Forecasting systemic financial risks is crucial for effectively preventing and mitigating such risks. This paper introduces an integrated framework for systemic financial risk forecasting, with its core innovation being the novel combination of phase space reconstruction with delay parameterization (RDPM) and a radial basis function neural network (RBFNN). Based on stock trading data from 41 listed companies in the China A-share stock market, a complex financial network is constructed.
Citation: Yulian An, Zhaojun Liu, Zhenkuan Sun, Boan Li. Prediction of systemic risk of China's financial industry based on phase space reconstruction and machine learning[J]. Networks and Heterogeneous Media, 2026, 21(3): 773-800. doi: 10.3934/nhm.2026033
Forecasting systemic financial risks is crucial for effectively preventing and mitigating such risks. This paper introduces an integrated framework for systemic financial risk forecasting, with its core innovation being the novel combination of phase space reconstruction with delay parameterization (RDPM) and a radial basis function neural network (RBFNN). Based on stock trading data from 41 listed companies in the China A-share stock market, a complex financial network is constructed.
| [1] |
W. Silva, H. Kimura, V. A. Sobreiro, An analysis of the literature on systemic financial risk: A survey, J. Financ. Stab., 17 (2017), 91–114. https://doi.org/10.1016/j.jfs.2016.12.004 doi: 10.1016/j.jfs.2016.12.004
|
| [2] |
K. Bluwstein, M. Buckmann, A. Joseph, S. Kapadia, Ö. Şimşek, Credit growth, the yield curve and financial crisis prediction: Evidence from a machine learning approach, J. Int. Econ., 145 (2023), 103773. https://doi.org/10.1016/j.jinteco.2023.103773 doi: 10.1016/j.jinteco.2023.103773
|
| [3] |
S. Zedda, G. Cannas, Analysis of banks' systemic risk contribution and contagion determinants through the leave-one-out approach, J. Banking Finance, 112 (2020), 105160. https://doi.org/10.1016/j.jbankfin.2017.06.008 doi: 10.1016/j.jbankfin.2017.06.008
|
| [4] |
G. Hale, T. Kapan, C. Minoiu, Shock transmission through cross-border bank lending: Credit and real effects, Rev. Financ. Stud., 33 (2020), 4839–4882. https://doi.org/10.1093/rfs/hhz147 doi: 10.1093/rfs/hhz147
|
| [5] |
M. V. Oordt, C. Zhou, Systemic risk and bank business models, J. Appl. Econom., 34 (2019), 365–384. https://doi.org/10.1002/jae.2666 doi: 10.1002/jae.2666
|
| [6] |
V. Elenev, T. Landvoigt, S. V. Nieuwerburgh, A macroeconomic model with financially constrained producers and intermediaries, Econometrica, 89 (2021), 1361–1418. https://doi.org/10.3982/ECTA16438 doi: 10.3982/ECTA16438
|
| [7] |
S. Benoit, J. E. Colliard, C. Hurlin, C. Pérignon, Where the risks lie: A survey on systemic risk, Rev. Finance, 21 (2017), 109–152. https://doi.org/10.1093/rof/rfw026 doi: 10.1093/rof/rfw026
|
| [8] |
C. Diks, C. Hommes, J. Wang, Critical slowing down as an early warning signal for financial crises, Empir. Econ., 57 (2019), 1201–1228. https://doi.org/10.1007/s00181-018-1527-3 doi: 10.1007/s00181-018-1527-3
|
| [9] |
M. Summer, Financial contagion and network analysis, Annu. Rev. Financ. Econ., 5 (2013), 277–297. https://doi.org/10.1146/annurev-financial-110112-120948 doi: 10.1146/annurev-financial-110112-120948
|
| [10] |
Z. Du, J. C. Escanciano, Backtesting expected shortfall: Accounting for tail risk, Manage. Sci., 63 (2017), 940–958. https://doi.org/10.1287/mnsc.2015.2342 doi: 10.1287/mnsc.2015.2342
|
| [11] |
V. V. Acharya, L. H. Pedersen, T. Philippon, M. Richardson, Measuring systemic risk, Rev. Financ. Stud., 30 (2017), 2–47. https://doi.org/10.1093/rfs/hhw088 doi: 10.1093/rfs/hhw088
|
| [12] |
T. Adrian, M. K. Brunnermeier, CoVaR, Am. Econ. Rev., 106 (2016), 1705–1741. http://dx.doi.org/10.1257/aer.20120555 doi: 10.1257/aer.20120555
|
| [13] |
E. Grant, J. Yung, The double-edged sword of global integration: Robustness, fragility, and contagion in the international firm network, J. Appl. Econom., 36 (2021), 760–783. https://doi.org/10.1002/jae.2839 doi: 10.1002/jae.2839
|
| [14] |
M. Billio, M. Getmansky, A. W. Lo, L. Pelizzon, Econometric measures of connectedness and systemic risk in the finance and insurance sectors, J. Financ. Econ., 104 (2012), 535–559. https://doi.org/10.1016/j.jfineco.2011.12.010 doi: 10.1016/j.jfineco.2011.12.010
|
| [15] |
G. Bostanci, K. Yilmaz, How connected is the global sovereign credit risk network, J. Bank. Finance, 113 (2020), 105761. https://doi.org/10.1016/j.jbankfin.2020.105761 doi: 10.1016/j.jbankfin.2020.105761
|
| [16] |
G. J. Wang, L. Wan, Y. Feng, C. Xie, G. S. Uddin, Y. Zhu, Interconnected multilayer networks: Quantifying connectedness among global stock and foreign exchange markets, Int. Rev. Financ. Anal., 86 (2023), 102518. https://doi.org/10.1016/j.irfa.2023.102518 doi: 10.1016/j.irfa.2023.102518
|
| [17] |
Z. Ouyang, X. Zhou, Multilayer networks in the frequency domain: Measuring extreme risk connectedness of Chinese financial institutions, Res. Int. Bus. Finance, 65 (2023), 101944. https://doi.org/10.1016/j.ribaf.2023.101944 doi: 10.1016/j.ribaf.2023.101944
|
| [18] | Ö. Akgüller, M. A. Balcı, Detecting financial contagion through higher-order networks: A deep learning approach to emerging market risk, Comput. Econ., (2026), 1–67. https://doi.org/10.1007/s10614-025-11287-3 |
| [19] |
F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, G. Monfardini, The graph neural network model, IEEE Trans. Neural Networks, 20 (2009), 61–80. https://doi.org/10.1109/TNN.2008.2005605 doi: 10.1109/TNN.2008.2005605
|
| [20] |
D. Parnes, A. Gormus, Prescreening bank failures with K-means clustering: Pros and cons, Int. Rev. Financ. Anal., 93 (2024), 103222. https://doi.org/10.1016/j.irfa.2024.103222 doi: 10.1016/j.irfa.2024.103222
|
| [21] |
K. K. Sharma, A. Seal, Spectral embedded generalized mean based k-nearest neighbors clustering with s-distance, Expert Syst. Appl., 169 (2021), 114326. https://doi.org/10.1016/j.eswa.2020.114326 doi: 10.1016/j.eswa.2020.114326
|
| [22] |
A. R. Benson, D. F. Gleich, J. Leskovec, Higher-order organization of complex networks, Science, 353 (2016), 163–166. https://doi.org/10.1126/science.aad9029 doi: 10.1126/science.aad9029
|
| [23] |
J. Duffy, A. Karadimitropoulou, M. Parravano, Financial contagion in the laboratory: Does network structure matter, J. Money Credit Bank., 51 (2019), 1097–1136. https://doi.org/10.1111/jmcb.12563 doi: 10.1111/jmcb.12563
|
| [24] |
C. Filippopoulou, E. Galariotis, S. Spyrou, An early warning system for predicting systemic banking crises in the Eurozone: A logit regression approach, J. Econ. Behav. Organ., 172 (2020), 344–363. https://doi.org/10.1016/j.jebo.2019.12.023 doi: 10.1016/j.jebo.2019.12.023
|
| [25] | Z. Gu, Y. Xu, Chaotic dynamics analysis based on financial time series, Complexity, (2021), 2373423. https://doi.org/10.1155/2021/2373423 |
| [26] | M. Pirani, P. Thakka, P. Jivrani, M. H. Bohara, D. Garg, A comparative analysis of ARIMA, GRU, LSTM and BiLSTM on financial time series forecasting, in 2022 IEEE International Conference on Distributed Computing and Electrical Circuits and Electronics (ICDCECE), Ballari, India, 2022, 1–6. https://doi.org/10.1109/ICDCECE53908.2022.9793213 |
| [27] | N. Nazareth, Y. V. R. Reddy, Financial applications of machine learning: A literature review, Expert Syst. Appl., (2023) 119640. https://doi.org/10.1016/j.eswa.2023.119640 |
| [28] |
S. Ahmed, M. M. Alshater, A. El Ammari, H. Hammami, Artificial intelligence and machine learning in finance: A bibliometric review, Res. Int. Bus. Finance, 61 (2022), 101646. https://doi.org/10.1016/j.ribaf.2022.101646 doi: 10.1016/j.ribaf.2022.101646
|
| [29] |
G. Kou, X. Chao, Y. Peng, F. E. Alsaadi, E. Herrera-Viedma, Machine learning methods for systemic risk analysis in financial sectors, Technol. Econ. Dev. Econ., 25 (2019), 716–742. https://doi.org/10.3846/tede.2019.8740 doi: 10.3846/tede.2019.8740
|
| [30] |
J. Beutel, S. List, G. von Schweinitz, Does machine learning help us predict banking crises, J. Financ. Stab., 45 (2019), 100693. https://doi.org/10.1016/j.jfs.2019.100693 doi: 10.1016/j.jfs.2019.100693
|
| [31] | V. Balmaseda, M. Coronado, G. C. Santiago, Predicting systemic risk in financial systems using Deep Graph Learning, Intell. Syst., (2023), 200240. https://doi.org/10.1016/j.iswa.2023.200240 |
| [32] |
T. Leonidas, T. Alexandra, B. Aristeidis, Big data in financial risk management: Evidence, advances, and open questions: A systematic review, Front. Artif. Intell., 8 (2025), 1658375. https://doi.org/10.3389/frai.2025.1658375 doi: 10.3389/frai.2025.1658375
|
| [33] |
L. J. Vásquez-Serpa, C. Rodríguez, J. R. Pérez-Núñez, C. Navarro, Challenges of Artificial Intelligence for the prevention and identification of bankruptcy risk in financial institutions: A systematic review, J. Risk Financ. Manage., 18 (2025), 26. https://doi.org/10.3390/jrfm18010026 doi: 10.3390/jrfm18010026
|
| [34] |
A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D: Nonlinear Phenom., 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9
|
| [35] |
M. D. Johansyah, S. Vaidyanathan, K. Benkouider, A. Sambas, C. Aruna, S. K. Annavarapu, et al., A chaotic butterfly attractor model for economic stability assessment in financial systems, Mathematics, 13 (2025), 1633. https://doi.org/10.3390/math13101633 doi: 10.3390/math13101633
|
| [36] |
K. He, J. Shi, H. Fang, Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology, Math. Comput. Simul., 223 (2024), 253–274. https://doi.org/10.1016/j.matcom.2024.04.013 doi: 10.1016/j.matcom.2024.04.013
|
| [37] |
X. Yan, H. Wang, Y. An, Forecasting systemic risk of China's banking industry by PDE model, J. Appl. Anal. Comput., 13 (2023), 1–27. https://doi.org/10.11948/20230306 doi: 10.11948/20230306
|
| [38] |
L. Yang, T. Gao, Y. Lu, J. Duan, T. Liu, Neural network stochastic differential equation models with applications to financial data forecasting, Appl. Math. Model., 115 (2023), 279–299. https://doi.org/10.1016/j.apm.2022.11.001 doi: 10.1016/j.apm.2022.11.001
|
| [39] |
W. Sun, Y. An, Y. Gao, Systemic risk contagion in China's financial-real estate network: Modeling and forecasting via fractional order PDEs, Fractal Fract., 9 (2025), 557. https://doi.org/10.3390/fractalfract9090557 doi: 10.3390/fractalfract9090557
|
| [40] |
X. Guo, W. Han, J. Ren, Design of a prediction system based on the dynamical feed-forward neural network, Sci. China Inf. Sci., 66 (2023), 1–17. https://doi.org/10.1007/s11432-020-3402-9 doi: 10.1007/s11432-020-3402-9
|
| [41] |
X. Guo, Y. Sun, J. Ren, Low dimensional mid-term chaotic time series prediction by delay parameterized method, Inf. Sci., 516 (2020), 1–19. https://doi.org/10.1016/j.ins.2019.12.021 doi: 10.1016/j.ins.2019.12.021
|
| [42] |
A. Zhang, Z. Xu, Chaotic time series prediction using phase space reconstruction based conceptor network, Cogn. Neurodyn., 14 (2020), 849–857. https://doi.org/10.1007/s11571-020-09612-7 doi: 10.1007/s11571-020-09612-7
|
| [43] |
H. Ma, S. Leng, K. Aihara, L. Chen, Randomly distributed embedding making short-term high-dimensional data predictable, Proc. Natl. Acad. Sci. U.S.A., 115 (2018), E9994-E10002. https://doi.org/10.1073/pnas.1802987115 doi: 10.1073/pnas.1802987115
|
| [44] |
C. Chen, R. Li, L. Shu, Z. He, J. Wang, C. Zhang, et al., Predicting future dynamics from short-term time series using an Anticipated Learning Machine, Natl. Sci. Rev., 7 (2020), 1079–1091. https://doi.org/10.1093/nsr/nwaa025 doi: 10.1093/nsr/nwaa025
|
| [45] |
J. Hou, H. Ma, D. He, J. Sun, Q. Nie, W. Lin, Harvesting random embedding for high-frequency change-point detection in temporal complex systems, Natl. Sci. Rev., 9 (2022), 92–104. https://doi.org/10.1093/nsr/nwab228 doi: 10.1093/nsr/nwab228
|
| [46] |
M. Lan, Energy control and chaos prediction of a fractional-order financial risk contagion system, Phys. Scr., 100 (2025), 085240. https://doi.org/10.1088/1402-4896/adf3ee doi: 10.1088/1402-4896/adf3ee
|
| [47] | Y. Wang, M. F. Ghazali, R. A. Razak, M. A. S. Zaidi, Complex system and PS-LSTM prediction of cryptocurrencies, stocks, bonds, exchange rates and commodities, Phys. A: Stat. Mech. Appl., (2025), 130976. https://doi.org/10.1016/j.physa.2025.130976 |
| [48] |
M. S. Murugan, Large-scale data-driven financial risk management & analysis using machine learning strategies, Meas.: Sens., 27 (2023), 100756. https://doi.org/10.1016/j.measen.2023.100756 doi: 10.1016/j.measen.2023.100756
|
| [49] |
X. Fan, Y. Wang, D. Wang, Network connectedness and China's systemic financial risk contagion analysis based on big data, Pac. Basin Finance J., 68 (2021), 101322. https://doi.org/10.1016/j.pacfin.2020.101322 doi: 10.1016/j.pacfin.2020.101322
|
| [50] |
P. Song, Y. Xiao, Estimating time-varying reproduction number by deep learning techniques, J. Appl. Anal. Comput., 12 (2022), 1077–1089. https://doi.org/10.11948/20220136 doi: 10.11948/20220136
|
| [51] |
H. Hu, L. Tang, S. Zhang, H. Wang, Predicting the direction of stock markets using optimized neural networks with Google trends, Neurocomputing, 285 (2018), 188–195. https://doi.org/10.1016/j.neucom.2018.01.038 doi: 10.1016/j.neucom.2018.01.038
|
| [52] |
I. Choi, W. C. Kim, Practical forecasting of risk boundaries for industrial metals and critical minerals via statistical machine learning techniques, Int. Rev. Financ. Anal., 94 (2024), 103252. https://doi.org/10.1016/j.irfa.2024.103252 doi: 10.1016/j.irfa.2024.103252
|
| [53] |
F. Takens, Detecting strange attractors in turbulence, Dyn. Syst. Turbul., 344 (1981), 366–381. https://doi.org/10.1007/BFb0091924 doi: 10.1007/BFb0091924
|
| [54] |
E. R. Deyle, G. Sugihara, Generalized theorems for nonlinear state space reconstruction, PLoS ONE, 6 (2011), 18295. https://doi.org/10.1371/journal.pone.0018295 doi: 10.1371/journal.pone.0018295
|
| [55] |
L. Zheng, X. Huang, X. Lu, Nonbank financial institutions and financial stability: Time series analysis, Financ. Res. Lett., 73 (2025), 106544. https://doi.org/10.1016/j.frl.2024.106544 doi: 10.1016/j.frl.2024.106544
|
| [56] | Y. Gao, R. Tan, C. Fu, S. Cai, Revealing stock market risk from information flow based on transfer entropy: The case of Chinese A-shares, Phys. A: Stat. Mech. Appl., (2023), 128982. https://doi.org/10.1016/j.physa.2023.128982 |
| [57] |
L. Barnett, A. B. Barrett, A. K. Seth, Granger causality and transfer entropy are equivalent for Gaussian variables, Phys. Rev. Lett., 103 (2009), 238701. https://doi.org/10.1103/PhysRevLett.103.238701 doi: 10.1103/PhysRevLett.103.238701
|
| [58] |
S. Thomas, Measuring information transfer, Phys. Rev. Lett., 85 (2000), 461. https://doi.org/10.1103/PhysRevLett.85.461 doi: 10.1103/PhysRevLett.85.461
|
| [59] |
I. Choi, W. C. Kim, A temporal information transfer network approach considering federal funds rate for an interpretable asset fluctuation prediction framework, Int. Rev. Econ. Finance, 96 (2024), 103562. https://doi.org/10.1016/j.iref.2024.103562 doi: 10.1016/j.iref.2024.103562
|
| [60] |
S. Battiston, M. Puliga, R. Kaushik, P. Tasca, G. Caldarelli, DebtRank: Too central to fail? Financial networks, the FED and systemic risk, Sci. Rep., 2 (2012), 541. https://doi.org/10.1038/srep00541 doi: 10.1038/srep00541
|
| [61] |
J. Sànchez García, S. Cruz Rambaud, Systemic risk in a macro-multiplex network, Soft Comput., 30 (2026), 1839–1851. https://doi.org/10.1007/s00500-023-09460-7 doi: 10.1007/s00500-023-09460-7
|
| [62] |
F. X. Diebold, K. Yilmaz, On the network topology of variance decompositions: Measuring the connectedness of financial firms, J. Econom., 182 (2014), 119–134. https://doi.org/10.1016/j.jeconom.2014.04.012 doi: 10.1016/j.jeconom.2014.04.012
|
| [63] |
H. Tzavellas, A multilayer view of systemic importance and aggregate fluctuations, Int. Econ. Rev., 64 (2023), 1023—1046. https://doi.org/10.1111/iere.12622 doi: 10.1111/iere.12622
|
| [64] |
D. Caldara, M. Iacoviello, Measuring geopolitical risk, Am. Econ. Rev., 112 (2022), 1194–1225. https://doi.org/10.1257/aer.20191823 doi: 10.1257/aer.20191823
|