A novel hybrid LMD-SPF forecasting framework for financial time series: Evidence from gold returns

Muhammad Aamir; Hasnain Iftikhar; Jawaria Nasir; Paulo Canas Rodrigues; Abdulmajeed Atiah Alharbi; Jeza Allohibi; Muhammad Aamir; Hasnain Iftikhar; Jawaria Nasir; Paulo Canas Rodrigues; Abdulmajeed Atiah Alharbi; Jeza Allohibi

doi:10.3934/math.2025974

AIMS Mathematics

2025, Volume 10, Issue 9: 21875-21901. doi: 10.3934/math.2025974

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A novel hybrid LMD-SPF forecasting framework for financial time series: Evidence from gold returns

1.
Department of Statistics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
2.
Department of Statistics, University of Peshawar, Peshawar 25120, Pakistan
3.
Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
4.
Department of Statistics, Federal University of Bahia, Salvador 40170-110, Brazil
5.
Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia

Received: 22 July 2025 Revised: 07 September 2025 Accepted: 16 September 2025 Published: 19 September 2025
MSC : 62M10, 91G70, 68T07, 62P05

Accurately forecasting gold returns is critical for investors, policymakers, and risk managers, yet it remains challenging due to the coexistence of deterministic cycles, stochastic volatility, and nonlinear dependencies. This study introduced a novel hybrid model, local mean decomposition (LMD), stochastic product function (SPF), autoregressive integrated moving average (ARIMA), and long short-term memory (LSTM) (LMD-SPF-ARIMA-LSTM), that integrates signal decomposition, statistical diagnostics, and machine learning to address these complexities. LMD first decomposes the return series into product functions (PFs). An SPF then applies formal tests (augmented Dickey-Fuller (ADF) for stationarity, Brock-Dechert-Scheinkman (BDS) for linearity, and correlation filtering) to classify PFs, allocating them to ARIMA for linear-stationary components and to LSTM for nonlinear-nonstationary ones. Using daily gold return data from June 2020 to May 2025, the proposed framework achieved substantially improves over benchmarks, reducing mean absolute error (MAE) by more than 55% compared to ARIMA, lowering root mean squared error (RMSE) by 57% relative to LSTM, and attaining 85.31% directional accuracy-over three percentage points higher than the best competing hybrid. Unlike previous LMD-ARIMA-LSTM approaches that treat all decomposed components uniformly, our method tailors the modeling strategy to the statistical properties of each PF, reducing redundancy, lowering computational cost, and enhancing generalization. These results not only demonstrate the methodological significance of combining decomposition with statistically informed model selection but also provide practical value by delivering more reliable and interpretable forecasts for financial decision-making in volatile markets.
- gold return forecasting,
- local mean decomposition,
- ARIMA,
- LSTM,
- hybrid forecasting models,
- financial time series,
- econometric modeling,
- deep learning in finance
Citation: Muhammad Aamir, Hasnain Iftikhar, Jawaria Nasir, Paulo Canas Rodrigues, Abdulmajeed Atiah Alharbi, Jeza Allohibi. A novel hybrid LMD-SPF forecasting framework for financial time series: Evidence from gold returns[J]. AIMS Mathematics, 2025, 10(9): 21875-21901. doi: 10.3934/math.2025974

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Abstract

Accurately forecasting gold returns is critical for investors, policymakers, and risk managers, yet it remains challenging due to the coexistence of deterministic cycles, stochastic volatility, and nonlinear dependencies. This study introduced a novel hybrid model, local mean decomposition (LMD), stochastic product function (SPF), autoregressive integrated moving average (ARIMA), and long short-term memory (LSTM) (LMD-SPF-ARIMA-LSTM), that integrates signal decomposition, statistical diagnostics, and machine learning to address these complexities. LMD first decomposes the return series into product functions (PFs). An SPF then applies formal tests (augmented Dickey-Fuller (ADF) for stationarity, Brock-Dechert-Scheinkman (BDS) for linearity, and correlation filtering) to classify PFs, allocating them to ARIMA for linear-stationary components and to LSTM for nonlinear-nonstationary ones. Using daily gold return data from June 2020 to May 2025, the proposed framework achieved substantially improves over benchmarks, reducing mean absolute error (MAE) by more than 55% compared to ARIMA, lowering root mean squared error (RMSE) by 57% relative to LSTM, and attaining 85.31% directional accuracy-over three percentage points higher than the best competing hybrid. Unlike previous LMD-ARIMA-LSTM approaches that treat all decomposed components uniformly, our method tailors the modeling strategy to the statistical properties of each PF, reducing redundancy, lowering computational cost, and enhancing generalization. These results not only demonstrate the methodological significance of combining decomposition with statistically informed model selection but also provide practical value by delivering more reliable and interpretable forecasts for financial decision-making in volatile markets.

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