The Fuzzy-Neural Network Traffic Prediction Framework with Wavelet Decomposition

Heng Xiao* Hongyu Sun** Department of Civil and Environmental Engineering University of Wisconsin at Madison 1415 Engineering Drive, Madison, WI 53706 Tel: (608) 262-2524, Fax: (608) 262-5199 *E-mail: [email protected] **E-mail: [email protected]

Bin Ran Department of Civil and Environmental Engineering University of Wisconsin at Madison 1415 Engineering Drive, Madison, WI 53706 Tel: (608) 262-0052, Fax: (608) 262-5199 E-mail: [email protected]

Submitted to the 82nd TRB Annual Meeting

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Paper revised from original submittal.

The Fuzzy-Neural Network Traffic Prediction Framework with Wavelet Decomposition Heng Xiao, Hongyu Sun and Bin Ran Department of Civil and Environmental Engineering University of Wisconsin at Madison, Madison, WI, USA

Abstract

This paper addressed a framework of a traffic prediction model which could eliminate the noises caused by random travel conditions. In the meantime, this model can also quantitatively calculate the influence of special factors. This framework combined several artificial intelligence technologies such as wavelet transform, neural network, and fuzzy logic. In addition to developing the prediction framework, the wavelet de-noising method is also emphasized and analyzed in this paper.

Key words: Traffic Prediction, Wavelet, Fuzzy Logic, Neural Network, Special Factor

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Paper revised from original submittal.

1. Introduction

Traffic prediction could provide useful traffic information to motorists and fleet operators. A good traffic prediction model is important to provide accurate traffic prediction. However, in the real world, transportation systems are very complex and difficult to predict since there are many impacting factors involved. Currently, there is no efficient model to integrate the factor influences to the prediction models.

This paper proposes a model to predict traffic speed based on a historical travel speed profile along a specific link. To conduct the prediction, the model finds a basic pattern in the historical data and applies it to the future prediction. In the meantime, the impacts of special factors are considered as well.

Traffic prediction models have been becoming more and more complex. Each model tries to include all considerable real-life conditions to obtain “better” prediction results. These models become extremely complicated just because there is too much “unexpected” information which should be integrated into these models. In other words, there are a large number of variables in the traffic data. The interaction should be simplified in these models.

Wavelet transform is a powerful tool which has the ability to analyze non-stationary signals to obtain their trends, and discover the interesting patterns to get their local details. Fuzzy logic is another tool which can reduce the complexity of the data. Neural Network is another tool which can do self-study to increase the accuracy of the prediction. These tools will be used in this study to develop the basic framework.

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Paper revised from original submittal.

This research is intended to establish a framework for analyzing the influences of special factors in traffic speed prediction and develop a general traffic speed prediction model which can be applied in many situations. The procedure developed in this model includes two parts. First, the basic traffic speed signal pattern recognition will be identified by using wavelet transform. In this step, the wavelet decomposition is used to eliminate the noise mixed in the historical profile. Then, a systematic traffic speed prediction and factor adjustment model will be structured and developed. The traffic speed prediction model will be calibrated and verified through case studies. It is noted that the model is a link-specific prediction model, which requires historical traffic speed data of a specific link in order to conduct the traffic prediction for the same link into the future.

2. Literature Review

This paper employs many Artificial Intelligence techniques and can be classified as a combination of the statistical approach and the heuristic approach. Since the turn of the century, Fourier transform is widely used almost everywhere, such as in mathematics and in engineering. Fourier analysis was not challenged until the 1950s, when the application of Fourier analysis was proven to be limited to stationary signals. Researchers started to look for alternative transform that could work with all sorts of signals including nonstationary signals.

Wavelet is a powerful tool which can process non-stationary signals efficiently. Local features can be described better with wavelets. In general, time series consist of a

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Paper revised from original submittal.

deterministic part with a superimposed stochastic component (non-deterministic part). Traffic data are no exception. A technique to separate both components has been proposed by Farge & Philipovitch (1993) and implemented in viable software by Wernik & Grzesiak (1997). In that method, through a kind of non-linear filtering, also called threshold filtering, a wavelet frequency spectrum of the time series is calculated.

Recently, the fuzzy-neural network model has been applied to transportation systems including incident detection (Adeli et al., 2000; Ishak et al., 1998; Xu et al., 1996), access control system (Adorni et al., 2001), pavement distress evaluation(Chou et al., 1995), transmission control(Hayashi et al., 1995), vehicle separation control (Jou et al., 1999), machine maintenance(Liu et al., 1997), behavioral mode choice modeling (Sayed et al., 2000), ramp metering algorithm(Taylor et al., 1994), vehicle assignment (Vukadinovic et al., 1999), and so on.

The neural network approaches have been commonly used for the traffic prediction problem during the past decade (Dougherty et al., 1994; Ledoux et al., 1997). More applications have been developed in the traffic flow forecast area recently (Yin et al., 2001; Yuan et al., 2000).

The fuzzy-neuro model consists of two modules: a gate network which uses a fuzzy approach to classify the input data into clusters and an expert network which employs a neural network to specify the input-output relationship within each cluster (Yin et al., 2001).

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3. Methodology

There are many unexpected conditions which can influence the traffic speed and only have the impact for a very short time. The objective of traffic prediction is to provide useful information to users and the key is to provide a reasonable average estimation of the future traffic. Therefore, the unexpected information can be treated as noise. To filter these noises, there are many existing methods. Within them, wavelet transform is one of the most efficient methods.

Wavelet transform has ma ny benefits when used in the de-noising application. It can retain the detail in the signal and does not need any assumption for the noise (compared with that, many other methods assume the noise is Gaussian distributed).

The basic de-noising procedure wavelet transform is described below: •

A deterministic "strong'' part is obtained by setting to zero all wavelet coefficients less than a certain threshold level. The inverse wavelet transform is used to calculate the corresponding time series.

•

A stochastic "weak'' part is obtained by setting to zero all wavelet coefficients greater than that threshold level. The inverse wavelet transform is also used here to calculate the corresponding time series.

•

New wavelet spectra are calculated for each partial time series.

The details of wavelet decomposition and de-noising will be discussed in the next section. The general de-noising procedure has three steps. The basic version of the procedure follows the steps described below.

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1. Decompose Choose a wavelet (here DB4 is used in this study), and then choose a level N (4 is used in this study). Compute the wavelet decomposition of the signal s at level N.

2. Threshold detail coefficients For each level from 1 to N, select a threshold and apply soft thresholding to the detail coefficients.

3. Reconstruct Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N.

After the de-noising procedure, the noises are eliminated. Therefore, the pattern can be recognized as the properties of signals. In general, the pattern can be identified from the “smooth” signal by specifying some characteristics such as the number of wave crest. By identifying the characteristics of the signal, the “standard” pattern is created for future use.

The next step is using a neuro-fuzzy network to identify the influence of special factors on the specific pattern. Consider data which are influenced by special factors: during the time period of special factors taking place, the characteristics of the signal are changed. The influence of the specific special factor can be identified by comparing the time series during the influenced time period and its “standard” pattern. Therefore, the

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influence can be recorded and can be used in future prediction when the situation of the same special factor happens.

The combination of special factors will result in a very large set. Therefore, it is impossible to record all these influences without letting the system become too huge. On the other hand, the influence of the same factor ma y be different from time to time, even to the same pattern. Therefore, it is important to balance them and give a reasonable estimate for the influences. Neuro-fuzzy network gives us the ability to reduce the system complexity and find convincing solutions.

Unlike common combined usage of neural network and fuzzy logic, the neurofuzzy network employed in this research implements fuzzy logic inside the neural network. In other words, the fuzzy logic is used in the hidden layer of the neural network. On the contrary, most combined models which use both fuzzy and neural network technologies either put the fuzzy logic before the input of the neural network or after the output of the neural network. The purpose of this design is to maximize the efficiency of the system: first, the neural network has the ability to self-study and gets the optimized result based on training data. Then, with the fuzzy logic used in the hidden layer of the neural network, the system has the ability to adjust the member function in the fuzzy set which can simplify the computation model without decreasing the efficiency of the system.

The discussion of the neuro-fuzzy network will be addressed in a separate paper. In this paper, the de-noising method will be discussed in the remaining part. The whole procedure is shown in Figure 1 and the data flow is shown in Figure 2.

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Paper revised from original submittal.

4. The Wavelet De-noising Method

In wavelet multiresolution analysis, the signal is investigated in different time scales and different solutions. On each level j, the signal is decomposed into two parts: an approximation and details. For example, at level 3, the original signal can be represented as: A0 = A1 + D1 = A2 + D2 + D1 = A3 + D3 + D2 + D1 ,

(1)

where A j is called j-level approximation, or the approximation at level j. D j is called j-level detail, or the detail at level j. It is noticeable that the original signal can be treated as the 0-level approximation. The approximations are represented by scaling functions φ i , j (t ) and details are represented by wavelets ψ i , j (t ) . Both φ i , j (t ) and ψ i , j (t ) are translated from the scaling function φ (t ) (also called father wavelet) and wavelet ψ(t) (also called mother wavelet), where φ j ,k ( t ) = 2 − j / 2 φ ( 2 − j t − k )

(2)

ψ j ,k ( t ) = 2 − j / 2 ψ ( 2 − j t − k )

(3)

It should be noted that φ = φ0, 0 andψ = ψ 0, 0 . Then, in the level J (or say: scale 2J) the time series can be written as:

x(t ) = ∑ a J ,k φ J , k (t ) + ∑∑ d j ,kψ j ,k (t ) k∈Z

(4)

j ≤ J k∈Z

Where the summation of a J , k is the approximation on scale 2J and the summation of

d j ,k represents the details of all scales. In the mean time, we have

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a j , k = ∑ hn*−2k a j −1,n

(5)

d j , k = ∑ g *n−2 k a j −1, n

(6)

n∈Z

n∈Z

Where hk is the scaling filter associated with the wavelet and g k is the wavelet filter associated with the wavelet, * represented the complex conjugates. The reconstruction can be represented as: a j −1,k = ∑ (hk −2 n a j ,k + g k −2n d j , k )

(7)

n∈Z

where +∞

hn = ∫ φ (t )φ −1, n (t )dt

(8)

g n = (−1) n h1*−n

(9)

−∞

Since in decomposition results, d j ,k mainly represent noises, the following method can be used as a threshold function: 0, T ( d j ,k , τ j ) = d j ,k − sign(d j , k )τ j

d j, k ≤ τ j otherwise

(10)

~ If the modified coefficients d j , k = T ( d j ,k τ k ) , then (7) becomes:

~ a~ j −1,k = ∑ (hk −2 n a~ j ,k + g k −2n d j , k )

(11)

a~J ,k = a J , k

(12)

n∈Z

with

There are several different methods for selecting the threshold. One of the most widely used methods was developed by Donoho and Johnstone (1995). In this method, the threshold is defined from this equation:

τ j = σ j 2 ln n j

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(13)

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where σ j is the estimate of the standard deviation and n j is the number of coefficients

d j , k on level j.

5. Experimental Design

In order to effectively utilize the collected data and produce a convincing result, the experimental design of this study includes four parts: data collection, data standardization, data processing and result explanation. These steps are described as follows.

5.1. Data Collection

Data used in this study are retrieved from several websites. Several programs are developed to complete this project. These programs can retrieve real-time speed, construction, incident information and weather information from different cities.

5.2. Data Standardization

Data retrieved from data collection have different formats and it is not convenient to put them in a unique model. The standardization step is integrated by several programs which can read different data formats and output a standard format to the prediction model. The unique format will benefit the further processing in the whole framework.

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5.3. Data Processing

In this study, all data collected will be divided into three data sets: the training set, tuning set, and testing set. The training set will be used in the model to recognize the travel speed signal pattern, and to obtain the estimates of its parameters and special factor influences. The training set will include two third of all data that will be used to train the whole system. After the training procedure is completed, the tuning set will be applied to tune the results, which are obtained from the fuzzy-neural network. The testing set is used to verify the results and measure the performance of the prediction system.

5.4. Results Explanation

The result of the experiments can be described from different points of view. In this step, the result data will be analyzed to identify the advantages and disadvantages of the model.

6. Experimental Result

The data used in this paper are collected from Houston. Figure 3 shows the one-day speed profile on January 14, 2002. In this one day profile, it is obvious that the peak hour happened around from 6am to 9:30am. The speed dramatically went down during this time. However, lowest point (which represents the time with heaviest traffic jam) happened around 8:00am. The sampling period between every two points is 5 minutes. Therefore, we take the decomposition up to the level j=5, since 2 5 = 32 * 5 = 160 minutes is close to 3 hours and

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we are then able to study the components of the signal for which the period is less than 3 hours. The approximation of A5, which is corresponding to time scale of 160 minutes, is a crude estimation. It is easy to see that there is a lag for the estimation of the peak hour and the speed of the peak hour is overestimated. Most important information is missing in this level. Therefore, we should investigate lower levels. The result of the de-noising procedure is shown in Figure 5.

7. Conclusion

The framework developed in this paper provided a new procedure for short-time traffic prediction. This procedure is fast and easy to implement. It considered the tradeoff between computing time and the accuracy. The wavelet decomposition method employed in this paper is a fast and efficient way to smooth the historical traffic data and provide fully comprehensive input to the next step of the prediction framework.

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References [1] Adeli, H., and Karim, A., Fuzzy-wavelet RBFNN model for freeway incident detection, Journal of Transportation Engineering. v 126 n 6 Nov 2000, p 464-471, 2000.

[2] Adorni, G., Cagnoni, S., Gori, M., and Mordonini, M., Access control system with neuro-fuzzy supervision, IEEE – Conference on Intelligent Transportation Systems, Proceedings, -ITSC. n 01TH8585), p 472-477, 2001.

[3] Chou, J., O'Neill, W. A., and Cheng, H., Pavement distress evaluation using fuzzy logic and moment invariants, Transportation Research Record. n 1505, Jul 1995. p 39-46, 1995.

[4] Donoho, D. L., and Johnstone, I. M., Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, vol. 90, no. 432, p. 1200-1224, 1995.

[5] Dougherty, M., Kirby, H, and R., Boyce, Using neural networks to recognize, predict and model traffic. Artificial Intelligence Applications to Traffic Engineering, P.235-250, 1994.

[6] Farge M., and Philipovitch T., Coherent structure analysis and extraction using wavelets, in Progress in Wavelet Analysis and Applications. Éditions Frontières, Gif-surYvette, 1993.

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[7] Hayashi, K., Shimizu, Y., Dote, Y., Takayama, A., and Hirako, A., Neuro fuzzy transmission control for automobile with variable loads, IEEE – Transactions on Control Systems Technology. v 3 n 1 Mar 1995, p 49-52, 1995.

[8] Ishak, S. S., and Al, D. M., Fuzzy ART neural network model for automated detection of freeway incidents, Transportation Research Record -1634. Nov 1998, p 5663, 1998.

[9] Jou, I. C., Chang, C. J., and Chen, H. K., Hybrid neuro-fuzzy system for adaptive vehicle separation control, Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology. v 21 n 1 1999, p 15-29, 1999.

[10] Ledoux, C., An urban traffic flow model integrating neural network, Transportation Research 5C, P.287-300, 1997.

[11] Liu, J. N.K., Fuzzy neural networks for machine maintenance in mass transit railway system, IEEE - Transactions on Neural Networks. v 8 n 4 Jul 1997, p 932-941, 1997.

[12] Sayed, T., Razavi, A., Comparison of neural and conventional approaches to mode choice analysis, Journal of Computing in Civil Engineering. v 14 n 1 Jan 2000, p 23-30, 2000.

[13] Taylor, C., and Meldrum, D., Freeway traffic data prediction via artificial neural networks for use in a fuzzy logic ramp metering algorithm, Intelligent Vehicles

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Symposium, Proceedings. 1994, IEEE, Piscataway, NJ, USA, 94TH8011. p 308-313, 1994.

[14] Vukadinovic, K., Teodorovic, D., and Pavkovic, G., Application of neurofuzzy modeling: The vehicle assignment problem, European Journal of Operational Research. v 114 n 3 May 1 1999, p 474-488, 1999.

[15] Xu, H., Kwan, C.M., Haynes, L., and Pryor, J.D., Real-time adaptive on-line traffic incident detection, IEEE – International Symposium on Intelligent Control Proceedings. 1996, IEEE, Piscataway, NJ, USA, 96CH35855. p 200-205, 1996.

[16] Yin, H., Wong, S.C., et al., Urban traffic flow prediction using a fuzzy-neural approach, Transportation Research Part C, 2001.

[17] Yuan, Z., Li, W., and Liu, H., Forecast of dynamic traffic flow, Proceedings of the Conference on Traffic and Transportation Studies, -ICTTS. 2000, ASCE, Reston, VA, USA. p 507-512, 2000.

[18] Wernik A.W., and Grzesiak M., 1997, in Sadowski M. and Rothkaehl H. (eds.), Proc. Int. Symp. "Plasma 97'', Jarnoltowek, Space Research Center, Polish Academy of Sciences, Vol. 1, p. 391, 1997.

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Develop a Signal Processing Model (SPM) based on Historical Data

Apply the SPM Model to sample area

Develop a Factor Influence Model (FIM) to get the relation of the speed changes

Apply FIM Model in Sample Area

Calibration and Simulation Analysis of SPM and FIM

Reliability analysis of SPM and FIM

Figure 1. Overview of the Whole Procedure

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Create data signals

Wavelet Transform: Decomposition

Get level 3 approximation of base signal

Reconstruct the base signal

Compare target signal with base signal to get the influence data

Fuzzify the influence data and use Neural Network to train the data

Defuzzify the result

Figure 2. The Detail Dataflow of the Whole Procedure

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22:30 23:15

21:45

18:45 19:30 20:15 21:00

15:45 16:30 17:15 18:00

14:15 15:00

12:45 13:30

9:45 10:30 11:15 12:00

6:45 7:30 8:15 9:00

6:00

3:45 4:30 5:15

0:45 1:30 2:15 3:00

80 70 60 50 40 30 20 10 0 0:00

Speed

Jan 14 Inbound

Time Jan 14 Inbound

Figure 3. The One-Day Speed Profile for Houston Inbound on Jan/14/2002

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Figure 4. The Analysis of the One-Day Data

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Figure 5. The Result of Wavelet Decomposition

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Heng Xiao* Hongyu Sun** Department of Civil and Environmental Engineering University of Wisconsin at Madison 1415 Engineering Drive, Madison, WI 53706 Tel: (608) 262-2524, Fax: (608) 262-5199 *E-mail: [email protected] **E-mail: [email protected]

Bin Ran Department of Civil and Environmental Engineering University of Wisconsin at Madison 1415 Engineering Drive, Madison, WI 53706 Tel: (608) 262-0052, Fax: (608) 262-5199 E-mail: [email protected]

Submitted to the 82nd TRB Annual Meeting

TRB 2003 Annual Meeting CD-ROM

1

Paper revised from original submittal.

The Fuzzy-Neural Network Traffic Prediction Framework with Wavelet Decomposition Heng Xiao, Hongyu Sun and Bin Ran Department of Civil and Environmental Engineering University of Wisconsin at Madison, Madison, WI, USA

Abstract

This paper addressed a framework of a traffic prediction model which could eliminate the noises caused by random travel conditions. In the meantime, this model can also quantitatively calculate the influence of special factors. This framework combined several artificial intelligence technologies such as wavelet transform, neural network, and fuzzy logic. In addition to developing the prediction framework, the wavelet de-noising method is also emphasized and analyzed in this paper.

Key words: Traffic Prediction, Wavelet, Fuzzy Logic, Neural Network, Special Factor

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Paper revised from original submittal.

1. Introduction

Traffic prediction could provide useful traffic information to motorists and fleet operators. A good traffic prediction model is important to provide accurate traffic prediction. However, in the real world, transportation systems are very complex and difficult to predict since there are many impacting factors involved. Currently, there is no efficient model to integrate the factor influences to the prediction models.

This paper proposes a model to predict traffic speed based on a historical travel speed profile along a specific link. To conduct the prediction, the model finds a basic pattern in the historical data and applies it to the future prediction. In the meantime, the impacts of special factors are considered as well.

Traffic prediction models have been becoming more and more complex. Each model tries to include all considerable real-life conditions to obtain “better” prediction results. These models become extremely complicated just because there is too much “unexpected” information which should be integrated into these models. In other words, there are a large number of variables in the traffic data. The interaction should be simplified in these models.

Wavelet transform is a powerful tool which has the ability to analyze non-stationary signals to obtain their trends, and discover the interesting patterns to get their local details. Fuzzy logic is another tool which can reduce the complexity of the data. Neural Network is another tool which can do self-study to increase the accuracy of the prediction. These tools will be used in this study to develop the basic framework.

TRB 2003 Annual Meeting CD-ROM

3

Paper revised from original submittal.

This research is intended to establish a framework for analyzing the influences of special factors in traffic speed prediction and develop a general traffic speed prediction model which can be applied in many situations. The procedure developed in this model includes two parts. First, the basic traffic speed signal pattern recognition will be identified by using wavelet transform. In this step, the wavelet decomposition is used to eliminate the noise mixed in the historical profile. Then, a systematic traffic speed prediction and factor adjustment model will be structured and developed. The traffic speed prediction model will be calibrated and verified through case studies. It is noted that the model is a link-specific prediction model, which requires historical traffic speed data of a specific link in order to conduct the traffic prediction for the same link into the future.

2. Literature Review

This paper employs many Artificial Intelligence techniques and can be classified as a combination of the statistical approach and the heuristic approach. Since the turn of the century, Fourier transform is widely used almost everywhere, such as in mathematics and in engineering. Fourier analysis was not challenged until the 1950s, when the application of Fourier analysis was proven to be limited to stationary signals. Researchers started to look for alternative transform that could work with all sorts of signals including nonstationary signals.

Wavelet is a powerful tool which can process non-stationary signals efficiently. Local features can be described better with wavelets. In general, time series consist of a

TRB 2003 Annual Meeting CD-ROM

4

Paper revised from original submittal.

deterministic part with a superimposed stochastic component (non-deterministic part). Traffic data are no exception. A technique to separate both components has been proposed by Farge & Philipovitch (1993) and implemented in viable software by Wernik & Grzesiak (1997). In that method, through a kind of non-linear filtering, also called threshold filtering, a wavelet frequency spectrum of the time series is calculated.

Recently, the fuzzy-neural network model has been applied to transportation systems including incident detection (Adeli et al., 2000; Ishak et al., 1998; Xu et al., 1996), access control system (Adorni et al., 2001), pavement distress evaluation(Chou et al., 1995), transmission control(Hayashi et al., 1995), vehicle separation control (Jou et al., 1999), machine maintenance(Liu et al., 1997), behavioral mode choice modeling (Sayed et al., 2000), ramp metering algorithm(Taylor et al., 1994), vehicle assignment (Vukadinovic et al., 1999), and so on.

The neural network approaches have been commonly used for the traffic prediction problem during the past decade (Dougherty et al., 1994; Ledoux et al., 1997). More applications have been developed in the traffic flow forecast area recently (Yin et al., 2001; Yuan et al., 2000).

The fuzzy-neuro model consists of two modules: a gate network which uses a fuzzy approach to classify the input data into clusters and an expert network which employs a neural network to specify the input-output relationship within each cluster (Yin et al., 2001).

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Paper revised from original submittal.

3. Methodology

There are many unexpected conditions which can influence the traffic speed and only have the impact for a very short time. The objective of traffic prediction is to provide useful information to users and the key is to provide a reasonable average estimation of the future traffic. Therefore, the unexpected information can be treated as noise. To filter these noises, there are many existing methods. Within them, wavelet transform is one of the most efficient methods.

Wavelet transform has ma ny benefits when used in the de-noising application. It can retain the detail in the signal and does not need any assumption for the noise (compared with that, many other methods assume the noise is Gaussian distributed).

The basic de-noising procedure wavelet transform is described below: •

A deterministic "strong'' part is obtained by setting to zero all wavelet coefficients less than a certain threshold level. The inverse wavelet transform is used to calculate the corresponding time series.

•

A stochastic "weak'' part is obtained by setting to zero all wavelet coefficients greater than that threshold level. The inverse wavelet transform is also used here to calculate the corresponding time series.

•

New wavelet spectra are calculated for each partial time series.

The details of wavelet decomposition and de-noising will be discussed in the next section. The general de-noising procedure has three steps. The basic version of the procedure follows the steps described below.

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Paper revised from original submittal.

1. Decompose Choose a wavelet (here DB4 is used in this study), and then choose a level N (4 is used in this study). Compute the wavelet decomposition of the signal s at level N.

2. Threshold detail coefficients For each level from 1 to N, select a threshold and apply soft thresholding to the detail coefficients.

3. Reconstruct Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N.

After the de-noising procedure, the noises are eliminated. Therefore, the pattern can be recognized as the properties of signals. In general, the pattern can be identified from the “smooth” signal by specifying some characteristics such as the number of wave crest. By identifying the characteristics of the signal, the “standard” pattern is created for future use.

The next step is using a neuro-fuzzy network to identify the influence of special factors on the specific pattern. Consider data which are influenced by special factors: during the time period of special factors taking place, the characteristics of the signal are changed. The influence of the specific special factor can be identified by comparing the time series during the influenced time period and its “standard” pattern. Therefore, the

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Paper revised from original submittal.

influence can be recorded and can be used in future prediction when the situation of the same special factor happens.

The combination of special factors will result in a very large set. Therefore, it is impossible to record all these influences without letting the system become too huge. On the other hand, the influence of the same factor ma y be different from time to time, even to the same pattern. Therefore, it is important to balance them and give a reasonable estimate for the influences. Neuro-fuzzy network gives us the ability to reduce the system complexity and find convincing solutions.

Unlike common combined usage of neural network and fuzzy logic, the neurofuzzy network employed in this research implements fuzzy logic inside the neural network. In other words, the fuzzy logic is used in the hidden layer of the neural network. On the contrary, most combined models which use both fuzzy and neural network technologies either put the fuzzy logic before the input of the neural network or after the output of the neural network. The purpose of this design is to maximize the efficiency of the system: first, the neural network has the ability to self-study and gets the optimized result based on training data. Then, with the fuzzy logic used in the hidden layer of the neural network, the system has the ability to adjust the member function in the fuzzy set which can simplify the computation model without decreasing the efficiency of the system.

The discussion of the neuro-fuzzy network will be addressed in a separate paper. In this paper, the de-noising method will be discussed in the remaining part. The whole procedure is shown in Figure 1 and the data flow is shown in Figure 2.

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Paper revised from original submittal.

4. The Wavelet De-noising Method

In wavelet multiresolution analysis, the signal is investigated in different time scales and different solutions. On each level j, the signal is decomposed into two parts: an approximation and details. For example, at level 3, the original signal can be represented as: A0 = A1 + D1 = A2 + D2 + D1 = A3 + D3 + D2 + D1 ,

(1)

where A j is called j-level approximation, or the approximation at level j. D j is called j-level detail, or the detail at level j. It is noticeable that the original signal can be treated as the 0-level approximation. The approximations are represented by scaling functions φ i , j (t ) and details are represented by wavelets ψ i , j (t ) . Both φ i , j (t ) and ψ i , j (t ) are translated from the scaling function φ (t ) (also called father wavelet) and wavelet ψ(t) (also called mother wavelet), where φ j ,k ( t ) = 2 − j / 2 φ ( 2 − j t − k )

(2)

ψ j ,k ( t ) = 2 − j / 2 ψ ( 2 − j t − k )

(3)

It should be noted that φ = φ0, 0 andψ = ψ 0, 0 . Then, in the level J (or say: scale 2J) the time series can be written as:

x(t ) = ∑ a J ,k φ J , k (t ) + ∑∑ d j ,kψ j ,k (t ) k∈Z

(4)

j ≤ J k∈Z

Where the summation of a J , k is the approximation on scale 2J and the summation of

d j ,k represents the details of all scales. In the mean time, we have

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a j , k = ∑ hn*−2k a j −1,n

(5)

d j , k = ∑ g *n−2 k a j −1, n

(6)

n∈Z

n∈Z

Where hk is the scaling filter associated with the wavelet and g k is the wavelet filter associated with the wavelet, * represented the complex conjugates. The reconstruction can be represented as: a j −1,k = ∑ (hk −2 n a j ,k + g k −2n d j , k )

(7)

n∈Z

where +∞

hn = ∫ φ (t )φ −1, n (t )dt

(8)

g n = (−1) n h1*−n

(9)

−∞

Since in decomposition results, d j ,k mainly represent noises, the following method can be used as a threshold function: 0, T ( d j ,k , τ j ) = d j ,k − sign(d j , k )τ j

d j, k ≤ τ j otherwise

(10)

~ If the modified coefficients d j , k = T ( d j ,k τ k ) , then (7) becomes:

~ a~ j −1,k = ∑ (hk −2 n a~ j ,k + g k −2n d j , k )

(11)

a~J ,k = a J , k

(12)

n∈Z

with

There are several different methods for selecting the threshold. One of the most widely used methods was developed by Donoho and Johnstone (1995). In this method, the threshold is defined from this equation:

τ j = σ j 2 ln n j

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where σ j is the estimate of the standard deviation and n j is the number of coefficients

d j , k on level j.

5. Experimental Design

In order to effectively utilize the collected data and produce a convincing result, the experimental design of this study includes four parts: data collection, data standardization, data processing and result explanation. These steps are described as follows.

5.1. Data Collection

Data used in this study are retrieved from several websites. Several programs are developed to complete this project. These programs can retrieve real-time speed, construction, incident information and weather information from different cities.

5.2. Data Standardization

Data retrieved from data collection have different formats and it is not convenient to put them in a unique model. The standardization step is integrated by several programs which can read different data formats and output a standard format to the prediction model. The unique format will benefit the further processing in the whole framework.

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5.3. Data Processing

In this study, all data collected will be divided into three data sets: the training set, tuning set, and testing set. The training set will be used in the model to recognize the travel speed signal pattern, and to obtain the estimates of its parameters and special factor influences. The training set will include two third of all data that will be used to train the whole system. After the training procedure is completed, the tuning set will be applied to tune the results, which are obtained from the fuzzy-neural network. The testing set is used to verify the results and measure the performance of the prediction system.

5.4. Results Explanation

The result of the experiments can be described from different points of view. In this step, the result data will be analyzed to identify the advantages and disadvantages of the model.

6. Experimental Result

The data used in this paper are collected from Houston. Figure 3 shows the one-day speed profile on January 14, 2002. In this one day profile, it is obvious that the peak hour happened around from 6am to 9:30am. The speed dramatically went down during this time. However, lowest point (which represents the time with heaviest traffic jam) happened around 8:00am. The sampling period between every two points is 5 minutes. Therefore, we take the decomposition up to the level j=5, since 2 5 = 32 * 5 = 160 minutes is close to 3 hours and

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we are then able to study the components of the signal for which the period is less than 3 hours. The approximation of A5, which is corresponding to time scale of 160 minutes, is a crude estimation. It is easy to see that there is a lag for the estimation of the peak hour and the speed of the peak hour is overestimated. Most important information is missing in this level. Therefore, we should investigate lower levels. The result of the de-noising procedure is shown in Figure 5.

7. Conclusion

The framework developed in this paper provided a new procedure for short-time traffic prediction. This procedure is fast and easy to implement. It considered the tradeoff between computing time and the accuracy. The wavelet decomposition method employed in this paper is a fast and efficient way to smooth the historical traffic data and provide fully comprehensive input to the next step of the prediction framework.

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References [1] Adeli, H., and Karim, A., Fuzzy-wavelet RBFNN model for freeway incident detection, Journal of Transportation Engineering. v 126 n 6 Nov 2000, p 464-471, 2000.

[2] Adorni, G., Cagnoni, S., Gori, M., and Mordonini, M., Access control system with neuro-fuzzy supervision, IEEE – Conference on Intelligent Transportation Systems, Proceedings, -ITSC. n 01TH8585), p 472-477, 2001.

[3] Chou, J., O'Neill, W. A., and Cheng, H., Pavement distress evaluation using fuzzy logic and moment invariants, Transportation Research Record. n 1505, Jul 1995. p 39-46, 1995.

[4] Donoho, D. L., and Johnstone, I. M., Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, vol. 90, no. 432, p. 1200-1224, 1995.

[5] Dougherty, M., Kirby, H, and R., Boyce, Using neural networks to recognize, predict and model traffic. Artificial Intelligence Applications to Traffic Engineering, P.235-250, 1994.

[6] Farge M., and Philipovitch T., Coherent structure analysis and extraction using wavelets, in Progress in Wavelet Analysis and Applications. Éditions Frontières, Gif-surYvette, 1993.

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[7] Hayashi, K., Shimizu, Y., Dote, Y., Takayama, A., and Hirako, A., Neuro fuzzy transmission control for automobile with variable loads, IEEE – Transactions on Control Systems Technology. v 3 n 1 Mar 1995, p 49-52, 1995.

[8] Ishak, S. S., and Al, D. M., Fuzzy ART neural network model for automated detection of freeway incidents, Transportation Research Record -1634. Nov 1998, p 5663, 1998.

[9] Jou, I. C., Chang, C. J., and Chen, H. K., Hybrid neuro-fuzzy system for adaptive vehicle separation control, Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology. v 21 n 1 1999, p 15-29, 1999.

[10] Ledoux, C., An urban traffic flow model integrating neural network, Transportation Research 5C, P.287-300, 1997.

[11] Liu, J. N.K., Fuzzy neural networks for machine maintenance in mass transit railway system, IEEE - Transactions on Neural Networks. v 8 n 4 Jul 1997, p 932-941, 1997.

[12] Sayed, T., Razavi, A., Comparison of neural and conventional approaches to mode choice analysis, Journal of Computing in Civil Engineering. v 14 n 1 Jan 2000, p 23-30, 2000.

[13] Taylor, C., and Meldrum, D., Freeway traffic data prediction via artificial neural networks for use in a fuzzy logic ramp metering algorithm, Intelligent Vehicles

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Symposium, Proceedings. 1994, IEEE, Piscataway, NJ, USA, 94TH8011. p 308-313, 1994.

[14] Vukadinovic, K., Teodorovic, D., and Pavkovic, G., Application of neurofuzzy modeling: The vehicle assignment problem, European Journal of Operational Research. v 114 n 3 May 1 1999, p 474-488, 1999.

[15] Xu, H., Kwan, C.M., Haynes, L., and Pryor, J.D., Real-time adaptive on-line traffic incident detection, IEEE – International Symposium on Intelligent Control Proceedings. 1996, IEEE, Piscataway, NJ, USA, 96CH35855. p 200-205, 1996.

[16] Yin, H., Wong, S.C., et al., Urban traffic flow prediction using a fuzzy-neural approach, Transportation Research Part C, 2001.

[17] Yuan, Z., Li, W., and Liu, H., Forecast of dynamic traffic flow, Proceedings of the Conference on Traffic and Transportation Studies, -ICTTS. 2000, ASCE, Reston, VA, USA. p 507-512, 2000.

[18] Wernik A.W., and Grzesiak M., 1997, in Sadowski M. and Rothkaehl H. (eds.), Proc. Int. Symp. "Plasma 97'', Jarnoltowek, Space Research Center, Polish Academy of Sciences, Vol. 1, p. 391, 1997.

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Develop a Signal Processing Model (SPM) based on Historical Data

Apply the SPM Model to sample area

Develop a Factor Influence Model (FIM) to get the relation of the speed changes

Apply FIM Model in Sample Area

Calibration and Simulation Analysis of SPM and FIM

Reliability analysis of SPM and FIM

Figure 1. Overview of the Whole Procedure

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Create data signals

Wavelet Transform: Decomposition

Get level 3 approximation of base signal

Reconstruct the base signal

Compare target signal with base signal to get the influence data

Fuzzify the influence data and use Neural Network to train the data

Defuzzify the result

Figure 2. The Detail Dataflow of the Whole Procedure

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22:30 23:15

21:45

18:45 19:30 20:15 21:00

15:45 16:30 17:15 18:00

14:15 15:00

12:45 13:30

9:45 10:30 11:15 12:00

6:45 7:30 8:15 9:00

6:00

3:45 4:30 5:15

0:45 1:30 2:15 3:00

80 70 60 50 40 30 20 10 0 0:00

Speed

Jan 14 Inbound

Time Jan 14 Inbound

Figure 3. The One-Day Speed Profile for Houston Inbound on Jan/14/2002

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Figure 4. The Analysis of the One-Day Data

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Figure 5. The Result of Wavelet Decomposition

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