Highway Traffic Flow Nonlinear Character Analysis and Prediction

In order to meet the highway guidance demand, this work studies the short-term traffic flow prediction method of highway. The Yu-Wu highway which is the main road in Chongqing, China, traffic flow time series is taken as the study object. It uses phase space reconstruction theory and Lyapunov exponent to analyze the nonlinear character of traffic flow. A new Volterra prediction method based on model order reduction via quadratic-linear systems (QLMOR) is applied to predict the traffic flow. Compared with Taylor-expansion-based methods, these QLMOR-reduced Volterra models retain more information of the system and more accuracy. The simulation results using this new Volterra model to predict short time traffic flow reveal that the accuracy of chaotic traffic flow prediction is enough for highway guidance and could be a new reference for intelligent highway management.


Introduction
The prediction of highway traffic flow is a crucial part of transportation planning, traffic control, and intelligent transportation systems [1].In particular, short-term traffic flow predicting can provide real time information for highway guidance.Also accurate predictions of traffic flow are essential part of efficient operations in advanced traffic management systems.In recent years, the short-term predicting technologies have gotten the attention of many researchers.
Short-term traffic flow is a typical nonlinear time series in a time horizon of 15 min or less [2].During the decades a variety of techniques have been applied to forecast the shortterm traffic flow, such as fuzzy theory [3], neural networks [4], Kalman filter [5], and wavelet analysis [6] et al.But the models generated by these methods could not capture some strongly nonlinear characteristics of short-term traffic flow data.The prediction results may have big error.Chaos theory is an effective tool to study nonlinear system.This tool has been applied to forecast short-term traffic flow time series [7,8].The forecasting results are better than other methods applied before.However, the accuracy of chaotic forecasting result is not enough for real time highway guidance.There remains room for proposing new prediction technology to make the prediction more accurate.
To use chaos theory we should judge whether the traffic flow data is a chaotic time series or not.The largest Lyapunov exponent is usually used to achieve this goal.Wolf algorithm [9], Jacobin algorithm [10], and small data method [11] are three algorithms often used to calculate Lyapunov exponent.If the traffic flow data is a chaotic time series, the phase space reconstruction theory could be applied to predict the traffic flow.
In this paper, the traffic flow data of Yu-Wu highway in Chongqing, China, is taken as study object.We study the general day traffic flow character of highway and the holiday traffic flow model is not contained in this paper.Firstly, the approved small data method is used to judge whether the traffic flow data is a chaotic time series.Secondly, phase space reconstruction theory is used to reconstruct the traffic flow system.Thirdly, a new predicting model based on model order reduction via quadratic-linear systems is used to predict the chaotic traffic flow.This new prediction model has no truncation error and contains more information of original system.It is more accurate than other prediction models ever used before.

Traffic Flow Data Phase Space Reconstruction
2.1.Traffic Flow Data.All the traffic flow data are collected at Yu-Wu highway in Chongqing, China.Sampling time begins at 0:00 November 20, 2013, and ends at 0:00 November 27, 2013.Time interval is 5 minutes.Figure 1 shows the traffic data.

Mutual Information Method.
Takens theorem [12] is the mathematical base of time series phase space reconstruction.Choosing optimal time delay  and embedding dimension  is the key of phase space reconstruction.Mutual information method is an effective tool to calculate .
Assuming chaotic time series  1 ,  2 , . . .time delay is  and embedding dimension is .The reconstruction phase space  of this time series is where   () = ( + ).The average information of variable  is called information entropy which can be described as where   (  ) is the probability of event  in .Let [, ] = [ () ,  ( + )], if system (, ) is a coupling system and  is given as   the uncertainty of  is where  | (  |   ) is the probability that a measurement of  yields   , given that the measured value of  is   .Given that  has been measured at time  the average uncertainty in a measurement of  at time  +  is where (, ) is the uncertainty of  in isolation.( | ) is the uncertainty of  in the measurement of .The amount of a measurement of  reduces the uncertainty of .The mutual information is

𝐼 (𝑈, 𝑅) = 𝐻 (𝑈) − 𝐻 (𝑈 | 𝑅)
=  () +  () −  (, ) =  (, ) . ( The mutual information is a function of   which is the joint distribution probability of the event  and .If a vector is a phase space reconstruction while the mutual information  first reaches the minimum the time delay could be the delay time of phase space reconstruction.
Figure 2 shows the traffic flow delay time calculated by the mutual information method.When  = 2 the system reaches the minimum so the optimal delay time  = 2.
For time series data from a random set of numbers,  1 () will never attain a saturation value as  increases in principle.But in practical computations, it is difficult to resolve whether  1 () is slowly increasing or has stopped changing if  is sufficiently large.In fact, since available observed data samples are limited, it may happen that  1 () stops changing at some dimension although the time series is random.Therefore, we define For random series,  2 () = 1.For deterministic series, there must exist  making  2 () ̸ = 1. Figure 3 shows the embedding dimension calculated by CAO method.The optimal embedding dimension  = 10.

Chaotic Character of Traffic Flow
Correlation dimension and largest Lyapunov exponent are two main indexes to distinguish chaotic system from others.

Correlation Dimension.
G-P algorithm is the most popular method to calculate correlation dimension for its simplicity.Figure 4 shows the ln (, )-ln  curves.These curves got through increasing embedding dimension until the slope of the curve's linear part is almost constant.Then, the correlation dimension estimation can be obtained using least square regression.
Through Figure 4 we can get the correlation dimension () = 5.4116; its fractal reveals that the system is a chaotic system.

The Largest Lyapunov Exponent of Traffic Flow.
In the reconstructed phase space the distance between each point   and its nearest neighbor point    is expressed as where  is the mean period.Based on the study in [14] the largest Lyapunov exponent is defined as where Δ is sampling period and   () is the distance between the th pair of nearest neighbors after  discrete-time steps.Sato also gives an alternate form of ( 13) where  is constant.Combining ( 13) and ( 14) yields By taking the logarithm of both sides of above equation, we obtain ln   () = ln   +  1 (Δ) . (16) The largest Lyapunov exponent equals the slope of ( 16).
We can obtain the largest Lyapunov exponent of traffic flow data  = 0.1760.Its a positive value which means the system exist chaos phenomena.

The Volterra Prediction Model
For single-input-multiple-output system in the form of where  ∈   are the state variables,  =  cos(),  ∈  is the input signal.Assume an autonomous system as follows: , V ∈  2 .The system matrix  could be decomposition is unitary; the close form solution to (18) is Also the input  =  cos() can be taken as Combining ( 17)-( 21) yields If the eigenvalue   () −   () ̸ = 0, ( = 1, 2, . . ., ,  = 1, 2), the Sylvester equation has unique solution.Then let   denote the th order identity matrix; we get from which the solution to ( 22) is Equation (25) shows the system solution dependent on initial condition  (0)−ΠV (0) and steady state ΠV().If (0) = ΠV(0) the system is driven into period steady state regardless of the fact that  is stable or not.
The study [14] reveals that nonlinear ordinary differential equations can be rewritten equivalently in a special representation, quadratic-linear differential algebraic equations (QLDAEs).The QLDAEs take the form as Similarly, any generalized polynomial system can be converted into generalized QLDAEs as follows: The system (27) is equivalent representation of the original system; this step makes no approximation of original system.Based on the Volterra theory that the response of a nonlinear system can be decomposed into responses of a series of homogeneous nonlinear systems, the solution of system can be written as follows: where . ., and ( −   ) 1 , . . .,   .ℎ  ( 1 , . . .,   ) is th order Volterra kernel.If the input of system (17) is  the response of system is Setting  = 1 and substituting (29) into (26) we can get and so on.Substituting (21) into (30) the steady state solution of system is  1 () = Π 1 V(), and Π 1 ∈  ×2 solves the Sylvester equation A unique Π 1 always exists since the eigenvalues of  1 and the imaginary eigenvalues in  never add to zero.When the initial condition  1 (0) = Π 1 V(0) and input  =  cos() the steady state solution of system is  1 () = Π 1 V().Therefore, the time argument in the steady state solution can be omitted without ambiguity; namely, Moreover, the derivative of V B is Equations ( 34) and (35) could construct another Sylvester equation like (23) with unknown Π 2 ∈  ×4 .The new Sylvester equation is The steady state solution of (36) is 1 , the third Sylvester equation is The steady state solution of (37 Based on the analysis we could get the pattern of the steady state response   = Π  V u , Π  ∈  ×2  ,  = 1, 2, . .., and the steady state solution of system is The initial condition (0) = ∑ ∞ =1   (0).Equation ( 38) is exactly the same desired solution sought by SN algorithm.
In order to simplify the algorithm programing, the traffic flow data is normalized as follows: where () is the normalization time series, () is the original time series, () is average value of original time series, and Max[()] and Min[()] are the maximum and minimum value of original time series.

Simulation Results
The effectiveness of the Volterra prediction algorithm is verified by the traffic flow shown in Figure 1.The 1500 data of 2500 traffic flow data are used to train the Volterra algorithm.The last 2 hours and 4 hours traffic data of November 27, 2013, are used to verify the effectiveness of the prediction algorithm mentioned in this paper.Also comparing with RBFNN prediction algorithm and old Volterra algorithm, the results show that the accuracy of new Volterra algorithm is better than RBFNN algorithm and old Volterra algorithm.The prediction error is shown in Table 1.The new Volterra

Mathematical Problems in Engineering
Real traffic flow data RBFNN method predicting data

Figure 2 :
Figure 2: Mutual information method calculates delay time.

Table 1 :
Error of two algorithms.