributions.

In this section, data fitting will use the expectation maximization (EM) algorithm


, which assigns posterior probabilities to each component density with respect to each observation, to get the best

for each in (9).

4.4 Procedure

A flowchart of the proposed GMM method, including a training stage and a testing stage is presented in Fig. 4. The details are as below.

Figure 4: Flowchart of the proposed GMM method.

A. Training Stage

Step 1: Input inlier training data.
Step 2: Fit each data into individual identical Gaussian distribution model.
Step 3: Mixture all models into a whole GMM model (9).
Step 4: Set the criteria region for different criterion method and different significance level [(i) 0.10, (ii) 0.05, (iii) 0.01].

B. Testing Stage

Step 1: Input testing data set.
Step 2: Label data as an outlier if the data is out of the criteria region

Also, Algorithm 2 is given.

1:   The inliers data set and testing data set
2:procedure Employ Confidence Region
3:     define outliers set
4:      every data point
5:     form a Gaussian model with
8:     fixed the significant level
9:     construct GMM x confidence region X with:
11:     construct GMM y confidence region Y with:
13:     for all in
14:     if  then
15:         if  then
17:         else
19:     else
21:     output results
Algorithm 2 GMM with 1 mixture classifier in confidence region method

4.5 Experimental results

Experimental results based on the GMM method are listed in Table 2. The performances in terms of FPR, PPV and Acc. are gradually better with the decrease of significant level. The highest accuracies in both AM (96.65%) and PM (94.50%) sessions are achieved when the significant level is 0.01. A ROC plot on the GMM method is shown in Fig. 5. By observation, CR in the 0.01 level is the closest point to the perfect classification point. Inlier data points lying in the centroid of the GMM model and the CR cover at least  significant level built by training data set. Therefore, the CR criterion is good at describing the inlier group which is centered.

Figure 5: ROC plot for the GMM method based on four kernels.
Level 0.10 0.05 0.01
AM Acc. 89.47 93.78 96.65
PPV 9.09 14.29 50.00
NPV 97.70 97.99 98.09
Sen. 33.33 33.33 33.33
FPR 9.04 4.50 1.44
PM Acc. 82.30 86.12 92.34
PPV 18.83 28.02 46.30
NPV 96.26 96.52 96.11
Sen. 55.50 55.50 53.50
FPR 16.55 11.81 4.55
All Acc. 85.89 89.95 94.50
PPV 13.96 21.16 48.15
NPV 96.98 97.26 97.10
Sen. 44.42 44.42 43.42
FPR 12.80 8.16 3.00
Table 2: Performance of the GMM Method in Various Significant Levels.

5 Conclusion

In this paper, we present two OD methods, kernel smoothing NB and GMM, to detect outliers in large-scale traffic data. The kernel smoothing NB method utilizes the trained kernel smoothing NB classifiers to detect outliers, in which the classifiers set the best bandwidth value with the inlier region. Any data points out of that region are classified as outliers. In the GMM method, the rectangular confidence region is formed in Bonferroni adjustment way. True  significant level region of the GMM can be constructed as the inlier region for the more accuracy. Experimental results show that the two algorithms can achieve pleasing detection accuracies compared with the OD methods in our previous studies [12], [13], including Gaussian mixture model, one-class SVM, S-estimator and kernel density estimation.

6 Acknowledgments

This research is supported by Hong Kong RGC GRF: 12201814 and HKBU FRG2/14-15/075.


  • [1] C. Aggarwal, Outlier Analysis, Springer, 2013.
  • [2] M. Aitkin and G.T. Wilson, “Mixture models, outliers, and the EM algorithm," Technometrics, 22, 325-331, 1980.
  • [3]

    V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: A survey,"

    ACM Computing Surveys, 41(3), pp.1-58, 2009.
  • [4] V. Hautamaki , I. Karkkainen and P. Franti, “Outlier Detection Using k-Nearest Neighbour Graph," Proc. IEEE ICPR, vol. 3, pp.430-433, 2004.
  • [5] V. J. Hodge, J. Austin, “A Survey of Outlier Detection Methodologies," Artificial Intelligence Review, vol. 22 no. 2, pp.85-126, 2004.
  • [6] G.S.D.S. Jayakumar and B. J. Thomas “A New Procedure of Clustering Based on Multivariate Outlier Detection", Science, 11, pp. 69-84, 2013.
  • [7] G.H. John and P. Langley, “Estimating continuous distributions in Bayesian classifiers," In: Hanks, P. B. eds., Proc. 11th Conf. Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA, pp. 338-345, 1995.
  • [8] H. Kriegel, P. Kroger and A. Zimek, “Outlier Detection Techniques," Tutorial at the 13th Pacific-Asia Conference on Knowledge Discovery and Data Mining, 2009.
  • [9] V. Kumar, “Parallel and Distributed Computing of Cybersecurity," IEEE Distributed Systems Online, vol. 6, no. 10, 2005.
  • [10] M. Minsky and S. Papert, “An Introduction to Computational Geometry," MIT Press, ISBN 0-262-63022-2, 1969.
  • [11] H.Y.T. Ngan, N.H.C. Yung and A.G.O. Yeh, “Detection of Outliers in Traffic Data based on Dirichlet Process Mixture Model," Proc. IEEE CASE, pp 224-229, 2012.
  • [12] H.Y.T. Ngan, N.H.C. Yung and A.G.O. Yeh, “Detection of Outliers in Traffic Data based on Dirichlet Process Mixture Model," IET Intelligent Transportation Systems, vol. 9, no. 7, pp. 773-781, 2015.
  • [13] H.Y.T. Ngan, N.H.C. Yung and A.G.O. Yeh, “A Comparative Study of Outlier Detection for Large-scale Traffic Data by One-class SVM and Kernel Density Estimation," IS&T/SPIE Electronic Imaging, 94050I-94050I-10, 2015.
  • [14] D.A. Reynolds, “Gaussian Mixture Models," Encyclopedia of Biometric Recognition, Springer, Heidelberg, 2008.
  • [15] S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach, 2nd ed., Prentice Hall, ISBN 978–0137903955, 2003.
  • [16] M. Sahami, S. Dumais, D. Heckerman and E. Horvitz, “A Bayesian approach to filtering junk e-mail," AAAI Workshop on Learning for Text Categorization, AAAI Technical Report, WS-98-05, 1998.
  • [17] D.W. Scott, “Outlier detection and clustering by partial mixture modelling," COMPSTAT 2004 Symposium, pp. 453-465, Heidelberg, 2005.
  • [18] M. N. Stuttle, “A Gaussian mixture model spectral representation for speech recognition," Ph.D. dissertation, University of Cambridge, 2003.
  • [19]

    K-A. Yoon, O-S. Kwon, and D-H. Bae, “An Approach to Outlier Detection of Software Measurement Data Using the K-means Clustering Method,"

    IEEE 1st Int’l Sym. ESEM, pp. 443-445 , 2007.