Research on short-term traffic flow prediction based on the tensor decomposition algorithm

Abstract

In view of the uncertainties in short-time traffic flows and the multimode correlation of traffic flow data, a grey prediction model for short-time traffic flows based on tensor decomposition is proposed. First, traffic flow data are expressed as tensors based on the multimode characteristics of traffic flow data, and the principle of the tensor decomposition algorithm is introduced. Second, the Verhulst model is a classic grey prediction model that can effectively predict saturated S-type data, but traffic flow data do not have saturated S-type data. Therefore, the tensor decomposition algorithm is applied to the Verhulst model, and then, the Verhulst model of the tensor decomposition algorithm is established. Finally, the new model is applied to short-term traffic flow prediction, and an instance analysis shows that the model can deeply excavate the multimode correlation of traffic flow data. At the same time, the effect of the new model is superior to five other grey prediction models. The predicted results can provide intelligent transportation system planning, control and optimization with reliable real-time dynamic information in a timely manner.

Keywords

Intelligent transportation short-term traffic flow forecasting grey model tensor decomposition

1 Introduction

With the continued population growth, economic development, and urbanization occurring around the world, urban transport facilities cannot meet the increasing demand for road use, and traffic congestion and its resulting social problems have become a major bottleneck in urban development. At present, intelligent transportation systems can effectively alleviate traffic congestion to a certain extent and prevent traffic accidents. Short-term traffic flow analysis is a core component of intelligent transportation systems and an important basis for traffic management and control systems in terms of traffic guidance measures. Short-term traffic flow prediction can obtain real-time, dynamic and accurate effective traffic information, provide travellers with real-time traffic conditions, and realize route guidance for congestion avoidance.

A traffic flow system is a complete, integral network system, and the structural pattern of traffic flow is affected by various factors such as personal travel habits, meteorological factors, environmental factors and traffic development; therefore, the system has a high degree of uncertainty. In terms of time and space, traffic flow data show multidimensional pattern characteristics [1], including weeks, days, hours and so on, and have a strong time correlation [2, 3]. The time scale of short-term traffic flow prediction data generally falls within 15 minutes due to weather and other reasons. Remote data will lose freshness, and if calculated every 5 minutes, there will be 12 data points in an hour; therefore, short-term traffic flow data can be characterized as small sample data. In addition, the traffic flow data of the next section are closely related to the previous section; therefore, traffic flow has obvious grey system characteristics [4, 5].

Grey systems theory was initially proposed by Professor Deng [6] and mainly includes the grey forecasting model, grey correlation model, grey decision-making model and grey clustering model. The grey prediction model is an important part of grey theory and has been widely used in industry, agriculture, economics, transportation, energy, and many other fields [7 –16]. Studies on grey forecast models include traditional GM (1,1), discrete DGM (1,1), NDGM (1,1) and Verhulst models and other single variable models [6, 17 to 20] along with GM (1,N), GM (2,1) and other multivariable models [21 –26]. Researchers have optimized the existing model [27 –32] according to the construction of background values, data accumulation methods, model solution methods, parameter estimation and other machine learning methods with the purpose of promoting the development and improvement of the theoretical system of grey prediction.

Short-term traffic flow prediction is an important application. Guo et al. [33] built a short-term traffic flow nonlinear delay GM (1,1) model. Hsuet et al. [34] proposed a kind of adaptive GM (1,1) model that is used for traffic prediction at nondetector intersections. Bezuglov et al. [35] established the GM (1,1) model and the grey Verhulst model for Fourier error correction, in which the speed and travel time prediction of short-term traffic flows obtained a good prediction effect. Xiao et al. [36] proposed a dynamic grey prediction model and successfully applied it to short-term traffic flow prediction. Lu et al. [37] obtained a grey prediction model using the nonlinear grey Bernoulli equation for traffic flow prediction, achieving good results. Duan et al. [4, 5] proposed an inertial grey prediction model based on the mechanical properties of the data to determine the traffic flow state and short-term traffic flow prediction. Duan et al. [21, 22], based on the multimode characteristics of traffic flow data, established a multimode dynamic grey prediction tensor model, made dynamic predictions of traffic data streams, optimized the grey GM(1,1) model by using the tensor least squares algorithm, and applied it to short-term traffic flow prediction.

Most grey forecast models for short-term traffic flows exist in vector form; however, there are a few grey prediction models that use traffic parameter properties. This paper explores the characteristics of traffic data in several patterns, such as week, day and time; by using the Tucker decomposition algorithm, we can fully explore the multimode characteristics and integrity of traffic data. At the same time, according to the characteristics of the unsaturated S-type data in traffic flow data, the grey Verhulst model is established by using the tensor decomposition algorithm. The new model is applied to short-term traffic flow prediction; the experimental analysis results show that the new model can improve short-term traffic flow prediction accuracy, and the prediction results can be applied to intelligent traffic management systems.

The arrangement of this paper is as follows: Part 2 introduces the Verhulst model; Part 3 presents the algebraic basis of the tensor, the theory of the Tucker decomposition model and the analysis of approximate tensors; Part 4 presents an example analysis and comparison discussion of the new model. Part 5 offers conclusions.

2 Verhulst model

This section introduces the definition and related properties of the traditional grey Verhulst model, analyses the characteristics of traffic flow data that do not satisfy saturated S-type data, and introduces the definition.

Definition 2.1. Set the original sequence as follows: $X^{(0)} = (x^{(0)} (1), x^{(0)} (2), \dots, x^{(0)} (n)),$ (1)

The one-time accumulation generation sequence (1-ago) is as follows: $X^{(1)} = (x^{(1)} (1), x^{(1)} (2), \dots, x^{(1)} (n)),$ (2)

where $x^{(1)} (k) = AGO (x^{(0)} (i)) = \sum_{i = 1}^{k} x^{(0)} (i), k = 1, 2, \dots, n .$ (3)

Let Z⁽¹⁾ = (z⁽¹⁾ (2) , x⁽¹⁾ (3) , ⋯ , z⁽¹⁾ (n)) be the adjacent mean equal weight-generating sequence of X⁽¹⁾, where $z^{(1)} (k) = 0.5 x^{(1)} (k) + 0.5 x^{(1)} (k - 1), k = 2, 3, \dots, n .$ (4)

Definition 2.2 Let the sequences X⁽⁰⁾, X⁽¹⁾, and Z⁽¹⁾ be the sequences in (1),(2), and (3). Thus, $x^{(0)} (k) + {az}^{(1)} (k) = b [z^{(1)} (k)]^{2},$ (5)

is referred to as the grey Verhulstmodel. $\frac{{dx}^{(1)} (t)}{dt} + {ax}^{(1)} (t) = b (x^{(1)} (t))^{2} .$ (6)

This is referred to as the whitening differential equation of Equation (5). Solving Equation (6) gives. $x^{(1)} (t) = \frac{{ax}^{(0)} (1)}{{bx}^{(0)} (1) + (a - {bx}^{(0)} (1)) e^{- a t}},$

Let ${\hat{x}}^{(1)} (1) = x^{(1)} (1) = x^{(0)} (1)$ be the initial value. The time response sequence of the grey Verhulst model is ${\hat{x}}^{(1)} (k + 1) = \frac{{ax}^{(0)} (1)}{{bx}^{(0)} (1) + (a - {bx}^{(0)}) e^{- a k}}, k = 1, 2, \dots, n .$ (7)

For the grey Verhulst model, when the original data are saturated S type data, the effect of the simulation and forecast data is better [20], but generally not for saturated S type data, so the raw data of traffic flow directly derived using the traditional Verhulst model prediction effect is not ideal. To improve the precision of the model, we use a tensor decomposition algorithm based on traffic flow data tensor multi-mode correlation characteristics and establish a tensor decomposition algorithm with the grey prediction model, which can improve the precision of the model.

3 Optimization of the tucker decomposition algorithm tucker decomposition algorithm

A row or column vector isafirst-order tensor, a matrixis a second-order tensor, and third-order tensors and above are higher-order tensors. Therefore, the traffic flow data stream refers to the time series of row or column vectors, the matrix data stream refers to the panel data or section data in the form of a matrix, and the tensor data stream refers to the traffic flow data of high-dimensional and multi-mode data.

Definition 3.1. [40] Mode-n expansion: The mode-n expansion of a tensor is also called the matrix of a tensor. The mode-n expansion is used to quantify the tensor as a matrix. The matrix of mode-n expansion is X_(n).

Definition 3.2. [40] Set two N-order tensors as χ₁ ∈ R^{I₁×I₂×⋯×I_N} and χ₂ ∈ R^{I₁×I₂×⋯×I_N}; then, $< χ_{1}, χ_{2} > = \sum_{i_{1} = 1}^{I_{1}} \sum_{i_{2} = 2}^{I_{2}} \dots \sum_{i_{N} = 1}^{I_{N}} x_{i_{1} i_{2} \dots i_{N}} y_{i_{1} i_{2} \dots i_{N}}$

is the inner product of the two N-order tensors.

Definition 3.3. [40] For matrix M_m₁×m₂ and matrix N_n₁×n₂, the Kronecker product of M and N is $M \otimes N = [\begin{matrix} m_{11} N & m_{12} N & . . . & m_{1 m_{2}} N \\ m_{21} N & m_{22} N & . . . & m_{2 m_{2}} N \\ . . . & . . . & . . . & . . . \\ m_{m_{1} 1} N & m_{m_{1} 2} N & . . . & m_{m_{1} m_{2}} N \end{matrix}]$

From the result calculated above, it can be seen that the number of rows and columns for the matrix M ⊗ N is (m₁n₁) × (m₂n₂), which represents the number of rows and columns of the calculated matrix.

Definition 3.4. [40] Given a matrix A = (x₁, x₂, ⋯ , x_k) with m rows and k columns anda matrix B = (y₁, y₂, . . . , y_k) with n rows and k columns, the Khatri-Rao product of A and B is $A ⊙ B = (x_{1} \otimes y_{1}, x_{2} \otimes y_{2}, . . ., x_{k} \otimes y_{k})$

Definition 3.5. [40] The N order tensor χ ∈ R^{I₁×I₂×⋯×I_N} can be expressed as an exteriorproduct of N vectors, namely, $χ = x_{1} \otimes x_{2} \otimes \dots \otimes x_{N}, x_{k} \in R^{I_{k}} (k = 1, 2, \dots N),$

Let the tensor χ be a rank-one tensor.

Multiplication in tensors is generally represented as a pattern product, which is the multiplication of a tensor and a matrix (or vector) in the corresponding pattern. χ ∈ R^{I₁×I₂×⋯×I_n} For the tensor and matrix A ∈ R^J×I_n, the n pattern product is as follows: $(χ \times_{n} A)_{I_{1} \times \dots \times I_{n - 1} \times J \times I_{n + 1} \times \dots \times I_{N}} .$ (8)

Every element in the n pattern product is $(χ \times_{n} A)_{i_{1} \times \dots \times i_{n - 1} \times j \times i_{n + 1} \times \dots \times i_{N}} = \sum_{i_{n} = 1}^{I_{n}} x_{i_{1} i_{2} \dots i_{N}} u_{{ji}_{n}}$

The following introduces the Tucker decomposition algorithm: Tucker decomposition mainly decomposesa tensor into a core tensor and several factor matrices of the same dimension, and the tensor data are multiplied by the core tensor along each modulus by the corresponding factor matrix. Therefore, Tucker decomposition is also a high-order principal component analysis method.

Definition 3.6. [40] Set a third-order tensor as χ ∈ R^P×Q×R; using Tucker decomposition can be expressed as a product of a core tensor and three factor matrices: $χ = \sum_{l = 1}^{L} \sum_{m = 1}^{M} \sum_{n = 1}^{N} ϑ_{l, m, n} (u_{l} \circ v_{m} \circ w_{n}) = ϑ \times_{1} U \times_{2} V \times_{3} W$ where “∘” is called the symbol of exterior product operation, and its form is consistent with (8). ϑ ∈ R^L×M×N is called the core tensor, ₁U, ₂V, ₃W represent three factor matrices, ϑ × ₁U × ₂V × ₃W is consistent with the definition in (8), represents the n-mode product of the core tensor and matrix ₁U, ₂V, ₃W, and L, M, N are the ranks of the matrix for third-order tensor χ under the expansion of mode 1, mode 2, and mode 3, respectively.

The Tucker decomposition can also be expressed as χ = [-0.15em [ϑ : U, V, W] -0.15em]. Each element in the core tensor reflects the interrelation of factor matrices U, V, W. The Tucker decomposition model is analysed from the perspective of tensor elements, which can also be expressed as: $χ_{p, q, r} = \sum_{l = 1}^{L} \sum_{m = 1}^{M} \sum_{n = 1}^{N} ϑ_{l, m, n} (u_{p, l} \circ v_{q, m} \circ w_{r, n})$

For N-order tensor χ, the Tucker decomposition model can be written as: $χ = ϑ \times_{1} U^{(1)} \times_{2} U^{(2)} \times_{3} \dots \times_{N} U^{(N)}$

Then, the inverse transformation of the decomposition is $ϑ = χ \times_{1} (U^{(1)})^{T} \times_{2} (U^{(2)})^{T} \times_{3} \dots \times_{N} (U^{(N)})^{T}$

The specific steps of Tucker decomposition can be referred to in [39], and the specific steps can be represented by the following block diagram:

3.2 Example analysis of the tucker algorithm

The traffic flow data were obtained from the University of Alberta Transportation Research Centre [41] and comprise Whitemud Drive highway data from August 5 to 26, 2015—a total of 21 days of data. Choosing the higher correlation of traffic flow, eight groups of 5-minute data were selected for a 3-week working day from 18:00 p.m. to 18:40 p.m., for a total of 15 days and 120 5-minute data points. The specific steps for the Tucker decomposition are given below:

Set the initial tensor: Select the traffic flow data in the evening peak period of 21 days from 18:00 to 18:40 to establish the tensor χ^8×5×3, in which 8 represents the traffic flow data every 5 minutes from 18:00 to 18:40, for a total of 8 5-minute data points, 5 represents the traffic flow data for the same period from Monday to Friday, 3 represents the three weeks, and the established tensor χ^8×5×3 is the initial tensor, as shown in Fig. 2.

Fig. 1

The Tucker algorithm step diagram.

Fig. 2

The initial tensor.

The approximate tensor is obtained by the Tucker decomposition step, as shown in Fig. 3.

Fig. 3

The approximate tensor.

In Fig. 3, matrix val (:,:, 1) represents all traffic flow data from 18:00 to 18:40 in week 1, matrix val (:,:, 2) represents all traffic flow data from 18:00 to 18:40 in week 2, and matrix val (:,:, 3) represents all traffic flow data from 18:00 to 18:40 in week 3. The first column of each matrix represents traffic flow data on Monday, and so on, until the fifth column of traffic flow data on Friday.

The approximate tensor data are obtained by the tensor decomposition algorithm from the initial tensor data. According to [39], the original tensor data have a strong correlation. Whether the data of the approximate tensor satisfy the correlation can be determined by the following analysis. Choose the data of the approximate tensor in week 1 of val (:,:, 1) data from five days 18:00 to 18:40, week 2 of val (:,:, 2) data from five days 18:00 to 18:40, and week 3 of val (:,:, 3) data from five days 18:00 to 18:40. The trend charts of the three-week data are shown in Fig. 4, Figs. 5 and 6.

Fig. 4

Traffic flow trend for the first week from 18:00 to 18:40.

Fig. 5

Traffic flow trend for the second week from 18:00 to 18:40.

Fig. 6

Traffic flow trend for the third week from 18:00 to 18:40.

As seen in Fig. 4, Figs. 5 and 6, the approximate tensor data trends for the five-day working day dataover the three weeks are basically consistent, indicating that the approximate tensor data obtained by the tensor decomposition algorithm have a strong correlation. Furthermore, the trend chart of traffic flow data on Monday and Friday for the three weeks can be obtained, as shown in Figs. 7 and 8.

Fig. 7

Three-week Monday 18:00–18:40 traffic flow chart.

Fig. 8

Three-week Friday 18:00–18:40 traffic flow chart.

It can be seen in Fig. 7 and Fig. 8 that the traffic flow trend for Monday and Friday over the three-week period is very close, which also indicates that the traffic flow data for the same period of the same day but over different weeks are highly correlated, so the approximate tensor data can be directly applied to modelling.

4 Establishment of the ATVGM model and an instance analysis

In this section, we establish the grey prediction model of the tensor algorithm and then analyse the validity of the model through an example application.

4.1 Establishment of the ATVGM model

The Tucker decomposition algorithm is combined with the Verhulst model, which is specifically defined as follows:

Definition 4.1 Set a higher dimensional tensor as tensor χ; decompose χ with Tucker decomposition, update the initial factor matrix, and combine the updated factor matrix with the core tensor to restore the approximate tensor. The data obtained from the approximate tensor are simulated and predicted by the Verhulst model, which is called the approximate tensor Verhulst grey model (ATVGM).

The specific steps of the ATVGM are as follows:

Step 1. Set up the initial tensor.

Step 2. Using Tucker decomposition to decompose the initial tensors, the Tucker algorithm can obtain the corresponding core tensors and factor matrices and then verify the validity of the factor matrices.

Step 3. Combine the factor matrices and the core tensors obtained after the simulation to obtain the approximate tensors.

Step 4. The ATVGM is used to predict the time series of the same period of approximate tensors.

Step 5. Calculate the relative error and MAPE.

4.2 Instance analysis of the ATVGM

Select the data in the first column of the initial tensor in Fig. 2, that is, the data from 18:00 to 18:40 on Monday over three weeks, as shown in Table 1. Select the first column of the approximate tensor in Fig. 3 as the approximate tensor data for the three Mondays, as shown in Table 2. The first experiment compares the Verhulst model and the ATVGM. Second, the ATVGM is used to predict the data for the three Mondays and analyse the results. Finally, a comparative analysis between the ATVGM and five grey prediction models is performed.

Table 1
The original tensor traffic flow on Monday of the first week

time 18:00–18:05 18:05–18:10 18:10–18:15 18:15–18:20 18:20–18:25 18:25–18:30 18:30–18:35 18:30–18:35

tensor traffic flow 45.75 58.50 54.25 55.25 45.75 52.00 44.50 50.25

time	18:00–18:05	18:05–18:10	18:10–18:15	18:15–18:20	18:20–18:25	18:25–18:30	18:30–18:35	18:30–18:35
tensor traffic flow	45.75	58.50	54.25	55.25	45.75	52.00	44.50	50.25

Table 2

Three weeks of approximate tensor traffic flow on Mondays

time	18:00–18:05	18:05–18:10	18:10–18:15	18:15–18:20	18:20–18:25	18:25–18:30	18:30–18:35	18:30–18:35
traffic flow	53.346	50.311	51.769	54.634	52.172	48.002	47.866	47.397
traffic flow	58.650	55.313	56.917	60.066	57.358	52.774	52.625	52.109
traffic flow	61.629	58.123	59.808	63.117	60.271	55.455	55.298	54.756

The data in Table 1 are modelled by the Verhulst model, and the first row of Table 2 is the approximate tensor data from Table 1. Therefore, the first row of Table 2 is modelled by the ATVGM, and the specific results are shown in Table 3.

Table 3

Comparison between the Verhulst model of the original tensor data and the ATVGM of the approximate tensor data.

Number	Verhulst			ATVGM
	initial tensor	Simulation value	error value (%)	approximate tensor	Simulation value	error value (%)
K = 1	45.75	45.7500	0.0000	53.346	53.3460	0.0000
K = 2	58.50	47.3487	–19.0620	50.311	52.6743	4.6979
K = 3	54.25	48.6801	–10.2671	51.769	51.9429	0.3356
K = 4	55.25	49.7755	–9.9087	54.634	51.1486	–6.3788
K = 5	45.75	50.6677	10.7492	52.172	50.2886	–3.6073
K = 6	52.00	51.3887	–1.1755	48.002	49.3604	2.8309
K = 7	44.50	51.9675	16.7808	47.866	48.3620	1.0365
K = 8	50.25	52.4296	4.3374	47.397	47.2923	–0.2201
MAPE			10.3258			2.7296

As seen in Table 3, the simulation effect of the original tensor data using the Verhulst model is not as good as the approximate tensor using the ATVGM. In particular, the cumulative data are not applied here because the results of the Verhulst model are worse (the following calculation is the same). The MAPE of the Verhulst model is 10.3258%, while the MAPE of the ATVGM simulation is 2.7296%. The specific comparison results can be drawn in MATLAB with curve trend graphs and error graphs, as shown in Figs. 5 and 6.

As shown in Fig. 9, the volatility of the approximate tensor data is lower than that of the original tensor data and more suitable for modelling the grey forecasting model. Meanwhile, the simulation data of theVerhulst modelincrease slowly, but the simulation data of the ATVGM decrease slowly and then approach the approximate tensor data. As shown in Fig. 10, only the sixth data point of the Verhulst model has a smaller relative error than the ATVGM, and the rest are all larger, so the ATVGM model can effectively predict short-term traffic flow and provide real-time dynamic and reliable traffic information for intelligent transportation systems. The data in Table 2 are calculated by using the ATVGM, and the results are shown in Table 4.

Fig. 9

Curve trend diagram for two traffic flow data sets simulated by two models.

Fig. 10

Error graphs for two traffic flow data sets simulated by two models.

Table 4

Three weeks of Monday approximate tensor traffic flow calculation results.

order	approximate tensor 1	Simulation value	error value (%)	approximate tensor 2	Simulation value	error value (%)	approximate tensor 3	Simulation value	error value (%)
K = 1	53.346	53.3460	0.0000	58.650	58.6500	0.0000	61.629	61.629	0.0000
K = 2	50.311	52.6743	4.6979	55.313	57.9116	4.6980	58.123	60.853	4.6971
K = 3	51.769	51.9429	0.3356	56.917	57.1075	0.3347	59.808	60.008	0.3347
K = 4	54.634	51.1486	–6.3788	60.066	56.2342	–6.3793	63.117	59.091	–6.3793
K = 5	52.171	50.2886	–3.6073	57.358	55.2886	–3.6078	60.271	58.097	–3.607
K = 6	48.002	49.3604	2.8309	52.774	54.2681	2.8311	55.455	57.025	2.8305
K = 7	47.866	48.3620	1.0365	52.625	53.1704	1.0364	55.298	55.871	1.0367
K = 8	47.397	47.2923	–0.2201	52.109	51.9942	–0.2203	54.756	54.635	–0.2203
MAPE	2.730	2.729	2.729

As seen in Table 4, the results of Monday traffic flow data over three weeks simulated by the ATVGM are very close. The results of the first week and the second week are basically the same, and the average relative errors of the simulation over three weeks are all approximately 2.7295%, suggestingthe traffic flow data over the three Mondays during the same time have a strong correlation, and the approximate tensor simulation results are similar. In terms of model validity, the simulation results from the ATVGM show that the approximation of tensor data has a strong correlation, further illustrating that the approximate tensor data can maintain integrity and reflect the intrinsic characteristics of the traffic flow data. To further understand the relationship of the simulation value and the approximate value of the tensor, we show the curve and error variation trends in Figs. 11 and 12.

Fig. 11

Three-week Monday traffic flow data simulation curve trend.

Fig. 12

Error graphs for traffic flow data simulation for three weeks on Monday.

As seen in Fig. 11, the curve trends of approximate tensor data over the same three-day period are similar, and the curve trends simulated by the TAVGM are also similar, which also indicates that the approximate tensor data can fully consider the integrity of the data. It can be further seen in Fig. 12 that the relative errors at each point are basically consistent, leading to consistent average relative errors at the end.

Then, the data in Table 1 are taken as the original tensor data, and the data in the first row of Table 2 are taken as the approximate tensor data for the same period. The 5 data points of 18:00-18:25 are selected as the simulated values, and the 3 data points of 18:25-18:40 are selected as the predicted values. The grey prediction models for comparison are GM(1,1) [6], DGM (1,1) [17], ONGM(1,1) [42], ARGM(1,1) [43] and Verhulst [20]. The ATVGM uses the results of approximate tensors for calculation, while the other five models use the original tensor data for comparison. The predicted data are consistent, as shown in Table 5.

Table 5

Simulation values and errors for traffic flow on Thursday for six models.

order	original data	approximate tensor data	GM (1,1)	DGM (1,1)	ONGM (1,1)	ARGM (1,1)	Verhulst	ATVGM
k = 1	45.75	53.346	45.75	45.75	45.75	45.75	45.75	53.346
k = 2	58.50	50.311	59.02	59.09	56.45	57.54	46.31	52.966
k = 3	54.25	51.769	55.12	55.15	54.23	51.25	46.84	52.626
k = 4	55.25	54.634	51.48	51.47	46.67	54.60	47.34	52.323
k = 5	45.75	52.172	48.08	48.04	21.04	52.82	47.81	52.051
MAPE(%)			3.602	3.631	18.271	4.759	13.323	2.848
k = 7	52.00		44.91	44.84	–65.97	53.77	48.24	51.807
k = 8	44.50		41.94	41.85	–361.26	53.26	48.66	51.588
k = 9	50.25		39.17	39.05	–1363.5	53.53	49.02	51.391
MAPE(%)			13.813	14.007	1317.38	9.873	6.330	6.190

As seen in Table 5, the ATVGM has the best simulation and prediction effect among the six grey prediction models; the simulated MAPE is 2.848%, and the predicted MAPE is 6.190%. The simulation and prediction results of the ONGM(1,1) model are both poor; the simulation results of the GM(1,1) DGM (1,1) and ARGM(1,1) models are good, but their prediction results are not good; the simulation results of the Verhulst model are not good, but the prediction results are good. Specific results can be drawn from curve trend diagrams and error change diagrams, as shown in Figs. 13 and 14.

Fig. 13

Curve of traffic flow on Monday simulated and predicted by five models.

Fig. 14

Error graph of traffic flow on Monday simulated and predicted by six models.

Due to the poor effect of the ONGM(1,1), the renderings of this model are not shown in Figs. 13 and 14. It can be seen from the trend chart in Fig. 13 that the predicted value of the TAVGM slowly decreases and approaches the actual value, and its predicted value is the closest to the actual value. It can be seen in Fig. 14 that both the simulated and predicted MAPE of the TAVGM are the lowest, indicating that the model has good validity.

5 Conclusions

Accurate and reliable traffic forecast information can provide a theoretical basis and basic methods for intelligent transportation systems. To address the high uncertainty of the traffic flow system and the multimode correlation of traffic flow data, the grey prediction model of atensor decomposition algorithm is proposed by organically combining the modelling mechanism of the grey prediction model and the theory of the tensor decomposition algorithm. First, according to the multimode correlation of traffic flow data, the intrinsic characteristics of traffic data are fully mined by using the Tucker decomposition algorithm. Second, the combination of tensor theory and grey prediction theory is realized by combining the tensor algorithm and the grey prediction model, thus expanding the application scope of the grey prediction model. Finally, the new model is applied to short-term traffic flow prediction, and its effect is far better than five other grey prediction models. According to the prediction results, the proposed model can provide effective traffic flow data information for intelligent systems to solve the practical traffic system problems. However, this paper only focuses on the basic research of traffic flow parameters, tensors and grey prediction models, and further research on the combination of the three theories will be a future development direction.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Data Availability

The data used to support the findings of this study are included within the article. The data from the Figures.

Footnotes

Acknowledgments

The authors are grateful to the editor for the valuable comments. This work is supported by the Project of Humanities and Social Sciences Youth Fund of the Ministry of Education of China (19XJC630011); The Humanities and Social Sciences Research Program of Chong Qing Municipal Educational Committee (19SKGH043);The Science and Technology Research Program of Chong Qing Municipal Educational Committee (KJ1703057); The Chongqing Normal University Fund Project (18XWB017).

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Research on short-term traffic flow prediction based on the tensor decomposition algorithm

Abstract

Keywords

1 Introduction

2 Verhulst model

4.1 Establishment of the ATVGM model

4.2 Instance analysis of the ATVGM

Table 1 The original tensor traffic flow on Monday of the first week time 18:00–18:05 18:05–18:10 18:10–18:15 18:15–18:20 18:20–18:25 18:25–18:30 18:30–18:35 18:30–18:35 tensor traffic flow 45.75 58.50 54.25 55.25 45.75 52.00 44.50 50.25

Conflict of interest

Data Availability

Footnotes

Acknowledgments

References

Table 1
The original tensor traffic flow on Monday of the first week

time 18:00–18:05 18:05–18:10 18:10–18:15 18:15–18:20 18:20–18:25 18:25–18:30 18:30–18:35 18:30–18:35

tensor traffic flow 45.75 58.50 54.25 55.25 45.75 52.00 44.50 50.25