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A Comparison of Traditional and Deep Learning Methods for Parameter Estimation of the Ornstein-Uhlenbeck Process (2404.11526v3)

Published 17 Apr 2024 in q-fin.CP and cs.LG

Abstract: We consider the Ornstein-Uhlenbeck (OU) process, a stochastic process widely used in finance, physics, and biology. Parameter estimation of the OU process is a challenging problem. Thus, we review traditional tracking methods and compare them with novel applications of deep learning to estimate the parameters of the OU process. We use a multi-layer perceptron to estimate the parameters of the OU process and compare its performance with traditional parameter estimation methods, such as the Kalman filter and maximum likelihood estimation. We find that the multi-layer perceptron can accurately estimate the parameters of the OU process given a large dataset of observed trajectories and, on average, outperforms traditional parameter estimation methods.

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References (9)
  1. Valdivieso, L., Schoutens, W., Tuerlinckx, F.: Maximum likelihood estimation in processes of ornstein-uhlenbeck type. Statistical Inference for Stochastic Processes 12(1), 1–19 (2009) https://doi.org/10.1007/s11203-008-9021-8 Yuecaia and Dingwen [2023] Yuecaia, H., Dingwen, Z.: Least squares estimators for reflected ornstein–uhlenbeck processes. Communications in Statistics - Theory and Methods 0(0), 1–14 (2023) https://doi.org/10.1080/03610926.2023.2273204 https://doi.org/10.1080/03610926.2023.2273204 Jesica and Poznyak [2018] Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Yuecaia, H., Dingwen, Z.: Least squares estimators for reflected ornstein–uhlenbeck processes. Communications in Statistics - Theory and Methods 0(0), 1–14 (2023) https://doi.org/10.1080/03610926.2023.2273204 https://doi.org/10.1080/03610926.2023.2273204 Jesica and Poznyak [2018] Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  2. Yuecaia, H., Dingwen, Z.: Least squares estimators for reflected ornstein–uhlenbeck processes. Communications in Statistics - Theory and Methods 0(0), 1–14 (2023) https://doi.org/10.1080/03610926.2023.2273204 https://doi.org/10.1080/03610926.2023.2273204 Jesica and Poznyak [2018] Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  3. Jesica, E., Poznyak, A.: Parameter estimation in continuous-time stochastic systems with correlated noises using the kalman filter and least squares method. IFAC-PapersOnLine 51(13), 309–313 (2018) https://doi.org/10.1016/j.ifacol.2018.07.296 . 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2018 Shumway and Stoffer [1982] Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  4. Shumway, R.H., Stoffer, D.S.: An approach to time series smoothing and forecasting using the em algorithm. Journal of Time Series Analysis 3(4), 253–264 (1982) https://doi.org/10.1111/j.1467-9892.1982.tb00349.x https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1982.tb00349.x Kumar [2023] Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  5. Kumar, K.: Machine learning in parameter estimation of nonlinear systems (2023) Cantarutti [2019] Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  6. Cantarutti, N.: Financial Models Numerical Methods. https://github.com/cantaro86/Financial-Models-Numerical-Methods Kingma and Ba [2017] Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  7. Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization (2017) Glorot and Bengio [2010] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  8. Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html Hornik [1991] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991) Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)
  9. Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Networks 4, 251–257 (1991)

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