Modified K-Means Clustering for Travel Time Prediction Based on ...

Modified K-Means Clustering for Travel Time Prediction Based on Historical Traffic Data Rudra Pratap Deb Nath1, Hyun-Jo Lee2, Nihad Karim Chowdhury1, and Jae-Woo Chang2 1

Department of Computer Science & Engineering, University of Chittagong, Bangladesh [email protected], [email protected] 2 Department of Computer Engineering, Chonbuk National University, South Korea {o2near,jwchang}@chonbuk.ac.kr

Abstract. Prediction of travel time has major concern in the research domain of Intelligent Transportation Systems (ITS). Clustering strategy can be used as a powerful tool of discovering hidden knowledge that can easily be applied on historical traffic data to predict accurate travel time. In our Modified K-means Clustering (MKC) approach, a set of historical data is portioned into a group of meaningful sub-classes (also known as clusters) based on travel time, frequency of travel time and velocity for a specific road segment and time group. With the use of same set of historical travel time estimates, comparison is also made to the forecasting results of other three methods: Successive Moving Average (SMA), Chain Average (CA) and Naïve Bayesian Classification (NBC) method. The results suggest that the travel times for the study periods could be predicted by the proposed method with the minimum Mean Absolute Relative Error (MARE). Keywords: Intelligent Transportation System (ITS), K-means Clustering, Successive Moving Average (SMA), Chain Average (CA), Naïve Bayesian Classification (NBC).

1 Introduction Recently, accurate estimation of travel times has been central for traffic data analysis to various Advanced Travelers Information System (ATIS) and ITS applications such as trip planning, vehicular navigation systems and dynamic route guidance systems. Moreover, Travel time prediction is also becoming increasingly important with the development of ATIS [1]. In addition, Travel time forecasting provides information that helps travelers to decide whether they should change their routes, travel mode, starting time or even cancel their trip [2]. So, the reliable and accurate travel time prediction on road network plays an important role in any kind of dynamic route guidance systems to fulfill the users’ desires. On top of that, the importance of travel time information is also indispensable to find the fastest path (i.e. shortest path according to travel time) that connects the origin and destination. Besides, accurate travel time information also helps delivery industries to progress their service quality by delivering on time. R. Setchi et al. (Eds.): KES 2010, Part I, LNAI 6276, pp. 511–521, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Travel time prediction is based on vehicle speed, traffic flow and occupancy which are extremely sensitive to external event like weather condition and traffic incident [3]. Addressing the uncertainty on the road network is also a crucial issue in the research domain. Prediction on uncertain situation is very complex, so it is important to reach optimal accuracy. Yet, the structure of the traffic flow of a specific road network fluctuates based on daily, weekly and occasional events. For example, the traffic condition of weekend may differ from that of weekday. So, time-varying feature of traffic flow is one of the major issues to estimate accurate travel time [12]. In this study, we focus a new method that is able to predict travel time reliably and accurately. Generally this effort is the extension of our previous works. In this research, we have tried to combine the advantages of our previous methods namely NBC [12], SMA and CA [13] by eliminating the shortcomings of those methods. Proposed MKC method is able to address the arbitrary route on road networks that is given by user. Furthermore proposed method flushes a functional relationship between traffic data as input variables and predicted travel time as the output variables. According to the experimental result, our method exhibits satisfactory performance in terms of prediction accuracy. At the same time, the result is considered to be superior rather than other prediction methods like NBC, SMA and CA. The format of the remaining portions of this paper is depicted as follows: Section 2 introduces some related research in this field. An outline of MKC method with example is illustrated in section 3. Section 4 exhibits a concise experimental evaluation. Finally, the conclusion statement and direction of future research is discussed in section 5.

2 Literature Review and Motivation Numerous researchers have paid their attention on the accurate travel time prediction as it is one of the major issues for effective dynamic route guidance systems. Various methodologies have been investigated till date for computation and prediction of travel times with varying degree of successes. In this section, a historical background on the topic of travel time prediction is discussed briefly. Park et al [5], [6] proposed Artificial Neural Network (ANN) models for forecasting freeway corridor travel time rather than link travel time. One model used a Kohonen Self Organizing Feature Map (SOFM) whereas the other utilized a fuzzy c-means clustering technique for traffic pattern classification. Lint et al [7], [8] proposed a state-space neural network based approach to provide robust travel time predictions in the presence of gaps in traffic data. Kwon et al [9] focused their research on linear regression method. A linear predictor consisting of a linear combination of the current times and the historical means of the travel times was proposed by Rice et al [10]. They proposed a method to predict the time that would be needed to traverse a given time in the future. Wu et al [3] applied support vector regression (SVR) for travel time predictions and compared its results to other baseline travel-time prediction methods using real highway traffic data. Most recent research in this field has been proposed by Erick et al [11]. They investigated a switching model consisting of two linear predictors for travel time prediction.

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An efficient method for predicting travel time by using NBC was proposed by Lee et al [12] which had also been scalable to road networks with arbitrary travel routes. The main idea of NBC was that it would give probable velocity level for any road segment based on historical traffic data. It was shown from experiments that NBC could reduce MARE significantly rather than the other predictors. In our previous research, we formulated two completely new methods namely SMA and CA that were based on moving average. In that research, we eliminated the drawbacks of conventional moving average approach such as unwanted fluctuation in data set. These methods were also scalable to large network with arbitrary travel routes. Moreover, both methods were less expensive in terms of computational time. Consequently, it was revealed that these proposed methods can reduce error significantly, compared with existing methods [13]. Travel time prediction forms an integral part of any ATIS. The grouping style of whole day is efficiently and effectively done by NBC. But a significant problem will arise when we calculate velocity level for a particular route. Moreover, this method emphasize on those data whose probabilities are higher i.e. it does not concern with all data. Although NBC is capable to predict more accurately ordinary, usually it doesn’t give significant result in the uncertain situation. Nevertheless, we have to compute velocity class prior probability and velocity class posterior probability. So, it takes more computational time. On the other hand, SMA and CA compute all data which are not based on probability theory. Though they provide an almost accurate travel time, they are also failed to find out uncertain data from the available traffic data. There are various knowledge based techniques that can be used in the traffic data. Clustering is one of the leading tools for discovering hidden knowledge that can be applied in the large historical traffic data set. To address the uncertain situation and predict the travel time more accurately, we propose MKC method. In this study, we try to eliminate the shortcomings of traditional K-means Clustering approach as well as NBC, SMA and CA. The key challenges of this research are to reduce prediction error as well as to predict the uncertain situation. At the same time, proposed method can also be scalable to large network with arbitrary travel routes. To motivate the presentation of proposed method, the following sections will explore complete scenario of our proposed method.

3 Proposed Travel Time Prediction Methods In this section, a new method for foretelling travel time from historical traffic data using MKC method is depicted. Cluster analysis or clustering is an assignment of separating the set of observations into subset. A cluster is therefore a collection of objects which are similar between themselves and are dissimilar to the objects belonging to other clusters. K-means (MacQueen, 1967) is one of the simplest unsupervised learning algorithms that solve the well known clustering problems. The main advantages of K-means algorithm are its simplicity and speed which allows it to run on large datasets. The procedure follows a simple and easy way to classify a given data set through a certain number of clusters (assume K clusters) fixed a priori. The main concept is to define K centroids, one for each cluster. These centroids should be positioned in a cunning way because location verification causes result verification. So,

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the better choice is to place them as much as possible far away from each other. In our MKC method, we incorporate a technique so that centroids of different clusters maintain a sufficient difference. The main disadvantage of K-means clustering is that it doesn’t yield the same result with each run, since the resulting clusters depend on the initial random assignments. We have formulated MKC method in a cunning way such that we can eliminate this kind of shortcoming of well-defined K-means algorithm. Initially, an origin with start time and destination is initialized by user. A route may consist of several road segments from origin to destination. First of all, we apply our MKC method on the data set of the first road segment to calculate the end time of first road segment which in turn becomes the start time of the next road segment. Finally, applying successive repetition approximate travel time from origin to destination can be measured. Again, the road environment of the same road network for running vehicles on the different time periods of a day is different. So, for our convenience, the whole day time is separated into several groups according to the time as shown in Table 1. Table 1. Time Group definition Start_time_range 06:01~10:00 10:01~11:00 11:01~12:00 12:01~14:00 14:01~16:00

Time_group 1 2 3 4 5

Start_time_range 16:01~18:00 18:01~22:00 22:01~00:00 00:01~06:00

Time_group 6 7 8 9

Table 2. Sample historical traffic data Vehicle_ID

Road_ID

Time_group

Start_time

End_time

1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1 1

6 6 6 6 6 6 6 6 6 6

16:50 17:20 17:43 16:02 16:16 16:05 17:03 17:11 17:35 16:09

16:57 17:31 17:56 16:11 16:32 16:18 17:10 17:18 17:46 16:16

Travel_time (min) 7 11 13 9 16 13 7 7 11 7

Velocity (km/min) 1.8725 1.1916 1.0082 1.456 0.8192 1.0082 1.8725 1.8725 1.1916 1.8725

For example: if a vehicle starts from any road segment `between 16:01 and 18:00, its Time_group will be 6. Table 2 illustrates the sample snapshot of historical traffic data for any road segment. Each record of the table contains seven attributes. The value of Time_group is calculated from the Start_time. Travel_time is the difference from End_time to Start_time. Dividing length of road segment by Travel_time, Velocity is measured. To calculate approximate travel time for any road segment, we introduce MKC method in the following section with appropriate example.

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3.1 Modified K-Means Clustering Method Step by step, MKC procedure is given below to predict travel time. PROCEDURE Step 1: Frequency for each travel time is measured by counting the repetition of that travel time in different records. Step 2: Define Prediction relation that contains three attributes namely Frequency, Travel_time and Velocity. Each tuple of Prediction relation must contain distinct travel time. Step 3: Find the greatest value from the Frequency attribute (fmax). If two or more tuples contain the greatest value then find the greatest Travel_time for available highest frequencies. A tuple P (xp , yp , zp) is chosen as a centroid of Cluster1, where xp is the maximum frequency, yp is the corresponding maximum travel_time associated with xp and zp is the velocity associated with travel_time yp. Step 4: Compare each tuple Ti (xi, yi, zi) of relation Prediction with the centroid P (xp, yp, zp) of Cluster1 by using the following formula: COST (P, Ti) = |xp-xi| + |yp-yi| + |zp-zi|

(1)

Choose tuple Q (xq, yq, zq) as the centroid of Cluster2, where COST(P,Q) is maximum. Step 5: Build two clusters where the centroid of Cluster1 is tuple P (xp, yp, zp) and that of Cluster2 is tuple Q (xq, yq, zq). Step 6: Define the cluster memberships of tuples by assigning them to the nearest cluster representative tuple. The cost is given by Eq.1. Step 7: Re-estimate the cluster centre (we consider arithmetic mean) by assuming the memberships found above are correct. Step 8: Step 6 and Step 7 are repeated until no change in clusters Step 9: After complete preparation of clusters, desired predicted time is calculated separately for each cluster by using the following formula: N

τi =

∑f i =1

i

* ti (2)

N

∑f i =1

i

Where, τi is the travel time obtained from i-th cluster, N is the total number of tuple in associated cluster, fi is the Frequency of the i-th tuple, and ti is the Travel_time of the i-th tuple. Step 10: If τ1 and τ2 are desired travel times calculated from Cluster1 and Cluster2 respectively, then the final predicted approximate travel time, T for the road segment in the specific time group is the arithmetic mean of τ1 and τ2. i.e. T= (τ1 + τ2) /2

(3)

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3.2 Explanation of MKC Method with Example Considering the sample historical traffic data of Table 2 that contains data Road_id =1 and Time_group=6, the steps of MKC procedure are explained below: Step 1: There are 10 records in Table 2 where Road_id and Time_group are common. First step of MKC reveals to find the frequency of each distinct travel time. If we observe Table 2, then we find that the frequency of Travel_time 7 is four (4) because the number of repetition of Travel_time 7 in different records is four. Similarly, frequencies of Travel_time 16,9,13, and 11 are 1, 1, 2, and 2 respectively. Step 2: Prediction relation is illustrated in Table 3. Each tuple in relation has three attributes namely Frequency, Travel_time and Velocity. The relation also reveals that it contains only those tuples that have distinct travel time. Table 3. Prediction relation of Table 2 Frequency

Travel_time(min)

1 1 2

16 9 13

Velocity (km/min) 0.8192 1.456 1.0082

Frequency

Travel_time(min)

2 4

11 7

Velocity (km/min) 1.1916 1.8725

Step 3: The Frequency column of relation Prediction represents that the maximum value of it is 4. No more than one tuple contain the highest frequency. So, the centroid for Cluster1 is the tuple P (xp, yp, zp) = (4, 7, 1.8725). Step 4: Table 4 calculated the cost of each tuple Ti (xi, yi, zi) from the seed of Cluster1 by using Eq. 1. Table 4. Comparison of each tuple with the centroid of Cluster1 Frequency

Travel_time (min)

Velocity (km/min)

1

16

0.8192

1 2 2 4

9 13 11 7

1.456 1.0082 1.1916 1.8725

Distance from ( 4,7,1.8725 ) | 4-1 | + | 7-16 | + | 1.8725 - 0.8192 | = 3 + 9 + 1.0533 = 13.0533 3 + 2 + 0.4165 = 5.4165 2 + 6 + 0.8643 = 8.8643 2 + 4 + 0.6809 = 6.6809 0

The maximum cost (13.0553) from centroid of Cluster1 is marked as block in the Distance column of Table 4. So, the tuple Q (xq, yq, zq) = (1, 16, 0.8192) is selected as the centroid of Cluster2. Step 5: Two clusters are built where the centroid of Cluster1 is the tuple P (xp, yp, zp) = (4, 7, 1.8725) and that of Cluster2 is the tuple Q (xq, yq, zq) = (1, 16, 0.8192). Step 6: Table 5 decides the cluster memberships of tuples by assigning them to the nearest cluster representative tuple. The numbers marked as block indicate the lowest cost comparison to other. Eq. 1 is also used to find cost. 1st scenario of both clusters is shown in Table 6.

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Table 5. Deciding cluster memberships

Frequency

Travel _time (min) 16 9 13 11 7

1 1 2 2 4

Velocity (km/min) 0.8192 1.456 1.0082 1.1916 1.8725

Distance from Cluster1 centroid ( 4,7,1.8725 ) 3 + 9 + 1.0533 = 13.053 3 + 2 + 0.4165 = 5.4165 2 + 6 + 0.8643 = 8.8643 2 + 4 + 0.6809 = 6.6809 0

Distance from Cluster2 centroid ( 1,16,0.8192 ) 0 0+7+0.6368=7.6368 1+3+0.189=4.189 1+5+0.3724=6.3724 3+9+1.0533=13.0533

Table 6. 1st scenario of both clusters with their members Frequency 4 1 1 2 2

Cluster1

Cluster2

Travel_time(min) 7 9 16 13 11

Velocity(km/min) 1.8725 1.456 0.8192 1.0082 1.1916

Step 7: Re-estimating of new centroid for each cluster. New centroid for Cluster1 P1 (xp , yp , zp) = ((4+1) /2, (7+9) /2, (1.8725+1.456) /2) = (5 /2, 16 /2, 3.3285 /2) = (2.5, 8, 1.664). New centroid for Cluster2 Q1 (xq , yq , zq) = ( (1 +2 + 2) /3, (16 + 13 + 11) /3, (0.8192 + 1.0082 + 1.1926) /3 ) = (5 /3, 40 /3, 3.02 /3) = (1.6, 13.3, 1.006). Step 8: Repetition of Step 6 with new centroids of both clusters. Blocking numbers indicate lowest cost comparing to other. Detail description illustrates in Table 7. Table 7. Deciding cluster memberships with new centroids Frequency

1

Travel _time (min) 16

0.8192

Distance from Cluster1 new centroid ( 2.5,8,1.664 ) 1.5+8+0.84=10.34

1

9

1.456

1.5+1+0.208=2.708

0.6+4.3+0.45=5.35

2

13

1.0082

0.5+5+0.655=6.155

0.4+0.3+0.0022=0.7022

2

11

1.1916

0.5+3+0.4724=3.9724

0.4+2.3+0.1856=2.8856

4

7

1.8725

1.5+1+0.2085=2.7085

2.4+6.3+0.8665=9.5665

Velocity (km/min)

Distance from Cluster2 new centroid ( 1.6,13.3,1.006) 0.6+2.7+0.189=3.489

Re-estimating the cluster memberships from Table 7, 2nd scenario of both clusters has been represented in Table 8. After repetition of step 7 we get that the most recent centroids of Cluster1 P2 (xp, yp, zp) and Cluster2 Q2 (xq, yq, zq) are (2.5, 8, 1.664) and (1.6, 13.3, 1.006) respectively. The most recent centroids of both clusters are similar to the 2nd most recent centroids. So, the need of repetition of step 6 and step 7 again and again are unnecessary. Table 8 shows the final clusters.

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Cluster2

Frequency 4 1 1 2 2

Travel_time(min) 7 9 16 13 11

Velocity(km/min) 1.8725 1.456 0.8192 1.0082 1.1916

Step 9: By using Eq. 2, desired travel time from Cluster1 and Cluster2 can be measured Expected Travel Time from Cluster1 Expected Travel Time from Cluster2 Here, N=2 Here, N=3 So, τ1 = (4*7+1*9)/ (4+1) So, τ2 = (1*16+2*13+2*11) / (1+2+2) = (28 +9) / 5 = 7.4 = (16+26+22) / 5 = 12.8 So, expected travel time from Cluster1, τ1 =7 min (applying round operation) and expected travel time from Cluster2, τ2 =13 min (applying round operation) Step 10: The final approximate travel time, T (for Road_id=1 and Time_ group= 6) is predicted by using Eq. 3 such as the simple arithmetic mean of τ1 and τ2. So, the final approximate travel time is T = ((7+13)/2) min =10 min.

4 Simulation Results We used Pusan National University (PNU) generator [12] to measure the performance of different predictors. This generator is based on real traffic situation in Pusan city, South Korea. Global Positioning System (GPS) sensor is used to collect real traffic delay for building this well-organized PNU generator. Traffic pattern of Pusan city was extracted from this data. According to this traffic pattern, generator simulates and generates trajectory data which is almost same as real data. By using this generator, 167,669 trajectories are generated. Every trajectory may be composed of several road segments. This data organization format sufficiently reflects real traffic situations. For computing easily and efficiently and accurate evaluation of performance of the algorithms, data is divided into two categories, namely training data and test data sets. 365 days traffic data are used as training data set and 30 days traffic data are used as testing data set. Testing data sets are chronologically after 365 days data used for training. Data from 365 training days are used for fitting the model. However, 30 days test data are used to measure prediction performance for all methods. The prediction error indices, MARE are used to compare the accuracy among all prediction methods. MARE is the simplest and well-known method for measuring overall error in travel time prediction. MARE measures the magnitude of the relative error over the desired time range. The MARE is measured by the following formula:

MARE =

1 N | x(t ) − x * (t ) | ∑ x(t ) N t =1

(4)

where, x(t ) is the observation value, x * (t ) is the predicted value and N is the number of samples. In experimental evaluation, proposed methods are tested against other predictors like NBC, SMA and CA. Prediction errors of all predictors from 8 AM to 6 PM are examined. There are 11 test cases evaluated between 8 AM and 6 PM.

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The line chart shown in Fig.1 illustrates relative performance of all travel time predictors. From the overall point of view, proposed method performs much better than NBC, SMA and CA method. In case of MKC method, it is shown that seven test cases exhibit errors less than 0.40. At 10.00 AM, 3.00 PM and 6.00 PM our method MKC predicted more accurately than others and datasets of those period included uncertain data. By contrary, NBC, SMA and CA outperform our method in one, two and one cases respectively but that are slight differences. Summarized result of MARE for different travel time predictors are shown in Fig. 2. MARE of MKC, NBC, SMA and CA are 3.96, 4.891, 4.99 and 4.76 respectively. Thus, our proposed method reduces MARE from NBC, SMA and CA method by 19%, 20% and 17% respectively. MARE ANALYSIS 0.9 MKC

MARE

0.8

NBC

0.7

SMA

0.6

CA

0.5 0.4 0.3 0.2 0.1 0 8:00 AM

9:00 AM

10:00 11:00 12:00 AM AM PM

1:00 PM

2:00 PM

3:00 PM

4:00 PM

5:00 PM

6:00 PM

Departure Time

Fig. 1. MARE of each method during different time interval

MARE

SUMMARIZED MARE ANALYSIS

5.10

MKC method

4.90

NBC method

4.70

SMA method

4.50

CA method

4.30 4.10 3.90 3.70

Fig. 2. Summarized MARE of each prediction method

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5 Conclusion In our research, an efficient and scalable method for predicting travel time with arbitrary routes in road network is focused. The K-means algorithm is one of the simplest clustering techniques. One of the significant disadvantages of K-means clustering is that it doesn’t yield the same result with each run, since the resulting clusters depend on the initial random assignments. Fixing two centroids by a selective procedure our MKC method eliminates this drawback of traditional K-means clustering. Moreover, the centroids are placed in a cunning way so that they maintain as much as possible far way from each other. In our clustering approach, our first cluster amalgamates those data whose possibilities are higher and second cluster congregates those data that probabilities are not significant but help to address the uncertain situation. Another shortcoming of standard clustering methods is that they ignore measurement errors or uncertainty associated with data. If these errors are available, they can play a significant role in improving the clustering decision. If we take output only from the first cluster then the results are very good in most cases but in uncertain situations the results are worse. So, we take the predicted travel time by analyzing both clusters such that the algorithm can be able to address uncertain situation. However, our method is able to predict in uncertain situations more accurately comparing with other methods. Moreover performance analysis portion of this research reveals that our proposed method outperforms other methods in most cases. The superiority of MKC is that the more the historical traffic data set increases the more the predictor is able to predict accurately. As our future, we will extend our clustering approach considering not only day time but also week days and seasonal patterns. This may help us to address uncertain situations more efficiently. We will also pay attention in the relationship between the length of roadways and accuracy of the prediction. We also try to improve our algorithm by addressing which data in the historical data are associated with uncertain situation.

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