Table 1.
Core components and operational phases of the IAO.
Fig 1.
Workflow of the IAO algorithm, including population initialization, information acquisition, fitness evaluation, and best-guided updating.
Fig 2.
Overall framework of the proposed IVMD-CNN-SVM-LSCV-B runoff prediction model.
Fig 3.
Monthly hydrological characteristics of the Yangtze River Basin at two stations.
(A) Monthly runoff series at the Hankou station, including training and testing datasets, linear trend, and identified flood months. (B) Monthly runoff series at the Luoshan station, including training and testing datasets, linear trend, and identified flood months. (C) Monthly variations in multi-year mean water level and flow rate at the Hankou station, with warning and defense thresholds indicated. (D) Monthly variations in multi-year mean water level and flow rate at the Luoshan station, with warning and defense thresholds indicated.
Fig 4.
Sensitivity analysis of VMD parameters at the Luoshan station.
(A) Heatmap of test RMSE values under different combinations of the number of modes K and the penalty factor . (B) Test RMSE as a function of K with
fixed at different values. (C) Test RMSE as a function of
with K fixed at different values.
Table 2.
Optimized VMD parameters and objective function values at Hankou and Luoshan stations.
Fig 5.
Optimization performance comparison of different algorithms for VMD and the corresponding IMF results.
(A) Convergence curves of different optimization algorithms for VMD parameter optimization at the Hankou station. (B) Convergence curves of different optimization algorithms for VMD parameter optimization at the Luoshan station. (C) IMFs obtained by IVMD at the Hankou station. (D) IMFs obtained by IVMD at the Luoshan station.
Fig 6.
Iterative trajectories of optimized VMD parameters at two hydrological stations.
(A) Iteration curve of the number of modes K at the Hankou station. (B) Iteration curve of the penalty factor α at the Hankou station. (C) Iteration curve of the number of modes K at the Luoshan station. (D) Iteration curve of the penalty factor α at the Luoshan station.
Table 3.
Performance metrics of different models on the training dataset with VMD and IVMD preprocessing.
Table 4.
Performance metrics of different models on the testing dataset with VMD and IVMD preprocessing.
Fig 7.
Comparison of runoff point prediction results under IVMD and VMD preprocessing.
(A) Runoff point prediction curve at the Hankou station using an IVMD-based model. (B) Runoff point prediction curve at the Luoshan station using an IVMD-based model. (C) Runoff point prediction curve at the Hankou station using a VMD-based model. (D) Runoff point prediction curve at the Luoshan station using a VMD-based model.
Fig 8.
Local enlarged comparison of runoff point predictions under IVMD and VMD preprocessing.
(A) Local enlarged runoff point prediction curve at the Hankou station using an IVMD-based model. (B) Local enlarged runoff point prediction curve at the Luoshan station using an IVMD-based model. (C) Local enlarged runoff point prediction curve at the Hankou station using a VMD-based model. (D) Local enlarged runoff point prediction curve at the Luoshan station using a VMD-based model.
Table 5.
KDE bandwidth comparison for IVMD-CNN-SVM: error fit (EMAE, ERMSE, ER2) and 90% interval metrics (PICP, PINAW, F) at two stations.
Fig 9.
Comparison of interval prediction performance using different KDE bandwidth selection methods.
(A) Empirical and fitted CDFs of normalized prediction errors at the Hankou station. (B) Empirical and fitted CDFs of normalized prediction errors at the Luoshan station. (C) Prediction intervals obtained using different KDE bandwidth selection methods at the Hankou station. (D) Prediction intervals obtained using different KDE bandwidth selection methods at the Luoshan station.
Table 6.
Comparison of interval prediction performance between IVMD-CNN-SVM-LSCV-B and Bootstrap methods under different confidence levels.
Fig 10.
Comparison of interval prediction performance between Bootstrap and LSCV-B methods at two stations.
(A) Bootstrap-based prediction intervals at the Hankou station. (B) Bootstrap-based prediction intervals at the Luoshan station. (C) LSCV-B-based prediction intervals at the Hankou station. (D) LSCV-B-based prediction intervals at the Luoshan station.