State filtering for hidden Markov models

Kalman and friends

Kalman-Bucy filter and variants, recursive estimation, predictive state models, Data assimilation. A particular sub-field of signal processing for models with hidden state.

In statistics terms, the state filters are a kind of online-updating hierarchical model for sequential observations of a dynamical system where the random state is unobserved, but you can get an optimal estimate of it based on incoming measurements and known parameters.

A unifying feature of all these is by assuming a sparse influence graph between observations and dynamics, that you can estimate behaviour using efficient message passing.

This is a twin problem to optimal control. If I wish to tackle this problem from the perspective of observations rather than true state, perhaps I could do it from the perspective of Koopman operators.

Linear dynamical systems

In Kalman filters per se the default problem is usually concerned with multivariate real vector signals representing different axes of some telemetry data. In the degenerate case, where there is no observation noise, we can just design a linear filter which solves the target problem.

The classic Kalman filter (R. E. Kalman 1960) assumes a linear model with Gaussian noise, although it might work with not-quite Gaussian, not-quite linear models if you prod it. You can extend this flavour to somewhat more general dynamics. For that, see later.

NB I’m conflating linear observation and linear process models, for now. We can relax that when there are some concrete examples in play.

There are a large number of equivalent formulations of the Kalman filter. The notation of Fearnhead and KΓΌnsch (2018) is representative. They start from the usual state filter setting: The state process \(\left(\mathbf{X}_{t}\right)\) is assumed to be Markovian and the \(i\)-th observation, \(\mathbf{Y}_{i}\), depends only on the state at time \(i, \mathbf{X}_{i}\), so that the evolution and observation variates are defined by \[ \begin{aligned} \mathbf{X}_{t} \mid\left(\mathbf{x}_{0: t-1}, \mathbf{y}_{1: t-1}\right) & \sim P\left(d \mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right), \quad \mathbf{X}_{0} \sim \pi_{0}\left(d \mathbf{x}_{0}\right) \\ \mathbf{Y}_{t} \mid\left(\mathbf{x}_{0: t}, \mathbf{y}_{1: t-1}\right) & \sim g\left(\mathbf{y}_{t} \mid \mathbf{x}_{t}\right) d \nu\left(\mathbf{y}_{t}\right) \end{aligned} \] with joint distribution \[ \left(\mathbf{X}_{0: s}, \mathbf{Y}_{1: t}\right) \sim \pi_{0}\left(d \mathbf{x}_{0}\right) \prod_{i=1}^{s} P\left(d \mathbf{x}_{i} \mid \mathbf{x}_{i-1}\right) \prod_{j=1}^{t} g\left(\mathbf{y}_{j} \mid \mathbf{x}_{j}\right) \nu\left(d \mathbf{y}_{j}\right), \quad s \geq t. \]

Integrating out the path of the state process, we obtain that \[\begin{aligned} \mathbf{Y}_{1: t} &\sim p\left(\mathbf{y}_{1: t}\right) \prod_{j} \nu\left(d \mathbf{y}_{j}\right)\text{, where}\\ p\left(\mathbf{y}_{1: t}\right) &=\int \pi_{0}\left(d \mathbf{x}_{0}\right) \prod_{i=1}^{s} P\left(d \mathbf{x}_{i} \mid \mathbf{x}_{i-1}\right) \prod_{j=1}^{t} g\left(\mathbf{y}_{j} \mid \mathbf{x}_{j}\right). \end{aligned} \] We wish to find the distribution \(\pi_{0: s \mid t}=\frac{p(\mathbf{y}_{1: t},\mathbf{x}_{0:s})}{p(\mathbf{y}_{1: t})}\) (by Bayes’ rule). We deduce the recursion \[ \begin{aligned} \pi_{0: t \mid t-1}\left(d \mathbf{x}_{0: t} \mid \mathbf{y}_{1: t-1}\right) &=\pi_{0: t-1 \mid t-1}\left(d \mathbf{x}_{0: t-1} \mid \mathbf{y}_{1: t-1}\right) P\left(d \mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right) &\text{ prediction}\\ \pi_{0: t \mid t}\left(d \mathbf{x}_{0: t} \mid \mathbf{y}_{1: t}\right) &=\pi_{0: t \mid t-1}\left(d \mathbf{x}_{0: t} \mid \mathbf{y}_{1: t-1}\right) \frac{g\left(\mathbf{y}_{t} \mid \mathbf{x}_{t}\right)}{p\left(\mathbf{y}_{t} \mid \mathbf{y}_{1: t-1}\right)} &\text{ correction} \end{aligned} \] where \[ p\left(\mathbf{y}_{t} \mid \mathbf{y}_{1: t-1}\right)=\frac{p\left(\mathbf{y}_{1: t}\right)}{p\left(\mathbf{y}_{1: t-1}\right)}=\int \pi_{t \mid t-1}\left(d \mathbf{x}_{t} \mid \mathbf{y}_{1: t-1}\right) g\left(\mathbf{y}_{t} \mid \mathbf{x}_{t}\right) . \] Integrating out all but the latest states \(\mathbf{x}_{0: t-1}\) gives us the one-step recursion \[ \begin{aligned} \pi_{t \mid t-1}\left(d \mathbf{x}_{t} \mid \mathbf{y}_{1: t-1}\right) &=\int \pi_{t-1}\left(d \mathbf{x}_{t-1} \mid \mathbf{y}_{1: t-1}\right) P\left(d \mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right) &\text{ prediction}\\ \pi_{t}\left(d \mathbf{x}_{t} \mid \mathbf{y}_{1: t}\right) &=\pi_{t \mid t-1}\left(d \mathbf{x}_{t} \mid \mathbf{y}_{1: t-1}\right) \frac{g\left(\mathbf{y}_{t} \mid \mathbf{x}_{t}\right)}{p_{t}\left(\mathbf{y}_{t} \mid \mathbf{y}_{1: t-1}\right)}&\text{ correction} \end{aligned} \]

If we approximate the filter distribution \(\pi_t\) with a Monte Carlo sample, we are doing particle filtering, which Fearnhead and KΓΌnsch (2018) refer to as bootstrap filtering.

TODO: implied Kalman gain etc.

Non-linear dynamical systems

Cute exercise: you can derive the analytic Kalman filter for any noise and process dynamics of with Bayesian conjugate, and this leads to filters of nonlinear behaviour. Multivariate distributions are a bit of a mess for non-Gaussians, though, and a beta-Kalman filter feels contrived.

Upshot is, the non-linear extensions don’t usually rely on non-Gaussian conjugate distributions and analytic forms, but rather do some Gaussian/linear approximation, or use randomised methods such as particle filters.

For some examples in Stan see Sinhrks’ stan-statespace.

As errors-in-variables models

see, e.g. Bagge Carlson (2018).

Discrete state Hidden Markov models

πŸ— Viterbi algorithm.

Unscented Kalman filter

i.e. using the unscented transform.

Variational state filters

See variational state filters.

Kalman filtering Gaussian processes

See filtering Gaussian processes.

Ensemble Kalman filters

See Ensemble Kalman filters.

State filter inference

How about learning the parameters of the model generating your states? Ways that you can do this in dynamical systems include basic linear system identification, general system identification, .


Aasnaes, H., and T. Kailath. 1973. β€œAn Innovations Approach to Least-Squares Estimation–Part VII: Some Applications of Vector Autoregressive-Moving Average Models.” IEEE Transactions on Automatic Control 18 (6): 601–7.
Alliney, S. 1992. β€œDigital Filters as Absolute Norm Regularizers.” IEEE Transactions on Signal Processing 40 (6): 1548–62.
Alzraiee, Ayman H., Jeremy T. White, Matthew J. Knowling, Randall J. Hunt, and Michael N. Fienen. 2022. β€œA Scalable Model-Independent Iterative Data Assimilation Tool for Sequential and Batch Estimation of High Dimensional Model Parameters and States.” Environmental Modelling & Software 150 (April): 105284.
Ansley, Craig F., and Robert Kohn. 1985. β€œEstimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions.” The Annals of Statistics 13 (4): 1286–316.
Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp. 2002. β€œA Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking.” IEEE Transactions on Signal Processing 50 (2): 174–88.
Bagge Carlson, Fredrik. 2018. β€œMachine Learning and System Identification for Estimation in Physical Systems.” Thesis/docmono, Lund University.
Battey, Heather, and Alessio Sancetta. 2013. β€œConditional Estimation for Dependent Functional Data.” Journal of Multivariate Analysis 120 (September): 1–17.
Batz, Philipp, Andreas Ruttor, and Manfred Opper. 2017. β€œApproximate Bayes Learning of Stochastic Differential Equations.” arXiv:1702.05390 [Physics, Stat], February.
Becker, Philipp, Harit Pandya, Gregor Gebhardt, Cheng Zhao, C. James Taylor, and Gerhard Neumann. 2019. β€œRecurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces.” In International Conference on Machine Learning, 544–52.
Berkhout, A. J., and P. R. Zaanen. 1976. β€œA Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*.” Geophysical Prospecting 24 (1): 141–97.
Bilmes, Jeff A. 1998. β€œA Gentle Tutorial of the EM Algorithm and Its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models.” International Computer Science Institute 4 (510): 126.
Bishop, Adrian N., and Pierre Del Moral. 2016. β€œOn the Stability of Kalman-Bucy Diffusion Processes.” SIAM Journal on Control and Optimization 55 (6): 4015–47.
β€”β€”β€”. 2023. β€œOn the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering.” Mathematics of Control, Signals, and Systems, May.
Bishop, Adrian N., Pierre Del Moral, and Sahani D. Pathiraja. 2017. β€œPerturbations and Projections of Kalman-Bucy Semigroups Motivated by Methods in Data Assimilation.” arXiv:1701.05978 [Math], January.
BretΓ³, Carles, Daihai He, Edward L. Ionides, and Aaron A. King. 2009. β€œTime Series Analysis via Mechanistic Models.” The Annals of Applied Statistics 3 (1): 319–48.
Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. 2016. β€œDiscovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences 113 (15): 3932–37.
Campbell, Andrew, Yuyang Shi, Tom Rainforth, and Arnaud Doucet. 2021. β€œOnline Variational Filtering and Parameter Learning.” In.
Carmi, Avishy Y. 2013. β€œCompressive System Identification: Sequential Methods and Entropy Bounds.” Digital Signal Processing 23 (3): 751–70.
β€”β€”β€”. 2014. β€œCompressive System Identification.” In Compressed Sensing & Sparse Filtering, edited by Avishy Y. Carmi, Lyudmila Mihaylova, and Simon J. Godsill, 281–324. Signals and Communication Technology. Springer Berlin Heidelberg.
Cassidy, Ben, Caroline Rae, and Victor Solo. 2015. β€œBrain Activity: Connectivity, Sparsity, and Mutual Information.” IEEE Transactions on Medical Imaging 34 (4): 846–60.
Cauchemez, Simon, and Neil M. Ferguson. 2008. β€œLikelihood-Based Estimation of Continuous-Time Epidemic Models from Time-Series Data: Application to Measles Transmission in London.” Journal of The Royal Society Interface 5 (25): 885–97.
Charles, Adam, Aurele Balavoine, and Christopher Rozell. 2016. β€œDynamic Filtering of Time-Varying Sparse Signals via L1 Minimization.” IEEE Transactions on Signal Processing 64 (21): 5644–56.
Chen, Bin, and Yongmiao Hong. 2012. β€œTesting for the Markov Property in Time Series.” Econometric Theory 28 (01): 130–78.
Chen, Y., and A. O. Hero. 2012. β€œRecursive β„“1,∞ Group Lasso.” IEEE Transactions on Signal Processing 60 (8): 3978–87.
Chung, Junyoung, Kyle Kastner, Laurent Dinh, Kratarth Goel, Aaron C Courville, and Yoshua Bengio. 2015. β€œA Recurrent Latent Variable Model for Sequential Data.” In Advances in Neural Information Processing Systems 28, edited by C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, 2980–88. Curran Associates, Inc.
Clark, James S., and Ottar N. BjΓΈrnstad. 2004. β€œPopulation Time Series: Process Variability, Observation Errors, Missing Values, Lags, and Hidden States.” Ecology 85 (11): 3140–50.
Commandeur, Jacques J. F., and Siem Jan Koopman. 2007. An Introduction to State Space Time Series Analysis. 1 edition. Oxford ; New York: Oxford University Press.
Cox, Marco, Thijs van de Laar, and Bert de Vries. 2019. β€œA Factor Graph Approach to Automated Design of Bayesian Signal Processing Algorithms.” International Journal of Approximate Reasoning 104 (January): 185–204.
Cressie, Noel, and Hsin-Cheng Huang. 1999. β€œClasses of Nonseparable, Spatio-Temporal Stationary Covariance Functions.” Journal of the American Statistical Association 94 (448): 1330–39.
Cressie, Noel, Tao Shi, and Emily L. Kang. 2010. β€œFixed Rank Filtering for Spatio-Temporal Data.” Journal of Computational and Graphical Statistics 19 (3): 724–45.
Cressie, Noel, and Christopher K. Wikle. 2011. Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics 2.0. John Wiley and Sons.
Deisenroth, Marc Peter, and Shakir Mohamed. 2012. β€œExpectation Propagation in Gaussian Process Dynamical Systems.” In Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 2, 25:2609–17. NIPS’12. Red Hook, NY, USA: Curran Associates Inc.
Del Moral, P., A. Kurtzmann, and J. Tugaut. 2017. β€œOn the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman-Bucy Filters.” SIAM Journal on Control and Optimization 55 (1): 119–55.
Doucet, Arnaud, Pierre E. Jacob, and Sylvain Rubenthaler. 2013. β€œDerivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models.” arXiv:1304.5768 [Stat], April.
Durbin, J., and S. J. Koopman. 1997. β€œMonte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models.” Biometrika 84 (3): 669–84.
β€”β€”β€”. 2012. Time Series Analysis by State Space Methods. 2nd ed. Oxford Statistical Science Series 38. Oxford: Oxford University Press.
Duttweiler, D., and T. Kailath. 1973a. β€œRKHS Approach to Detection and Estimation Problems–IV: Non-Gaussian Detection.” IEEE Transactions on Information Theory 19 (1): 19–28.
β€”β€”β€”. 1973b. β€œRKHS Approach to Detection and Estimation Problems–V: Parameter Estimation.” IEEE Transactions on Information Theory 19 (1): 29–37.
Easley, Deanna, and Tyrus Berry. 2020. β€œA Higher Order Unscented Transform.” arXiv:2006.13429 [Cs, Math], June.
Eddy, Sean R. 1996. β€œHidden Markov Models.” Current Opinion in Structural Biology 6 (3): 361–65.
Eden, U, L Frank, R Barbieri, V Solo, and E Brown. 2004. β€œDynamic Analysis of Neural Encoding by Point Process Adaptive Filtering.” Neural Computation 16 (5): 971–98.
Edwards, David, and Smitha Ankinakatte. 2015. β€œContext-Specific Graphical Models for Discrete Longitudinal Data.” Statistical Modelling 15 (4): 301–25.
Eleftheriadis, Stefanos, Tom Nicholson, Marc Deisenroth, and James Hensman. 2017. β€œIdentification of Gaussian Process State Space Models.” In Advances in Neural Information Processing Systems 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5309–19. Curran Associates, Inc.
Fearnhead, Paul, and Hans R. KΓΌnsch. 2018. β€œParticle Filters and Data Assimilation.” Annual Review of Statistics and Its Application 5 (1): 421–49.
Finke, Axel, and Sumeetpal S. Singh. 2016. β€œApproximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models.” arXiv:1606.08650 [Stat], June.
FΓΆll, Roman, Bernard Haasdonk, Markus Hanselmann, and Holger Ulmer. 2017. β€œDeep Recurrent Gaussian Process with Variational Sparse Spectrum Approximation.” arXiv:1711.00799 [Stat], November.
Fraccaro, Marco, SΓΈ ren Kaae SΓΈ nderby, Ulrich Paquet, and Ole Winther. 2016. β€œSequential Neural Models with Stochastic Layers.” In Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 2199–2207. Curran Associates, Inc.
Fraser, Andrew M. 2008. Hidden Markov Models and Dynamical Systems. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Freitas, J. F. G. de, Mahesan Niranjan, A. H. Gee, and Arnaud Doucet. 1998. β€œSequential Monte Carlo Methods for Optimisation of Neural Network Models.” Cambridge University Engineering Department, Cambridge, England, Technical Report TR-328.
Freitas, JoΓ£o FG de, Arnaud Doucet, Mahesan Niranjan, and Andrew H. Gee. 1998. β€œGlobal Optimisation of Neural Network Models via Sequential Sampling.” In Proceedings of the 11th International Conference on Neural Information Processing Systems, 410–16. NIPS’98. Cambridge, MA, USA: MIT Press.
Friedlander, B., T. Kailath, and L. Ljung. 1975. β€œScattering Theory and Linear Least Squares Estimation: Part II: Discrete-Time Problems.” In 1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes, 57–58.
Frigola, Roger, Yutian Chen, and Carl Edward Rasmussen. 2014. β€œVariational Gaussian Process State-Space Models.” In Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 3680–88. Curran Associates, Inc.
Frigola, Roger, Fredrik Lindsten, Thomas B SchΓΆn, and Carl Edward Rasmussen. 2013. β€œBayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC.” In Advances in Neural Information Processing Systems 26, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, 3156–64. Curran Associates, Inc.
Friston, K. J. 2008. β€œVariational Filtering.” NeuroImage 41 (3): 747–66.
Gevers, M., and T. Kailath. 1973. β€œAn Innovations Approach to Least-Squares Estimation–Part VI: Discrete-Time Innovations Representations and Recursive Estimation.” IEEE Transactions on Automatic Control 18 (6): 588–600.
Gorad, Ajinkya, Zheng Zhao, and Simo SΓ€rkkΓ€. 2020. β€œParameter Estimation in Non-Linear State-Space Models by Automatic Differentiation of Non-Linear Kalman Filters.” In, 6.
Gottwald, Georg A., and Sebastian Reich. 2020. β€œSupervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation.” arXiv:2007.07383 [Physics, Stat], July.
Gourieroux, Christian, and Joann Jasiak. 2015. β€œFiltering, Prediction and Simulation Methods for Noncausal Processes.” Journal of Time Series Analysis, January, n/a–.
Gu, Albert, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher RΓ©. 2021. β€œCombining Recurrent, Convolutional, and Continuous-Time Models with Linear State Space Layers.” In Advances in Neural Information Processing Systems, 34:572–85. Curran Associates, Inc.
Haber, Eldad, Felix Lucka, and Lars Ruthotto. 2018. β€œNever Look Back - A Modified EnKF Method and Its Application to the Training of Neural Networks Without Back Propagation.” arXiv:1805.08034 [Cs, Math], May.
Hamilton, Franz, Tyrus Berry, and Timothy Sauer. 2016. β€œKalman-Takens Filtering in the Presence of Dynamical Noise.” arXiv:1611.05414 [Physics, Stat], November.
Hartikainen, J., and S. SΓ€rkkΓ€. 2010. β€œKalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In 2010 IEEE International Workshop on Machine Learning for Signal Processing, 379–84. Kittila, Finland: IEEE.
Harvey, A., and S. J. Koopman. 2005. β€œStructural Time Series Models.” In Encyclopedia of Biostatistics. John Wiley & Sons, Ltd.
Harvey, Andrew, and Alessandra Luati. 2014. β€œFiltering With Heavy Tails.” Journal of the American Statistical Association 109 (507): 1112–22.
He, Daihai, Edward L. Ionides, and Aaron A. King. 2010. β€œPlug-and-Play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study.” Journal of The Royal Society Interface 7 (43): 271–83.
Hefny, Ahmed, Carlton Downey, and Geoffrey Gordon. 2015. β€œA New View of Predictive State Methods for Dynamical System Learning.” arXiv:1505.05310 [Cs, Stat], May.
Hong, X., R. J. Mitchell, S. Chen, C. J. Harris, K. Li, and G. W. Irwin. 2008. β€œModel Selection Approaches for Non-Linear System Identification: A Review.” International Journal of Systems Science 39 (10): 925–46.
Hou, Elizabeth, Earl Lawrence, and Alfred O. Hero. 2016. β€œPenalized Ensemble Kalman Filters for High Dimensional Non-Linear Systems.” arXiv:1610.00195 [Physics, Stat], October.
Hsiao, Roger, and Tanja Schultz. 2011. β€œGeneralized Baum-Welch Algorithm and Its Implication to a New Extended Baum-Welch Algorithm.” In In Proceedings of INTERSPEECH.
Hsu, Daniel, Sham M. Kakade, and Tong Zhang. 2012. β€œA Spectral Algorithm for Learning Hidden Markov Models.” Journal of Computer and System Sciences, JCSS Special Issue: Cloud Computing 2011, 78 (5): 1460–80.
Huber, Marco F. 2014. β€œRecursive Gaussian Process: On-Line Regression and Learning.” Pattern Recognition Letters 45 (August): 85–91.
Ionides, E. L., C. BretΓ³, and A. A. King. 2006. β€œInference for Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences 103 (49): 18438–43.
Ionides, Edward L., Anindya Bhadra, Yves AtchadΓ©, and Aaron King. 2011. β€œIterated Filtering.” The Annals of Statistics 39 (3): 1776–1802.
Johansen, Adam, Arnaud Doucet, and Manuel Davy. 2006. β€œSequential Monte Carlo for Marginal Optimisation Problems.” Scis & Isis 2006: 1866–71.
Johnson, Matthew James. 2012. β€œA Simple Explanation of A Spectral Algorithm for Learning Hidden Markov Models.” arXiv:1204.2477 [Cs, Stat], April.
Julier, S.J., J.K. Uhlmann, and H.F. Durrant-Whyte. 1995. β€œA New Approach for Filtering Nonlinear Systems.” In American Control Conference, Proceedings of the 1995, 3:1628–1632 vol.3.
Kailath, T. 1971. β€œRKHS Approach to Detection and Estimation Problems–I: Deterministic Signals in Gaussian Noise.” IEEE Transactions on Information Theory 17 (5): 530–49.
β€”β€”β€”. 1974. β€œA View of Three Decades of Linear Filtering Theory.” IEEE Transactions on Information Theory 20 (2): 146–81.
Kailath, T., and D. Duttweiler. 1972. β€œAn RKHS Approach to Detection and Estimation Problems– III: Generalized Innovations Representations and a Likelihood-Ratio Formula.” IEEE Transactions on Information Theory 18 (6): 730–45.
Kailath, T., and R. Geesey. 1971. β€œAn Innovations Approach to Least Squares Estimation–Part IV: Recursive Estimation Given Lumped Covariance Functions.” IEEE Transactions on Automatic Control 16 (6): 720–27.
β€”β€”β€”. 1973. β€œAn Innovations Approach to Least-Squares Estimation–Part V: Innovations Representations and Recursive Estimation in Colored Noise.” IEEE Transactions on Automatic Control 18 (5): 435–53.
Kailath, T., and H. Weinert. 1975. β€œAn RKHS Approach to Detection and Estimation Problems–II: Gaussian Signal Detection.” IEEE Transactions on Information Theory 21 (1): 15–23.
Kalman, R. 1959. β€œOn the General Theory of Control Systems.” IRE Transactions on Automatic Control 4 (3): 110–10.
Kalman, R. E. 1960. β€œA New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering 82 (1): 35.
Kalouptsidis, Nicholas, Gerasimos Mileounis, Behtash Babadi, and Vahid Tarokh. 2011. β€œAdaptive Algorithms for Sparse System Identification.” Signal Processing 91 (8): 1910–19.
Karvonen, Toni, and Simo SΓ€rkkΓ€. 2016. β€œApproximate State-Space Gaussian Processes via Spectral Transformation.” In 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), 1–6. Vietri sul Mare, Salerno, Italy: IEEE.
Kelly, D. T. B., K. J. H. Law, and A. M. Stuart. 2014. β€œWell-Posedness and Accuracy of the Ensemble Kalman Filter in Discrete and Continuous Time.” Nonlinearity 27 (10): 2579.
Kirch, Claudia, Matthew C. Edwards, Alexander Meier, and Renate Meyer. 2019. β€œBeyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis.” Bayesian Analysis 14 (4): 1037–73.
Kitagawa, Genshiro. 1987. β€œNon-Gaussian Stateβ€”Space Modeling of Nonstationary Time Series.” Journal of the American Statistical Association 82 (400): 1032–41.
β€”β€”β€”. 1996. β€œMonte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models.” Journal of Computational and Graphical Statistics 5 (1): 1–25.
Kitagawa, Genshiro, and Will Gersch. 1996. Smoothness Priors Analysis of Time Series. Lecture notes in statistics 116. New York, NY: Springer New York : Imprint : Springer.
Kobayashi, Hisashi, Brian L. Mark, and William Turin. 2011. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance. Cambridge University Press.
Koopman, S. J., and J. Durbin. 2000. β€œFast Filtering and Smoothing for Multivariate State Space Models.” Journal of Time Series Analysis 21 (3): 281–96.
Krishnan, Rahul G., Uri Shalit, and David Sontag. 2017. β€œStructured Inference Networks for Nonlinear State Space Models.” In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, 2101–9.
KulhavΓ½, Rudolf. 1990. β€œRecursive Nonlinear Estimation: A Geometric Approach.” Automatica 26 (3): 545–55.
β€”β€”β€”. 1996. Recursive Nonlinear Estimation. Vol. 216. Lecture Notes in Control and Information Sciences. London: Springer-Verlag.
Kutschireiter, Anna, Simone C Surace, Henning Sprekeler, and Jean-Pascal Pfister. 2015. β€œApproximate Nonlinear Filtering with a Recurrent Neural Network.” BMC Neuroscience 16 (Suppl 1): P196.
LΓ‘zaro-Gredilla, Miguel, Joaquin QuiΓ±onero-Candela, Carl Edward Rasmussen, and AnΓ­bal R. Figueiras-Vidal. 2010. β€œSparse Spectrum Gaussian Process Regression.” Journal of Machine Learning Research 11 (Jun): 1865–81.
Le Gland, FranΓ§ois, Valerie Monbet, and Vu-Duc Tran. 2009. β€œLarge Sample Asymptotics for the Ensemble Kalman Filter,” 25.
Lei, Jing, Peter Bickel, and Chris Snyder. 2009. β€œComparison of Ensemble Kalman Filters Under Non-Gaussianity.” Monthly Weather Review 138 (4): 1293–1306.
Levin, David N. 2017. β€œThe Inner Structure of Time-Dependent Signals.” arXiv:1703.08596 [Cs, Math, Stat], March.
Lindgren, Finn, HΓ₯vard Rue, and Johan LindstrΓΆm. 2011. β€œAn Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (4): 423–98.
Ljung, L., and T. Kailath. 1976. β€œBackwards Markovian Models for Second-Order Stochastic Processes (Corresp.).” IEEE Transactions on Information Theory 22 (4): 488–91.
Ljung, L., T. Kailath, and B. Friedlander. 1975. β€œScattering Theory and Linear Least Squares Estimation: Part I: Continuous-Time Problems.” In 1975 IEEE Conference on Decision and Control Including the 14th Symposium on Adaptive Processes, 55–56.
Loeliger, Hans-Andrea, Justin Dauwels, Junli Hu, Sascha Korl, Li Ping, and Frank R. Kschischang. 2007. β€œThe Factor Graph Approach to Model-Based Signal Processing.” Proceedings of the IEEE 95 (6): 1295–1322.
Manton, J. H., V. Krishnamurthy, and H. V. Poor. 1998. β€œJames-Stein State Filtering Algorithms.” IEEE Transactions on Signal Processing 46 (9): 2431–47.
Mattos, CΓ©sar Lincoln C., Zhenwen Dai, Andreas Damianou, Guilherme A. Barreto, and Neil D. Lawrence. 2017. β€œDeep Recurrent Gaussian Processes for Outlier-Robust System Identification.” Journal of Process Control, DYCOPS-CAB 2016, 60 (December): 82–94.
Mattos, CΓ©sar Lincoln C., Zhenwen Dai, Andreas Damianou, Jeremy Forth, Guilherme A. Barreto, and Neil D. Lawrence. 2016. β€œRecurrent Gaussian Processes.” In Proceedings of ICLR.
Meyer, Renate, Matthew C. Edwards, Patricio Maturana-Russel, and Nelson Christensen. 2020. β€œComputational Techniques for Parameter Estimation of Gravitational Wave Signals.” WIREs Computational Statistics n/a (n/a): e1532.
Micchelli, Charles A., and Peder Olsen. 2000. β€œPenalized Maximum-Likelihood Estimation, the Baum–Welch Algorithm, Diagonal Balancing of Symmetric Matrices and Applications to Training Acoustic Data.” Journal of Computational and Applied Mathematics 119 (1–2): 301–31.
Miller, David L., Richard Glennie, and Andrew E. Seaton. 2020. β€œUnderstanding the Stochastic Partial Differential Equation Approach to Smoothing.” Journal of Agricultural, Biological and Environmental Statistics 25 (1): 1–16.
Nickisch, Hannes, Arno Solin, and Alexander Grigorevskiy. 2018. β€œState Space Gaussian Processes with Non-Gaussian Likelihood.” In International Conference on Machine Learning, 3789–98.
Olfati-Saber, R. 2005. β€œDistributed Kalman Filter with Embedded Consensus Filters.” In 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05, 8179–84. Seville, Spain: IEEE.
Ollivier, Yann. 2017. β€œOnline Natural Gradient as a Kalman Filter.” arXiv:1703.00209 [Math, Stat], March.
Papadopoulos, Alexandre, FranΓ§ois Pachet, Pierre Roy, and Jason Sakellariou. 2015. β€œExact Sampling for Regular and Markov Constraints with Belief Propagation.” In Principles and Practice of Constraint Programming, 341–50. Lecture Notes in Computer Science. Switzerland: Springer, Cham.
Perry, T.S. 2010. β€œAndrew Viterbi’s Fabulous Formula [Medal of Honor].” IEEE Spectrum 47 (5): 47–50.
Picci, G. 1991. β€œStochastic Realization Theory.” In Mathematical System Theory: The Influence of R. E. Kalman, edited by Athanasios C. Antoulas, 213–29. Berlin, Heidelberg: Springer.
Psiaki, M. 2013. β€œThe Blind Tricyclist Problem and a Comparative Study of Nonlinear Filters: A Challenging Benchmark for Evaluating Nonlinear Estimation Methods.” IEEE Control Systems 33 (3): 40–54.
Pugachev, V. S., and I. N. SinitοΈ sοΈ‘yn. 2001. Stochastic systems: theory and applications. River Edge, NJ: World Scientific.
Pugachev, V.S. 1982. β€œConditionally Optimal Estimation in Stochastic Differential Systems.” Automatica 18 (6): 685–96.
QuiΓ±onero-Candela, Joaquin, and Carl Edward Rasmussen. 2005. β€œA Unifying View of Sparse Approximate Gaussian Process Regression.” Journal of Machine Learning Research 6 (Dec): 1939–59.
Rabiner, L., and B.H. Juang. 1986. β€œAn Introduction to Hidden Markov Models.” IEEE ASSP Magazine 3 (1): 4–16.
Rabiner, L.R. 1989. β€œA Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.” Proceedings of the IEEE 77 (2): 257–86.
Raol, J. R., and N. K. Sinha. 1987. β€œOn Pugachev’s Filtering Theory for Stochastic Nonlinear Systems.” In Stochastic Control, edited by N. K. Sinha and L. A. Telksnys, 183–88. IFAC Symposia Series. Oxford: Pergamon.
Reece, S., and S. Roberts. 2010. β€œAn Introduction to Gaussian Processes for the Kalman Filter Expert.” In 2010 13th International Conference on Information Fusion, 1–9.
Reller, Christoph. 2013. β€œState-Space Methods in Statistical Signal Processing: New Ideas and Applications.” Application/pdf. Konstanz: ETH Zurich.
Revach, Guy, Nir Shlezinger, Ruud J. G. van Sloun, and Yonina C. Eldar. 2021. β€œKalmannet: Data-Driven Kalman Filtering.” In ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3905–9.
Robertson, Andrew N. 2011. β€œA Bayesian Approach to Drum Tracking.” In.
Robertson, Andrew, and Mark Plumbley. 2007. β€œB-Keeper: A Beat-Tracker for Live Performance.” In Proceedings of the 7th International Conference on New Interfaces for Musical Expression, 234–37. NIME ’07. New York, NY, USA: ACM.
Robertson, Andrew, Adam M. Stark, and Mark D. Plumbley. 2011. β€œReal-Time Visual Beat Tracking Using a Comb Filter Matrix.” In Proceedings of the International Computer Music Conference 2011.
Robertson, Andrew, Adam Stark, and Matthew EP Davies. 2013. β€œPercussive Beat Tracking Using Real-Time Median Filtering.” In Proceedings of European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases.
Rodriguez, Alejandro, and Esther Ruiz. 2009. β€œBootstrap Prediction Intervals in State–Space Models.” Journal of Time Series Analysis 30 (2): 167–78.
Roth, Michael, Gustaf Hendeby, Carsten Fritsche, and Fredrik Gustafsson. 2017. β€œThe Ensemble Kalman Filter: A Signal Processing Perspective.” EURASIP Journal on Advances in Signal Processing 2017 (1): 56.
Rudenko, E. A. 2013. β€œOptimal Structure of Continuous Nonlinear Reduced-Order Pugachev Filter.” Journal of Computer and Systems Sciences International 52 (6): 866–92.
SΓ€rkkΓ€, S., and J. Hartikainen. 2013. β€œNon-Linear Noise Adaptive Kalman Filtering via Variational Bayes.” In 2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP), 1–6.
SΓ€rkkΓ€, Simo. 2007. β€œOn Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.” IEEE Transactions on Automatic Control 52 (9): 1631–41.
β€”β€”β€”. 2013. Bayesian Filtering and Smoothing. Institute of Mathematical Statistics Textbooks 3. Cambridge, U.K. ; New York: Cambridge University Press.
SΓ€rkkΓ€, Simo, and Jouni Hartikainen. 2012. β€œInfinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.” In Artificial Intelligence and Statistics.
SΓ€rkkΓ€, Simo, and A. Nummenmaa. 2009. β€œRecursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations.” IEEE Transactions on Automatic Control 54 (3): 596–600.
SΓ€rkkΓ€, Simo, A. Solin, and J. Hartikainen. 2013. β€œSpatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine 30 (4): 51–61.
Schein, Aaron, Hanna Wallach, and Mingyuan Zhou. 2016. β€œPoisson-Gamma Dynamical Systems.” In Advances In Neural Information Processing Systems, 5006–14.
Schmidt, Jonathan, Nicholas KrΓ€mer, and Philipp Hennig. 2021. β€œA Probabilistic State Space Model for Joint Inference from Differential Equations and Data.” arXiv:2103.10153 [Cs, Stat], June.
Segall, A., M. Davis, and T. Kailath. 1975. β€œNonlinear Filtering with Counting Observations.” IEEE Transactions on Information Theory 21 (2): 143–49.
Ε indelΓ‘Ε™, Jan, Igor Vajda, and Miroslav KΓ‘rnα»³. 2008. β€œStochastic Control Optimal in the Kullback Sense.” Kybernetika 44 (1): 53–60.
Sorenson, H.W. 1970. β€œLeast-Squares Estimation: From Gauss to Kalman.” IEEE Spectrum 7 (7): 63–68.
StΓ€dler, Nicolas, and Sach Mukherjee. 2013. β€œPenalized Estimation in High-Dimensional Hidden Markov Models with State-Specific Graphical Models.” The Annals of Applied Statistics 7 (4): 2157–79.
Surace, Simone Carlo, and Jean-Pascal Pfister. 2016. β€œOnline Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes.” In.
Tavakoli, Shahin, and Victor M. Panaretos. 2016. β€œDetecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics.” Journal of the American Statistical Association, March, 1–31.
Thrun, Sebastian, and John Langford. 1998. β€œMonte Carlo Hidden Markov Models.” DTIC Document.
Thrun, Sebastian, John Langford, and Dieter Fox. 1999. β€œMonte Carlo Hidden Markov Models: Learning Non-Parametric Models of Partially Observable Stochastic Processes.” In Proceedings of the International Conference on Machine Learning. Bled, Slovenia.
Turner, Ryan, Marc Deisenroth, and Carl Rasmussen. 2010. β€œState-Space Inference and Learning with Gaussian Processes.” In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 868–75.
Wikle, Christopher K., and L. Mark Berliner. 2007. β€œA Bayesian Tutorial for Data Assimilation.” Physica D: Nonlinear Phenomena, Data Assimilation, 230 (1): 1–16.
Wikle, Christopher K., L. Mark Berliner, and Noel Cressie. 1998. β€œHierarchical Bayesian Space-Time Models.” Environmental and Ecological Statistics 5 (2): 117–54.
Zhao, Yiran, and Tiangang Cui. 2023. β€œTensor-Based Methods for Sequential State and Parameter Estimation in State Space Models.” arXiv.
Zoeter, Onno. 2007. β€œBayesian Generalized Linear Models in a Terabyte World.” In 2007 5th International Symposium on Image and Signal Processing and Analysis, 435–40. Istanbul, Turkey: IEEE.

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