Performance Comparison of Deep and Shallow Artificial Neural Networks for the S&P 500 Index

Year 2026, Volume: 10 Issue: 1, 8 - 19, 12.03.2026

Mete Özdemir

https://doi.org/10.34110/forecasting.1867852

https://izlik.org/JA44FH89PD

Abstract

In recent years, artificial neural network architectures have been increasingly proposed for time series forecasting and have become a major focus of research. A key discussion in this field concerns the selection of an appropriate network architecture for modeling complex time series structures. This study comparatively evaluates the forecasting performance of six artificial neural network architectures (SMN, DNM, MLP, Pi-Sigma, GRU, and LSTM) using S&P 500 index data characterized by high volatility and noise. To address the local minimum problem during model training, the Particle Swarm Optimization (PSO) algorithm, which has strong global search capability, is employed. The results indicate that high-order (multiplicative) neuron models, such as Pi-Sigma, SMN and DNM, exhibit greater robustness and stability against noise in financial data compared to deep learning models like LSTM and GRU. As the dataset size increases from 100 to 250 observations, the learning capacity of all models improves, leading to a substantial reduction in Root Mean Square Error (RMSE) values. According to the Friedman test results, the Deep Dendritic Neuron Model (DNM) and the Multilayer Perceptron (MLP) achieve the lowest mean ranks and emerge as the most successful models. In addition, SMN and Pi-Sigma models demonstrate extremely low standard deviation values, particularly in long-term datasets, highlighting their reliability in financial forecasting. Overall, the findings show that multiplicative architectures, which can capture complex relationships with fewer parameters, provide more efficient and stable forecasting performance than conventional deep learning approaches in financial time series analysis.

Keywords

Forecasting , Artificial Neural Networks , Shallow Neural Networks , Deep Neural Networks

Ethical Statement

Ethics committee approval is not required for this study as it is based on public historical data.

References

• Akdeniz, E., Egrioglu, E., Bas, E., & Yolcu, U. (2018). An ARMA type pi-sigma artificial neural network for nonlinear time series forecasting. Journal of Automation, Mobile Robotics and Intelligent Systems, 8(2), 121–132.
• Akram, U., Ghazali, R., & Mushtaq, M. F. (2017). A comprehensive survey on Pi-Sigma neural network for time series prediction. International Journal on Advanced Science, Engineering and Information Technology, 7(6), 2193–2199.
• Aksakal, S. S., & Egrioglu, E. (2025). A hybrid forecasting method based on exponential smoothing and multiplicative neuron model artificial neural network. Turkish Journal of Forecasting, 9(2), 37–43.
• Bas, E., Egrioglu, E., & Cansu, T. (2024). Robust training of median dendritic artificial neural networks for time series forecasting. Expert Systems with Applications, 238, 122080.
• Dahmani, M. (2025). Price prediction of S&P 500 using machine learning. The Journal of Prediction Markets, 19(1), 53–64.
• Egrioglu, E., & Bas, E. (2022). A new automatic forecasting method based on a new input significancy test of a single multiplicative neuron model artificial neural network. Network: Computation in Neural Systems, 33(1–2), 1–16.
• Egrioglu, E., & Bas, E. (2023). Modified Pi-Sigma artificial neural networks for forecasting. Granular Computing, 8, 131–135.
• Egrioglu, E., & Bas, E. (2024). A new deep neural network for forecasting: Deep dendritic artificial neural network. Artificial Intelligence Review, 57, 171.
• Fischer, T., & Krauss, C. (2018). Deep learning with long short-term memory networks for financial market predictions. European Journal of Operational Research, 270(2), 654–669.
• Ghashami, F., Kamyar, K., & Riazi, S. A. (2021). Prediction of stock market index using a hybrid technique of artificial neural networks and particle swarm optimization. Applied Economics and Finance, 8(3), 1–10.
• Jia, D., Zheng, S., Yang, L., Todo, Y., & Gao, S. (2018). A dendritic neuron model with nonlinearity validation on Istanbul stock and Taiwan futures exchange indexes prediction. Proceedings of CCIS 2018, 1–13.
• Jiang, M., Liu, J., Zhang, L., & Liu, C. (2020). An improved stacking framework for stock index prediction by leveraging tree-based ensemble models and deep learning algorithms. Physica A, 541, 122272.
• Kara, M. A., & Bayyurt, D. (2025). Stock price forecasting using single multiplicative neuron model artificial neural network. İşletme, 6(2), 213–225.
• Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, 4, 1942–1948.
• Khashei, M., & Bijari, M. (2011). A novel hybridization of artificial neural networks and ARIMA models for time series forecasting. Applied Soft Computing, 11(2), 2664–2675.
• Kocak, C., Egrioglu, E., & Bas, E. (2021). A new deep intuitionistic fuzzy time series forecasting method based on long short-term memory. The Journal of Supercomputing, 77, 6178–6196.
• Kolemen, E., Egrioglu, E., Bas, E., & Turkmen, M. (2025). A new hybrid neural network based on gated recurrent unit and simple exponential smoothing for forecasting. Computing, 107, 196.
• Lin, S. L., & Huang, H. W. (2020). Improving deep learning for forecasting accuracy in financial data. Discrete Dynamics in Nature and Society, 2020, 1–12.
• Muncharaz, J. O. (2020). Comparing classic time series models and the LSTM recurrent neural network: An application to S&P 500 stocks. Finance, Markets and Valuation, 6(2), 137–148.
• Pirani, M., Thakkar, P., Jivrani, P., Bohara, M. H., & Garg, D. (2022). A comparative analysis of ARIMA, GRU, LSTM and BiLSTM on financial time series forecasting. IEEE ICDCECE, 1–6.
• Sako, K., Mpinda, B. N., & Rodrigues, P. C. (2022). Neural networks for financial time series forecasting. Entropy, 24(5), 657.
• Smyl, S. (2020). A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting. International Journal of Forecasting, 36(1), 75–85.
• Wu, G., Zhang, J., & Xue, H. (2023). Long-term prediction of hydrometeorological time series using a PSO-based combined model composed of EEMD and LSTM. Sustainability, 15(17), 13209.
• Yadav, R. N., Mishra, S., & Kalra, P. K. (2007). Time series prediction using single multiplicative neuron model. Applied Soft Computing, 7(3), 1047–1055.
• Zeng, X., Liang, C., Yang, Q., Wang, F., & Cai, J. (2025). Enhancing stock index prediction: A hybrid LSTM-PSO model. PLoS ONE, 20(1), e0310296.
• Zhang, G. P. (2003). Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing, 50, 159–175.
• Zhang, C., Sjarif, N. N. A., & Ibrahim, R. (2023). Deep learning models for price forecasting of financial time series: A review of recent advancements. WIREs Data Mining and Knowledge Discovery, 14(1), e1519.
• Zhao, L., & Yang, Y. (2009). PSO-based single multiplicative neuron model for time series prediction. Expert Systems with Applications, 36(2), 2805–2812.
• Zhou, T., Gao, S., Wang, J., Chu, C., Todo, Y., & Tang, Z. (2016). Financial time series prediction using a dendritic neuron model. Knowledge-Based Systems, 105, 214–224.