A decomposition-integration framework coupling singular spectrum analysis with a volatility-adaptive LSTM for stock price forecasting

Xin-Rui Lu; Da-Qing Zhang

cs.CEOct 17, 2026

A decomposition-integration framework coupling singular spectrum analysis with a volatility-adaptive LSTM for stock price forecasting

Authors: Xin-Rui Lu, Da-Qing Zhang

Organizations: School of Science, University of Science and Technology Liaoning, Anshan, 114051, Liaoning, PR China

Abstract

Financial time series exhibit multi-scale, high-noise, and non-stationary volatility dynamics. Decomposition--integration frameworks built on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) have advanced this field, yet existing CEEMDAN hybrids remain unstable: high-frequency mode mixing contaminates downstream predictors, and the heterogeneous volatility of decomposed components is not matched by any volatility-adaptive mechanism. To address these gaps, this paper proposes CEEMDAN-SE-SSA-VMGL. CEEMDAN adaptively decomposes the original series, and sample-entropy-driven K-means clustering groups the intrinsic mode functions into high-, mid-, and low-frequency and residual components. Singular spectrum analysis (SSA) is then applied to the high-frequency group, whose singular-value decomposition of the trajectory matrix and principal-component reconstruction decouple intertwined pseudo-oscillatory modes and alleviate mode mixing. We further design a plug-and-play robust LSTM variant, the volatility-modulated GARCH-LSTM (VMGL), which embeds a differentiable GARCH(1,1) layer inside the LSTM cell as a risk gate; the estimated conditional volatility multiplicatively scales the forget-gate activation, letting memory decay adapt to the instantaneous volatility while remaining fully end-to-end trainable. Component predictions are linearly combined. On five US and Chinese financial series (CSI 300, CSI 500, AAPL, GOOGL, S&P 500), the proposed model is compared with eleven benchmark architectures, lowering the forecasting error by 9% to 24% relative to the best benchmark and by 60% to 70% relative to a standard LSTM. Diebold--Mariano and model-confidence-set (MCS) tests confirm the statistical significance and stability of the gains, spectral analyses verify the SSA denoising efficacy, and backtesting demonstrates the framework's economic utility.

Figures & tables

Figure 1: Flowchart of the proposed CEEMDAN-SE-SSA-VMGL model.

Figure 2: Schematic diagram of LSTM structure.

Figure 3: Schematic diagram of VMGL structure.

Metric	CSI300	CSI500	AAPL	GOOGL	S&P500
Sample Count	3401	3401	3522	3521	3521
Mean	3493.11	5403.82	52.59	52.68	221.78
Min	2086.97	2749.13	$-3.17$	9.47	72.26
Max	5807.72	11545.89	195.83	148.41	465.31
25%	2850.30	4368.61	10.07	20.50	124.00
50%	3463.64	5549.67	23.66	39.58	193.95

Table 1: Descriptive statistics of the five datasets.

Metric	CSI300	CSI500	AAPL	GOOGL	S&P500
Sample Count	3401	3401	3522	3521	3521
ARCH LM Statistic	361.789	527.848	1047.014	95.469	1354.502
ARCH LM p-value	$1.25\times 10^{-71}$ ***	$4.92\times 10^{-107}$ ***	$1.39\times 10^{-218}$ ***	$4.38\times 10^{-16}$ ***	$6.59\times 10^{-285}$ ***
Ljung–Box Statistic	859.26	1451.08	3135.80	155.02	4571.61
Ljung–Box p-value	$3.72\times 10^{-178}$ ***	$9.29\times 10^{-306}$ ***	$<1.00\times 10^{-4}$ ***	$3.46\times 10^{-28}$ ***	$<1.00\times 10^{-4}$ ***
Heteroscedasticity	Yes	Yes	Yes	Yes	Yes

Table 2: Heteroscedasticity test results.

Figure 4: CEEMDAN decomposition results of the CSI 300 index.

IMF	CSI 300	CSI 500	AAPL	GOOGL	S&P 500
IMF1	1.846552 ∗	1.781471 ∗	2.022148 ∗	1.962428 ∗	2.019072 ∗
IMF2	2.092127 ∗	2.007787 ∗	2.061346 ∗	2.039358 ∗	2.062140 ∗
IMF3	0.639570 ∗∗	0.580142 ∗∗	0.587209 ∗∗	0.551567 ∗∗	0.633861 ∗∗
IMF4	0.521673 ∗∗	0.303984 ∗∗	0.276680 ∗∗∗	0.306324 ∗∗	0.357532 ∗∗
IMF5	0.293600 ∗∗∗	0.144551 ∗∗∗	0.107043 ∗∗∗	0.178634 ∗∗∗	0.158862 ∗∗∗
IMF6	0.090415 ∗∗∗	0.116054 ∗∗∗	0.064036 ∗∗∗	0.127954 ∗∗∗	0.101392 ∗∗∗

Table 3: Sample entropy of each sub-series from the CEEMDAN decomposition.

Component	CSI 300	CSI 500	AAPL	GOOGL	S&P 500
HF	1.765321	1.708424	1.121385	1.169481	1.441444
MF	0.606492	0.472504	0.587209	0.471241	0.559882
LF	0.054099	0.040794	0.075098	0.029009	0.046345
Res	0.001972	0.001048	0.000993	0.001269	0.001341

Table 4: Sample entropy of the compressed HF, MF, LF, and residual (Res) components.

Group	CSI 300	CSI 500	AAPL	GOOGL	S&P 500
HF	24.02	19.25	23.70	23.21	23.63
MF	15.46	12.84	16.47	16.35	13.07
LF	0.05	0.08	0.54	0.05	0.02
Res	0.00	0.00	0.00	0.00	0.00

Table 5: Annualized volatility of frequency-grouped co-IMF components across five datasets.

Metric	Model	CSI300	CSI500	AAPL	GOOGL	S&P500
RMSE	LSTM	36.35	50.99	2.8464	2.6203	4.2406
	VMGL	33.69	49.88	2.3481	2.3308	3.5458
	EMD-SE-LSTM	55.94	32.29	1.9680	2.3258	2.0771
	EMD-SE-VMGL	25.98	27.13	1.9512	2.1187	1.9009
	EEMD-SE-LSTM	60.58	127.11	3.3011	2.6965	6.9613
	EEMD-SE-VMGL	23.87	40.94	1.8358	1.5530	2.7535

Table 6: Performance comparison of different models across five datasets.

Figure 5: Scatter plots of actual versus predicted closing prices for the proposed CEEMDAN-SE-SSA-VMGL model across five datasets.

Figure 6: Percentage improvement of the proposed CEEMDAN-SE-SSA-VMGL model over eleven benchmark architectures.

Figure 7: SSA-enhanced processing suppresses mode mixing.

Dataset	Variance_Before	Variance_After	Variance_Change(%)	ACF1_Before	ACF1_After	ACF1_Change
CSI300	968.1809	491.2525	-49.26	0.0517	0.3543	0.3026
CSI500	3343.4916	1794.3963	-46.33	0.0846	0.3747	0.2901
AAPL	0.9611	0.4482	-53.37	0.1234	0.4409	0.3174
GOOGL	0.6352	0.2902	-54.31	0.0867	0.4434	0.3567
S&P500	3.3180	1.4070	-57.60	0.0968	0.4940	0.3972

Table 7: SSA denoising performance comparison across all datasets.

Figure 8: Performance comparison of different models across five datasets in terms of

R^{2}

, RMSE, MAE, and MAPE.

Model	S&P500		AAPL		CSI300		CSI500		GOOGL
Model	Stat.	p-value	Stat.	p-value	Stat.	p-value	Stat.	p-value	Stat.	p-value
LSTM	-10.67***	0.0000	-9.36***	0.0000	-10.33***	0.0000	-8.91***	0.0000	-6.92***	0.0000
VMGL	-11.70***	0.0000	-8.88***	0.0000	-9.46***	0.0000	-8.96***	0.0000	-6.06***	0.0000
EMD-SE-LSTM	-7.74***	0.0000	-10.26***	0.0000	-17.73***	0.0000	-6.81***	0.0000	-9.19***	0.0000
EMD-SE-VMGL	-4.49***	0.0000	-9.92***	0.0000	-9.99***	0.0000	-5.83***	0.0000	-7.77***	0.0000
EEMD-SE-LSTM	-11.33***	0.0000	-11.64***	0.0000	-9.31***	0.0000	-9.95***	0.0000	-10.22***	0.0000

Table 8: Diebold–Mariano test results of the proposed CEEMDAN-SE-SSA-VMGL model versus other benchmark models.

Model	S&P500		AAPL		CSI300		CSI500		GOOGL
Model	$T_{R}$	$T_{SQ}$	$T_{R}$	$T_{SQ}$	$T_{R}$	$T_{SQ}$	$T_{R}$	$T_{SQ}$	$T_{R}$	$T_{SQ}$
LSTM	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
VMGL	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
EMD-SE-LSTM	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
EMD-SE-VMGL	0.0010	0.0010	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
EEMD-SE-LSTM	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Table 9: Model Confidence Set (MCS) test results.

Dataset	Metric	$H=5$	$H=10$	$H=15$	$H=20$
CSI 300	$R^{2}$	0.996932	0.997338	0.997163	0.997034
	RMSE	12.7805	11.9096	12.1404	12.2485
	MAE	10.3809	9.6850	9.9169	9.9396
	MAPE (%)	0.2714	0.2534	0.2591	0.2607
CSI 500	$R^{2}$	0.996156	0.996237	0.995996	0.995863
	RMSE	20.0087	19.9813	20.6772	20.8890

Table 10: Best-run metrics per dataset across sliding window sizes

H

Model	CSI 300	CSI 500	AAPL	GOOGL	S&P 500
LSTM	100.4	98.4	27.6	75.7	104.0
VMGL	72.1	91.0	64.2	77.6	95.5
EMD-SE-LSTM	599.9	658.4	686.5	653.2	597.6
EMD-SE-VMGL	763.9	831.4	752.5	801.2	775.8
EEMD-SE-LSTM	699.4	732.5	675.1	634.0	682.0
EEMD-SE-VMGL	852.4	898.2	767.5	793.8	768.5

Table 11: Total running time in seconds.

Strategy	CSI300	CSI500	AAPL	GOOGL	S&P500
Buy-and-Hold
Buy-and-Hold	$-$ 0.0959	$-$ 0.0633	0.1205	0.0924	0.1083
Prediction-driven directional strategy
CEEMDAN-SE-SSA-VMGL	0.5357	0.6065	0.6630	0.6214	0.6612
CEEMDAN-SE-SSA-LSTM	0.5153	0.5783	0.6115	0.5931	0.6193
VMD-SE-VMGL	0.4411	0.4228	0.5986	0.5366	0.5035

Table 12: Sharpe ratio performance by investment strategy.

Figure 9: Cumulative log-returns of the signal-based long/flat strategy across five datasets.

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Learning volatility-adaptive forgetting via GARCH-modulated LSTM for stock price forecasting

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