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Fig 1.

A diagram of a digraph in two dimensions.

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Fig 2.

Diagram of a four-dimensional digraph.

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Fig 3.

Structure diagram of four-dimensional directed GCN model.

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Fig 4.

Structure diagram of four-dimensional directed GCN-LSTM model.

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Table 1.

Yangtze River Delta urban agglomeration data set.

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Table 2.

Beijing-Tianjin-Hebei urban agglomeration data set.

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Fig 5.

Monte Carlo Time Series Cross-Validation data sets during the first 330 days.

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Table 3.

Typical correlation analysis of test data.

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Fig 6.

Main view of 3D visual curve of the typical-load coefficient of xr and X.

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Fig 7.

Left view of 3D visual curve of the typical-load coefficient of xr and X.

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Fig 8.

Main view of 3D visual curve of the typical-load coefficient of yr and Y.

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Fig 9.

. Left view of 3D visual curve of the typical-load coefficient of yr and Y.

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Fig 10.

Main view of 3D visual curve of the cross-load coefficient of yr and X.

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Fig 11.

Left view of the 3D visual curve of the cross-load coefficient of yr and X.

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Fig 12.

Main view of 3D visual curve of the cross-load coefficient of xr and Y.

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Fig 13.

Left view of the 3D visual curve of the cross-load coefficient of xr and Y.

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Table 4.

Evaluation metrics for various models using the data set of Yangtze River Delta City Cluster.

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Table 5.

Evaluation metrics for various models using the data set of Yangtze River Delta City Cluster.

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Fig 14.

Curve of predicted and true PM2.5 concentration in Suzhou by five methods.

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Fig 15.

Curve of predicted and true PM2.5 concentration in Beijing by five methods.

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