The Living Thing / Notebooks :

Feedback system identification, not necessarily linear

System Identification
Model parameter estimation for stochastic processes/dynamical systems, in the argot of signal processing people.
Non-linear system identification
dropping the assumption that the model has a nice linear impulse response, but you still want to know what the conditional distribution is at the next timestep.

After all, if you have a system whose future evolution is important to predict, why not try to infer a plausible model instead of a convenient one?


I am in the process of taxonomising here. Stuff which fits the particular (likelihood) model of recursive estimation and so on will be kept there. Miscellaneous other approaches here,

A compact overview is inserted incidentally in Cosma’s review of Fan and Yao —- FaYa03 —- (wherein he also recommends Bosq98, TaKa00 and BoBl07.)

To reconstruct the state, as opposed to the parameters, you do state filtering. There can be interplay between these steps, if you are doing simulation-based online parameter inference, as in recursive estimation.

Anyway, for what kind of systems can you infer parameters? Mutually exciting point processes? Yep, EFBS04 do that.

From an engineering/control perspective, we have BrPK16, who give a sparse regression version. Generally it seems it can be done by indirect inference, or recursive hierarchical generalised linear models, generalising the process for linear time series.

There are many highly general formulations; Kita96 gives a Bayesian “smooth” one.

See e.g. the HeDG15 paper:

We address […] these problems with a new view of predictive state methods for dynamical system learning. In this view, a dynamical system learning problem is reduced to a sequence of supervised learning problems. So, we can directly apply the rich literature on supervised learning methods to incorporate many types of prior knowledge about problem structure. We give a general convergence rate analysis that allows a high degree of flexibility in designing estimators. And finally, implementing a new estimator becomes as simple as rearranging our data and calling the appropriate supervised learning subroutines.

[…] More specifically, our contribution is to show that we can use much-more- general supervised learning algorithms in place of linear regression, and still get a meaningful theoretical analysis. In more detail:

  1. we point out that we can equally well use any well-behaved supervised learning algorithm in place of linear regression in the first stage of instrumental-variable regression;
  2. for the second stage of instrumental-variable regression, we generalize ordinary linear regression to its RKHS counterpart;
  3. we analyze the resulting combination, and show that we get convergence to the correct answer, with a rate that depends on how quickly the individual supervised learners converge

Also, sparsely or unevenly observed series are tricky. I’m looking at those at the moment.

Awaiting filing


Andrews, D. W. K.(1994) Empirical process methods in econometrics. In R. F. E. and D. L. McFadden (Ed.), Handbook of Econometrics (Vol. 4, pp. 2247–2294). Elsevier
Antoniano-Villalobos, I., & Walker, S. G.(2016) A Nonparametric Model for Stationary Time Series. Journal of Time Series Analysis, 37(1), 126–142. DOI.
Arulampalam, M. S., Maskell, S., Gordon, N., & Clapp, T. (2002) A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing, 50(2), 174–188. DOI.
Ben Taieb, S., & Atiya, A. F.(2016) A Bias and Variance Analysis for Multistep-Ahead Time Series Forecasting. IEEE Transactions on Neural Networks and Learning Systems, 27(1), 62–76. DOI.
Bengio, S., Vinyals, O., Jaitly, N., & Shazeer, N. (2015) Scheduled sampling for sequence prediction with recurrent neural networks. In Advances in Neural Information Processing Systems 28 (pp. 1171–1179). Cambridge, MA, USA: Curran Associates, Inc.
Bosq, D. (1998) Nonparametric statistics for stochastic processes: estimation and prediction. (2nd ed.). New York: Springer
Bosq, D., & Blanke, D. (2007) Inference and prediction in large dimensions. . Chichester, England ; Hoboken, NJ: John Wiley/Dunod
Bretó, C., He, D., Ionides, E. L., & King, A. A.(2009) Time series analysis via mechanistic models. The Annals of Applied Statistics, 3(1), 319–348. DOI.
Brunton, S. L., Proctor, J. L., & Kutz, J. N.(2016) Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937. DOI.
Bühlmann, P., & Künsch, H. R.(1999) Block length selection in the bootstrap for time series. Computational Statistics & Data Analysis, 31(3), 295–310. DOI.
Carmi, A. Y.(2014) Compressive System Identification. In A. Y. Carmi, L. Mihaylova, & S. J. Godsill (Eds.), Compressed Sensing & Sparse Filtering (pp. 281–324). Springer Berlin Heidelberg DOI.
Cassidy, B., Rae, C., & Solo, V. (2015) Brain Activity: Connectivity, Sparsity, and Mutual Information. IEEE Transactions on Medical Imaging, 34(4), 846–860. DOI.
Chan, N. H., Lu, Y., & Yau, C. Y.(2016) Factor Modelling for High-Dimensional Time Series: Inference and Model Selection. Journal of Time Series Analysis, n/a-n/a. DOI.
Chevillon, G. (2007) Direct Multi-Step Estimation and Forecasting. Journal of Economic Surveys, 21(4), 746–785. DOI.
Clark, J. S., & Bjørnstad, O. N.(2004) Population time series: process variability, observation errors, missing values, lags, and hidden states. Ecology, 85(11), 3140–3150. DOI.
Cook, A. R., Otten, W., Marion, G., Gibson, G. J., & Gilligan, C. A.(2007) Estimation of multiple transmission rates for epidemics in heterogeneous populations. Proceedings of the National Academy of Sciences, 104(51), 20392–20397. DOI.
Doucet, A., Jacob, P. E., & Rubenthaler, S. (2013) Derivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models. ArXiv:1304.5768 [Stat].
Durbin, J., & Koopman, S. J.(1997) Monte Carlo maximum likelihood estimation for non-Gaussian state space models. Biometrika, 84(3), 669–684. DOI.
Durbin, J., & Koopman, S. J.(2012) Time series analysis by state space methods. (2nd ed.). Oxford: Oxford University Press
Eden, U., Frank, L., Barbieri, R., Solo, V., & Brown, E. (2004) Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering. Neural Computation, 16(5), 971–998. DOI.
Fan, J., & Yao, Q. (2003) Nonlinear time series: nonparametric and parametric methods. . New York: Springer
Fearnhead, P., & Künsch, H. R.(2018) Particle Filters and Data Assimilation. Annual Review of Statistics and Its Application, 5(1), 421–449. DOI.
Finke, A., & Singh, S. S.(2016) Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models. ArXiv:1606.08650 [Stat].
Flunkert, V., Salinas, D., & Gasthaus, J. (2017) DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks. ArXiv:1704.04110 [Cs, Stat].
Fraser, A. M.(2008) Hidden Markov models and dynamical systems. . Philadelphia, PA: Society for Industrial and Applied Mathematics
Harvey, A., & Koopman, S. J.(2005) Structural Time Series Models. In Encyclopedia of Biostatistics. John Wiley & Sons, Ltd
He, D., Ionides, E. L., & King, A. A.(2010) Plug-and-play inference for disease dynamics: measles in large and small populations as a case study. Journal of The Royal Society Interface, 7(43), 271–283. DOI.
Hefny, A., Downey, C., & Gordon, G. (2015) A New View of Predictive State Methods for Dynamical System Learning. ArXiv:1505.05310 [Cs, Stat].
Hong, X., Mitchell, R. J., Chen, S., Harris, C. J., Li, K., & Irwin, G. W.(2008) Model selection approaches for non-linear system identification: a review. International Journal of Systems Science, 39(10), 925–946. DOI.
Hong, Y., & Li, H. (2005) Nonparametric Specification Testing for Continuous-Time Models with Applications to Term Structure of Interest Rates. Review of Financial Studies, 18(1), 37–84. DOI.
Ionides, E. L., Bretó, C., & King, A. A.(2006) Inference for nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 103(49), 18438–18443. DOI.
Ionides, Edward L., Bhadra, A., Atchadé, Y., & King, A. (2011) Iterated filtering. The Annals of Statistics, 39(3), 1776–1802. DOI.
Kantz, H., & Schreiber, T. (2004) Nonlinear time series analysis. (2nd ed.). Cambridge, UK ; New York: Cambridge University Press
Kass, R. E., Amari, S.-I., Arai, K., Brown, E. N., Diekman, C. O., Diesmann, M., … Kramer, M. A.(2018) Computational Neuroscience: Mathematical and Statistical Perspectives. Annual Review of Statistics and Its Application, 5(1), 183–214. DOI.
Kemerait, R., & Childers, D. (1972) Signal detection and extraction by cepstrum techniques. IEEE Transactions on Information Theory, 18(6), 745–759. DOI.
Kendall, B. E., Ellner, S. P., McCauley, E., Wood, S. N., Briggs, C. J., Murdoch, W. W., & Turchin, P. (2005) Population cycles in the pine looper moth: Dynamical tests of mechanistic hypotheses. Ecological Monographs, 75(2), 259–276.
Kitagawa, G. (1987) Non-Gaussian State—Space Modeling of Nonstationary Time Series. Journal of the American Statistical Association, 82(400), 1032–1041. DOI.
Kitagawa, G. (1996) Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models. Journal of Computational and Graphical Statistics, 5(1), 1–25. DOI.
Kitagawa, G., & Gersch, W. (1996) Smoothness Priors Analysis of Time Series. . New York, NY: Springer New York : Imprint : Springer
Lamb, A., Goyal, A., Zhang, Y., Zhang, S., Courville, A., & Bengio, Y. (2016) Professor Forcing: A New Algorithm for Training Recurrent Networks. In Advances In Neural Information Processing Systems.
Levin, D. N.(2017) The Inner Structure of Time-Dependent Signals. ArXiv:1703.08596 [Cs, Math, Stat].
Nerrand, O., Roussel-Ragot, P., Personnaz, L., Dreyfus, G., & Marcos, S. (1993) Neural Networks and Nonlinear Adaptive Filtering: Unifying Concepts and New Algorithms. Neural Computation, 5(2), 165–199. DOI.
Pereyra, M., Schniter, P., Chouzenoux, É., Pesquet, J. C., Tourneret, J. Y., Hero, A. O., & McLaughlin, S. (2016) A Survey of Stochastic Simulation and Optimization Methods in Signal Processing. IEEE Journal of Selected Topics in Signal Processing, 10(2), 224–241. DOI.
Pham, T., & Panaretos, V. (2016) Methodology and Convergence Rates for Functional Time Series Regression. ArXiv:1612.07197 [Math, Stat].
Pillonetto, G. (2016) The interplay between system identification and machine learning. ArXiv:1612.09158 [Cs, Stat].
Plis, S., Danks, D., & Yang, J. (2015) Mesochronal Structure Learning. Uncertainty in Artificial Intelligence : Proceedings of the … Conference. Conference on Uncertainty in Artificial Intelligence, 31.
Robinson, P. M.(1983) Nonparametric Estimators for Time Series. Journal of Time Series Analysis, 4(3), 185–207. DOI.
Routtenberg, T., & Tabrikian, J. (2010) Blind MIMO-AR System Identification and Source Separation with Finite-alphabet. IEEE Transactions on Signal Processing, 58(3), 990–1000. DOI.
Runge, J., Donner, R. V., & Kurths, J. (2015) Optimal model-free prediction from multivariate time series. Physical Review E, 91(5). DOI.
Särkkä, S. (2007) On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems. IEEE Transactions on Automatic Control, 52(9), 1631–1641. DOI.
Sjöberg, J., Zhang, Q., Ljung, L., Benveniste, A., Delyon, B., Glorennec, P.-Y., … Juditsky, A. (1995) Nonlinear black-box modeling in system identification: a unified overview. Automatica, 31(12), 1691–1724. DOI.
Städler, N., & Mukherjee, S. (2013) Penalized estimation in high-dimensional hidden Markov models with state-specific graphical models. The Annals of Applied Statistics, 7(4), 2157–2179. DOI.
Stark, J., Broomhead, D. S., Davies, M. E., & Huke, J. (2003) Delay Embeddings for Forced Systems II Stochastic Forcing. Journal of Nonlinear Science, 13(6), 519–577. DOI.
Tallec, C., & Ollivier, Y. (2017) Unbiasing Truncated Backpropagation Through Time. ArXiv:1705.08209 [Cs].
Taniguchi, M., & Kakizawa, Y. (2000) Asymptotic theory of statistical inference for time series. . New York: Springer
Tanizaki, H. (2001) Estimation of unknown parameters in nonlinear and non-Gaussian state-space models. Journal of Statistical Planning and Inference, 96(2), 301–323. DOI.
Unser, M. A., & Tafti, P. (2014) An introduction to sparse stochastic processes. . New York: Cambridge University Press
Wen, R., Torkkola, K., & Narayanaswamy, B. (2017) A Multi-Horizon Quantile Recurrent Forecaster. ArXiv:1711.11053 [Stat].
Werbos, P. J.(1988) Generalization of backpropagation with application to a recurrent gas market model. Neural Networks, 1(4), 339–356. DOI.
Williams, R. J., & Zipser, D. (1989) A Learning Algorithm for Continually Running Fully Recurrent Neural Networks. Neural Computation, 1(2), 270–280. DOI.