A Multimode Dynamic Short-Term Traffic Flow Grey Prediction Model of High-Dimension Tensors

Short-term traffic flow prediction is an important theoretical basis for intelligent transportation systems, and traffic flow data contain abundant multimode features and exhibit characteristic spatiotemporal correlations and dynamics. To predict the traffic flow state, it is necessary to design a model that can adapt to changing traffic flow characteristics. Thus, a dynamic tensor rolling nonhomogeneous discrete grey model (DTRNDGM) is proposed. This model achieves rolling prediction by introducing a cycle truncation accumulated generating operation; furthermore, the proposed model is unbiased, and it can perfectly fit nonhomogeneous exponential sequences. In addition, based on the multimode characteristics of traffic flow data tensors and the relationship between the cycle truncation accumulated generating operation and matrix perturbation to determine the cycle of dynamic prediction, the proposedmodel compensates for the periodic verification of the RSDGMand SGMgrey predictionmodels. Finally, traffic flow data from themain route of Shaoshan Road, Changsha, Hunan, China, are used as an example.The experimental results show that the simulation and prediction results of DTRNDGM are good.


Introduction
Intelligent transportation systems (ITSs), which are traffic control management information systems, have been developed over the past few years.Traffic control systems can provide accurate real-time traffic information for traffic management and traffic guidance systems and consequently represent important components of intelligent traffic control networks.Short-term traffic flow predictions based on realtime information constitute preconditions for implementing real-time traffic control and management, and thus, they form an important theoretical basis for ITSs.Accordingly, traffic data provide an important foundation for the research and development of ITSs, and such data contain rich multimode features.However, determining how to fully use the multimodal features of traffic data under a unified framework remains a challenge.
Tensor models are generalizations of both vector models and matrix models as multidimensional models.In recent years, the use of tensors to represent multidimensional data with multimode features [1] has been shown to overcome the deficiencies of vector and matrix data forms with which it is difficult to characterize multidimensional features [2].However, the multimode information of traffic flow can be analyzed in a tensor framework.The multiple modes can reflect the characteristics of the traffic system in many respects, although traditional time series and matrix modeling analysis methods are limited when analyzing the features of traffic data in multiple modes (e.g., at various temporal and spatial scales) in a unified framework.By constructing a "week-day-time" model of traffic data, better results can be obtained than those yielded by vector and matrix data formats [3][4][5][6][7].This approach is conducive to maintaining the structure and characteristics of traffic data, which can be observed at different spatial and temporal scales and exhibit remarkable interconnected patterns.Accordingly, researchers have found that traffic data can exhibit multiple strong correlations among these patterns [8][9][10][11].Additionally, the inherent correlations in traffic data can be analyzed according to the multimode tensor characteristics of traffic, 2 Complexity and the potential trends of traffic data can be determined to provide a basis upon which traffic systems process traffic data.
To date, many forecasting methods have been used to predict resource problems.These methods are divided into three categories: short-term traffic flow prediction techniques based on vector data flow (such as the wavelet analysis [12], support vector machine [13], the chaotic prediction model [14,15], and neural network [16,17]), short-term traffic flow prediction techniques based on matrix data flow [18][19][20][21] (such as the multivariate time series prediction model [18] and the Kalman filtering method [20]), and short-term traffic flow prediction techniques based on tensor data flow (such as the seasonal autoregressive integrated moving average + generalized autoregressive conditional heteroscedasticity (SARIMA + GARCH) model [22] and seasonal selfvector regression (Seasonal-SVR) prediction model [23]).The abovementioned prediction models are usually based on a large sample size and thus cannot be used to solve smallscale problems.However, short-term traffic flow prediction, which is employed to forecast the traffic flow state in a future period, is based on the dynamic data of road traffic flow states.The time interval and prediction period of these data are generally within 15 minutes.Consequently, the traffic flow data in the preceding time period will have the greatest impact on the subsequent period.The sampling rate also has a considerable impact on the prediction; for example, if the interval is one hour and the sampling interval is 5 minutes, only 12 groups of data will be collected in one hour, which constitutes a small sample of data.In contrast, the grey model (GM) involves a small sample size and very poor information, while the prediction model is adaptable and capable of more effectively handling changes in parameters.
Since grey theory was initially proposed in 1982 [24], the grey prediction model has formed the core component of grey system theory, and grey theory has been widely investigated.Accordingly, the development of GM (1,1), which constitutes the core grey prediction model, has been constantly improved and optimized [25][26][27][28][29][30].This model has been widely applied in various fields [31][32][33][34][35][36][37][38], among which short-term traffic flow prediction is particularly important.For example, Hsu et al. [39] proposed an adaptive GM (1, 1) model for traffic prediction in nondetector intersections.In addition, Wen et al. [40] used the GM (1,1) model to predict air traffic flow and showed that the prediction capability of the GM (1,1) model is better than those of the ARIMA and multiple regression models through an experimental comparison.Guo et al. [41] established a grey nonlinear delay GM (l, l) model to predict short-term traffic flow on urban roads, and Lu et al. [42] used the nonlinear grey Bernoulli equation to obtain a grey prediction model for traffic flow prediction and achieved good results.Furthermore, Mao et al. [43] constructed a grey triangular GM (l, l) model to predict traffic flow fluctuations, while Xiao et al. [44] proposed a seasonal grey GM (1,1) rolling prediction model based on the cycle truncation accumulated generating operation (CTAGO).Yang et al. [45] established a coupling model with a close grey relational degree for weight determination based on the ARIMA model and the seasonal grey DGM (1,1) rolling model that obtained time sequence data and cross-section data at the intersection point.Bezuglov and Comert [46] used the Fu Liye error to improve the coupled prediction using the GM (1, l) and grey Verhulst models and applied the combined model to traffic flow prediction.Additionally, Ren et al. [47] established an improved grey GM (1,4) model for road traffic safety predictions in Germany.
The abovementioned grey models, which are based on either vector data flow (i.e., single-variable prediction models) or matrix data flow (i.e., multivariable prediction models), are used to predict short-term traffic flow.Unfortunately, it is often difficult to characterize traffic data in a unified framework with these linear or planar representations due to the limitations of data dimensionality (e.g., space, week, day, and time).Moreover, the prediction of future short-term traffic flow needs to take advantage of data collected in real time, and it requires traffic data that are acquired continuously (i.e., rolled over from preceding prediction periods) to predict traffic flow during future periods.In addition, data such as social network data streams and video and network data streams are often dynamic and have multiple modes.Consequently, researchers often construct these data sets as dynamic tensors.Dynamic tensors are used to characterize dynamic data with multiple patterns; furthermore, they can retain the original multimode characteristics of the data set and reflect the dynamic characteristics of the data.Dynamic tensors have achieved good results in practical applications.
Therefore, to better extract the multimode characteristics (e.g., the week, day, and time) of traffic data, this paper presents a rolling approximate nonhomogeneous discrete grey model (DTRNDGM model) based on the characteristics of dynamic tensor data and the characteristics of traffic flow data.The purpose of this paper is to utilize the dynamic tensor.Under the proposed framework, we can fully exploit the multimodal and dynamic characteristics of traffic data and improve the accuracy of short-term traffic flow forecasting.This model can fully employ the characteristics of real data; accordingly, affine transformation is utilized to study the unbiased prediction of an approximate nonhomogeneous exponential sequence by using the proposed model and to verify that the proposed model is completely fit for nonhomogeneous exponential sequences.The information priority principle and the optimal rolling cycle according to a correlation analysis of the "week-day-time" model are used in dynamic tensor mode, thereby promoting the accuracy of the rolling cycle of the model composing the rolling grey SGM model [44] and the RSDGM [45] model cycles through data testing and promoting a model with a higher accuracy through calculations and predictions.Therefore, the research results in this paper are of great significance for the data structure of traditional grey prediction models and for improving their performance.
In the following chapters, we use abbreviations for different grey prediction models; these abbreviations and their meanings are listed in Table 1.
The remainder of this paper is arranged as follows.In the third chapter, the rolling nonhomogeneous discrete grey model (RNDGM) is established.In the third chapter, the multimode DTRNDGM is established.In the fourth chapter, the dynamic cycle of the new model is studied with the

Rolling Nonhomogeneous Discrete Grey Model (RNDGM)
In this section, a rolling NDGM is constructed based on the periodic truncated accumulating sequences of the original sequence, and the properties of the model are studied.
Because the NDGM model has a good prediction effect on the approximate nonhomogeneous exponential series, an affine transformation is performed to investigate the parametric properties of the RNDGM, which can fully fit the sequences of nonhomogeneous exponential growth.The rolling NDGM model has the same effect on the approximate nonhomogeneous exponential series.
Lemma 5 (see [44]).e primary accumulated sequence of CTAGO transform sequence  (0) () of the original sequence  (0) () is associated with a grey index trend and has a positive grey index value of .
(2) is known from the definition of the sequence. ( The least squares estimation from Definition 6 can directly get Property 9.
In this section, the parameter characteristics of RNDGM starting from the affine transformation are investigated.Then, according to the intrinsic links among the model parameters and the sequence forms, the simulation prediction accuracy of the model is determined.The specific properties of this method are given by the following theorem.

Dynamic Tensor Rolling Nonhomogeneous Discrete Grey Model (DTRNDGM)
In practical applications, multimode correlations exist among traffic data at various scales.Thus, multimode correlations of traffic data can be used to construct different traffic data tensor models, which should fully consider the multimode information associated with traffic data.
. .Establishing a Traffic Data Tensor Model.Because traffic data exhibit strong spatiotemporal correlations, multiple traffic flow data tensor models can be constructed in a combined model.For example, the traffic volume of a certain road segment can be expressed as tensor data in a "collectorweek-day-time" model as follows: ∈  퐿×푊×퐷×푇 ,  = 11,  = 11,  = 7,  = 288 (33) where L represents a loop/collector (i.e., the model has 11 collectors), W represents the number of weeks of data, D represents the number of days with data per week, and T represents the time.In this case, 288 traffic flows per day are analyzed.
If the above model is composed of traffic data collected by a single collector, it can be expressed in a "week-daytime" format as a third-order tensor:  ∈  푊×퐷×푇 .If the tensor  = 288 = 24 × 12 described above includes data collected every five minutes over the course of a day (i.e., 12 times an hour), then a fifth-order tensor  ∈  11×11×7×24×12 model can be constructed.For a time series, different tensor models can be constructed based on different acquisition times.For example, for data collected once every 10 minutes over the course of a day, a fifth-order tensor  ∈  11×11×7×24×12 model can be built.The time series can also be expanded to two or three weeks.To intuitively describe the traffic flow tensor model, a typical third-order tensor can be constructed based on a combination of transverse time series, longitudinal time series, and spatial time series.A schematic diagram for a third-order tensor is shown in Figure 1.Thus, different tensor models can be constructed according to different requirements.
. .Establishing DTRNDGM.As discussed above, traffic data exhibit multimodal correlations.The correlation of the weekly model is the highest in the traffic tensor model.However, the data for the RSDGM require that the length of the data sequence be kept constant (i.e., when new data are introduced, old data must be eliminated).Replacing old information with new information allows a one-step dynamic rolling prediction with the grey model.Furthermore, the length of the model is closely related to the prediction accuracy, and the correlation of the cycle model in the traffic tensor model is the highest.Accordingly, a higher precision is associated with a higher correlation when the cycle length is During the prediction and simulation processes with DTRNDGM, real-time observations of traffic flow data over an interval of  = 2 affect the simulation and prediction accuracy of the model.According to the strongest correlation of the tensor data, we can select a cycle length of  = 7.If parameter  is small, then the resulting lack of information will distort the prediction.In contrast, if parameter  is large, then it may cause data redundancy, following which an optimal prediction cannot be obtained.From the characteristics of the model itself, we will illustrate the prediction effect of parameter  and introduce new information priority and modeling concepts derived mainly from the perturbation [44,45] of the matrix.First, we introduce several related lemmas [48].
To satisfy the new information priority principle of the grey model,  = 2 − 1.
The following subsection discusses whether the corresponding weights of the original data in DTRNDGM satisfy the weight priority principle of the data corresponding to the previous period.The following theorems are presented.
Because x(0) (2) = ŷ(0) ( + 1) −  (0) () +  (0) (), Theorem 17 shows that, in the case of the same disturbance, the parameter perturbation bounds are the largest when the time data in  (0) () correspond to the time data in the previous period of x(0) (2), indicating that the influence of the parameter estimation in  (0) () is greater and can be understood as the maximum weight of the data.In brief, DTRNDGM considers the new information priority principle of the grey model and the periodic correlation of traffic flow data.Therefore, the number of original data points in the sequence is kept unchanged at  = 2 = 14, and new data are introduced continuously while old data are excluded.The DTRNDGM process can be described as follows.
Step .Select the tensor model based on the raw data.
Step .Construct matrices B and Y and compute parameter values.
Step .For the first time, model and calculate the above sequence, and then use DTRNDGM to predict x(0) (2 + 1).
Step .Repeat Step 3 for all data points that need to be predicted.
Step .Calculate the error of the simulated predictions.Substitute the values in Steps 2-5 to calculate the simulated and predicted values.The sequences x(0) () and  (0) () constitute the raw data of Step 1.
The above rolling prediction process is shown schematically in Figure 2.

An Empirical Analysis of RNDGM in the Tensor Framework
. .Data Analysis Using the Traffic Tensor Model.The data used in this study were obtained from the Institute of Urban Transport, School of Transportation Engineering, Central South University [49].The data were collected from the four straight lanes (from south to north) of Shaoshan Road in Changsha, China, by collectors every five minutes.The traffic flow data were acquired from 8:00 to 9:00 for 22 days from October 14, 2013, to November 4, 2013.The details are presented in Table 2.
(I) Data Processing.Obtain the original sequence  (0) () based on the characteristics of the model data: The CTAGO sequence  (0) () is as follows: = (9333, 9300, 9243, 9181, 9170, 9011, 9110, 9220) Similarly, the 1-AGO sequence of CTAGO sequence  (1) () can be written as follows: The above modeling process is repeated to forecast the sequences x(0) (8), x(0) (9), . . .x(0) (21).(MRSPE), also known as the mean relative prediction percentage error (MRPPE), is calculated as follows: where () is the value of traffic flow, x() is the predicted value of traffic flow, and N is the number of data points.The predictive values and errors for traffic flow based on DTRNDGM, RSDGM, SGM, and NDGM are given in Table 3.
For the third week of the data shown in Table 3, with regard to the predictions during this period, the accuracy of DTRNDGM is much better than those of RSDGM [45], SGM [44], and classic NDGM.Moreover, the rolling predictions of RSDGM and SGM yield similar simulation results; the simulation results of RSDGM and SGM are consistent, and these models perform best in simulations of exponential sequenced data.According to the prediction for November 4, DTRNDGM performs much better than the other three models.Based on Table 1, the absolute MRSPE and MRPPE values for the above four models of traffic flow and the real data are illustrated in Figures 4(a)-4(d).
Figure 4 reveals that the simulation and prediction performances of DTRNDGM, RDGM, SGM, and NDGM are the best among the three models, which demonstrates that the optimization and reform of DTRNDGM, RDGM, SGM, and NDGM are effective.
To determine the prediction capabilities of DTRNDGM with the three tensor data models ( As presented in Tables 4-6, the DTRNDGM results are the best among the three tensor models for two-day rolling prediction.Additionally, the prediction results of RSDGM, SGM, and NDGM are extremely similar, thereby illustrating the consistency of the predictions of these models and the factors that influence each model.DTRNDGM is therefore better for simulating nonhomogeneous sequences.

Conclusions
The multimode traffic flow data prediction technique proposed in this paper is characterized by a high-dimensional tensor.According to the temporal correlations of the traffic flow data and the nonhomogeneous exponential properties and dynamics of the data, a dynamic tensor rolling nonhomogeneous discrete grey forecasting model (DTRNDGM) suitable for multimode state traffic flow data is proposed.Accordingly, the properties of the model are studied, and a comparative experiment is performed.The following points summarize the findings of this research: (1) According to the approximately nonhomogeneous exponential properties and dynamics of traffic flow data, a new dynamic and approximate nonhomogeneous grey prediction model is proposed.According to an affine transformation, the model is completely fitted to the nonhomogeneous exponential sequence, and the cycle truncation accumulated generating operation is introduced to achieve the prediction or simulation of dynamic traffic flow data.
(2) The multimode characteristics of traffic data are expressed by a high-dimensional tensor, and a grey

Figure 1 :
Figure 1: Structure of a third-order tensor model of traffic flow.

(
V) Computing and Comparing the Forecasted Values and Errors.The mean relative simulation percentage error

Figure 3 :
Figure 3: Four tensor models with different combinations of traffic data.
Simulated and forecasted effects of SGMThe real data The simulated and forecasted data Simulated and forecasted effects of NDGM

Figure 4 :
Figure 4: Simulated and forecasted effects of the four models.

Table 1 :
Abbreviations and corresponding definitions for grey prediction models.

Table 3 :
Simulated or forecasted values and errors of the traffic flow in the  ∈  3×7×24×1 model.