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Traffic Flow Characteristics mining of Time Series Data Based on Phase Space Reconstruction Using Complex Networks

Traffic flow time series is vital for mining the traditional statistical characteristics by using the theory of statistics and machine learning when its identity is a special time series. The network analysis of the traffic flow time series, who uses the complex network of time series analysis method, is designed to inquire into the special law of traffic flow time series which uses its visualization characteristics. Through the network analysis of traffic data flow, the connotation of traffic data flow can be revealed, and the relationship between all data and some data can be further studied. Therefore, it is constructed by combination with the phase space reconstruction theory. The phase space trajectory may be squeezed and the structure of attractor may change. We need to use C-C method to estimate the time delay according to the characteristics of integral parameters, and use G-P algorithm to estimate the embedding dimension to avoid it. This study can effectively reveal the motion law of the system. After constructing the complex network of traffic flow time series with various traffic parameters, the degree distribution, clustering coefficient and modularization of the representative critical threshold corresponding network are statistically analysed. The analysis results show that the new networked structure of traffic flow time series proposed in this study has strong advantages, and its core is phase space reconstruction, which can well reflect the information space of traffic dynamic fluctuation. The time series networking method based on phase space reconstruction has become a new approach to inquire into the characteristics of traffic flow time series. The degree distribution of the actual multi-traffic parameter time series construction network satisfies the characteristics of a Gaussian distribution. Their average clustering coefficients have attenuation characteristics, and their modularization degree is obvious.


I. Introduction
The research of traffic flow time series has always been in the basic position in the dynamic analysis of traffic networks. With the increasing demand for transportation, the demand for intelligent transportation is also increasing. Therefore, it is necessary to deeply mine the temporal features in the traffic system. Previous studies on traffic flow analysis principally included traffic flow prediction as well as nonlinear analysis. In terms of prediction, we mainly focus on the application of statistical methods and machine learning methods to analyse traffic flow [1][2][3]. Previously, nonlinear analysis, researchers basically utilize chaos theory will investigation the nonlinear characteristics for traffic flow sequences. Traffic flow has many influencing factors and dynamic characteristics. How its nonlinear characteristics affect the traffic state has become a deep-seated problem in the field of transportation engineering. The research of time series tends to use spatial structure to analyze time series relationship. Reference [4,5] lists three methods, which can construct complex networks from time series: cyclic time series construction method, visual method and phase space reconstruction (PSR) method. The method to construct the periodic time series is mainly applied in the field of life sciences. Reference [6] improved the method of mapping heart rhythm time series to complex networks and found the invisible sequence structure in traditional The network analysis of traffic data flow can present the connotation of traffic data flow, grasp the relationship between the whole and part of the data and excavate the hidden rules in big data to reduce the frequency of malignant traffic accidents and road congestion rates. Unlike the traditional methods of studying the traffic flow sequence, based on the complex network method, it opens a new perspective for exploring the characteristics of the traffic flow timing data. This paper analyzes the traffic flow in view of the aspect of phase space, and obtains the traffic characteristics from the characteristics of time sequence structure. Phase space reconstruction (PSR) based on time series provides a new analytical approach for traffic engineering.

II. Time Series Network Method Based on PSR Theory
As one of the methods to construct complex networks in time series, the complex network structured by view method retains some features of the original time series [14][15][16], but breaks the time series relationship of the original sequence. In addition, since the construction characteristics of the periodic method are mainly applied in the field of life sciences and the characteristics of various methods to construct complex networks in the time series are considered, this study selects the PSR theory to study the time series of traffic flow. PSR theory recommends how to recover the attractor of dynamic system to determine the time series, which stretches and folds the trajectory in high-dimensional space, and analyzes its dynamic properties [17][18][19]. The most widely used PSR method at this stage is the time delay state space reconstruction method. In a one-dimensional time series, the timedelay embedding method is used to construct a m-dimensional phase space by embedding dimension m and the delay time τ. Equation (1) is shown. Each point of the multidimensional state space is correspondingly mapped by a vector composed of delay points. A new approximate phase space can be rebuilded by repeating multiple times. The dimension of phase space is determined by the number of delay points of time series. This method is an extension of Takens theorem, i.e., that is to say, the condition of recovering the corresponding space is to find an appropriate embedding dimension m, which is also the premise of PSR. ( 1) { , ,..., }, 1, 2,..., Selecting the appropriate delay time τ and embedding dimension m is the core of the method application. According to the assumption of Takens theorem, an infinite one-dimensional time series without noise-free leads to the arbitrariness of τ and m.The premise of the analysis is that the infinite length and noise-free time series are not consistent with reality. The key to the high quality and good application effect of chaotic time series is to apply reasonable time delay and embedding dimension. At present, the main methods to estimate the time delay parameters include the complex autocorrelation method, mutual information method and C-C method [16,20]. For the embedding dimension m, if m is sufficiently large, the motion law of the chaotic system can be revealed. However, an m that is too large will cause many operations, so the optimal embedding dimension should be the minimum value that satisfies Takens theorem.

Network model of traffic flow timing data
According to aforementioned PSR theory, a time series phase space is reconstructed to construct a network. The core of the construction is to determine the three elements of the network: the nodes of the network, edges between nodes, and connection relationship between nodes. Reconstructed sequence Y i is used as the node, and the connection relationship between the nodes determined by reconstructed sequences Y i and Y j is determined by correlation coefficient C ij between reconstructed sequences Y i and Y j . The correlation coefficient is defined in formula (2).
In the formula, L is the dimension of the vector, The connection relationship of network node is determined by the relationship between correlation coefficient C ij and connection threshold r c . If correlation coefficient C ij is within the threshold control range, the nodes determined by reconstructed sequences Yi and Y j have edge associations; otherwise, the node determined by reconstructed sequences Y i and Y j have no associated edges. Thus, network adjacency matrix M is determined, and the composition of the association matrix is shown in formula (3).
Considering that the main parameters of the estimated traffic state are the traffic flow, time occupancy rate and regional average speed obtained from traffic detection, the actual time series of the traffic flow, time occupancy time series, and regional average speed obtained through data preprocessing are used. The time series is networked. The construction process of traffic flow time series network will be illustrated by taking time series of traffic flow as an example. The following figure can directly and clearly illustrate the method of constructing the traffic flow time series network based on PSR, as shown in Fig.1.

Parameter estimation for PSR
Because the time series obtained in the actual detection must have errors and noise, the selection of the time delay must prevent the phase space trajectory from being too small and avoid structural changes of the attractor caused by a time delay that is too large. Among various methods to estimate the delay-time, the C-C method proposes the concept of related integrals and estimates the time delay based on the integral parameters. It is vital to determine the parameters of phase space reconstruction of series, which will be determined by C-C method.
Based on the principle that the divided time series cannot intersect, t time series will be divided, denoted by . Compute the test statistics for ( , , , ) S m N r t to estimate time delay t. First, define ( , , , ) C m N r t , which mainly counts the number of phase space vector distances in the vicinity of the radius, as shown in formula (4).
In the formula, ()  is the Heaviside step function, which is ;m represents the embedding dimension under the premise of phase space; N denotes the length of time series; The number of vectors in phase space is represented by ( 1) M N m t    ; R denotes the neighborhood radius.We define ( , , , ) S m N r t as the test statistic of the time series, which is shown in formula (5). The first zero point of ( , , , )S m N r t t , which can intuitively reflect some characteristics of sequence or time point, primarily autocorrelation with the smallest amount of appropriate distance within radius r, is taken as the most time-delayed point. At present, the points representing the rebuilded phase space are nearest to a symmetric probability distribution, that is, uniform distribution. Therefore, the difference between maximum and minimum radii is selected as the statistical quantity, as shown in formula (6) In the formula, () S C  denotes the correlation score obtained after the S-th subsequence of the experiment.
( , ) max{ ( , , )} min{ ( , , )}, In the formula, the mean value of , and the average value of ( , , ) For the embedding dimension m, if m is sufficiently large, the motion law of the chaotic system can be revealed. However, if m is too large, it will cause many calculations.Therefore, the determination of optimal embedding dimension is consistent with the purpose of Grassberger and procaceia G-P algorithm. Inspired by this, this method is used to obtain the the minimum value satisfying Takens theorem.
Before calculating the correlation dimension, first, we define the correlation integral, as shown in formula (8).

IV. Case Analysis
The data source of the experiment was selected from the traffic demonstration area of Xiamen City on January 24, 2015. The multisource traffic detection data (including induction coil detection data, geomagnetic detection data, and bayonet video detection data) on typical roads in the period from 00:00 to 23:00. The time series of multiple traffic parameter flows is shown in Fig. 2.

Characteristic analysis of experimental data
The main function of phase space reconstruction is to recover the attractor of dynamic system in high-dimensional space, so it is also a key method to analyze nonlinear time series. With the participation of time delay estimation  and embedding dimension m, the actual traffic flow sequence can be reconstructed in phase space.According to the phase-space reconstructed traffic flow time-series network construction process, connection threshold r c determines the network structure to be constructed. By comparing the network matrices with different thresholds, it can be intuitively found that a higher the threshold value corresponds to more prominent highly associated nodes, and the research finds that a reasonable connection threshold helps analyse the original clustering characteristics of the network.
ISSN: 0010-8189 © CONVERTER 2020 www.converter-magazine.info  The degree distribution of network nodes is analysed for each traffic flow time series to determine the threshold range of each type of traffic parameter time series network feature analysis. The network node degree distribution map of the traffic flow time occupancy sequence with different thresholds r c is selected, as shown in Fig. 7. Select the network node degree distribution map of the traffic flow time occupancy sequence with different threshold r c , as shown in Fig. 8. Select the node speed distribution of the average speed sequence network with different thresholds r c , as shown in Fig. 9.
The degree distribution of network nodes is analysed for each traffic flow time series to determine the threshold range of each type of traffic parameter time series network feature analysis. The network node degree distribution map of the traffic flow time occupancy sequence with different thresholds r c is selected, as shown in Fig. 7. Select the network node degree distribution map of the traffic flow time occupancy sequence with different threshold r c , as shown in Fig. 8. Select the node speed distribution of the average speed sequence network with different thresholds r c , as shown in Fig. 9.
In the traffic flow time occupancy sequence network, the fitting of the node degree distribution of the network with the threshold r c = 0.60 conforms to the Gaussian distribution; the fitting line of the node degree distribution is y 1 =0.0167*exp(- ((d 1 -128

V. Conclusions
The complex network is one of the most vital ways to analyze traffic flow sequence, which is different from the traditional traffic flow sequence analysis method. The major challenge of this study is how to deconstruct the time series to get its statistical characteristics. We use the phase space method to solve this problem. The chief idea of this method is to analyze the traffic time series network under different thresholds from the actual road network. The experiment provides a valuable perspective for studying the characteristics of traffic flow time series. This paper aims to reconstruct the time series to explore the attributes of traffic flow and determine the optimal critical threshold.
(1) Considering the phase space reconstruction requirements of the single-parameter traffic flow time series, the time delay and embedding dimension m of the three types of traffic parameter time series in the example are estimated. The network characteristics statistics (such as the modularity of the network structure, the average clustering coefficient, and the degree distribution) are performed on the PSR network of the three types of traffic ISSN: 0010-8189 © CONVERTER 2020 www.converter-magazine.info 160 parameters with reasonable estimated parameters.
(2) The new structure which is reconstructed by the network phase space of traffic flow time series is the information space in traffic flow dynamics. The case analysis shows that the degree distribution of the traffic flow time series network with different parameters has Gaussian distribution characteristics, and the average clustering coefficient has attenuation characteristics and high modularity characteristics. At this stage, phase-space reconstruction is used to perform network analysis on the traffic flow time series. Subsequent research will explore the relationship between the visual method and this method to build a traffic flow time series network to obtain a deeper understanding of the traffic flow state.