Hybrid Neuro-Fuzzy Application in Short-Term Freeway Traffic Volume

■ Transportation Research Record 1802 Paper No. 02-2921

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Hybrid Neuro-Fuzzy Application in Short-Term Freeway Traffic Volume Forecasting Byungkyu “Brian” Park A hybrid neuro-fuzzy application for short-term freeway traffic volume forecasting was developed. The hybrid model consists of two components: a fuzzy C-means (FCM) method, which classifies traffic flow patterns into a couple of clusters, and a radial-basis-function (RBF) neural network, which develops forecasting models associated with each cluster. The new hybrid model was compared with previously developed clustering-based RBF models. In addition, the dynamic linear model was studied for comparison. The study results showed that the clusteringbased hybrid method did not produce time-lag phenomena, whereas the dynamic linear model and the RBF model without clustering revealed apparent time-lag phenomena. The forecasting performance for freeway traffic volumes from the San Antonio, Texas, TransGuide system shows that even though the hybrid of the FCM and RBF models appears to be promising, additional research efforts should be devoted to achieving more reliable traffic forecasting.

With the implementation of advanced traffic management systems, most traffic management centers are able to monitor freeway traffic conditions. These centers implement various traffic management systems, such as incident detection and management, ramp metering, and route guidance using changeable message signs. Among these, ramp metering and route guidance systems require a reliable forecasting algorithm. For the last two decades, a great deal of research has been dedicated to the application of various algorithms and techniques in traffic volume forecasting. For instance, the time series model (1–4), Kalman filtering model (5–7 ), adaptive technique based on the least-meansquares algorithm (8), nonparametric regression model (9,10), eventbased model (11), expert-system-based model (12), and neural network model (13–17 ) have been widely applied. As for the application of the neural network model, most approaches were based on the multilayer feed-forward neural network model, also known as the backpropagation neural network. In a recent study by Park et al. (17), the backpropagation neural network model was compared with the radial-basis-function (RBF) neural network model. The study showed that the RBF neural network model is more reasonable because it required less network training time and showed better performance. Several hybrid methods have emerged. The ATHENA method (18) employed a mathematical clustering technique to categorize the data. This method developed a linear regression model for each cluster. Although the forecasting results were superior to those of other

Department of Civil Engineering, University of Virginia, Thornton Hall, 351 McCormick Road, P.O. Box 400742, Charlottesville, VA 22904-4742.

methods, including the backpropagation neural network method and the Box-Jenkins method, it required up to 192 different clusters for a single geographical forecasting point. Despite requiring such a large number of clusters, it is clear that the ATHENA method showed the advantage of a combined model. Van Der Voort et al. (19) utilized a combined model of the Kohonen neural network and the ARIMA model. The ARIMA model was used for forecasting, and the Kohonen model was used for clustering. Park (20, pp. 280–287) also applied a clustering-based RBF neural network model. He compared two clustering algorithms: the Kohonen neural network and the K-means method in conjunction with the RBF neural-networkbased forecasting algorithm. Data collection is described next, followed by presentation of the models used in this study: clustering models, the RBF neural network model, and the dynamic linear model. Forecasting model designs are presented and performance analysis follows. Discussion of results is provided, and conclusions and recommendations are given.

DATA COLLECTION The newly opened TransGuide system in San Antonio, Texas, is one of the most significant efforts toward implementing intelligent transportation systems in America. It was designed to improve freeway operations in and around San Antonio. The system collects traffic data from 836 loop detectors including freeways and ramps, which were installed in 26 mi (Phase I) of freeway sections throughout San Antonio. In Figure 1, the basic network of the TransGuide system and the data collection location are shown. The traffic volume data for this paper were collected from the TransGuide system between 7:00 a.m. and 8:00 p.m. from February 1 to February 29, 1996. Traffic volume data on weekends and holidays were excluded from the study because they showed significantly different patterns from those on weekdays. For this study, 5-min traffic volume data were aggregated from the 20-s raw data, and the resulting traffic data were divided into training and test data sets for model development and testing. Each day consisted of 156 time intervals. Among the 18 days of weekday traffic volume data, the first 10 days were used for training and the remaining 8 days were used for testing. The training data set consisted of 1,560 time intervals, and the test data set consisted of 1,248 time intervals. The 5-min aggregation time period is assumed to be reasonable for a traffic control system, although other aggregation time periods could be used as well.

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US 281

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training) the most similar (closest) cluster unit is selected as the winning unit by comparing the Euclidean distance between input pattern and cluster units. The winning unit and its neighboring units update their weights. Since the weights physically represent the X–Y coordinates for the two-dimensional case, the coordinates of the winning unit (cluster center i) move toward to a new input ( j). The algorithm is summarized as follows:

FIGURE 1 TransGuide network and data collection site (CBD  central business district).

CLUSTERING MODELS In a previous study, Park (20, pp. 280–287) utilized two clustering models: the K-means method and the Kohonen neural network. In this paper, the fuzzy C-means (FCM) method was examined as well. A brief description of these algorithms follows.

K-Means Clustering

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∑∑

xn − µ j

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µj =

− xj )

ij

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wij ( new) − wij (old ) <  objective function, number of centers, K disjoint subsets containing Nj data points, N data points to be clustered, and mean of data points in set Sj and is given by 1 N

∑x

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where = = = = =

∑ (w

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The number of centers, K, has to be determined in advance. This algorithm employs a simple reestimation procedure. There are N data points xn, and it is desired to find centers µj, where j = 1, . . . , K, to minimize the following function: J =

Kohonen neural network model.

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The algorithm is summarized as follows: Step 1. Initially assign the points (xn) at random to K sets. Step 2. Compute the mean vectors of the points in each set. Step 3. Reassign each point according to the differences between the point vector and the mean vector. Step 4. Recompute the mean vector. Step 5. Check stopping criteria. If not satisfied, go to Step 3.

( 4)

Fuzzy Clustering Fuzzy clustering is a procedure of clustering data into possibly “overlapping” clusters, such that each of the data examples may belong to each of the clusters to a certain degree. The procedure aims at finding the cluster centers Ci (i = 1, 2, . . . , c) and the cluster membership functions µi, which define to what degree each of the n examples belong to the ith cluster. The number of clusters c is either defined a priori (supervised type of clustering) or chosen by the clustering procedure (unsupervised type of clustering). A widely used algorithm for fuzzy clustering is the C-means algorithm as suggested by Bezdek and Pal (22). The FCM algorithm is summarized as follows: Step 1. Initialize c fuzzy cluster centers C1, C2, . . . , Cc arbitrarily and calculate the membership degrees mi, k i = 1, 2, . . . , c, k = 1, 2, . . . , n such that the general conditions are met. Step 2. Calculate the next values for cluster centers:

Kohonen Neural Network

∑ (µ ) x ∑ (µ ) 2

The Kohonen neural network (21), shown in Figure 2, is one of the famous unsupervised neural networks. It is often called a selforganizing neural network. During the self-organizing process (or

i, k

Ci =

k =1

2

i, k

k =1

k

for i = 1, 2, . . . , c

(5)

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Step 3. Update the fuzzy degree of membership: µ i, k =

1 d ∑j dij,,kk

(6)

c

of nonlinearity for hidden nodes is not crucial, and a uniform width for every hidden node is sufficient for universal approximation (23). The network output is defined as follows: Vˆ (ti +1 ) =

where di, k = (xk − Ci)2 and dj, k = (xk − Ck)2 (Euclidean distance). Step 4. If the currently calculated values Ci for the cluster centers are not different from the values calculated at the previous step (subject to a small error ), stop.

RBF NEURAL NETWORK MODEL The RBF neural network is employed to develop traffic volume forecasting models. The RBF methods originated from techniques developed for performing an exact interpolation of a set of data points in a multidimensional space (23). As shown in Figure 3, the RBF neural network is a feed-forward type of neural network with a single hidden layer. Each node of the hidden layer has a basis function, which is used to compare with the network input vector and to produce an output that has a radial type of symmetrical response. The response depends on the type of radial function. Responses of the hidden layer are then linearly combined with the synaptic weights of the output layer. Finally, the network output is produced through the activation function. To predict the target signal value, the RBF network input vector is In = [V (ti ), V (ti −1 ), . . . , V (ti − m −1 )]

( 7)

where In is the input vector, and V(ti) is the observed traffic volume for the time interval ti. In is an m-dimensional vector consisting of historic data samples. The m is often referred to as the data vector length. In this study, the Gaussian function was chosen as the basis function of the hidden nodes. The response of the jth hidden node to xi is given by

K

∑φ h ij

j

( 9)

j =1

where hj is the connection weights between the hidden and output nodes, and Vˆ (ti + 1) is the forecast traffic volume for the time interval ti + 1. It is assumed that N samples of the input data samples, [In]Nn = 1, are available for training. The centers, cj, 1 < j < K, can be selected from the network training input Ii, 1 < I < N. The weights can be obtained from the least-squares regression method (24). A constructive approach is to use the ordinary least-squares algorithm to simultaneously determine RBF centers and weights.

DYNAMIC LINEAR MODEL The Kalman-filtering-based dynamic linear model is used to forecast freeway traffic volumes. It is an optimal state estimation process applied to a dynamic system that involves random perturbation. More precisely, it yields a linear, unbiased, and minimum error variance recursive algorithm to optimally estimate the unknown state of a dynamic system from noisy data taken at discrete real-time intervals (25). The dynamic linear model consists of the following two equations; the state equation and the observed equation. The state equation performs an a priori estimate, and the observed equation updates a posterior estimation based on real observations. State equation: Xk +1 = Ak Xk + Gk wk

(10)

Observed equation:

2  I−c  φ ij = exp −  2  2σ 

(8)

where

Zk = Hk Xk + vk

(11)

where

cj = m-dimensional center vector, σ = standard deviation of Gaussian function, and φij = output of hidden node. Equation 8 determines the width of the symmetric response of the hidden node. Theoretical investigations have shown that the choice

outputs Output Layer

Xk Ak Gk Hk

= = = =

process state vector at time k, transition matrix from time k to time k + 1, transition matrix from time k to time k + 1, matrix giving ideal (noiseless) connection between measurement and state vector at time k, wk = assumed white noise sequence with known covariance structure, Zk = vector measurement at time k, and vk = measurement error assumed to be white sequence covariance structure and having zero cross-correlation with sequence.

Hidden Layer

FORECASTING MODEL DESIGN

bias basis functions

Input Layer inputs

FIGURE 3

RBF neural network.

In this section, model design issues including number of clusters, variable selection, and the RBF neural network architecture are discussed. Following this discussion, the dynamic linear model design and calculation loop are provided.

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results were used for clustering. For example, if three previous traffic volume counts were used, the input elements of the clustering model were as follows:

Clustering Model Design The traffic patterns of training and test data are plotted in Figure 4. The traffic fluctuations vary severely and they are mixed up together. It is noted that for a clear demonstration, only 20 input traffic patterns were randomly selected and plotted. Each line contains four traffic volumes at times of t − 2, t − 1, t, and t + 1. In order to capture the fluctuations in traffic flow patterns, the input vectors are classified into a few clusters. First, the traffic volumes themselves were fed into the clustering model. The clustering was done by traffic volume level such that no traffic flow fluctuations were captured. Second, the variation (slope) of the traffic flow patterns was considered. After the first differencing of input data, the

Input vector {v(t − 2) − v(t − 1), v(t − 1) − v(t )} where v(t) is the traffic volume during time interval t. As shown in Figure 5, the traffic flow data were categorized into five clusters. Although one could use a different number of clusters, five was arbitrarily chosen. It is noted that only three previous traffic volume counts were utilized in this clustering, and v(t + 1) is plotted along with the three input counts for comparison purposes. That is, the traffic volume data at time t + 1 is not used in

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the clustering analysis, since in real time the control traffic volume at time t + 1 is to be forecast. It is apparent that each cluster has a different type of slope. RBF Neural Network Design Five different RBF neural network models that correspond to the five clusters were developed. There are several network parameters to be considered in developing the RBF neural network models: the number of hidden layer nodes, the initialization of centers, and their standard deviations. Although the number of centers is determined arbitrarily, the standard deviation of each center is randomly assigned at approximately a third of the maximum input minus the minimum input, as a rule of thumb. Sometimes the standard deviations are fixed in order to increase network convergence and reduce the time required for learning. In this study, the standard deviation of each center was fixed at 1.0 after a significant amount of trial and error. After tests and investigations were conducted, a neural network architecture having 3 input nodes, 20 centers, and a learning rate of 0.05 was selected. Dynamic Linear Model Design When the dynamic linear model is applied, since the prior estimate is usually not available, the mean value is used for an approximation (26). In this study, the time series analysis was performed to determine the initial value of the dynamic linear model. Through time series analysis, the AR(2) model was derived. The transition matrix, Ak, was determined based on the AR(2) parameters. The calculation process of the Kalman filter method is graphically described in Figure 6. Most of the notation is shown in the section on the dynamic linear model. Q and R, shown in Figure 6, are covariance matrices with zero means. Interested readers should refer to the discussion by Brown and Hwang (26 ). PERFORMANCE ANALYSIS For model comparison, the mean absolute percentage error (MAPE), variance of absolute percentage error (APE), and probability of percentage errors were employed. The MAPE is defined as follows:

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[

] 

 abs V (ti +1 ) − Vˆ (ti +1 )  ∑ V (ti +1 ) i =1   MAPE(%) = N N

DISCUSSION OF RESULTS It is noted that the RBF neural network forecasting models outperformed the dynamic linear model. Among the RBF neural network models, RBF with FCM showed the lowest MAPE. It seems that the RBF model without clustering algorithms and the dynamic linear model show apparent time-lag phenomena, even though they are more sensitive to trace afternoon peak traffic volumes. The hybrid of the clustering algorithm and the RBF neural network model is not as sensitive as the RBF alone or the dynamic linear model to trace peak volumes. However, they seemed to follow up-and-down patterns without showing any time-lag phenomena. The reason is that the hybrid models categorized traffic flow patterns into a couple of clusters.

INITIAL INPUT • •

Error variance matrix of the prior estimate Pk

−

CALCULATION OF KALMAN GAIN

K k = Pk− H kT ( H k Pk− H kT + Rk )

−1

FORECAST FUTURE ESTIMATE •

Xˆ k−+1 = Ak Xˆ k

•

Pk−+1 = Ak Pk AkT + Qk

UPDATE OUTPUT WITH OBSERVED VALUE •

FIGURE 6

Pk = ( I − K k H k ) Pk −

Dynamic linear model loop (26).

(

)

Xˆ k = Xˆ k− + K k Zk − H k Xˆ k−

UPDATE COVARIANCE MATRIX

•

(12)

where V(ti+1_) is the observed traffic volume for the time interval ti+1, and V (ti + 1) is the forecast traffic volume for the time interval ti+1. The forecasting results of five different models were plotted in Figures 7 through 11, and summary results are shown in Table 1.

− Prior estimate Xˆ k

•

 × 100

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FIGURE 9 Forecasting results of RBF neural network with FCM model (Day 6).

FIGURE 11

Forecasting results of dynamic linear model (Day 6).

It is also observed that the FCM method provides better forecasting performance. This finding is understandable since the FCM method classifies traffic flow fluctuations into a number of clusters in better way than the other clustering methods do. The K-means method initially makes a fixed number of clusters by assigning traffic flow patterns at random. Then it calculates the centers of corresponding clusters and updates those centers. Therefore, if the initial random assignments were made incorrectly for some of the traffic flow patterns, some of those traffic flow patterns are not assigned to an appropriate cluster. The K-means method does not update the centers of corresponding clusters until all traffic flow patterns are considered. Furthermore, it just updates cluster centers by averaging all the corresponding cluster inputs. The Kohonen neural network method has similar deficiencies. The Kohonen method is considered to be a somewhat better clustering method since it updates the cluster centers whenever a new input is applied. However, it also initially assigns the centers randomly and only the closest center is updated among those centers. As a consequence, no consideration is given to the second-closest center or others. The FCM method is able to accommodate this deficiency since it considers not only the first-closest center but also the second, third, and the other ranked centers by using membership function values. Therefore it has better chance of assigning traffic flow patterns into

an appropriate cluster even though those patterns are initially assigned incorrectly. CONCLUSIONS AND RECOMMENDATIONS In this study, a hybrid model of the RBF neural network and the FCM method was developed and evaluated with freeway traffic volume data from the TransGuide system of San Antonio, Texas. Comparisons were made with two different previously developed clustering algorithms (the K-means method and the Kohonen neural network) and the Kalman-filtering-based dynamic linear model developed in this study. The study results indicated that the FCM method outperformed the other models. However, as shown in Table 1, approximately 5% of the forecast volumes result in more than 20% forecasting errors. The results are somewhat discouraging, and better methods are needed. In this research, an arbitrary number of both clusters and input elements, five and three, respectively, was used. More research efforts should be undertaken on deciding what the optimum number of clusters and input elements should be. The models proposed here were only tested with normal traffic volume cases. Further study should be conducted with abnormal volume cases to analyze traffic conditions during accidents, on special days, and with road construction, weather conditions, and so on.

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Transportation Research Record 1802

Performance Analysis of Forecasting Models

MAPE (%) Var. of APE Std. Dev. of APE Prob. (error < ± 0.1) Prob. (error < ± 0.2)

RBF w/o Clustering 8.8260 53.744 7.3311 0.6481 0.9139

RBF w/ Kohonen 7.4408 36.882 6.0731 0.7205 0.9526

RBF w/ FCM 7.2278 38.525 6.2069 0.7410 0.9517

RBF w/ K-means 8.3840 37.851 6.1523 0.6372 0.9624

Dynamic Linear Model 9.3341 56.946 7.5463 0.6291 0.9044

NOTE: Numbers represent an average of 8 days of test data.

ACKNOWLEDGMENT Traffic volume data used in this study were provided by the TransGuide traffic management center located in San Antonio, Texas. REFERENCES 1. Ahmed, M. S., and A. R. Cook. Analysis of Freeway Traffic Time-Series Data by Using Box-Jenkins Techniques. In Transportation Research Record 722, TRB, National Research Council, Washington, D.C., 1979, pp. 1–9. 2. Nihan, N. L., and K. O. Holmesland. Use of Box and Jenkins Time Series Techniques in Traffic Engineering. Transportation, 1980, pp. 125–143. 3. Moorthy, C. K., and B. G. Ratcliffe. Short-Term Traffic Forecasting Using Time Series Methods. Transportation Planning and Technology, Vol. 12, 1988, pp. 45–46. 4. Kyte, M., J. Marek, and D. Firth. Using Multivariate Time Series Analysis to Model Freeway Traffic Flow. Presented at the 68th Annual Meeting of the Transportation Research Board, Washington, D.C., 1989. 5. Okutani, I., and Y. J. Stephanedes. Dynamic Prediction of Traffic Volume Through Kalman Filtering Theory. Transportation Research, Vol. 18B, 1984, pp. 1–11. 6. Okutani, I. The Kalman Filtering Approach in Some Transportation and Traffic Problems. Proc., 10th International Symposium on Transportation and Traffic Flow Theory, Massachusetts Institute of Technology, Cambridge, July 8–10, 1987, pp. 387–416. 7. Bhattacharjee, D., and K. C. Sinha. Application of Dynamic Linear Model in Short-Term Freeway Traffic Flow Prediction. Presented at 76th Annual Meeting of the Transportation Research Board, Washington, D.C., 1997. 8. Lu, J. Prediction of Traffic Flow by an Adaptive Prediction System. In Transportation Research Record 1287, TRB, National Research Council, Washington, D.C., 1990, pp. 54–61. 9. Davis, G. A., and N. L. Nihan. Nonparametric Regression and Short-Term Freeway Traffic Forecasting. Journal of Transportation Engineering, Vol. 177, No. 2, 1991, pp. 178–188. 10. Smith, B. L., and M. J. Demetsky. Multiple-Interval Freeway Traffic Flow Forecasting. In Transportation Research Record 1554, TRB, National Research Council, Washington, D.C., 1996, pp. 136–141. 11. Head, K. L. Event-Based Short-Term Traffic Flow Prediction Model. In Transportation Research Record 1510, TRB, National Research Council, Washington, D.C., 1995, pp. 45–52. 12. Rahman, S., and R. Bhatnagar. An Expert System-Based Algorithm for Short-Term Load Forecast. IEEE Transactions on Power Systems, Vol. 3, No. 2, 1988, pp. 329–399.

13. Dougherty, M. S., H. R. Kirby, and R. D. Boyle. The Use of Neural Networks to Recognize and Predict Traffic Congestion. Traffic Engineering and Control, Vol. 34, No. 6, 1993, pp. 311–314. 14. Kwon, E., and Y. J. Stephanedes. Comparative Evaluation of Adaptive and Neural-Network Exit Demand Prediction for Freeway Control. In Transportation Research Record 1446, TRB, National Research Council, Washington, D.C., 1994, pp. 66–76. 15. Gilmore, J. F., and N. Abe. Neural Network Models for Traffic Control and Congestion Prediction. Intelligent Vehicle-Highway System (IVHS) Journal, Vol. 2, No. 3, 1995, pp. 231–252. 16. Smith, B. L., and M. J. Demetsky. Short-Term Traffic Flow Prediction: Neural Network Approach. In Transportation Research Record 1453, TRB, National Research Council, Washington, D.C., 1994, pp. 98–104. 17. Park, B., C. J. Messer, and T. Urbanik II. Short-Term Freeway Traffic Volume Forecasting Using Radial Basis Function Neural Network. In Transportation Research Record 1651, TRB, National Research Council, Washington, D.C., 1998, pp. 39–47. 18. Danech-Pajouh, M., and M. Aron. ATHENA, A Method for Short-Term Inter-urban Traffic Forecasting. Report 177. Institut National de Recherche sur les Transports, Arcueil, France, 1991. 19. Van Der Voort, M., M. Dougherty, and S. Watson. Combining Kohonen Maps with ARIMA Time Series Models to Forecast Traffic Flow. Transportation Research, Vol. 4C, 1996, pp. 307–318. 20. Park, B. Clustering-Based RBF Neural Network Model for Short-Term Freeway Traffic Volume Forecasting. In Applications of Advanced Technologies in Transportation Engineering (C. T. Hendrickson and S. G. Ritchie, eds.), School of Civil Engineering, Purdue University, West Lafayette, Indiana, 1988. 21. Kohonen, T. Self-Organized Formation of Topologically Correct Feature Maps. Biological Cybernetics, Vol. 43, 1992, pp. 59–69. 22. Bezdek, J. C., and S. K. Pal. Fuzzy Models for Pattern Recognition: Methods That Search for Structures in Data. IEEE Press, New York, 1992. 23. Powell, M. J. D. Radial Basis Functions for Multivariable Interpolation: A Review. In Algorithms for Approximation (J. C. Mason and M. G. Cox, eds.), Oxford University Press, England, 1987. 24. Golub, G. H., and C. Reinsch. Singular Value Decomposition and Least Squares Solutions. Numerical Mathematics, Vol. 14, 1970, pp. 403–420. 25. Chui, C. K., and G. Chen. Kalman Filtering with Real-Time Applications, 2nd ed. Springer-Verlag, Berlin, 1991. 26. Brown, R. G., and P. Y. C. Hwang. Introduction to Random Signals and Applied Kalman Filtering, 2nd ed. John Wiley & Sons, Inc., New York, 1992.

Publication of this paper sponsored by Committee on Traffic Flow Theory and Characteristics.