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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Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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

Xin-Rui Lu, Da-Qing Zhang

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

Time Series Forecasting Time Series Modeling Long Short-Term Memory Networks Recurrent Neural Networks

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Stock price forecasting is hindered by the heteroscedastic nature of financial time series, in which turbulent and tranquil regimes alternate. Although the Long Short-Term Memory (LSTM) network is widely used for this task, its gates are oblivious to such volatility variation, so memory retention remains fixed regardless of market conditions. A common remedy is to supply volatility estimates as an additional input feature, yet this leaves the internal memory dynamics untouched. Instead, volatility can serve as a direct regulatory signal over the gate activation itself. This paper presents the Gsigmoid-LSTM model, which embeds Generalized Autoregressive Conditional Heteroscedasticity (GARCH(1,1))-derived volatility estimates into the LSTM forgetting gate through the Gsigmoid activation function, a volatility-adaptive variant of the sigmoid that combines a volatility adjustment factor with slope regulation via a learnable parameter. The model thereby automatically prioritizes recent information during turbulent periods while preserving long-term dependencies in stable markets. In the engineering application of daily stock price forecasting, extensive experiments, including activation function comparisons, baseline evaluations, and ablation studies, demonstrate that this structural gating modification effectively improves forecasting accuracy and robustness. The proposed model significantly outperforms a feature-level fusion baseline carrying the same volatility signal on all five datasets, as confirmed by Diebold-Mariano tests with all p<0.01, attains the best average rank of 1.00 in the Friedman test across nine models, and achieves the lowest MSE and MAE on all five datasets.

Abstract

Time Series Forecasting

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