A regional multi-site PM2.5 prediction method based on semi-dynamic graph neural network

Abstract

The present invention relates to a regional multi-site PM based on a semi-dynamic graph neural network. 2.5 Prediction methods include: obtaining historical wind fields and corresponding PM 2.5 Data, through data cleaning and preprocessing, construct site PM 2.5 and the corresponding wind farm data set; using geographic and wind farm information, dynamic and static relationship diagrams between sites are constructed respectively, and the PM 2.5 Predicted semi-dynamic directed graph; using the spatial information provided by the semi-dynamic directed graph, combined with the time information, to build a spatiotemporal convolutional graph model, using site PM 2.5 The spatiotemporal convolutional graph model is trained and verified with the corresponding wind field dataset to obtain the spatiotemporal PM 2.5 Prediction model; the current wind field and PM 2.5 Data input space-time PM 2.5 In the prediction model, the output is the site PM within the future set time period 2.5 Compared with the existing technology, the present invention can complete spatiotemporal prediction for dynamic graphs, and finally realize PM prediction for multiple sites in the region. 2.5 More accurate prediction of concentration.

Description

Technical Field

[0001] The present invention relates to the field of air quality monitoring technology, and in particular to a regional multi-site PM2.5 prediction method based on a semi-dynamic graph neural network. Background Art

[0002] In recent years, with the improvement of economic level and the rapid development of industrial manufacturing, air pollution has become one of the most concerned issues in the world. It is an important component of atmospheric pollutants. With the development of economy, especially in developing countries, Air pollution is becoming an increasingly serious problem. Studies have shown that exposure to the environment can cause serious health problems such as heart and lung damage, cancer, cardiovascular disease, and a variety of other adverse effects. Some studies have also shown that prenatal exposure to Therefore, with the increasing problem of air pollution, It is particularly important to effectively predict the concentration and study the diffusion trend, to carry out environmental pollution control in advance and to prevent the occurrence of pollution incidents.

[0003] Concentration prediction and diffusion trend prediction have become a research hotspot in the academic community. There are many factors that influence the generation, diffusion and dissipation of air pollution, including natural factors such as weather, wind and geographical conditions, as well as human factors such as social economy and production life. Concentration prediction models face significant challenges. Traditional methods often rely on researchers' understanding of aerodynamics, meteorology, and atmospheric physics and chemistry, combined with real-world emission data. However, these approaches face challenges such as high modeling difficulty, high computational cost, and difficulty in feature extraction.

[0004] In recent years, deep learning methods have shown great advantages in many fields such as image processing, natural language processing, and target detection. Due to its powerful ability to automatically learn features and process large-scale data, deep learning methods have also been used in many fields related to real physics. It has also become the most studied method in air pollution prediction and has achieved relatively good results. Considering the advantages of graphs in modeling sites in non-Euclidean space, more and more studies are using graphs to model the spatial relationships of pollutants. However, graph neural network methods in recent years often have some defects: wind field algorithms do not fully utilize information on regional wind fields, ignore the geographical relationship characteristics between sites, and the dynamic graph neural network model is complex, which seriously affects the model's performance at sites. Therefore, how to improve the model and integrate wind field and geographic information into the graph neural network model to achieve more accurate and interpretable prediction. Prediction is a theoretical and practical engineering problem that needs to be solved urgently. Summary of the Invention

[0005] The purpose of the present invention is to provide a regional multi-site based on semi-dynamic graph neural network in order to overcome the defects of the above-mentioned prior art. Prediction method to achieve multiple sites in the region Accurate prediction of concentration.

[0006] The purpose of the present invention can be achieved by the following technical solutions: A regional multi-site based on semi-dynamic graph neural network The prediction method comprises the following steps:

[0007] S1. Obtain historical wind fields and corresponding Data, through data cleaning and preprocessing, build the site and the corresponding wind field dataset;

[0008] S2. Use geographic and wind farm information to construct dynamic and static relationship diagrams between sites, and then integrate them to construct the site Predicted semi-dynamic directed graph;

[0009] S3, using the spatial information provided by the semi-dynamic directed graph, combined with the time information, to build a spatiotemporal convolutional graph model, using the site The spatiotemporal convolutional graph model is trained and verified with the corresponding wind field dataset to obtain the spatiotemporal Predictive models;

[0010] S4, the current wind field and Data input time and space In the prediction model, the output is the site within the set time period in the future Prediction results.

[0011] Furthermore, the step S1 specifically crawls the data published by the meteorological website to obtain the historical wind field and the corresponding data.

[0012] Furthermore, the specific process of step S2 is as follows:

[0013] S21. Construct a static directed graph, which is a geographic information graph and is initialized with distance information between sites;

[0014] S22. Construct a dynamic directed graph, which is a wind field graph. Use a wind field aggregation algorithm to first divide the wind zones and then aggregate the wind data. The wind zone division area is a rectangular area with the station line as the center line. The aggregated wind data is a weighted addition of the u and v components (i.e., the east-west component and the north-south component) of each grid point along the station line.

[0015] S23. Using the Hadamard product method, the static directed graph and the dynamic directed graph are superimposed to obtain a semi-dynamic directed graph that integrates the effects of geographical and meteorological factors on spatial diffusion.

[0016] Furthermore, the specific process of step S21 is: using a directed graph To describe the geographical topology of the site, Indicates the research Monitoring stations, Indicates how geographical conditions favor A measure of the spread between sites, where edges are represented by the adjacency matrix express, The elements in are obtained through learning. It is initialized to the inverse of the distance and will be reset to zero if it is less than a preset threshold.

[0017] Furthermore, the specific process of aggregating wind data in step S22 is as follows:

[0018] Considering wind speed and direction, calculate the wind volume in the straight line between two stations, and obtain the wind field relationship between stations by weighted aggregation based on the distance between grid points:

[0019] ,

[0020] ,

[0021] in, 、 and represents the distance between (A, B), (A, i) and (B, i), is a grid point The wind field, is the direction vector from station A to station B. By using the dot product of the vectors, we get the wind components in the directions of the two nodes, which are calculated from the u and v components of each grid point:

[0022] ,

[0023] in, Indicates the azimuth from A to B.

[0024] Furthermore, the semi-dynamic directed graph in step S23 is specifically:

[0025] ,

[0026] in, is the site adjacency matrix at time t, which is composed of the wind field matrix and geographic matrix For a specific time node, both sizes are , N is the number of sites, and then add a self-loop term:

[0027] ,

[0028] in, is the modified site adjacency matrix at time t, and I is the size of The identity matrix, is a hyperparameter, It is the self-loop term in the graph convolutional network.

[0029] Furthermore, the spatiotemporal convolutional graph model in step S3 includes three modules, namely, a spatial convolution module, a temporal aggregation module and an output module, wherein the spatial convolution module is composed of a semi-dynamic directed graph convolutional network, the temporal aggregation module is composed of a long short-term memory neural network, and the output module is composed of a fully connected layer.

[0030] Furthermore, the spatial convolution module takes the semi-dynamic graph adjacency matrix as input and uses random walk normalization to complete the matrix normalization, which is expressed as:

[0031] ,

[0032] in, is the inverse matrix of the degree matrix, Site History Input, is the weight matrix of the graph neural network, and the input data of the spatial convolution module is and , The size of , N is the number of sites, t is the length of the input historical information, The size is , Initialize using the distance between sites. When the distance between sites i and j exceeds a preset threshold, is 0, otherwise it is initialized to the inverse of the distance.

[0033] Furthermore, the process of training the spatiotemporal convolutional graph model in step S3 includes:

[0034] S31. Use a semi-dynamic graph as the adjacency matrix, input to a graph convolutional neural network, and use random walk normalization to normalize the adjacency matrix to maintain the utility of a directed graph with edge weights.

[0035] For a sequence input of length l, there is also a dynamic graph of length l and its corresponding adjacency matrix. The input and adjacency matrix are matched one-to-one based on the time step, processed separately by the spatial convolution module, and concatenated into a complete output of the same length:

[0036] ,

[0037] S32, the hidden layer obtained Input into the long short-term memory neural network to extract time information, and finally get the site through the fully connected layer Prediction output of concentration;

[0038] S33, use the Pytorch framework to build the model, set the neural network parameters of the spatiotemporal convolutional graph model, and The corresponding wind field dataset is input into the spatiotemporal convolutional graph model and trained with the loss function. The parameters with the highest prediction accuracy are retained as the ideal parameters of the spatiotemporal convolutional graph model to obtain the spatiotemporal Predictive model.

[0039] Furthermore, in step S32, the number of layers of the long short-term memory neural network is set to 2, the batch_first parameter is set to true, and the input size is , N is the batch size, L is the hidden layer sequence length, is the hidden layer embedding size, and the output size is , is the size of the output layer;

[0040] In the fully connected layer, the end of the sequence output by the long short-term memory neural network is extracted and input, and a one-dimensional output is obtained through the linear layer. , whose size is the product of the number of sites and the number of prediction steps, and then reshapes the one-dimensional output into an output of the same shape , N is the number of sites, and pred is the number of prediction steps.

[0041] Compared with the prior art, the present invention has the following advantages:

[0042] The present invention obtains historical wind fields and corresponding Data to build the site and the corresponding wind farm data set; then, using geographic and wind farm information, the dynamic relationship diagram and static relationship diagram between sites are constructed respectively, and the site The predicted semi-dynamic directed graph is then used to combine the spatial information provided by the semi-dynamic directed graph with the time information to construct a spatiotemporal convolutional graph model. The spatiotemporal convolutional graph model is trained and verified with the corresponding wind field dataset to obtain the spatiotemporal On the one hand, a new wind field calculation method is proposed to predict the wind speed that may affect the site All wind farm data in the propagation area are aggregated, which solves the problem of incomplete utilization of regional wind farm information by traditional methods and can better capture the spatial dynamic relationship between sites. On the other hand, the geographical features between sites are extracted by constructing a static geographic map and added to the In the site relationship of the propagation process, the problem of traditional methods ignoring geographical information other than distance is overcome; in addition, a simple semi-dynamic graph neural network architecture is constructed, which can complete spatiotemporal prediction for dynamic graphs, and finally realize the prediction of multiple sites in the region. More accurate prediction of concentration.

[0043] The present invention uses geographic and wind field information to construct dynamic and static relationship diagrams between sites, respectively. The dynamic directed graph is a wind field graph, which is implemented by a wind field aggregation algorithm. The area is divided into rectangular areas with the site connection line as the center line. The aggregation method is to perform weighted addition of the u and v components of each grid point along the site connection line direction. The static directed graph is a geographic information graph, which is initialized with the distance information between sites, and the geographic information is further automatically trained during the model training process. The dynamic directed graph and the static directed graph are then superimposed by the Hadamard product to obtain the fusion for the site. The predicted semi-dynamic directed graph fully considers the impact of geographical and meteorological factors on spatial diffusion, and can ensure the accuracy of the semi-dynamic directed graph.

[0044] The present invention uses the spatial information provided by the semi-dynamic directed graph and combines it with the time information to construct a spatiotemporal convolutional graph model, which includes three modules: a spatial convolution module, a temporal aggregation module, and an output module. The spatial convolution module is composed of a semi-dynamic directed graph convolutional network, which takes the obtained semi-dynamic graph adjacency matrix as input and uses random walk normalization to complete the matrix normalization; the temporal aggregation module is composed of a long short-term memory neural network, which uses the long short-term memory network to complete the extraction of time information, and the output module is composed of a fully connected layer. In this way, spatiotemporal prediction can be completed for dynamic graphs, and multiple sites in the region can be predicted. Accurately predict concentration. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 Schematic diagram of the method flow of the present invention;

[0046] Figure 2 Schematic diagram of the construction process of a semi-dynamic directed graph;

[0047] Figure 3 This is a schematic diagram of wind zone division;

[0048] Figure 4 The time and space established in the embodiment Schematic diagram of the prediction model;

[0049] Figure 5 For time and space Schematic diagram of the semi-dynamic directed graph convolution module in the prediction model;

[0050] Figure 6 For time and space Schematic diagram of the long short-term memory network in the prediction model. DETAILED DESCRIPTION

[0051] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0052] Example

[0053] like Figure 1 As shown, a regional multi-site based on semi-dynamic graph neural network The prediction method comprises the following steps:

[0054] S1. Obtain historical wind fields and corresponding Data, through data cleaning and preprocessing, build the site and the corresponding wind field dataset;

[0055] S2. Use geographic and wind farm information to construct dynamic and static relationship diagrams between sites, and then integrate them to construct the site Predicted semi-dynamic directed graph;

[0056] S3, using the spatial information provided by the semi-dynamic directed graph, combined with the time information, to build a spatiotemporal convolutional graph model, using the site The spatiotemporal convolutional graph model is trained and verified with the corresponding wind field dataset to obtain the spatiotemporal Predictive models;

[0057] S4, the current wind field and Data input time and space In the prediction model, the output is the site within the set time period in the future Prediction results.

[0058] This embodiment applies the above solution, and its main contents are:

[0059] Step 1: Use the wind field and Data is preprocessed. First, the prediction area is divided according to longitude and latitude, and the stations that meet the conditions (stations with fewer missing historical observations) are selected. The original data is organized into matrix data that can be used as neural network input, and the missing values ​​are linearly filled. Wind field data is downloaded from the public meteorological data website, and the missing wind field data is also linearly filled. Finally, the original wind field and The dataset, in this example, The dataset contains data from 2020 to 2022, with 50 sites.

[0060] Step 2, reference Figure 2 , using geographic and wind farm information, we construct dynamic and static relationship diagrams between sites, and further integrate them to construct site Predicted semi-dynamic directed graph.

[0061] The first step is to build a static directed graph. Using a directed graph To describe the geographical topology of the site. In our research Monitoring stations, Indicates how geographical conditions favor A measure of the spread between sites. Edges are represented by the adjacency matrix express. The elements in are obtained through learning. Initialized to the inverse of the distance, and reset to zero if it is less than a certain value.

[0062] The second step is to construct a dynamic directed graph. The wind direction and intensity are constantly changing compared to the short-term geographic invariance. In order to extract the wind relationship between sites, this solution uses a method to calculate the wind relationship from the original grid data. The algorithm can be divided into two steps. For two interacting sites, the reference Figure 3 First, the wind zones are divided according to geographical location. Then, the wind data of the nodes in the wind zone are aggregated. By considering wind speed and direction, the wind volume on the straight line between the two stations is calculated. The wind field relationship between the stations is obtained by weighted aggregation based on the distance between the grid points.

[0063] ,

[0064] ,

[0065] in 、 and represents the distance between (A, B), (A, i) and (B, i), is a grid point The wind field, is the direction vector from station A to station B. By taking the dot product of the vectors, we get the wind components in the directions of the two nodes. It can be calculated from the u and v components:

[0066] ,

[0067] in Indicates the azimuth from A to B.

[0068] In the third step, the static and dynamic site spatial relationship diagrams are superimposed to obtain a semi-dynamic directed graph that integrates the effects of geographical and meteorological factors on spatial diffusion.

[0069] ,

[0070] in is the site adjacency matrix at time t, which is composed of the wind field matrix and geographic matrix The combination of . The input is calculated from real-world wind data obtained in the second step. It is acquired through training. The elements in are used to measure the weakening or strengthening effect of the inter-station geography on the wind field. The final adjacency matrix is ​​obtained by the following formula:

[0071] ,

[0072] It is a self-loop term commonly used in graph convolutional networks, where I is the identity matrix.

[0073] Step 3, reference Figure 4 、 Figure 5 、 Figure 6 , using the spatial information provided by the semi-dynamic directed graph and combining it with the time information to construct a space-time The prediction model (consists of three modules: spatial convolution module, temporal aggregation module and output module. The spatial convolution module is composed of a semi-dynamic directed graph convolutional network, the temporal aggregation module is composed of a long short-term memory neural network, and the output module is composed of a fully connected layer) is trained using the training set data to obtain the optimal weights.

[0074] First, we build a spatial convolution module. We use a semi-dynamic graph as the adjacency matrix and input it into the improved graph convolutional neural network. We use random walk normalization to normalize the adjacency matrix to maintain the utility of a directed graph with edge weights.

[0075] ,

[0076] in and is the adjacency matrix obtained in the previous step that represents the spatial relationship between sites, is a self-loop term, Site History The input, and is the weight matrix of the graph neural network. The input data of this module is and ,in The size is , N is the number of sites and t is the length of the input history information, and The size is . Initialize using the distance between sites. When the distance between sites i and j exceeds 200 is 0, otherwise it is initialized to the inverse of the distance. In this embodiment, N is the number of sites, 50, and the length of the historical information t is 12.

[0077] The historical data of a certain length are convolved and concatenated separately to obtain the complete hidden layer output of the same length.

[0078] ,

[0079] Then the obtained hidden layer Input into the long short-term memory (LSTM) neural network to extract time information, and finally get the site through the fully connected layer The predicted output of the concentration. The number of LSTM layers is set to 2, the batch_first parameter is set to true, and the input size is , where N is the batch size, L is the hidden layer sequence length 12 obtained previously, is the hidden layer embedding size, which is 50 in the model. The output size is ,in The size of the output layer is set to 128 in the model. The data obtained by LSTM is then input into the fully connected layer and reshaped into an output of the same shape , where N is the number of stations and pred is the number of prediction steps. In the model, the number of stations is 50 and pred is 36, which means the stations for the next 36 hours are predicted. concentration.

[0080] Finally, the constructed spatiotemporal semi-dynamic graph model is trained. The training set is input into the neural network, the results are output through the neural network, and the weights of the neural network are updated using the gradient descent method.

[0081] In this embodiment, the Loss function is defined as:

[0082]

[0083] Among them, RMSE and MAE represent the root mean square error and absolute mean error respectively, and their calculation formulas are:

[0084] ,

[0085] ,

[0086] in, is the site observation value, is the model predicted value.

[0087] Step 4: Regional site The concentration is predicted. Through the obtained model, the test set or the latest acquired data is input into the trained time and space In the prediction model, the site The prediction results of this embodiment are verified by experiments. The prediction performance of the prediction model was higher than that of the baseline model.

[0088] In summary, this proposal designs a new wind field calculation method to All wind field data in the propagation area are aggregated and a dynamic directed graph is constructed as a wind field representation. A geographical relationship graph between sites is also constructed to In addition, based on the constructed semi-dynamic directed site graph, a semi-dynamic graph neural network model is proposed to learn the geographical factors in the process of transmission. Graph convolution is performed on the data and its corresponding graph at the same time point to extract the spatial relationship, and the output is spliced ​​to obtain a hidden layer containing time series information. The hidden layer is input into the time convolution module to extract the time relationship, and finally the output prediction is obtained through the fully connected layer.

[0089] This proposal designs a new wind field calculation method to Aggregating all wind farm data in the propagation area solves the problem of incomplete utilization of regional wind farm information in previous methods and can better capture the spatial dynamic relationship between sites. By constructing a static geographic map, the geographic features between sites are extracted and added to the In the site relationship of the propagation process, it can overcome the problem of ignoring geographical information other than distance in the past methods. A simple semi-dynamic graph neural network architecture is constructed to complete spatiotemporal prediction for dynamic graphs, and finally realize the prediction of multiple sites in the region. More accurate prediction of concentration.

Claims

1. A regional multi-site model based on semi-dynamic graph neural network The prediction method is characterized in that The following steps are involved: S1. Obtain historical wind fields and corresponding data, and build a site through data cleaning and preprocessing. and the corresponding wind field dataset; S2. Use geographic and wind farm information to construct dynamic and static relationship diagrams between sites, and then integrate them to construct the site Predicted semi-dynamic directed graph; S3, using the spatial information provided by the semi-dynamic directed graph, combined with the time information, to build a spatiotemporal convolutional graph model, using the site The spatiotemporal convolutional graph model is trained and verified with the corresponding wind field dataset to obtain the spatiotemporal Predictive models; S4, the current wind field and Data input time and space In the prediction model, the output is the site within the set time period in the future Prediction results.

2. A regional multi-site based semi-dynamic graph neural network according to claim 1 The prediction method is characterized in that The step S1 specifically crawls the data published on the meteorological website to obtain the historical wind field and the corresponding data.

3. A regional multi-site based semi-dynamic graph neural network according to claim 1 The prediction method is characterized in that The specific process of step S2 is: S21. Construct a static directed graph, which is a geographic information graph and is initialized with distance information between sites; S22. Construct a dynamic directed graph, which is a wind field graph. Use a wind field aggregation algorithm to first divide the wind zone and then aggregate the wind data. The wind zone division area is a rectangular area with the station line as the center line. The aggregated wind data is weighted addition of the u and v components of each grid point along the station line. S23. Using the Hadamard product method, the static directed graph and the dynamic directed graph are superimposed to obtain a semi-dynamic directed graph that integrates the effects of geographical and meteorological factors on spatial diffusion.

4. A regional multi-site based semi-dynamic graph neural network according to claim 3 The prediction method is characterized in that The specific process of step S21 is: using a directed graph To describe the geographical topology of the site, Indicates the research Monitoring stations, Indicates how geographical conditions favor A measure of the spread between sites, where edges are represented by the adjacency matrix express, The elements in are obtained through learning. It is initialized to the inverse of the distance and will be reset to zero if it is less than a preset threshold.

5. A regional multi-site based semi-dynamic graph neural network according to claim 4 The prediction method is characterized in that The specific process of aggregating wind data in step S22 is as follows: Considering wind speed and direction, calculate the wind volume in the straight line between two stations, and obtain the wind field relationship between stations by weighted aggregation based on the distance between grid points: ， ， in, 、 and represents the distance between (A, B), (A, i) and (B, i), is a grid point The wind field, is the direction vector from station A to station B. By using the dot product of the vectors, we get the wind components in the directions of the two nodes, which are calculated from the u and v components of each grid point: ， in, Indicates the azimuth from A to B.

6. A regional multi-site based semi-dynamic graph neural network according to claim 5 The prediction method is characterized in that The semi-dynamic directed graph in step S23 is specifically: ， in, is the site adjacency matrix at time t, which is composed of the wind field matrix and geographic matrix For a specific time node, both sizes are , N is the number of sites, and then add a self-loop term: ， in, is the modified site adjacency matrix at time t, and I is the size of the identity matrix, is a hyperparameter, It is the self-loop term in the graph convolutional network.

7. A regional multi-site based semi-dynamic graph neural network according to claim 6 The prediction method is characterized in that The spatiotemporal convolutional graph model in step S3 includes three modules, namely a spatial convolution module, a temporal aggregation module and an output module, wherein the spatial convolution module is composed of a semi-dynamic directed graph convolutional network, the temporal aggregation module is composed of a long short-term memory neural network, and the output module is composed of a fully connected layer.

8. A regional multi-site based semi-dynamic graph neural network according to claim 7 The prediction method is characterized in that The spatial convolution module takes the semi-dynamic graph adjacency matrix as input and uses random walk normalization to complete the matrix normalization, which is expressed as: ， in, is the inverse matrix of the degree matrix, Site History Input, is the weight matrix of the graph neural network, and the input data of the spatial convolution module is and , The size is , N is the number of sites, t is the length of the input historical information, The size is , Initialize using the distance between sites. When the distance between sites i and j exceeds a preset threshold, is 0, otherwise it is initialized to the inverse of the distance.

9. A regional multi-site based semi-dynamic graph neural network according to claim 8 The prediction method is characterized in that The process of training the spatiotemporal convolutional graph model in step S3 includes: S31. Use a semi-dynamic graph as the adjacency matrix, input to a graph convolutional neural network, and use random walk normalization to normalize the adjacency matrix to maintain the utility of a directed graph with edge weights. For a sequence input of length l, there is also a dynamic graph of length l and its corresponding adjacency matrix. The input and adjacency matrix are matched one-to-one based on the time step, processed separately by the spatial convolution module, and concatenated into a complete output of the same length: ， S32, the hidden layer obtained Input into the long short-term memory neural network to extract time information, and finally get the site through the fully connected layer Prediction output of concentration; S33, use the Pytorch framework to build the model, set the neural network parameters of the spatiotemporal convolutional graph model, and The corresponding wind field dataset is input into the spatiotemporal convolutional graph model and trained with the loss function. The parameters with the highest prediction accuracy are retained as the ideal parameters of the spatiotemporal convolutional graph model to obtain the spatiotemporal Predictive model.

10. A regional multi-site based semi-dynamic graph neural network according to claim 9 The prediction method is characterized in that In step S32, the number of layers of the long short-term memory neural network is set to 2, the batch_first parameter is set to true, and the input size is , N is the batch size, L is the hidden layer sequence length, is the hidden layer embedding size, and the output size is , is the size of the output layer; In the fully connected layer, the end of the sequence output by the long short-term memory neural network is extracted and input, and a one-dimensional output is obtained through the linear layer. , whose size is the product of the number of sites and the number of prediction steps, and then reshapes the one-dimensional output into an output of the same shape , N is the number of sites, and pred is the number of prediction steps.

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