F10.7 Index Prediction: A Multiscale Decomposition Strategy with Wavelet Transform for Performance Optimization

Xuran Ma; Xuebao Li; Yanfang Zheng; Yongshang Lv; Xiaojia Ji; Jiancheng Xu; Hongwei Ye; Zixian Wu

astro-ph.IMFeb 24, 2026

F10.7 Index Prediction: A Multiscale Decomposition Strategy with Wavelet Transform for Performance Optimization

Authors: Xuran Ma, Xuebao Li, Yanfang Zheng, Yongshang Lv, Xiaojia Ji, Jiancheng Xu, Hongwei Ye, Zixian Wu,

Organizations: School of Computer Science, Jiangsu University of Science and Technology, Zhenjiang 212100, People′s Republic of China · State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, People′s Republic of China · Yunnan Astronomical Observatory, Chinese Academy of Sciences, Kunming 650216, China · Radio Cosmology Lab, Centre for Astronomy and Astrophysics Research, Department of Physics, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia ·

Abstract

In this study, we construct Dataset A for training, validation, and testing, and Dataset B to evaluate generalization. We propose a novel F10.7 index forecasting method using wavelet decomposition, which feeds F10.7 together with its decomposed approximate and detail signals into the iTransformer model. We also incorporate the International Sunspot Number (ISN) and its wavelet-decomposed signals to assess their influence on prediction performance. Our optimal method is then compared with the latest method from S. Yan et al. (2025) and three operational models (SWPC, BGS, CLS). Additionally, we transfer our method to the PatchTST model used in H. Ye et al. (2024) and compare our method with theirs on Dataset B. Key findings include: (1) The wavelet-based combination methods overall outperform the baseline using only F10.7 index. The prediction performance improves as higher-level approximate and detail signals are incrementally added. The Combination 6 method integrating F10.7 with its first to fifth level approximate and detail signals outperforms methods using only approximate or detail signals. (2) Incorporating ISN and its wavelet-decomposed signals does not enhance prediction performance. (3) The Combination 6 method significantly surpasses S. Yan et al. (2025) and three operational models, with RMSE, MAE, and MAPE reduced by 18.22%, 15.09%, and 8.57%, respectively, against the former method. It also excels across four different conditions of solar activity. (4) Our method demonstrates superior generalization and prediction capability over the method of H. Ye et al. (2024) across all forecast horizons. To our knowledge, this is the first application of wavelet decomposition in F10.7 prediction, substantially improving forecast performance.

Figures & tables

Figure 1: Daily F10.7 index of DRAO data from 1957 to 2020. The training set, validation set and testing set are divided according to solar cycles. The blue part in the data is used for model training, the green part for model validation, and the yellow part for model testing.

Figure 2: Comparison of Langfang data and DRAO data for the time period 2024-2025.

Figure 3: Wavelet multi-level decomposition. X AL and X DL respectively represent the approximate signals and the detail signals of the L-th level.

Figure 4: The prediction scheme of the F10.7 index for future periods based on the iTransformer model.

Figure 5: The schematic diagram for F10.7 prediction method based on the iTransformer model and wavelet decomposition. The lower section of the diagram displays the integrated inputs to the model, which include the F10.7 index, ISN, their respective approximate signals and detail signals derived from wavelet decomposition, along with the original data. The upper section presents the iTransformer model architecture, which primarily consists of an embedding layer, a projection layer, and multiple inverted Transformer Blocks.

Hyperparameter	Concrete content
Model dimension	256
Training epochs	50
Number of heads	4
Number of encoder layers	2
Loss function	MSE
Feed-forward network dimension	512

Table 1: Hyperparameter settings of the iTransformer model that integrates wavelet-decomposed signals.

Evaluation indicators	Formulas	Descriptions
RMSE	$\text{RMSE}=\sqrt{\frac{1}{T\times S}\sum_{t=1}^{T}\sum_{s=1}^{S}(\hat{Y}_{t,s}-Y_{t,s})^{2}}$	Better closer to zero
MAE	$\text{MAE}=\frac{1}{T\times S}\sum_{t=1}^{T}\sum_{s=1}^{S}\|\hat{Y}_{t,s}-Y_{t,s}\|$	Better closer to zero
MAPE	$\text{MAPE}=\frac{1}{T\times S}\sum_{t=1}^{T}\sum_{s=1}^{S}\left\|\frac{\hat{Y}_{t,s}-Y_{t,s}}{Y_{t,s}}\right\|\times 100\%$	Better closer to zero

Table 2: Formulas and descriptions of evaluation indicators.

Figure 6: The wavelet-decomposed approximate signals of F10.7 index.

Figure 7: The wavelet-decomposed detail signals of F10.7 index.

Figure 8: The wavelet-decomposed approximate signals of ISN.

Figure 9: The wavelet-decomposed detail signals of ISN.

Combination	Number of	The content of the combination
number	variables
1	1	Original F10.7 index
2	3	The original F10.7 index with its first level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 )
3	5	The original F10.7 index with its first to second level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , F10.7 A2 , F10.7 D2 )
4	7	The original F10.7 index with its first to third level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , $\cdots$ , F10.7 A3 , F10.7 D3 )
5	9	The original F10.7 index with its first to fourth level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , $\cdots$ , F10.7 A4 , F10.7 D4 )

Table 3: Different combinations that integrates the approximate signals and detail signals of the F10.7 index.

Evaluation indicators	Combination	Prediction length (days)
Evaluation indicators	Combination	1	2	3	5	10	15	20	25	27	30	35	40	45
RMSE	Combination 1	5.236	6.089	6.919	8.397	11.875	11.768	12.349	13.633	13.143	13.298	13.775	14.320	14.495
	Combination 2	4.547	5.416	6.618	8.758	11.189	12.063	13.197	13.298	13.639	13.947	13.307	14.550	13.868
	Combination 3	4.204	4.695	5.700	7.348	10.469	11.565	12.987	11.663	14.076	12.004	13.497	12.666	13.708
	Combination 4	4.165	4.445	5.290	5.604	8.083	10.090	10.889	11.305	12.519	12.689	12.442	13.781	14.573
	Combination 5	4.232	4.484	5.189	6.027	7.974	10.056	10.352	11.606	11.975	12.421	12.438	13.126	13.658

Table 4: Performance results for each combination method on the corresponding testing datasets under different prediction lengths.

Figure 10: Bar chart comparing the prediction performance results for Combinations 1 to 6 methods under different prediction lengths.

Figure 11: Bar chart comparing the prediction performance of methods that incorporate different level of wavelet-decomposed signals when the prediction length is 27 days.

Figure 12: Bar chart comparing the prediction performance results for the methods of Combination 6, Combination A and Combination D under different prediction lengths.

Prediction length (days)	Combination 6			Combination A			Combination D
Prediction length (days)	RMSE	MAE	MAPE	RMSE	MAE	MAPE	RMSE	MAE	MAPE
1	4.526	2.225	2.062	4.525	2.295	2.134	4.505	2.246	2.089
2	4.587	2.381	2.223	4.711	2.523	2.353	4.908	2.577	2.401
3	5.173	2.644	2.458	5.487	2.949	2.727	5.608	3.088	2.871
5	6.125	3.327	3.100	6.404	3.552	3.290	6.691	3.688	3.420
10	8.039	4.562	4.220	8.753	4.946	4.545	8.276	4.645	4.343

Table 5: Performance results for the methods of Combination 6, Combination A and Combination D on the corresponding testing datasets under different prediction lengths.

Combination	Number of	The content of the combination
number	variables
7	12	Combination 6, ISN
8	14	Combination 6, as well as the ISN and its first level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 )
9	16	Combination 6, as well as the ISN and its first to second level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , ISN A2 , ISN D2 )
10	18	Combination 6, as well as the ISN and its first to third level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , $\cdots$ , ISN A3 , ISN D3 )
11	20	Combination 6, as well as the ISN and its first to fourth level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , $\cdots$ , ISN A4 , ISN D4 )

Table 6: Different combinations that integrates ISN along with its approximate signals and detail signals.

Figure 13: Bar chart comparing the prediction performance results of the Combination 6 method and the Combination 7 to 12 methods under different prediction lengths.

Method	Evaluation Indicator	2018	2019	2020	2021	2022	2023	Total
Combination 6	RMSE	1.357	1.847	4.649	7.050	18.735	23.206	10.774
	MAE	1.047	1.291	2.409	4.353	12.986	15.318	5.573
	MAPE	1.448	1.827	2.952	5.288	9.632	8.771	5.057
S. Yan et al. [32]	RMSE	1.400	1.998	4.862	8.714	21.379	24.278	13.175
	MAE	1.063	1.343	2.581	5.319	14.923	15.428	6.563
	MAPE	1.526	1.891	3.176	5.883	11.063	9.131	5.531

Table 7: The annual and total prediction performance results of the Combination 6 method, S. Yan et al. [32] , and the other three operational models in the prediction length of 27 days.

Figure 14: Bar chart of the total prediction performance of the Combination 6 method, S. Yan et al. [32] , and the other three operational models.

Condition	Model	RMSE	MAE	MAPE
Low (F10.7 $\leq$ 75)	Combination 6	2.121	1.459	2.062
	BGS	2.748	1.929	2.728
	CLS	2.977	2.219	3.165
	SWPC	2.470	1.726	2.437
Moderate (75 $<$ F10.7 $\leq$ 150)	Combination 6	11.836	8.696	7.807
	BGS	16.521	11.837	10.878

Table 8: Prediction performance of the Combination 6 method and the three operational models across four conditions in the prediction length of 27 days.

Figure 15: On the DRAO data and Langfang data, the line graphs showing the prediction performance of the Combination 6 method and the method of H. Ye et al. [35] under different prediction lengths.

Prediction length (days)	The method of H. Ye et al. [35]			Combination 6
Prediction length (days)	RMSE	MAE	MAPE	RMSE	MAE	MAPE
1	9.568	6.854	3.884	4.144	3.176	1.850
2	11.582	8.466	4.855	6.698	4.956	2.829
3	13.748	9.947	5.719	8.529	6.320	3.601
5	17.108	12.600	7.377	11.783	8.795	5.010
10	21.933	16.807	9.996	18.515	13.870	7.764

Table 9: Prediction performance results of the Combination 6 method and the method of H. Ye et al. [35] on the DRAO data under different forecast lengths.

Prediction length (days)	The method of H. Ye et al. [35]			Combination 6
Prediction length (days)	RMSE	MAE	MAPE	RMSE	MAE	MAPE
1	11.645	8.978	5.167	7.851	6.065	3.457
2	14.202	10.488	6.046	9.322	7.037	3.997
3	16.535	12.367	7.116	10.824	8.16 4	4.635
5	19.551	14.766	8.646	13.693	10.343	5.867
10	24.138	18.712	11.113	20.058	15.225	8.479

Table 10: Prediction performance results of the Combination 6 method and the method of H. Ye et al. [35] on the Langfang data under different forecast lengths.

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The F10.7 and F30 solar indices are the solar radio fluxes measured at wavelengths of 10.7 cm and 30 cm, respectively, which are key indicators of solar activity. F10.7 is valuable for explaining the impact of solar ultraviolet (UV) radiation on the upper atmosphere of Earth, while F30 is more sensitive and could improve the reaction of thermospheric density to solar stimulation. In this study, we present a new deep learning model, named the Solar Index Network, or SINet for short, to predict daily values of the F10.7 and F30 solar indices. The SINet model is designed to make medium-term predictions of the index values (1-60 days in advance). The observed data used for SINet training were taken from the National Oceanic and Atmospheric Administration (NOAA) as well as Toyokawa and Nobeyama facilities. Our experimental results show that SINet performs better than five closely related statistical and deep learning methods for the prediction of F10.7. Furthermore, to our knowledge, this is the first time deep learning has been used to predict the F30 solar index.

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