A method and system for predicting demand for shared bicycles
By constructing the initial network model that includes the time attention layer, map attention network, and door control time convolution network, the limitations of the demand prediction method of bicycle sharing in existing technology in capturing dynamic time dependence are solved, and the sharing is realized Precise prediction of bicycle demand.
Patent Information
- Application Number
- CN202411550359.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-01
- Publication Date
- 2025-01-28
- Estimated Expiration
- 2044-11-01
AI Technical Summary
The existing methods of shared bicycle demand prediction have limitations in capturing dynamic time dependencies within road networks and failing to adequately capture temporal and spatial characteristics.
Build the initial network model, including multiple spaces-time blocks and prediction layers with coding layers, in series, each space-time block contains time attention layer, diagram attention network, and door control time convolution network, capture data through these modules The space and time -dependence in the middle and generate the prediction results of the needs of shared bicycles.
Effectively capture complex time and space characteristics, improve the precision predictive ability of shared bicycle demand, and improve resource utilization and user satisfaction.
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Figure CN119048159B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of intelligent transportation technology, and in particular to a method and system for predicting demand for shared bicycles. Background Art
[0002] As a convenient and environmentally friendly way of travel in modern intelligent transportation systems (ITS), shared bicycles are increasingly favored by the public. The deployment and management of shared bicycles have the problem of unbalanced vehicle distribution, that is, bicycles are in short supply in some areas, while bicycles are piled up and have low utility in other areas. This imbalance seriously affects user experience and operational efficiency. The efficient operation of the bike sharing system (BSS) requires dynamic rebalancing of bicycle deployment to better match the changing travel needs of users. Therefore, accurate demand forecasting for shared bicycle travel is crucial. By predicting the bicycle demand in different areas, vehicles can be dispatched and configured in a timely manner to improve resource utilization and user satisfaction.
[0003] The demand forecasting of shared bicycles involves complex spatiotemporal dynamic changes and has certain technical difficulties. In the spatial dimension, different areas (such as city centers and suburbs) show significant differences in demand patterns due to different population densities and activity types. However, for stations (shared bicycle stops) with similar spatial locations, there are similar demands. In the temporal dimension, demand patterns are subject to multiple influences of daily, weekly, and seasonal cycles, such as the commuting peak on weekdays and special demand on holidays. These changes require the model to be able to handle long-term and short-term demand fluctuations. In addition, external factors such as weather, traffic conditions, and special events in the city also have an important impact on demand. Extreme weather and traffic congestion can significantly change users' willingness to ride, making the forecasting task more complicated. Therefore, the demand forecasting of shared bicycles not only needs to capture the dynamic changes in the spatiotemporal dimension, but also needs to flexibly respond to the influence of multiple external factors.
[0004] Recently, graph convolutional networks (GCNs) have gradually become an important tool for traffic network analysis, which can well represent data with non-Euclidean structures. Although the introduction of GCNs into spatiotemporal prediction models has achieved significant progress, these models also have some problems. First, existing models usually only rely on predefined spatial adjacency matrices to model graphs (such as the distance of nodes), while ignoring the similarity between different nodes in the time dimension, that is, temporal similarity. Second, GCN assumes that all neighbors have the same influence on the central node, which may not fully capture the complex relationships between nodes in graphs with large differences in node features and edge weights. In addition, most existing methods lack attention to the simultaneous use of short-term and long-term temporal correlations. Therefore, existing shared bicycle demand prediction methods are limited in capturing dynamic temporal dependencies within road networks. Summary of the invention
[0005] In order to solve the technical problem that the existing shared bicycle demand prediction method has limitations in capturing the dynamic time dependency within the road network, the embodiments of the present invention provide a shared bicycle demand prediction method and system.
[0006] The technical solution of the embodiment of the present invention is achieved as follows:
[0007] An embodiment of the present invention provides a method for predicting demand for shared bicycles, the method comprising: constructing an initial network model; the initial network model comprises a coding layer, a plurality of space-time blocks connected in series, and a prediction layer; the coding layer is used to convert input data into a high-dimensional feature representation; the plurality of space-time blocks connected in series are used to capture spatial and temporal dependencies in the data from the data represented by the high-dimensional feature; the prediction layer is used to connect the output of each space-time block in series in a residual connection manner and convert it into a prediction result; each space-time block comprises a time attention layer, a graph attention network, and a gated time convolutional network; the time attention The force layer is used to capture the dynamic correlation between different time steps in the high-dimensional feature representation; the graph attention network is used to model complex spatial interactions by adaptively aggregating the neighbor information between shared bicycle stations in the high-dimensional feature representation; the gated temporal convolutional network is used to effectively identify the long-term and short-term pattern changes of the time series in the high-dimensional feature representation through gating mechanism and convolution operation; obtain the historical demand data, weather and holiday data of shared bicycles; use the historical demand data of shared bicycles and the weather and holiday data to construct an undirected graph; calculate based on the undirected graph and the initial network model to obtain the shared bicycle demand prediction result.
[0008] In one embodiment, calculations are performed based on the undirected graph and the initial network model, including: obtaining a time series based on the undirected graph; calculating the minimum distance metric value of the time series using a dynamic time warping algorithm; inverting and normalizing the minimum distance metric value, and then comparing it with a preset threshold to determine the adjacency matrix in the undirected graph.
[0009] In one embodiment, calculating the minimum distance metric value of the time series using a dynamic time warping algorithm includes:
[0010] The minimum distance metric of the time series is calculated using the dynamic time warping algorithm by the following calculation formula:
[0011] Formula (1)
[0012] in, express and The minimum distance metric between , and For two time series, = ,express and The distance measure between .
[0013] In one embodiment, the minimum distance metric value is inverted and normalized, and then compared with a preset threshold to determine an adjacency matrix in the undirected graph, including:
[0014] The minimum distance metric value is inverted and normalized using the following calculation formula:
[0015] Formula (2)
[0016] in, Indicates the site and The similarity between express and The minimum distance metric between is a scale parameter used to control the range of similarity;
[0017] The adjacency matrix in the undirected graph is determined using the following calculation formula:
[0018] Formula (3)
[0019] in, represents the adjacency matrix, Indicates the site and The similarity between Indicates the preset threshold.
[0020] In one embodiment, the calculation is performed based on the undirected graph and the initial network model, including:
[0021] Based on the undirected graph, obtaining an input sequence of each space-time block;
[0022] The input sequence is converted into a query, a key and a value using the following calculation formula;
[0023] Formula (4)
[0024] Formula (5)
[0025] Formula (6)
[0026] in, 、 、 Respectively queries, keys and values for space-time chunk transformations, For the An input sequence of space-time blocks, is the number of time steps, It is The feature dimensions of the spatial-temporal block outputs, is the number of nodes, It is used for learning Parameters, is the projective dimension of attention;
[0027] The following calculation formula is used to perform parallel attention calculation on the query, key and value of multiple attention heads;
[0028] Formula (7)
[0029] in, For the h The output of an attention head is , The function is a function that converts similarity into probability distribution. 、 、 Respectively queries, keys and values for space-time chunk transformations, is the scaling factor, is a weight matrix independent of each head;
[0030] Use the following formula to concatenate the attention calculation results of multiple attention heads;
[0031] Formula (8)
[0032] in, is the multi-head attention output, Indicates i The attention calculation results of the attention heads are: H is the number of attention heads;
[0033] The multi-head attention output and input are residually connected, linearly transformed and normalized using the following calculation formula and then output:
[0034] Formula (9)
[0035] in, For the The output of the temporal attention layer in a spatial-temporal block, is the multi-head attention output, For the An input sequence of space-time blocks, represents the shape transformation function, represents the residual connection, represents a linear function, Represents the normalization function.
[0036] In one embodiment, the calculation is performed based on the undirected graph and the initial network model, including:
[0037] Based on the undirected graph, the attention coefficient of each node and all neighbor nodes in the undirected graph is calculated using the following calculation formula;
[0038] Formula (10)
[0039] in, is the attention coefficient of each node, is a feed-forward neural network. represents transpose, Represents the concatenation operation of vectors, is the weight matrix, is the shared parameter;
[0040] Regularize the attention coefficient using the following calculation formula to obtain the processed attention coefficient;
[0041] Formula (11)
[0042] in, is the attention coefficient after regularization, is the attention coefficient before regularization, ∈ N Represents this node, N Indicates the number of nodes, Indicates adjacent nodes;
[0043] Based on the processed attention coefficient, the updated features of the node are obtained using the following calculation formula;
[0044] Formula (12)
[0045] in, Indicates The updated features of the nodes are is the number of attention heads, For the k The attention coefficient after regularization of the attention heads, For the k The attention weight matrix, is a scale parameter used to control the range of similarity. Indicates adjacent nodes;
[0046] Use the following calculation formula to obtain the updated features of all nodes;
[0047] Formula (13)
[0048] in, represents the updated features of all nodes, N Indicates the number of nodes, Indicates The updated features of each node.
[0049] In one embodiment, the gated temporal convolutional network includes three gated units with different receptive fields; calculation based on the undirected graph and the initial network model includes:
[0050] Based on the undirected graph, obtaining an input vector of the gated temporal convolutional network;
[0051] The output of each gate control unit is obtained by using the following calculation formula;
[0052] Formula (14)
[0053] in, is the output of each of the gating units, is the convolution kernel of the gated unit, is a gated convolution operation, represents the input vector, express Activation function, express Activation function, Indicates that and The result of element-wise multiplication is represents the first half of the input vector, F Represents the second half of the input vector;
[0054] Based on the output of each of the gating units, the output of the gated temporal convolutional network is obtained using the following calculation formula;
[0055] Formula (15)
[0056] in, represents the output of the gated temporal convolutional network, represents the input vector, , , They are the convolution kernels of 3 gated units, with sizes respectively , , , represents the concatenation function, represents pooling processing, express Activation function.
[0057] An embodiment of the present invention also provides a shared bicycle demand prediction system, comprising: a processor and a memory for storing a computer program that can be run on the processor; wherein the processor executes the steps of the above-mentioned method when running the computer program.
[0058] An embodiment of the present invention further provides a storage medium, in which a computer program is stored. When the computer program is executed by a processor, the steps of the above method are implemented.
[0059] An embodiment of the present invention also provides a computer program product, which includes computer instructions. The computer instructions are stored in a computer-readable storage medium. A processor of a computer device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device executes the steps of the above-mentioned method.
[0060] The shared bicycle demand prediction method and system provided by the embodiment of the present invention constructs an initial network model; obtains shared bicycle historical demand data, weather and holiday data; uses the shared bicycle historical demand data and the weather and holiday data to construct an undirected graph; and performs calculations based on the undirected graph and the initial network model to obtain a shared bicycle demand prediction result. The solution provided by the present invention fully considers the dynamic spatiotemporal correlation of traffic data, effectively captures complex temporal and spatial characteristics, and realizes accurate prediction of shared bicycle demand. BRIEF DESCRIPTION OF THE DRAWINGS
[0061] Figure 1 A schematic diagram of a flow chart of a method for predicting demand for shared bicycles according to an embodiment of the present invention;
[0062] Figure 2 This is a schematic diagram of the initial network model structure of an embodiment of the present invention;
[0063] Figure 3 Schematic diagram of the attention mechanism of an embodiment of the present invention; wherein, Figure 3 (a) Schematic diagram of attention coefficient calculation for a single attention head; Figure 3 (b) Schematic diagram of attention coefficient calculation for multiple attention heads;
[0064] Figure 4 Schematic diagram of the comparison of prediction errors of different methods in different time ranges according to the embodiments of the present invention; wherein, Figure 4 (a) is a schematic diagram showing the comparison of mean absolute errors of different methods in different time ranges; Figure 4 (b) is a schematic diagram showing the comparison of the root mean square error of different methods in different time ranges;
[0065] Figure 5 1 is a diagram showing the internal structure of a computer device according to an embodiment of the present invention. DETAILED DESCRIPTION
[0066] The present invention will be described in further detail below with reference to the accompanying drawings and embodiments.
[0067] The embodiment of the present invention provides a method for predicting demand for shared bicycles. Figure 1 As shown, the method includes:
[0068] Step 101: construct an initial network model; the initial network model includes an encoding layer, a plurality of space-time blocks connected in series, and a prediction layer; the encoding layer is used to convert input data into a high-dimensional feature representation; the plurality of space-time blocks connected in series are used to capture the spatial and temporal dependencies in the data from the data represented by the high-dimensional feature; the prediction layer is used to connect the outputs of each space-time block in series in a residual connection manner and convert them into prediction results; each space-time block includes a time attention layer, a graph attention network, and a gated time convolution network; the time attention layer is used to capture the dynamic correlation between different time steps in the high-dimensional feature representation; the graph attention network is used to model complex spatial interactions by adaptively aggregating the neighbor information between shared bicycle stations in the high-dimensional feature representation; the gated time convolution network is used to effectively identify the long-term and short-term pattern changes of the time series in the high-dimensional feature representation through a gating mechanism and a convolution operation;
[0069] Step 102: Obtain historical demand data, weather and holiday data for shared bicycles;
[0070] Step 103: construct an undirected graph using the historical demand data for shared bicycles and the weather and holiday data;
[0071] Step 104: Perform calculations based on the undirected graph and the initial network model to obtain a shared bicycle demand forecast result.
[0072] This embodiment proposes a shared bicycle demand prediction method based on graph attention network, which mines the real dependency relationship between shared bicycle stations, captures the dynamic spatiotemporal correlation of shared bicycle demand, improves the perception ability of the dynamic time dependency of the shared bicycle station network, and considers the impact of external factors such as weather and social events on travel demand.
[0073] Specifically, for the demand forecast of shared bicycles, the problem can be defined as follows:
[0074] In this embodiment, the shared bicycle station network is defined as an undirected graph ,in Indicates the quantity N The node set corresponding to N bike-sharing stations; E is the edge set, which represents the connectivity between nodes; For the picture The adjacency matrix of and hour, This embodiment uses To represent the state of the i-th node at the t-th time step, where Represents the shared bicycle data characteristics of each node (such as historical demand, etc.). The states of all nodes at time step t are .
[0075] The goal of this embodiment is to obtain the historical demand, weather and holiday data of shared bicycles from the past. time step data Learn a model , to predict the future The demand for shared bicycles in time steps .
[0076] Formula (1)
[0077] Next, based on the above problem definition, the model structure (i.e., the initial network model in this embodiment) will be introduced.
[0078] like Figure 2 As shown, the structure of the initial network model (also called DSTGAT model) proposed in this embodiment is as follows Figure 2 As shown in Figure 2. The model consists of an encoding layer, stacked space-time blocks (ST blocks), and a prediction layer (fully connected layer). The encoding layer is responsible for converting the input raw data into a high-dimensional feature representation suitable for subsequent processing. The ST Block is the core part of the model and is responsible for capturing both spatial and temporal dependencies in the data.
[0079] Each ST Block contains a temporal attention layer, a graph attention network, and a gated temporal convolutional network. In each ST Block, the temporal attention layer is used to capture the dynamic correlation between different time steps, the graph attention network models complex spatial interactions by adaptively aggregating neighbor information between shared bicycle stations, and the gated temporal convolutional network effectively identifies long-term and short-term pattern changes in time series through gating mechanisms and convolution operations. The output of each ST block is concatenated and sent to the prediction layer in a manner similar to a residual connection. It converts high-dimensional features into prediction results through a fully connected layer.
[0080] The above model proposed in this embodiment has the following innovative features:
[0081] 1) Based on the Dynamic Time Warping (DTW) algorithm, an adjacency matrix is generated through a data-driven approach to mine and model the real dependency relationships between shared bicycle stations;
[0082] 2) A novel spatiotemporal attention mechanism, which combines the temporal multi-head attention mechanism and GAT, is proposed to learn the dynamic spatiotemporal correlation of shared bicycle demand data. GAT is used to model the complex spatial correlation between different locations and perform graph convolution, and the temporal attention mechanism is used to capture the temporal information that has an important influence in the prediction task;
[0083] 3) Based on the M-GTU time-gated convolution algorithm, the temporal features are fused through multi-scale gated convolution, which further improves the model's ability to perceive the dynamic time dependency of the shared bicycle station network.
[0084] The above innovations will be introduced in detail below.
[0085] 1. Construction of adjacency matrix based on DTW
[0086] In graph neural networks, the adjacency matrix directly affects the model's ability to understand and predict the relationships between nodes and spatiotemporal dynamics. Traditional adjacency matrix generation methods are mainly based on geographical distance or simple time series similarity. However, the actual demand relationship is also affected by complex temporal dynamic changes, for example: ① The peak demand periods of each site may be different; ② The demand of multiple sites in similar areas has similar change patterns. In order to better capture complex spatiotemporal relationships, this embodiment introduces the DTW algorithm to generate the adjacency matrix.
[0087] The DTW algorithm is an algorithm that measures the similarity of two time series. It uses dynamic programming to find the optimal alignment path to minimize the difference between the two time series in the time dimension. This feature makes the DTW algorithm particularly suitable for processing time series data with deformations such as offset, scaling, and distortion.
[0088] Given two sequences and , assuming that this embodiment uses express and In this embodiment, = ,in To calculate the minimum distance metric value while ensuring that the alignment path is continuous and monotonically increasing, the DTW algorithm can be implemented through the following three steps:
[0089] (1) Initialize the start and end points of the alignment path to and , and define a two-dimensional matrix ,in express and The minimum distance metric between .
[0090] (2) Recursive calculation of matrix The value of each element in and select the minimum value on the path. The formula is as follows:
[0091] Formula (2)
[0092] (3) Finally, in matrix D, follow the optimal path from To proceed Traverse to get the sequence and The minimum distance metric .
[0093] As can be seen from formula (2), the DTW algorithm is an algorithm based on dynamic programming, and its core is to solve the warping path , that is, the sequence point and Warping Path ,in , whose elements express and of the match.
[0094] However, the computational complexity of the DTW algorithm is , when both time series are relatively long, the DTW algorithm is inefficient. Therefore, this embodiment uses a variant of DTW, fast-DTW, to calculate the adjacency matrix by limiting the search length T. The search space of the warped path is as follows:
[0095] Formula (3)
[0096] Then, using the calculated DTW distance Constructing a similarity matrix , where each element Indicates the site and This embodiment uses formula (4) to invert and normalize the DTW distance.
[0097] Formula (4)
[0098] in, is a scale parameter used to control the range of similarity. Finally, in order to obtain a sparse adjacency matrix, this embodiment uses a threshold method so that each site is only associated with sites with a similarity exceeding the threshold. The sites remain connected. The constructed adjacency matrix is as follows:
[0099] Formula (5)
[0100] 2. Time and attention
[0101] DTW distance can accurately measure the dependencies between nodes, but in order to adapt to the changes in real-time data, these dynamic dependency characteristics need to be further refined. The temporal multi-head self-attention mechanism can calculate multiple different attention weights in parallel through multiple parallel attention heads, thereby integrating various spatiotemporal features in a higher dimension. This mechanism enables this embodiment to effectively capture the dynamic temporal dependencies between nodes.
[0102] In the self-attention mechanism, each input vector is calculated through a linear transformation of the query (Query, Q), key (Key, K) and value (Value, V) to calculate the attention weights, which reflect the correlation between the input vectors. Input of ST block ,in, is the number of time steps, It is The feature dimension of the ST block output is is the number of nodes. The calculation steps of the self-attention mechanism are as follows:
[0103] (1) Convert the input sequence into query Q, key K, and value V through a linear layer:
[0104] Formula (6)
[0105] in, It is used for learning Parameters, is the projected dimension of attention.
[0106] (2) Use the dot product to calculate the similarity between the query and the key and apply a scaling factor Normalize:
[0107] Formula (7)
[0108] in, The function is used to convert similarity into probability distribution.
[0109] (3) Use H attention heads to perform parallel attention calculations and concatenate them together. For the hth attention head, we have:
[0110] Formula (8)
[0111] Formula (9)
[0112] in, is a weight matrix independent of each head, .
[0113] (4) Finally, apply formula (10) to output the multi-head attention With input Perform residual connection, combine linear transformation to adjust the feature space, and normalize the output through Layer Normalization to adapt to the processing requirements of the next layer.
[0114] Formula (10)
[0115] Among them, the output .
[0116] 3. Spatial Attention Network GAT
[0117] In the shared bicycle demand prediction model, the use of graph attention network (GAT) to simulate the spatial relationship between nodes has significant advantages in capturing dynamic and complex spatial features. Compared with the traditional graph convolutional network (GCN), GAT can assign different weights to adjacent nodes by introducing an attention mechanism, thereby focusing on more important neighbor nodes when updating features. In addition, GAT's multi-head attention mechanism allows different feature representations to be captured in different subspaces, which helps to deal with the noise and uncertainty of data in shared bicycle demand prediction and improves the robustness of the model.
[0118] like Figure 3 As shown, GAT is the feature vector of each node Train a weight matrix , and calculate the attention coefficient between it and all neighboring nodes :
[0119] Formula (11)
[0120] in, is a feed-forward neural network. represents transpose, Represents the concatenation operation of a vector. Adjacency matrix In order to make the attention coefficient easier to calculate and compare, this embodiment uses For all nodes ∈N adjacent nodes Regularize and get the attention coefficient :
[0121] Formula (12)
[0122] In order to stabilize the learning process of self-attention and improve the generalization ability, this embodiment uses a multi-head attention mechanism to obtain the first The updated features of the nodes :
[0123] Formula (13)
[0124] in, is the number of attention heads. Finally, loop through all nodes in the graph and get the node representation after feature update. .
[0125] Finally, the output of temporal attention and the output of GAT Putting it together, we get:
[0126] Formula (14)
[0127] The temporal attention mechanism can dynamically model the temporal changes of demand, while GAT captures the spatial relationships between sites. By combining these two features, this embodiment can construct a feature matrix containing more comprehensive information, making the model more accurate in processing complex spatiotemporal patterns, thereby improving the accuracy and performance of predictions.
[0128] 4. Time-gated convolution
[0129] Time-gated convolution has a good effect in traffic flow prediction. This embodiment uses multi-scale gating The Multi-Scale Gated Temporal Unit (M-GTU) convolution module is used to improve the accuracy of shared bicycle demand prediction. Specifically, the M-GTU module captures the complex temporal dependencies in the data in a multi-scale manner. It is mainly composed of three gated temporal unit (GTU) modules with different receptive fields. Each GTU module is able to process features of different time spans, allowing the model to extract valuable information from multiple time scales from short-term to long-term. For example, short-term demand fluctuations may be related to weather changes or short-term traffic events, while long-term demand trends may be related to seasonal changes or long-term urban planning. The M-GTU module helps to build a more robust and accurate prediction model by integrating these different time scale information.
[0130] The input of the M-GTU module is Each GTU module first uses a convolution kernel size of The two-dimensional convolution of The activation function (i.e. )and The activation function (i.e. ) for the first half of the feature dimension and the second half The processing process is as follows:
[0131] Formula (15)
[0132] in, is a gated convolution operation, Operation will and The result of is element-wise multiplied to form the output of GTU. This operation controls which information is passed and which is suppressed through the gating mechanism, thus capturing the dynamic features in the input sequence well.
[0133] M-GTU expands the receptive field of the time dimension by using convolution kernels of different sizes in each GTU module, thereby enhancing the ability to extract multi-scale time-dependent features.
[0134] Formula (16)
[0135] in, , , The sizes are , , The convolution kernel. The operation concatenates the GTU output features with different convolution kernel sizes and performs residual connection with the input data to obtain The time dimension size is . Then, through a window size of The pooling layer processes the temporal features, thereby effectively extracting more representative temporal dynamic features, and the temporal dimension is reduced to Finally, use The activation function gets the output .
[0136] In M-GTU, this embodiment first performs the Max-Pooling operation and then concatenates the pooled features ( ) together. This strategy can be smoother and more natural when fusing information at different time scales, and only a single feature map is processed during Max-Pooling, reducing the computational complexity.
[0137] Based on the above initial network model, the whole process of the model from input to output will be introduced in detail below:
[0138] The original input data X is a four-dimensional matrix with dimensions [16, 986, 36, 6], where 16 represents batch_size, 986 is the number of shared bicycle stations, 36 is the length of the time series, and 6 is the number of features. Then,
[0139] ① Data preprocessing is performed through the Embedding Layer, which includes convolution, spatial encoding, and temporal encoding. Convolution is used to expand the feature dimension from 6 dimensions to 32 dimensions; spatial encoding is used to add spatial location information to the input data and perform normalization to better capture spatial dependencies. The specific operation is to create a corresponding learnable feature vector for each site and add it to the input data; the significance of temporal encoding is to explicitly inject temporal information into the input data so that the model can better understand and distinguish the dependencies between different time steps. The specific operation is to create a coding vector for each time step and add this coding to the input data.
[0140] The output dimension of this layer is [16, 986, 36, 32].
[0141] ② Use the temporal attention layer to process the output X' of the Embedding Layer or the output of the previous STBlock. First, input X' to merge the first two dimensions, and the dimension becomes [16 986, 36, 32], and then pass through three linear layers to get Q, K, V, and their dimensions become [16 986, 36, 3 32], where 3 represents the number of heads. Then the fourth dimension is split and the dimension becomes [16 986, 36, 3, 32]. Then apply equation (7) for calculation. During the calculation process, the attention calculation process of each head is independent, which allows each head to focus on capturing different features or patterns in the input sequence. After calculation, the dimension remains unchanged. Then change the dimension to [16 986, 36, 3 32], which is used to concatenate the information of the three heads. Then the result is residually connected with the result of merging the first two dimensions of the initial input data, and finally mapped back to the original feature dimension [16, 986, 36, 32] through a linear layer and normalized.
[0142] The output dimension of this layer is [16, 986, 36, 32].
[0143] ③ Use the graph attention network to process the output of the Embedding Layer or the output of the previous STBlock. Combine the adjacency matrix and X' for calculation.
[0144] The output dimension of this layer is [16, 986, 36, 32]. The output of this layer is concatenated with the last dimension (feature dimension) of the temporal attention layer output to obtain the dimension [16, 986, 36, 64].
[0145] ④ Use time-gated convolution to calculate the output results of the previous layer. First, transform the dimension to [16, 64,986, 36] to process the time dimension 36. The time-gated convolution module has 3 GTU modules. They have the same structure but different lengths of the convolution kernels, that is, n1, n2, and n3. Put the input data in the three GTU modules for calculation. Convolve first in each GTU module, then divide the convolution result into two halves and pass two activation functions respectively, and finally multiply the two results. Concatenate the output of each GTU module, process it through Max-Pooling, perform residual connection with the input data, and then pass the RELU activation function to get the result with a dimension of [16, 64, 986, 12].
[0146] The output of this layer is the output result of STBlock of this layer.
[0147] ⑤ Use two fully connected layers to calculate the output of STBlock. First, transform the dimension to [16,986, 12, 64], then pass through two fully connected layers (i.e. linear layers) to map the feature dimension to 1, the dimension is [16, 986,12, 1], and then transform it to [16, 986, 12] to get the output, that is, the number of vehicles used in the next 12 hours at 986 stations.
[0148] This embodiment proposes a network model based on dynamic time warping and spatiotemporal graph attention mechanism, which can more accurately predict the demand for shared bicycles. The model uses a data-driven method to construct an adjacency matrix to effectively reflect the real dependency relationship between bicycle stations; combines the time attention mechanism and the graph attention network to capture the dynamic spatiotemporal correlation of demand data; and applies multi-scale time-gated convolution to achieve short-term and long-term temporal feature fusion. Experimental results show that the model proposed in this embodiment is superior to other classic methods in terms of MAE, RMSE, and MSE evaluation indicators.
[0149] Specifically, this embodiment evaluates the prediction performance of the DSTGAT model through comparative experiments, and further verifies the effectiveness and contribution of the model through ablation experiments.
[0150] 1 Dataset
[0151] In order to verify the effectiveness of the model, this embodiment conducts prediction experiments on the New York CitiBike dataset and the RP5.ru meteorological dataset. The detailed information of the datasets is shown in Table 1.
[0152] (1) New York CitiBike: This example selects trip data from October 1, 2022 to September 30, 2023 (UTC) for analysis and prediction. The data set fields used include departure station, destination station, departure time, and arrival time. The time interval is set to 1 hour, and stations with less than 1 ride record per hour are filtered out, because stations with too low demand are not very meaningful in practical applications.
[0153] (2) RP5.ru: is a collection of global weather observation data provided by Russia. This dataset covers detailed weather information around the world, including temperature, relative humidity, wind speed, precipitation, air pressure, visibility, etc. This example selects the weather recorded by the Albany Weather Station in New York as the dataset of this example, and the sampling interval is 3 hours.
[0154] Table 1
[0155]
[0156] In data preprocessing, this embodiment first removes missing values and outliers to ensure the integrity and quality of the data. In the processing of the New York CitiBike dataset, this embodiment considers a total of 8760 time intervals. In terms of data partitioning, 60% of the samples are used for training, 20% for validation, and the remaining 20% for testing. When predicting the demand for shared bicycles, this embodiment uses the data of the first 36 time intervals (i.e., 36 hours) to predict the demand changes for each station in the subsequent 12 time intervals (i.e., 12 hours). This strategy helps the model capture the temporal characteristics of demand over a longer time window.
[0157] 2 Comparison method
[0158] This embodiment experimentally compares the model of this embodiment with the following models:
[0159] STGCN: This is a spatiotemporal graph convolution model based on a spatial method, which integrates graph convolution into a one-dimensional convolution unit and can capture both spatial and temporal features.
[0160] DCRNN: This model fuses diffuse graph convolution and recurrent neural network, combining graph convolution with gated recurrent units to effectively process the dynamic characteristics of spatiotemporal data.
[0161] ASTGCN: This is a spatiotemporal graph convolutional network model based on the attention mechanism. By introducing spatial and temporal attention mechanisms, it can better capture and model complex spatiotemporal information, especially in traffic flow prediction.
[0162] Seq2Seq: This model adopts an encoder-decoder structure and uses long short-term memory (LSTM) units to handle sequence-to-sequence prediction tasks.
[0163] GraphWaveNet: A hybrid architecture based on graph convolutional networks and temporal convolutions, combining adaptive adjacency matrices with one-dimensional dilated convolutions into graph convolutions.
[0164] STTN: A Transformer-based model that uses spatial attention to dynamically model directed spatial dependencies, while using temporal attention to model long-range bidirectional temporal dependencies across multiple time steps.
[0165] 3 Experimental setup
[0166] This example is based on Python 3.9.7 and Pytorch 2.0.1 framework for experiments (CPU: Intel(R) Xeon(R) Gold 5218R CPU @ 2.10GHz, GPU: NVIDIA GeForce RTX 3080ti 12GB video memory). In terms of parameter setting, this example sets the temporal attention mechanism and the number of heads of GAT multi-head attention. , the size of the convolution kernel of the time-gated convolution M-GTU module The Adam optimizer was used for model training, the mean absolute error (MAE) was used as the loss function, the number of epochs was 500, the batch size was 16, and the learning rate was 0.0005. The model structure used 3 STblocks. In order to evaluate the performance of the model, the experiment used the mean absolute error (MAE), the root mean square error (RMSE) and the mean square error (MSE) as the measurement indicators. The calculation formulas of each indicator are as follows:
[0167] Formula (17)
[0168] Formula (18)
[0169] Formula (19)
[0170] in, is the number of shared bike stations, is the predicted value, It is the true value.
[0171] 4 Experimental results and analysis
[0172] (1) Comparative experiment
[0173] The model of this embodiment is compared with 6 benchmark models. As can be seen from Table 2, the model of this embodiment (DSTGAT) outperforms other models in all evaluation indicators and achieves the best results. Compared with other models, the performance improvement of DSTGAT is mainly attributed to the following points: first, the DTW-based adjacency matrix can better reflect the true dependency relationship between nodes; second, the introduction of the temporal attention mechanism and GAT enables the model to focus on the data of key time points and key nodes, thereby effectively capturing complex spatiotemporal features; finally, the use of temporal gated convolution (M-GTU) can integrate multi-scale temporal features, thereby improving the accuracy and robustness of prediction.
[0174] It can be observed that: (1) Compared with Seq2Seq, which only considers temporal modeling, DSTGAT's MAE is reduced by 11.45%, which reflects the importance of spatial modeling in bike-sharing demand prediction. (2) Compared with spatiotemporal models such as STGCN, DCRNN, ASTGCN, GraphWaveNet, and STTN, DSTGAT has better prediction performance because these methods cannot capture the dynamic spatiotemporal correlation of the bike-sharing station network well, which illustrates the importance of modeling the dynamic temporal similarity of the adjacency matrix and the broader dynamic spatiotemporal characteristics.
[0175] Table 2
[0176]
[0177] (2) Multi-step comparative experiment
[0178] Figure 4 We further demonstrate the performance of different methods on the CitiBike dataset for different prediction time ranges (2, 4, 6, 8, 10, 12). Figure 4 As can be seen from the figure, as the prediction time range increases, the MAE and RMSE values of all models generally increase, indicating that the prediction accuracy decreases as the prediction time range increases. Observe:
[0179] (a) DSTGAT consistently outperforms other models at all time horizons, achieving the lowest MAE and RMSE. This indicates that DSTGAT can effectively capture spatiotemporal dependencies and maintain robust prediction accuracy when the prediction time horizon is extended.
[0180] (b) GraphWaveNet and ASTGCN show strong competitiveness, but they are still surpassed by DSTGAT. The higher MAE values of these models indicate that they are not as effective as DSTGAT in capturing complex dependencies in the data.
[0181] (c) Seq2Seq and STGCN have the highest MAE and RMSE values, especially at longer time ranges. The degradation of STGCN’s performance can be attributed to its lack of attention mechanism and reliance on predefined graph structures, which limits its adaptability in dynamic spatiotemporal scenarios.
[0182] (d) DCRNN and STTN have prediction performance close to DSTGAT when the prediction time range is 2, but their performance increases rapidly as the prediction time range increases, indicating that they have poor ability to capture and generalize spatiotemporal patterns over long ranges.
[0183] (3) Ablation experiment
[0184] In order to verify the effectiveness of each module in the DSTGAT model, this embodiment designs the following variants for comparative analysis: (1) Without-Weather: remove weather data from the input data and only use historical demand data for training; (2) Without-TA: remove the temporal attention mechanism; (3) Without-GAT: remove the GAT module; (4) Without-MGTU: remove 2 GTU units in the temporal gated convolution and only use 1 GTU unit. The experimental results are shown in Table 3. By analyzing the experimental results, the following conclusions can be drawn:
[0185] (a) The temporal attention mechanism significantly affects the prediction performance of the model. The temporal attention mechanism can dynamically identify and focus on important time points in the time series. After removing this mechanism, the model is unable to fully capture the key features in the time dimension, resulting in a decrease in prediction accuracy.
[0186] (b) The GAT module enhances the model’s ability to capture spatial dependencies by adaptively assigning weights to neighboring nodes. After removing the GAT module, the model’s ability to process complex relationships in spatial data is weakened, resulting in a decrease in performance.
[0187] (c) Weather factors are important variables that affect bicycle demand. Weather conditions such as temperature, rainfall, and wind speed directly affect people’s willingness and behavior to ride. Including these factors in the prediction model can improve the accuracy of the prediction.
[0188] (d) The variant using only one GTU unit in the temporal gated convolution shows better performance than the other variants, but still does not reach the performance of the full DSTGAT model. After removing the multi-scale features, although it can still capture some temporal dependencies, its advantage is weakened.
[0189] Table 3
[0190]
[0191] This embodiment proposes a new attention-based spatiotemporal graph convolution model DSTGAT for shared bicycle demand forecasting. The model introduces the DTW algorithm to construct an adjacency matrix in a data-driven manner to model the real dependencies between shared bicycle sites. At the same time, the model also combines the temporal attention mechanism and the graph attention network to capture the dynamic spatiotemporal correlation of demand data. The temporal attention mechanism can dynamically identify and focus on important time points in the time series, while GAT can adaptively assign weights to neighbor nodes. In addition, the model introduces multi-scale time-gated convolutions, which can extract important time features from different time spans. Experimental results on the New York CitiBike dataset show that DSTGAT outperforms existing baseline methods in all evaluation indicators. The superiority of each module of DSTGAT is verified through ablation experiments.
[0192] In order to implement the method of an embodiment of the present invention, an embodiment of the present invention also provides a shared bicycle demand prediction system, including: a processor and a memory for storing a computer program that can be run on the processor; wherein, when the processor is used to run the computer program, the steps of the above-mentioned method are executed.
[0193] The above-mentioned system provided in this embodiment belongs to the same concept as the above-mentioned method embodiment. The specific implementation process thereof is detailed in the method embodiment and will not be repeated here.
[0194] In order to implement the method of the embodiment of the present invention, the embodiment of the present invention also provides a computer program product, which includes computer instructions, and the computer instructions are stored in a computer-readable storage medium. The processor of the computer device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device performs the steps of the above method.
[0195] Based on the hardware implementation of the above program modules, and in order to implement the method of the embodiment of the present invention, the embodiment of the present invention further provides an electronic device (computer device). Specifically, in one embodiment, the computer device may be a terminal, and its internal structure diagram may be as follows: Figure 5As shown. The computer device includes a processor A01, a network interface A02, a display screen A04, an input device A05 and a memory (not shown in the figure) connected through a system bus. Among them, the processor A01 of the computer device is used to provide computing and control capabilities. The memory of the computer device includes an internal memory A03 and a non-volatile storage medium A06. The non-volatile storage medium A06 stores an operating system B01 and a computer program B02. The internal memory A03 provides an environment for the operation of the operating system B01 and the computer program B02 in the non-volatile storage medium A06. The network interface A02 of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor A01, the method of the above embodiment is implemented. The display screen A04 of the computer device can be a liquid crystal display screen or an electronic ink display screen, and the input device A05 of the computer device can be a touch layer covered on the display screen, or a key, trackball or touchpad set on the computer device housing, or an external keyboard, touchpad or mouse, etc.
[0196] Those skilled in the art will understand that Figure 5 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components.
[0197] The device provided by the embodiment of the present invention includes a processor, a memory, and a program stored in the memory and executable on the processor. When the processor executes the program, the method of the above embodiment is implemented.
[0198] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.
[0199] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.
[0200] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.
[0201] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.
[0202] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.
[0203] The memory may include non-permanent memory in a computer-readable medium, random access memory (RAM) and / or non-volatile memory in the form of read-only memory (ROM) or flash RAM. The memory is an example of a computer-readable medium.
[0204] Computer readable media include permanent and non-permanent, removable and non-removable media that can be implemented by any method or technology to store information. Information can be computer readable instructions, data structures, program modules or other data. Examples of computer storage media include, but are not limited to, phase change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technology, compact disk read-only memory (CD-ROM), digital versatile disk (DVD) or other optical storage, magnetic cassettes, magnetic tape disk storage or other magnetic storage devices or any other non-transmission media that can be used to store information that can be accessed by a computing device. As defined herein, computer readable media does not include temporary computer readable media (transitory media), such as modulated data signals and carrier waves.
[0205] It can be understood that the memory of the embodiment of the present invention can be a volatile memory or a non-volatile memory, and can also include both volatile and non-volatile memories. Among them, the non-volatile memory can be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a magnetic random access memory (FRAM), a flash memory, a magnetic surface memory, an optical disc, or a compact disc read-only memory (CD-ROM); the magnetic surface memory can be a disk memory or a tape memory. The volatile memory can be a random access memory (RAM), which is used as an external cache. By way of example and not limitation, many forms of RAM are available, such as static random access memory (SRAM), synchronous static random access memory (SSRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link dynamic random access memory (SLDRAM, SyncLink Dynamic Random Access Memory), and direct RAMbus random access memory (DRRAM, Direct Rambus Random Access Memory).The memories described in the embodiments of the present invention are intended to include, but are not limited to, these and any other suitable types of memories.
[0206] It should also be noted that the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, commodity or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, commodity or device. In the absence of more restrictions, the elements defined by the sentence "comprises a ..." do not exclude the existence of other identical elements in the process, method, commodity or device including the elements.
[0207] The above are only embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various changes and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included within the scope of the claims of the present application.
Claims
1. A method for predicting demand for shared bicycles, characterized in that: The method comprises: Construct an initial network model; the initial network model includes an encoding layer, a plurality of space-time blocks connected in series, and a prediction layer; the encoding layer is used to convert the input data into a high-dimensional feature representation; the plurality of space-time blocks connected in series are used to capture the spatial and temporal dependencies in the data from the data represented by the high-dimensional feature; the prediction layer is used to connect the outputs of each space-time block in series in a residual connection manner and convert them into prediction results; each space-time block includes a time attention layer, a graph attention network, and a gated time convolution network; the time attention layer is used to capture the dynamic correlation between different time steps in the high-dimensional feature representation; the graph attention network is used to model complex spatial interactions by adaptively aggregating the neighbor information between shared bicycle stations in the high-dimensional feature representation; the gated time convolution network is used to effectively identify the long-term and short-term pattern changes of the time series in the high-dimensional feature representation through gating mechanisms and convolution operations; Obtain historical demand data, weather and holiday data for shared bicycles; Using the historical demand data for shared bicycles and the weather and holiday data, an undirected graph is constructed; Calculating based on the undirected graph and the initial network model to obtain a shared bicycle demand forecast result; The calculation based on the undirected graph and the initial network model includes: Based on the undirected graph, obtaining a time series; Calculating the minimum distance metric of the time series using a dynamic time warping algorithm; After inverting and normalizing the minimum distance metric value, the minimum distance metric value is compared with a preset threshold value to determine an adjacency matrix in the undirected graph; The step of calculating the minimum distance metric value of the time series using a dynamic time warping algorithm includes: The minimum distance metric of the time series is calculated using the dynamic time warping algorithm by the following calculation formula: Formula (1) in, express and The minimum distance metric between , and For two time series, = ,express and The distance measure between them; The step of inverting and normalizing the minimum distance metric value and comparing it with a preset threshold to determine the adjacency matrix in the undirected graph includes: The minimum distance metric value is inverted and normalized using the following calculation formula: Formula (2) in, Indicates the site and The similarity between express and The minimum distance metric between is a scale parameter used to control the range of similarity; The adjacency matrix in the undirected graph is determined using the following calculation formula: Formula (3) in, represents the adjacency matrix, Indicates the site and The similarity between Indicates the preset threshold.
2. The method for predicting demand for shared bicycles according to claim 1, characterized in that: Calculating based on the undirected graph and the initial network model includes: Based on the undirected graph, obtaining an input sequence of each space-time block; The input sequence is converted into a query, a key and a value using the following calculation formula; Formula (4) Formula (5) Formula (6) in, 、 、 Respectively queries, keys and values for space-time chunk transformations, For the An input sequence of space-time blocks, is the number of time steps, It is The feature dimensions of the spatial-temporal block outputs, is the number of nodes, It is used for learning Parameters, is the projective dimension of attention; The following calculation formula is used to perform parallel attention calculation on the query, key and value of multiple attention heads; Formula (7) in, For the h The output of an attention head is , The function is a function that converts similarity into probability distribution. 、 、 Respectively queries, keys and values for space-time chunk transformations, is the scaling factor, is a weight matrix independent of each head; Use the following formula to concatenate the attention calculation results of multiple attention heads; Formula (8) in, is the multi-head attention output, Indicates i The attention calculation results of the attention heads are: H is the number of attention heads; The multi-head attention output and input are residually connected, linearly transformed and normalized using the following calculation formula and then output: Formula (9) in, For the The output of the temporal attention layer in a spatial-temporal block, is the multi-head attention output, For the An input sequence of space-time blocks, represents the shape transformation function, represents the residual connection, represents a linear function, Represents the normalization function.
3. The method for predicting demand for shared bicycles according to claim 1, characterized in that: Calculating based on the undirected graph and the initial network model includes: Based on the undirected graph, the attention coefficient of each node and all neighbor nodes in the undirected graph is calculated using the following calculation formula; Formula (10) in, is the attention coefficient of each node, is a feed-forward neural network. represents transpose, Represents the concatenation operation of vectors, is the weight matrix, is the shared parameter; Regularize the attention coefficient using the following calculation formula to obtain the processed attention coefficient; Formula (11) in, is the attention coefficient after regularization, is the attention coefficient before regularization, ∈ N Represents this node, N Indicates the number of nodes, Indicates adjacent nodes; Based on the processed attention coefficient, the updated features of the node are obtained using the following calculation formula; Formula (12) in, Indicates The updated features of the nodes are is the number of attention heads, For the k The attention coefficient after regularization of the attention heads, For the k The attention weight matrix, is a scale parameter used to control the range of similarity. Indicates adjacent nodes; Use the following calculation formula to obtain the updated features of all nodes; Formula (13) in, represents the updated features of all nodes, N Indicates the number of nodes, Indicates The updated features of each node.
4. The method for predicting demand for shared bicycles according to claim 1, characterized in that: The gated temporal convolutional network includes three gated units with different receptive fields; calculations are performed based on the undirected graph and the initial network model, including: Based on the undirected graph, obtaining an input vector of the gated temporal convolutional network; The output of each gate control unit is obtained by using the following calculation formula; Formula (14) in, is the output of each of the gating units, is the convolution kernel of the gated unit, is a gated convolution operation, represents the input vector, express Activation function, express Activation function, Indicates that and The result of element-wise multiplication is represents the first half of the input vector, F Represents the second half of the input vector; Based on the output of each of the gating units, the output of the gated temporal convolutional network is obtained using the following calculation formula; Formula (15) in, represents the output of the gated temporal convolutional network, represents the input vector, , , They are the convolution kernels of 3 gated units, with sizes respectively , , , represents the concatenation function, represents pooling processing, express Activation function.
5. A shared bicycle demand forecasting system, characterized in that: include: A processor and a memory for storing a computer program that can be run on the processor; wherein, when the processor is used to run the computer program, the steps of the method according to any one of claims 1 to 4 are performed.
6. A storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 4 are implemented.
7. A computer program product, characterized in that The computer program product includes computer instructions, which are stored in a computer-readable storage medium. A processor of a computer device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device performs the steps of the method according to any one of claims 1 to 4.
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