A traffic prediction method under data missing condition based on low-rank tensor dynamic mode decomposition

Abstract

The application provides a traffic prediction method based on low-rank tensor dynamic mode decomposition under data missing condition. First, a tensor form of dynamic mode decomposition is introduced, and dynamic changes of each time sequence of a day are expressed as a dynamic tensor composed of different state transition matrices, so as to capture dynamic characteristics in traffic data. Then, a mask operator is introduced, and errors between reconstructed observation tensors and original observation tensors with missing data in non-missing data parts are as small as possible. After that, considering time periodicity and spatial similarity of traffic data, global low-rank constraints and time constraints are applied to the dynamic tensor. Finally, the reconstructed observation tensor is solved to obtain a prediction result. Experiments prove that under the influence of data missing, the method provided by the application can effectively realize a traffic prediction task.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of intelligent transportation systems, in particular to a traffic prediction method based on low-rank tensor dynamic mode decomposition under data missing condition. BACKGROUND

[0002] Short-term traffic data prediction is an important part of intelligent transportation management, which can help traffic managers and travelers to grasp the trend of traffic state changes in time and make effective decisions. Due to hardware failure, weather interference, transmission failure and other reasons, traffic data is generally missing, which seriously affects the accuracy of traffic prediction. In order to reduce the impact of data missing, many methods fill the missing part of the data before prediction, but the error caused by data filling may cause further deviation in prediction. In addition, this strategy is difficult to guarantee the timeliness of prediction. Therefore, under the influence of missing data and the continuous complication of traffic network, how to complete accurate prediction under the condition of missing data is an important problem.

[0003] The prediction method based on matrix and tensor is a solution to the problem of traffic prediction under data missing condition, which mainly uses the idea of dimension reduction. The factor decomposition method organizes the time series into a matrix / tensor, projects it into a low-dimensional space to extract the main mode, and reconstructs the matrix / tensor as an approximation of the observed value, so as to update the anomaly and missing in the original data. Another trace norm minimization method usually assumes a low-rank structure on the matrix / tensor, and converts the non-convex rank minimization problem into a convex trace norm minimization problem. However, most of the above prediction methods only consider the characteristics of the observed data itself, ignoring the dynamic changes of time series, and it is difficult to capture the mutation.

[0004] On the other hand, deep learning methods have been gradually applied to the field of traffic prediction in recent years. They establish a prediction network by training a large amount of historical data. However, most deep learning methods do not consider data under missing condition, and serious missing of data may have a great impact on prediction performance. SUMMARY

[0005] The purpose of the present application is to solve the problem of traffic prediction under data missing condition, and a traffic prediction method based on low-rank tensor dynamic mode decomposition is proposed. The method is based on incomplete traffic data for traffic prediction, and the prediction problem is converted into a tensor completion problem, and the time series to be predicted is regarded as missing values, and data repair and prediction are completed at the same time, including the following steps:

[0006] Step 1, the complete traffic time series is constructed into a complete traffic data observation tensor according to the observation point, time point and day three dimensions respectively According to the dynamic mode decomposition method based on the Koopman operator, the dynamic change of the continuous time series in the observation tensor every day is expressed as different state transition matrices The state transition matrices are organized into a dynamic tensor The tensor form of dynamic mode decomposition is introduced

[0007] Step 2, introduce a mask operator for the missing problem in traffic data According to the original data actually collected, an original observation tensor is constructed Used for marking The position of the missing data in the middle, the size of the mask operator is the same as the original observation tensor and the complete observation tensor after completion, and the complete observation tensor and the original observation tensor with missing data have the following relationship:

[0008] Step 3, according to the time periodicity and spatial similarity of the traffic data, the dynamic tensor Has a low-rank structure, so a low-rank constraint is imposed on the dynamic tensor According to the tensor CP decomposition, the dynamic tensor is decomposed into factor matrices U, V with spatial characteristics and factor matrix W with time characteristics, and the low-rank constraint is further converted into Considering the time-varying characteristics of the dynamic tensor, the dynamic tensor is subjected to local time constraint Where D is a first-order difference matrix. Prevent the mode from changing too much between adjacent days. Obtain the prediction method based on low-rank tensor dynamic mode decomposition under the condition of data missing:

[0009]

[0010] Step 4, solve U, V, W, Get the update expression, then solve the last time series in the last forward matrix in Is the prediction result.

[0011] The traffic prediction method based on low-rank tensor dynamic mode decomposition under the condition of data missing has the following beneficial effects compared with the existing traffic prediction method:

[0012] 1. The method is suitable for traffic prediction of incomplete data, and is also suitable for data in abnormal conditions (sudden increase and decrease of traffic for a period of time), because the Koopman operator can be updated with dynamic changes.

[0013] ​​2. Extend the DMD algorithm from matrix to tensor, represent the change process of adjacent time series on different days with different Koopman operators. These Koopman operators constitute a dynamic tensor that records the state transition information of time series data, and can better capture the dynamic changes of traffic data.

[0014] 3. Apply low-rank constraints on dynamic tensors to represent global spatio-temporal correlations, and introduce time constraints to constrain the change trend of adjacent day time series, which well extracts spatio-temporal features. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 The process of introducing the tensor form of dynamic mode decomposition in the present application. DETAILED DESCRIPTION

[0016] The traffic prediction method based on low-rank tensor dynamic mode decomposition in the case of data missing will be further described and described in detail in combination with the drawings and examples.

[0017] Figure 1 The process diagram of introducing the tensor form of dynamic mode decomposition in the present application. The prediction task that the traffic prediction method proposed in the present application wants to achieve is: through capturing the dynamic changes in historical observation data, short-term prediction of future traffic conditions is carried out, that is, through fitting the dynamic tensor to predict the situation of the last time point in the observation tensor Figure 1 Tensor The sequence selected in the middle frame.

[0018] Step 1: Apply the traffic prediction method based on low-rank tensor dynamic mode decomposition in the case of data missing to the published real data set.

[0019] The present method has been verified in two real-world data sets, and the two data sets are specifically described as follows:

[0020] PeMS data set: The PeMS data set collects speed time series data of the California highway system. It contains working day data from May to June 2012, containing 44 days, 228 sites, and 288 time points per day (i.e. 5-minute frequency).

[0021] Guangzhou data set: The Guangzhou data set shows the speed data of 214 road segments in Guangzhou City from August to September 2016, containing 61 days, 144 time points per day (i.e. 10-minute frequency). There are 1.29% missing in the original data set, containing 24187 observation values.

[0022] For the two datasets, the parameters λ1, λ2, λ3, R of the model are set respectively. The alternating minimization algorithm is used to optimize the parameters. For λ1, while fixing other parameters, it is traversed in the range {0.0001, 0.001, 0.01, 0.1, 1, 10, 100, 1000} with exponential items of 10, and the parameter values corresponding to the best two experimental results are selected as the boundary values of the next round of search for the best parameters, thus obtaining a new search range. Continue to divide the appropriate search boundary, and search in the new range in this way until the best experimental result is obtained. λ2, λ3 are set in the same way as λ1. For the rank R of tensor CP decomposition, it reflects the global low-rank structure of the dynamic tensor and the dimension of the characteristic component, and different datasets have different optimal ranks. By gradually increasing R from 1 while fixing other parameters, the value of R is determined until the optimal prediction result is obtained.

[0023] According to the above parameter setting method, for the PeMS dataset, R = 94, λ1 = 0.01, λ2 = 0.33, λ3 = 140, and for the Guangzhou dataset, R = 87, λ1 = 0.004, λ2 = 0.33, λ3 = 85.

[0024] Since the original dataset has a low missing rate, in order to verify the prediction performance under the condition of data missing, missing data is manufactured in addition to the missing data provided by the original dataset. Two missing conditions are considered: (1) Random missing. That is, the missing points are random and have nothing to do with other variables or attributes. In this case, 20% and 40% of the elements in the row and column of the observation tensor are randomly taken out respectively. (2) Time continuous missing. That is, consecutive time points are missing. This kind of missing is usually caused by equipment failure. In the experiment, two hours of data are regarded as a continuous missing data segment, and a sufficient number of missing data segments are removed in the observation dataset to achieve two missing rates of 20% and 40%, and it is ensured that all missing parts do not overlap. For the PeMS dataset and the Guangzhou dataset, each missing data segment contains 24 consecutive time points and 10 consecutive time points respectively.

[0025] For the two datasets, the data of one time point is predicted each time, and a total of one day of time series is predicted. The observation dataset with missing data is constructed from the original dataset and the mask operator is obtained by multiplying the corresponding elements, and the mask operator comes from the two missing conditions under the two missing rates described above. In the experiment, the input data is the tensor constructed from the observation dataset with missing data is obtained by multiplying the corresponding elements, and the mask operator comes from the two missing conditions under the two missing rates described above. In the experiment, the input data is the tensor constructed from the observation dataset with missing data The last positive matrix (representing the data collected on the last day in the data set) is used as the M time series of N observations to be predicted. After all the time series are predicted, the prediction results are compared with the actual measured values. To ensure the reliability of the experiment, the comparison method inputs the same missing conditions. The y i represents the true value of the traffic data, represents the prediction result, and n represents the number of all observations. The evaluation indexes include the following two, which are used to measure the prediction error, and the smaller the value is, the better:

[0026] (1) Mean Absolute Percentage Error (MAPE):

[0027]

[0028] (2) Root Mean Square Error (RMSE):

[0029]

[0030] The comparison results are as shown in the following table:

[0031]

[0032]

[0033] As can be seen from the table, the present application (DMD-LRT) achieves the best prediction result compared with other methods in most cases, and the prediction accuracy is obviously improved.

[0034] Experimental results show that, under the condition of random missing data, the DMD-LRT method achieves the lowest error across both evaluation metrics on both datasets. On the other hand, HaLRTC, also based on a tensor low-rank constraint, performs poorly in the presence of random missing data. Compared to DMD-LRT, HaLRTC exploits the low-rank characteristics of the original observation tensor but lacks consideration of its dynamic properties. The dynamic tensor constructed using the dynamic pattern decomposition algorithm captures low-rank features from the dynamic tensor, effectively capturing the dynamic changes between time series and predicting future trends. For continuous missing data, the present invention demonstrates good results at a missing rate of 20%. When the missing rate rises to 40%, the MAPE of the present invention remains the lowest among all prediction models, but the RMSE is higher. The MAPE metric, as defined, describes the accuracy of the predictions. The RMSE metric, on the other hand, measures the dispersion of the predictions and is very sensitive to extreme values ​​(large or small). This suggests that, under high continuous missing data rates, the predictions of the present invention are generally accurate, but individual results deviate from the actual values. However, a missing rate of 40% or more is not common in real life, so the present invention can well adapt to common missing situations in the real world. When the missing rate is high, the model achieves better results.

[0035] The above description is only a preferred embodiment of the present invention, and the present invention should not be limited to the contents disclosed in the embodiment and the accompanying drawings. Any equivalent or modification completed without departing from the spirit disclosed in the present invention shall fall within the scope of protection of the present invention.

Claims

1. A traffic prediction method based on low-rank tensor dynamic mode decomposition, characterized in that: The specific process is as follows: Step 1: Introduce the tensor form of dynamic mode decomposition Dynamic pattern decomposition aims to extract the implicit dynamic features in the original data; assuming that the collected traffic time series observation matrix Where N is the number of row vectors in the matrix, which is the number of observation points in physical terms, and M is the number of column vectors in the matrix, which represents the number of time points; the element Z in the i-th row and j-th column of the observation matrix Z is i,j Represents the observation data of the i-th observation point at the j-th moment, then each column vector in Z, that is, each time series Contains the observation values ​​of N observation points; i = 1, 2, ..., M; according to the first M-1 time series and the last M-1 time series in Z, the observation matrix Z is split into two adjacent time snapshot matrices X and Y of size N×(M-1), the specific form is as follows: The purpose of dynamic mode decomposition based on Koopman operator is to find an optimal fitting matrix So that Y≈KX, that is represents the Moore-Penrose pseudo-inverse, K is the Koopman operator, and represents the transformation process of adjacent time series in the traffic system; The traffic time series data is organized into an observation tensor, and the dynamic pattern decomposition is extended to the tensor form to capture the subtle differences between each set of snapshot matrices and extract the dynamic characteristics of the traffic data; First, construct the observation tensor under the condition of complete data. The observation matrix composed of N observation points and L time series is divided into T continuous and non-overlapping time windows according to the same size of time length M. The size of each time window is N×M, L=M×T; the T sub-observation matrices, i.e., time windows, are arranged in chronological order to establish an observation tensor of size N×M×T. Here, the number of time series in a day is set to a time window length M, T is the number of time windows contained in the collected time series data, i.e., the number of days contained in the collected data, and N is the number of observation points. Thus, the observation tensor under the condition of complete data is established. and Represents tensors respectively The horizontal matrix, side matrix and front matrix of the , ":" in the subscript represents all elements of this dimension; Chinese elements Representing a tensor The data in the i-th row and j-th column of the k-th positive matrix is ​​physically the observation value of the i-th observation point at the j-th moment on the k-th day; The observation tensor is transformed into The kth positive matrix is ​​the kth day time series observation matrix Split into Y according to the first M-1 time series and the last M-1 time series k and X k Two data snapshot matrices, Assume that each set of snapshot matrix Y k and X k Corresponding to different dynamic fitting matrices A k , taking the kth group of snapshot matrices as an example, the time series matrices follow the following linear relationship: AND k ≈A k X k in is the observation tensor The Koopman operator of the kth positive observation matrix in records the state transition process of the adjacent time series on the kth day; since the tensor If there are T observation matrices in the time dimension, they are split into T pairs of snapshot matrices and corresponding T Koopman operators; all Koopman operators A1, A2, ..., A T Organize into tensor form and construct dynamic tensor but It records the state transition process of each day's time series and is a collection of state transition matrices. Contains the dynamic characteristics of Z in both time and space; assuming The last positive observation matrix in the Tth column is the last time series in the Mth column. For the time series to be predicted, the dynamic mode decomposition is performed by Get the prediction results when the data is complete; Use the least squares method to fit the dynamic tensor when the data is complete: Step 2: Introduce the mask operator When the collected traffic flow snapshots correspond to the movement of the real traffic flow, the Koopman operator can capture all the dynamic changes of the traffic flow system. Forecasting future traffic time series data Introducing the mask operator To mark the missing information in the original data; Due to the incompleteness of the original observation data, the traffic time series observation data collected in real life are organized into tensors according to the method in step 1 There are missing values ​​introduced in the actual acquisition process; the observation tensor organized in step 1 under the complete data condition regarded as The reconstructed complete tensor needs to be solved by To obtain the forecast value of the last time series on its last positive observation matrix Therefore, the objective function is not only to solve Still need to solve and The following relationship is satisfied: The mask operator and The same size, responsible for marking the location of missing data in the original observation data, as shown in the following formula, a, b, c are the position indexes on the three dimensions of the tensor respectively, Representing a tensor The element in row a and column b of the c-th positive matrix, Similarly: pass For two tensors and Perform the product operation of the corresponding position elements, and make the reconstructed tensor as close as possible to the original data of the corresponding non-missing part in the observation tensor; The following tensor-based dynamic pattern decomposition method is established to solve the traffic time series prediction problem under data missing conditions: Among them, ⊙ is the Hadamard product, that is, the multiplication of corresponding elements in the tensor; Step 3: Imposing low-rank and time constraints on dynamic tensors Since the original observation data Is an incomplete dataset with missing data, reconstructing the tensor and dynamic tensors is unknown, so the objective function is ill-posed and additional constraints need to be introduced; dynamic tensor The changing process of the time series of all observation points every day is recorded, which has temporal and spatial correlation; right Apply low-rank constraint, which is expressed as: where ||·|| * The nuclear norm of the tensor is the sum of the singular values ​​after the matrix SVD decomposition, which is used as a low-rank representation; according to The low-rank structure of the tensor CANDECAMP / PARAFAC decomposition, also known as CP decomposition, reduces the computational parameters. CP decomposition approximates a tensor into the form of a sum of R rank 1 tensors, which are further decomposed into vectors in three dimensions. The decomposition process is expressed as: Where R is the rank of the tensor; ο represents the outer product, u r ,v r ,w r They are tensors The factor matrix obtained after CP decomposition The rth component on U = [u1 u2 ... u R ],u r is the rth column vector, v r ,w r Similarly, for tensors The kth frontal slice matrix A in k Expressed as: TO k =UD (k) V T Among them D (k) =diag(W k,: ), is a diagonal matrix of size R×R composed of the k-th row vector of W, diag(·) represents the function of generating the diagonal matrix; the CP decomposition process reduces the number of parameters from N 2 T is reduced to (2N+T)R, which reduces the computational complexity. It also realizes the decomposition representation of the traffic data tensor in the physical space-time sense. U and V are regarded as factor matrices representing spatial features. W is the factor matrix representing temporal features, which records the state transition matrix A. k The temporal variation of Dynamic Tensors The low-rank constraint term is as follows: λ1 is the regularization parameter. When applied to real data sets, the alternating minimization algorithm is used to tune the parameter: ||·|| F is the Frobenius norm of the matrix, which is calculated by taking the square root of the sum of the squares of the matrix elements; Each row of the time feature matrix W represents the characteristics of the corresponding time window. Considering the temporal continuity of the data and the changes between different modes on weekdays and weekends, the data at adjacent time points are differentially processed. The matrix L1 norm ||·||1 is used, which represents the sum of the absolute values ​​of all elements in the matrix. Non-zero values ​​are penalized to enhance sparsity: λ2||DW||1 Among them, λ2 is the regularization parameter, and the setting method is the same as λ1; the matrix is the first-order difference matrix: Therefore, a dynamic pattern decomposition prediction method based on low-rank tensor is proposed in the case of data missing: According to the Lagrange multiplier method, the objective function is transformed into the following unconstrained optimization problem, then with respect to U, V, W, The objective function writing: λ3 is the regularization parameter, and the setting method is the same as λ1 and λ2; Step 4: Solution of the method The proposed method is solved using the alternating direction multiplier method, which updates one variable at a time while keeping the other variables fixed. The problem is divided into the following sub-problems: (1) When solving U, the objective function is transformed into a function about U: By taking the derivative of the function and setting it to 0, we can solve the normal equation to get the solution of U: Among them, due to D (k) is a diagonal matrix, D (k) =D (k)T ; (2) When solving V, the objective function is transformed into a function about V: Taking the derivative of the function we get: Setting the derivative to 0, the equation can be written in Sylvester equation form: EV+λ1VF -1 =GF -1 in F=D (k) U T UD (k) , The number of observation points in the real data set is large, so N is large. The conjugate gradient method is used to solve the above Sylvester equation to obtain V; (3) Due to D (k) is a diagonal matrix consisting of the elements in the kth row of W, through D (k) Update the W function. When solving W, the objective function is transformed into: The matrix L1 norm regularization term is convex but not differentiable. The proximal gradient method with Nesterov acceleration algorithm solves the value of W corresponding to the minimum of the above loss function. (4) Update When the objective function is transformed into Function: because Each column vector in the positive slice matrix of corresponds to a different objective function, so Divide into T frontal slice matrices, and solve each matrix column by column using the same method; the process of solving the first column, the second column to the second to last column and the last column of each matrix is ​​as follows: ① Solve the first column of each matrix The objective function is as follows: Setting the derivative of the objective function to 0, we get: ② Solve the second to the second to last columns of each matrix When , their objective functions are the same; assuming that we are solving the hth column, h=2,…M-1, the objective function is as follows: Setting the derivative of the objective function to 0, we get: ③Solve the last column of each matrix The objective function is as follows: Setting the derivative of the objective function to 0, we get:

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