Fractional order partial grey model short-time traffic flow prediction method based on two-dimensional advection diffusion equation

Abstract

The invention relates to a fractional order partial grey model short-time traffic flow prediction method based on a two-dimensional advection diffusion equation, and belongs to the technical field of intelligent traffic. The method comprises the following steps: constructing original traffic flow data into traffic flow data in a matrix sequence form; processing the original matrix sequence, and calculating an accumulation generation sequence and a mean value sequence and a partial derivative sequence corresponding to the accumulation generation sequence; establishing a fractional order partial grey short-time traffic flow prediction model AD-FPGM (2, 1) based on a two-dimensional advection diffusion equation; constructing matrixes B and Y to estimate model system parameters, and searching a fractional order accumulation order r and a parameter lambda in a time response formula by adopting a particle swarm algorithm; calculating a simulation value and a reduction value of the AD-FPGM (2, 1) model; and carrying out error test according to each evaluation index of the comparison model, if the error test is passed, predicting a future trend by using an AD-FPGM (2, 1) model, otherwise, reconstructing traffic flow data to carry out model optimization. The prediction result can provide reliable information for a traffic management system, and the research method of traffic flow prediction is enriched.

Description

Technical Field

[0001] This invention belongs to the field of intelligent transportation technology and relates to a short-term traffic flow prediction method based on a fractional gray model using a two-dimensional advection diffusion equation. Background Technology

[0002] In recent years, Intelligent Transportation Systems (ITS) have been recognized as an effective intelligent system for reducing traffic congestion. Traffic prediction is an essential component of ITS, providing scientific support for urban traffic management systems in road planning and play an indispensable role in traffic control and development. As an effective means of assisting traffic planning, management, and control, it is receiving increasing attention from researchers worldwide.

[0003] Traffic systems are complex, and traffic flow data exhibits spatiotemporal characteristics, periodicity, and uncertainty. Short-term road traffic flow prediction is a classic spatiotemporal research challenge. Clearly, observations of adjacent locations and adjacent times are not independent but rather have a close dynamic correlation. For example, from a spatial perspective, different locations have different impacts on a fixed location. Even when time changes, the same location will have different effects on a fixed location. From a temporal perspective, traffic flow data from different time periods has different impacts on the traffic state within a constant time period.

[0004] Traffic flow systems are complex dynamic systems characterized by human intervention, time-varying nature, and high uncertainty. They are influenced by uncertain real-world factors such as the environment, weather, driver psychology, and people's travel habits, exhibiting characteristics of grey systems. Furthermore, short-term traffic flow observations are typically conducted on a timescale of less than 15 minutes, with a traffic guidance cycle generally lasting 5 minutes, resulting in only about 12 data points per hour. Therefore, short-term traffic flow data represents small sample sizes and exhibits clear grey system characteristics.

[0005] Reliable and accurate short-term traffic forecasting is essential for providing information on recent traffic conditions and is key to traffic flow theory research. Therefore, the information provided by traffic forecasting can not only help traffic management departments plan and direct traffic flow in advance, improving traffic efficiency and road safety, alleviating traffic congestion to some extent, reducing traffic accidents, and reducing exhaust emissions, but also help travelers understand information in advance, rationally plan their trips, and choose the best routes. However, spatiotemporal traffic flow forecasting faces many difficulties and challenges. First, mining the nonlinearity and complexity of spatiotemporal traffic data is a demanding task; second, it is constrained by the complex interaction between actual roads and monitoring systems.

[0006] For a complex and uncertain system like a traffic system, it is essential to study information from multiple road segments to predict its overall development trend and evolution. Multivariate models can partially compensate for the shortcomings of univariate models, fully considering the influence of multiple variables on the changing trends of the traffic system during the modeling process, and multivariate models have better structural compatibility. Therefore, a new short-term traffic flow prediction method is urgently needed to solve the above problems. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a short-term traffic flow prediction method based on a fractional gray model using a two-dimensional advection diffusion equation. This method constructs a matrix of original data sequences from the original data of a one-dimensional point sequence. At the same time, it uses a particle swarm optimization algorithm to optimize the fractional cumulative order r and the parameter λ in the time response equation, thereby reducing model error and improving model applicability.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] A short-term traffic flow prediction method based on a fractional-order grey model using a two-dimensional advection diffusion equation, the method comprising:

[0010] The original traffic flow data of the one-dimensional point sequence is constructed into a matrix sequence of traffic flow data, resulting in a matrix sequence X consisting of n m×m matrices. (0) ;

[0011] The original matrix sequence is processed to calculate the accumulated sequence X. (r) and its corresponding mean sequence Z (r) and partial derivative sequence and

[0012] A fractional-order partial gray short-time traffic flow prediction model AD-FPGM(2,1) based on a two-dimensional advection diffusion equation is established; and matrices B and Y are constructed to estimate the model system parameters.

[0013] The particle swarm optimization algorithm is used to find the fractional-order cumulative order r and the parameter λ in the time response.

[0014] Calculate the simulated and restored values ​​of the AD-FPGM(2,1) model; calculate the simulated values ​​using the time response method and the restored values ​​using the restoration method.

[0015] The evaluation indicators of the AD-FPGM(2,1) model and the comparison model are calculated. Error tests are performed based on each evaluation indicator. If the error test is passed, the AD-FPGM(2,1) model is used to predict future trends. Otherwise, traffic flow data is reconstructed for model optimization.

[0016] Furthermore, the matrix sequence X (0) Represented as:

[0017] X (0) =(X (0) (1),X (0) (2),…,X (0) (n))

[0018] Among them, X (0) (k), k = 1, 2, ..., n is an m × m matrix, represented as:

[0019]

[0020] In the formula, X represents (0) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n.

[0021] Furthermore, the original sequence X (0) The r-order cumulative generation sequence X of (k) (r) Represented as:

[0022] X (r) =(X (r) (1),X (r) (2),...,X (r) (k),...,X (r) (n))

[0023] Where, r∈R + ;X (r) (k), k = 1, 2, ..., n is an m × m matrix. X represents (r) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n, as follows:

[0024]

[0025] and:

[0026]

[0027] Furthermore, the accumulated sequence X is generated. (r) The mean generation sequence Z (r) Represented as:

[0028] Z (r) =(Z (r) (2),…,Z (r) (n))

[0029] Among them, Z (r) (k), where k = 2, 3, ..., n is an m × m matrix. Z represents (r) (k) The value in row i, column j, where i,j=1,2,…,m; k=2,3,…,n, and Specifically as follows:

[0030]

[0031] Accumulated generation sequence Z (r) Horizontal and vertical partial derivative sequences and They respectively satisfy the following formulas:

[0032]

[0033] express The value in row i and column j, express The value in row i and column j, where i,j=1,2,…,m; k=2,…,n.

[0034] Furthermore, the expression for the AD-FPGM(2,1) model is:

[0035]

[0036] And its whitening equation is expressed as:

[0037]

[0038] Where A, B, a, b, c, m are real constant parameter variables, and t is the value of X. (r) The time variables, x and y, are X (r) The spatial variables, α0, are m×m constant matrices, as follows:

[0039]

[0040] α (1) X (r) (k) is the sequence X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + :

[0041]

[0042] α (2) X (r) (k) is the sequence α (1) X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + :

[0043]

[0044] Furthermore, the constructed matrix B is m 2 (n-1)×(n 2 A +3) order matrix, which is represented as:

[0045]

[0046] In matrix B, Z (i,j)x (r) (k), Z (i,j)y (r) (k) They are Z x (r) (k), Z y (r) (k), X (r) (k) is the value in row i and column j;

[0047] The constructed matrix Y is m 2 An (n-1)×1 matrix is ​​represented as:

[0048]

[0049] In matrix Y, It is X (r) (k) is the value in row i and column j;

[0050] For the AD-FPGM(2,1) model, the least squares parameter estimation vector satisfy:

[0051]

[0052] in,

[0053] Furthermore, the particle swarm optimization algorithm is used to find the optimal fractional-order cumulative order and the parameters in the time response expression. The process is as follows:

[0054] With the objective of minimizing the average relative error, and considering the relationship between model parameters and the range of parameter values, the following nonlinear optimization model is established:

[0055]

[0056] st.

[0057]

[0058] The optimization model was optimized using the particle swarm optimization algorithm to obtain the optimal fractional order r and the parameter λ in the time response.

[0059] Furthermore, the process of obtaining the simulated and restored values ​​based on the optimal fractional order r and parameter λ obtained through optimization is as follows:

[0060] The simulated values ​​are calculated using the time response formula, which is:

[0061]

[0062] In the formula, the constants λ and μ satisfy λ + μ = m, and κ is the number of equal-width subintervals in the interval [0, t]. ψ(y) and H(t) represent functions of x, y, and t, respectively, with the following specific forms:

[0063] for The function has the following possible values:

[0064] When Δ = a 2 When -4Aλ>0, denote but

[0065] When Δ = a 2 When -4Aλ=0, denote but

[0066] When Δ = a 2 When -4Aλ < 0, denote but

[0067] For the function ψ(y), its possible values ​​are as follows:

[0068] When Δ = b 2 When -4Bμ>0, record but

[0069] When Δ = b 2 When -4Bμ=0, record but

[0070] When Δ = b 2 When -4Bμ < 0, record but

[0071] For the function H(t), its possible values ​​are as follows:

[0072] When Δ = c 2 When -4m > 0, record but

[0073] When Δ = c 2 When -4m = 0, record but

[0074] When Δ = c 2 When -4m < 0, record but

[0075] The restored value is calculated using the reduction formula, and the final cumulative reduction formula is:

[0076]

[0077] In the formula, Γ(·) represents the Gamma function.

[0078] Furthermore, the evaluation indicators used for error testing include mean absolute percentage error (MAPE), mean absolute error (MAE), mean square error (MSE), root mean square percentage error (RMSPE), correlation coefficient (R), Theil U statistical coefficients (U1, U2), and absolute percentage error (APE).

[0079] The smaller the values ​​of MSE, MAPE, RMSPE, MAE, U1, and U2, the higher the model accuracy; and the closer the R value is to 1, the better the model performance.

[0080] The beneficial effects of this invention are as follows:

[0081] (1) This invention innovates upon the traditional grey prediction model by introducing a second-order partial differential equation based on the two-dimensional advection-diffusion equation, and simultaneously adding a fractional-order accumulation operator to more accurately predict short-term traffic flow. Through particle swarm optimization, this invention significantly improves the accuracy and stability of prediction. The AD-FPGM(2,1) model outperforms existing grey prediction models, statistical analysis methods, and deep learning methods in multiple experimental cases, demonstrating its practical value in intelligent transportation systems. The application of this model can provide more scientific and effective data support for urban traffic management and planning, and is particularly suitable for the spatiotemporal characteristic analysis and prediction of traffic flow data.

[0082] (2) This invention analyzed the effectiveness of the new model using measured traffic flow data and compared it with the analysis results of five comparative models. Data experiments showed that the new model's fitting performance was better than the five comparative models. Then, the new model effectively predicted the traffic flow development trend at different times during the five working days of the week, and the predicted trends at different times were consistent with the actual trends, achieving good results.

[0083] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0084] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0085] Figure 1 This is a flowchart of a short-time traffic flow prediction method based on a fractional gray model using a two-dimensional advection diffusion equation, according to an embodiment of the present invention.

[0086] Figure 2 This is a trend diagram of traffic flow fitting for different time periods using the AD-FPGM(2,1) model under an embodiment of the present invention;

[0087] Figure 3 This is a schematic diagram comparing the simulated and actual traffic flow values ​​of different models in embodiments of the present invention. Figure 3 (a) Figure 3 (b) Figure 3 (c) Figure 3 (d) Figure 3 (e) Figure 3 (f) shows a comparison of the simulated and actual values ​​of the AD-FPGM(2,1), CMPGM, WESPGM, PGM, ARIMA and LSTM models, respectively. Detailed Implementation

[0088] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0089] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures, and should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0090] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0091] Please see Figures 1-3 This is a short-term traffic flow prediction method based on a fractional gray model using a two-dimensional advection diffusion equation.

[0092] In this embodiment, the short-term traffic flow prediction method based on a fractional-order grey model using a two-dimensional advection diffusion equation proposed in this invention is described in detail. The grey prediction model is mainly established using a second-order advection diffusion equation, which is the whitening equation of the grey prediction model. Figure 1 As shown, the method specifically includes the following steps:

[0093] S1: Construct traffic flow data in the form of a matrix sequence from the original one-dimensional point sequence traffic flow data;

[0094] S2: Process the original matrix sequence and calculate the accumulated sequence X. (r) and its corresponding mean sequence Z (r) and partial derivative sequence and

[0095] S3: Establish a fractional-order partial gray short-term traffic flow prediction model AD-FPGM(2,1) based on a two-dimensional advection diffusion equation; and construct matrices B and Y to estimate the model system parameters, which are in vector form as follows:

[0096] S4: Use the particle swarm optimization algorithm to find the fractional-order cumulative order r and the parameter λ in the time response;

[0097] S5: Calculate the simulated and restored values ​​of the AD-FPGM(2,1) model; calculate the simulated values ​​using the time response method and the restored values ​​using the restoration method.

[0098] S6: Evaluation metrics for the calculation model AD-FPGM(2,1) and the comparison model, including: mean absolute percentage error MAPE, mean absolute error MAE, mean square error MSE, root mean square percentage error RMSPE, correlation coefficient R, Theil U statistical coefficients U1 and U2, and absolute percentage error APE.

[0099] S7: If all indicators pass the error test, use the AD-FPGM(2,1) model to predict future trends; otherwise, return to step S1.

[0100] In step S1 of this embodiment, the original traffic flow data of the one-dimensional point sequence is constructed into traffic flow data in the form of a matrix sequence, resulting in a matrix sequence X composed of n m×m matrices. (0) :

[0101] X (0) =(X (0) (1),X (0) (2),…,X (0) (n))

[0102] Among them, X (0) (k), k = 1, 2, ..., n is an m × m matrix, represented as:

[0103]

[0104] In the formula, X represents (0) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n.

[0105] In step S2 of this embodiment, the original sequence X (0) The r-order cumulative generation sequence X of (k) (r) Calculate as follows:

[0106] X (r) =(X (r) (1),X (r) (2),...,X (r) (k),...,X (r) (n))

[0107] Where, r∈R + ;X (r) (k), k = 1, 2, ..., n is an m × m matrix. X represents (r) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n, as follows:

[0108]

[0109] and:

[0110]

[0111] In step S2 of this embodiment, the sequence X is accumulated and generated. (r) The mean generation sequence Z (r) Represented as:

[0112] Z (r) =(Z (r) (2),…,Z (r) (n))

[0113] Among them, Z (r) (k), where k = 2, 3, ..., n is an m × m matrix. Z represents (r) (k) The value in row i, column j, where i,j=1,2,…,m; k=2,3,…,n, and Specifically as follows:

[0114]

[0115] Accumulated generation sequence Z (r) Horizontal and vertical partial derivative sequences and They respectively satisfy the following formulas:

[0116]

[0117] express The value in row i and column j, express The value in row i and column j, where i,j=1,2,…,m; k=2,…,n.

[0118] In step S3 of this embodiment, the expression for the AD-FPGM(2,1) model is:

[0119]

[0120] And its whitening equation is expressed as:

[0121]

[0122] Where A, B, a, b, c, m are real constant parameter variables, and t is the value of X. (r) The time variables, x and y, are X (r) The spatial variables, α0, are m×m constant matrices, as follows:

[0123]

[0124] α (1) X (r) (k) is the sequence X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + :

[0125]

[0126] α (2) X (r) (k) is the sequence α (1) X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + :

[0127]

[0128] In step S3 of this embodiment, the constructed matrix B is m 2 (n-1)×(n 2 A +3) order matrix, which is represented as:

[0129]

[0130] In matrix B, Z (i,j)x (r) (k), Z (i,j)y (r) (k) They are Z x (r) (k), Z y (r) (k), X (r) (k) is the value in row i and column j;

[0131] The constructed matrix Y is m 2 An (n-1)×1 matrix is ​​represented as:

[0132]

[0133] In matrix Y, It is X (r) (k) is the value in row i and column j;

[0134] For the AD-FPGM(2,1) model, the least squares parameter estimation vector satisfy:

[0135]

[0136] in,

[0137] In step S4 of this embodiment, the particle swarm optimization algorithm is used to find the optimal fractional-order cumulative order and the parameters in the time response expression. The process is as follows:

[0138] With the objective of minimizing the average relative error, and considering the relationship between model parameters and the range of parameter values, the following nonlinear optimization model is established:

[0139]

[0140] st.

[0141]

[0142] The optimization model was optimized using the particle swarm optimization algorithm to obtain the optimal fractional order r and the parameter λ in the time response.

[0143] In step S5, the time response is:

[0144]

[0145] Where the constants λ and μ satisfy λ + μ = m, and κ is the number of equal-width subintervals in the interval [0, t]. ψ(y) and H(t) represent functions of x, y, and t, respectively, with the following specific forms:

[0146] function It falls under one of the following circumstances:

[0147] When Δ = a 2 When -4Aλ>0, denote

[0148] When Δ = a 2 When -4Aλ=0, denote

[0149] When Δ = a 2 When -4Aλ < 0, denote

[0150] The function ψ(y) belongs to one of the following cases:

[0151] When Δ = b 2 When -4Bμ>0, record

[0152] When Δ = b 2 When -4Bμ=0, record

[0153] When Δ = b 2 When -4Bμ < 0, record

[0154] The function H(t) satisfies H(1)=0, H′(1)=α0, and belongs to one of the following cases:

[0155] When Δ = c 2 When -4m > 0, record

[0156] When Δ = c 2 When -4m = 0, record

[0157] When Δ = c 2 When -4m < 0, record

[0158] The final cumulative subtraction formula is:

[0159]

[0160] In the formula, Γ(·) represents the Gamma function.

[0161] In step S6 of this embodiment, the formula for calculating the mean square error (MSE) is:

[0162]

[0163] The formula for calculating the Mean Absolute Percentage Error (MAPE) is as follows:

[0164]

[0165] The formula for calculating the root mean square percentage error (RMSPE) is:

[0166]

[0167] The formula for calculating the Mean Absolute Error (MAE) is:

[0168]

[0169] The formula for calculating the correlation coefficient R is:

[0170]

[0171] In the formula, cov(·) represents the covariance, and Var(·) represents the variance.

[0172] The formula for calculating the statistical coefficient U1 is:

[0173]

[0174] The formula for calculating the statistical coefficient U² is:

[0175]

[0176] The smaller the values ​​of MSE, MAPE, RMSPE, MAE, and U1 and U2, the higher the model accuracy. The closer the R-value is to 1, the better the model performance.

[0177] This embodiment also compares the model of the present invention with other models. The MAPE trend chart of the prediction model of the present invention over multiple time periods is shown below. Figure 2 As shown in the figure, the trend comparison chart between the prediction model of this invention and other models is as follows. Figure 3 As shown.

[0178] The data in this embodiment comes from UK road traffic data, specifically traffic flow data from the M54 motorway, with traffic flow data intervals of 15 minutes. Traffic flow data from 5:59 to 15:29 for five consecutive days from November 7th to 11th was selected, and traffic flow data at 15-minute intervals were used for modeling. That is, the first matrix of the matrix sequence consists of 25 data points obtained every 15 minutes from 5:59 to 11:59 on November 7th. This pattern continues, with a total of 125 data points from 5:59 to 11:59 for five consecutive days from Monday to Friday forming the original sequence, numbered 1. Subsequent sequences are numbered similarly, with 15 sets for the morning peak, 15 sets for the evening peak, and a total of 30 sets.

[0179] Traffic flow simulations were performed for each five-day period. The calculated traffic flow results for November 7th-11th are shown in Table 1.

[0180]

[0181] Meanwhile, from these 30 sets of data, one set was selected, and five models were chosen as comparison models: the gray-based prediction PGM model, the gray-based model with control matrix CMPGM model, the second-order gray-based model WE-SPGM model, the traditional statistical analysis method ARIMA model, and the deep learning method LSTM model. The parameters of the AD-FPGM(2,1) model are λ=2.80208, r=0.0493203, c1=0.125969, c2=-0.0235746, c12=-2.17683, c22=-5.91765, c13=-37.1598, c23=

[0182] 39.9918, the ARIMA model is ARIMA(1,0,1). The seven error evaluation indices (Mape, MAE, RMSE, MSE, R, U1, U2) of the six models were calculated, and the results are shown in Table 2. The simulated values ​​and actual values ​​of the six models are plotted as follows. Figure 3 .

[0183] Table 2

[0184]

[0185]

[0186] Experimental results show that the overall error level of the model is low, with MAPE values ​​below 10% for most time periods, indicating that the new model has high fitting accuracy. Furthermore, the AD-FPGM(2,1) model exhibits the best performance across all seven error indices compared to other models, demonstrating the highest simulation accuracy. Therefore, AD-FPGM(2,1) is the optimal choice among the seven models.

[0187] observe Figure 2 As can be seen, the overall fluctuation of the image is relatively small. This indicates that the model can maintain stable performance under different time periods and data distributions, demonstrating good adaptability and strong generalization ability. The new model performs excellently in a week-long traffic flow prediction task, exhibiting small prediction errors and good temporal stability, effectively addressing traffic flow fluctuations across different time periods and work weeks.

[0188] Figure 3 The diagram illustrates a comparison between simulated and actual traffic flow values ​​from different models. Figure 3 (a) Figure 3 (b) Figure 3 (c) Figure 3 (d) Figure 3 (e) Figure 3 (f) shows a comparison of the simulated and actual values ​​of the AD-FPGM(2,1), CMPGM, WESPGM, PGM, ARIMA, and LSTM models. The simulated and predicted curves of the AD-FPGM(2,1) model are highly consistent with the actual data trend, especially in the high and low fluctuations of traffic flow, the curve is smooth and close to the actual value, demonstrating its excellent ability to capture periodicity and adaptability. In contrast, although the CMPGM model performs better on MAPE, its simulation of peak values ​​is significantly overestimated, resulting in excessive curve fluctuations and insufficient stability; the WESPGM model is highly sensitive to noise, and local fluctuations in the simulation stage are over-amplified, failing to accurately grasp the overall trend and deviating significantly from the actual data; the traditional PGM model appears too conservative in the fitting process, with simulated values ​​generally being too low, failing to fully reflect the actual changes in traffic flow; although the ARIMA model can grasp the general trend of the data, the predicted values ​​differ significantly from the actual data; the LSTM model's simulated values ​​show lag and insufficient ability to capture traffic flow fluctuations during troughs, the curve shows a flat trend, and it lacks sensitivity to short-term changes. The AD-FPGM(2,1) model outperforms the comparison models in both accuracy and learning of data trends.

[0189] In summary, the AD-FPGM(2,1) model accurately fits the process of traffic flow increase and peak change, demonstrating strong trend capture ability and short-term prediction accuracy. It maintains low prediction errors across different time periods, indicating its high accuracy. Therefore, this model can accurately provide a method for traffic flow prediction, possessing theoretical basis and practical value, and performs well in actual road traffic flow prediction tasks. Furthermore, based on the predicted traffic conditions, relevant traffic management departments can plan and direct traffic flow in advance, better improving traffic efficiency, alleviating traffic congestion, and reducing exhaust pollution. In addition, with this information, passengers can also plan their travel routes accordingly and make better route choices.

[0190] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A short-term traffic flow prediction method based on a fractional-order grey model using a two-dimensional advection-diffusion equation, characterized in that: The method includes: The original traffic flow data of the one-dimensional point sequence is constructed into a matrix sequence of traffic flow data, resulting in a matrix sequence X consisting of n m×m matrices. (0) ; The original matrix sequence is processed to calculate the accumulated sequence X. (r) and its corresponding mean sequence Z (r) and partial derivative sequence and A fractional-order partial gray short-time traffic flow prediction model AD-FPGM(2,1) based on a two-dimensional advection diffusion equation is established; and matrices B and Y are constructed to estimate the model system parameters. The particle swarm optimization algorithm is used to find the fractional-order cumulative order r and the parameter λ in the time response. Calculate the simulated and restored values ​​of the AD-FPGM(2,1) model; calculate the simulated values ​​using the time response method and the restored values ​​using the restoration method. The evaluation indicators of the AD-FPGM(2,1) model and the comparison model are calculated. Error tests are performed based on each evaluation indicator. If the error test is passed, the AD-FPGM(2,1) model is used to predict future trends. Otherwise, traffic flow data is reconstructed for model optimization.

2. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: Matrix sequence X (0) Represented as: X (0) =(X (0) (1),X (0) (2),…,X (0) (n)) Among them, X (0) (k), k = 1, 2, ..., n is an m × m matrix, represented as: In the formula, X represents (0) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n.

3. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: Original sequence X (0) The r-order cumulative generation sequence X of (k) (r) Represented as: X (r) =(X (r) (1),X (r) (2),...,X (r) (k),...,X (r) (n)) Where, r∈R + ;X (r) (k), k = 1, 2, ..., n is an m × m matrix. X represents (r) (k) is the value in row i and column j, where i,j = 1, 2, ..., m; k = 1, 2, ..., n, as follows: and:

4. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 3, characterized in that: Accumulated generation sequence X (r) The mean generation sequence Z (r) Represented as: WITH (r) =(Z (r) (2),…,Z (r) (n)) Among them, Z (r) (k), where k = 2, 3, ..., n is an m × m matrix. Z represents (r) (k) The value in row i, column j, where i,j=1,2,…,m; k=2,3,…,n, and Specifically as follows: Accumulated generation sequence Z (r) Horizontal and vertical partial derivative sequences and They respectively satisfy the following formulas: express The value in row i and column j, express The value in row i and column j, where i,j=1,2,…,m; k=2,…,n.

5. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: The expression for the AD-FPGM(2,1) model is: And its whitening equation is expressed as: Where A, B, a, b, c, m are real constant parameter variables, and t is the value of X. (r) The time variables, x and y, are X (r) The spatial variables, α0, are m×m constant matrices, as follows: α (1) X (r) (k) is the sequence X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + : α (2) X (r) (k) is the sequence α (1) X (r) The first-order cumulative decreasing generation sequence of (k), r∈R + :

6. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 5, characterized in that: The constructed matrix B is m 2 (n-1)×(n 2 A +3) order matrix, which is represented as: In matrix B, Z (i,j)x (r) (k), Z (i,j)y (r) (k) They are Z x (r) (k), Z y (r) (k), X (r) (k) is the value in row i and column j; The constructed matrix Y is m 2 An (n-1)×1 matrix is ​​represented as: In matrix Y, It is X (r) (k) is the value in row i and column j; For the AD-FPGM(2,1) model, the least squares parameter estimation vector satisfy: in, 7. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: The optimal fractional-order cumulative order and parameters in the time response are found using the particle swarm optimization algorithm, as follows: With the objective of minimizing the average relative error, and considering the relationship between model parameters and the range of parameter values, the following nonlinear optimization model is established: st. The optimization model was optimized using the particle swarm optimization algorithm to obtain the optimal fractional order r and the parameter λ in the time response.

8. The method for short-term traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: The process of obtaining the simulated and restored values ​​based on the optimal fractional order r and parameter λ obtained through optimization is as follows: The simulated values ​​are calculated using the time response formula, which is: In the formula, the constants λ and μ satisfy λ + μ = m, and κ is the number of equal-width subintervals in the interval [0, t]. ψ(y) and H(t) represent functions of x, y, and t, respectively, with the following specific forms: for The function has the following possible values: When Δ = a 2 When -4Aλ>0, denote but When Δ = a 2 When -4Aλ=0, denote but When Δ = a 2 When -4Aλ < 0, denote but For the function ψ(y), its possible values ​​are as follows: When Δ = b 2 When -4Bμ>0, record but When Δ = b 2 When -4Bμ=0, record but When Δ = b 2 When -4Bμ < 0, record but For the function H(t), its possible values ​​are as follows: When Δ = c 2 When -4m > 0, record but When Δ = c 2 When -4m = 0, record but When Δ = c 2 When -4m < 0, record but The restored value is calculated using the reduction formula, and the final cumulative reduction formula is: In the formula, Γ(·) represents the Gamma function.

9. The method for short-time traffic flow prediction based on a fractional-order grey model using a two-dimensional advection diffusion equation as described in claim 1, characterized in that: Evaluation metrics used for error testing include mean absolute percentage error (MAPE), mean absolute error (MAE), mean square error (MSE), root mean square percentage error (RMSPE), correlation coefficient (R), Theil U statistical coefficients (U1, U2), and absolute percentage error (APE). The smaller the values ​​of MSE, MAPE, RMSPE, MAE, U1, and U2, the higher the model accuracy; and the closer the R value is to 1, the better the model performance.

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