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EmrysFilter 0.1.0
State estimation utilities in C++23 using Eigen
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The sources the library implements. Each entry has an anchor named by first author and year, and a comment cites it as \ref Runnalls2007 "Runnalls (2007)". The library has nothing of its own to cite: every algorithm is one published here, and the comment that cites an entry says what the code takes from it, such as the equation numbers or the section, and where it departs.
An algorithm has two citations, often the same entry. The source that introduced it gets the credit. The source the code was implemented from is one that was read, in full, and every equation, section, or algorithm number in a comment is taken from that source. When the original was not read, the comment cites it for credit and names the source that was: \ref Rauch1965 "Rauch, Tung, and Striebel (1965)", as given in \ref Sarkka2013 "Särkkä (2013)", §8.2. The source read may be a free description, such as a Wikipedia section, linked inline. The entry for a source not read says so in a final sentence, Not read., and names the sources that describe its method; when the original is read, check the code against it and delete the sentence. Add an entry when a comment first cites it. Foundational concepts link to Wikipedia inline, not through this page.
Entries are in alphabetical order by first author.
D. L. Alspach and H. W. Sorenson. Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Transactions on Automatic Control, 17(4):439–448, 1972. Not read. Bergman (1999), §3.5.3, sketches the method; no free source read describes it fully.
I. Arasaratnam, S. Haykin, and T. R. Hurd. Cubature Kalman filtering for continuous-discrete systems: theory and simulations. IEEE Transactions on Signal Processing, 58(10):4977–4993,
D. Arthur and S. Vassilvitskii. k-means++: the advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1027–1035,
M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing, 50(2):174–188, 2002. Free: PDF, course copy.
M. Athans, D. Castañon, K.-P. Dunn, C. S. Greene, W. H. Lee, N. R. Sandell, Jr., and A. S. Willsky. The stochastic control of the F-8C aircraft using a multiple model adaptive control (MMAC) method—Part I: Equilibrium flight. IEEE Transactions on Automatic Control, 22(5):768–780, 1977. Free: PDF, scan.
D. Barber. Expectation correction for smoothed inference in switching linear dynamical systems. Journal of Machine Learning Research, 7:2515–2540, 2006. Section 2.2 is the backward pass; (12) and (13) give the smoothed switch given the next one, and the mean approximation evaluates the average in (12) at the mean of the smoothed Gaussian. Free: JMLR.
Y. Bar-Shalom and E. Tse. Tracking in a cluttered environment with probabilistic data association. Automatica, 11(5):451–460, 1975.
Y. Bar-Shalom, X. R. Li, and T. Kirubarajan. Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software. Wiley, 2001. Not read. The comments that cite it take their numbers from Chen et al. (2023) (NIS and NEES), Hashimzade et al. (2024) (IMM), Bar-Shalom and Tse (1975) (gating), and Discretization.
B. M. Bell and F. W. Cathey. The iterated Kalman filter update as a Gauss-Newton method. IEEE Transactions on Automatic Control, 38(2):294–297, 1993. Not read. Described in J. Huai and X. Gao, A quick guide for the iterated extended Kalman filter on manifolds, arXiv:2307.09237, §II and Algorithm 1.
N. Bergman. Recursive Bayesian Estimation: Navigation and Tracking Applications. PhD thesis, Linköping University, Linköping Studies in Science and Technology, Dissertation No. 579, 1999. Free: PDF.
A. Bhattacharyya. On a measure of divergence between two statistical populations defined by their probability distributions. Bulletin of the Calcutta Mathematical Society, 35:99–109, 1943. Free: scan.
G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, 1977 (Mathematics in Science and Engineering, volume 128; Dover reprint, 2006). Not read. The measurement update and the elimination of variables are as given in Ke et al. (2024), §II and §III-D; no free source read gives the time update's data array.
H. A. P. Blom and Y. Bar-Shalom. The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Transactions on Automatic Control, 33(8):780–783, 1988. Not read. Described in Hashimzade et al. (2024), §2.2.3.
R. S. Bucy and K. D. Senne. Digital synthesis of non-linear filters. Automatica, 7(3):287–298, 1971. Not read. Described in Bergman (1999), §2.3 and §5.2.
J. Carpenter, P. Clifford, and P. Fearnhead. Improved particle filter for nonlinear problems. IEE Proceedings – Radar, Sonar and Navigation, 146(1):2–7, 1999. Not read. Described in Douc et al. (2005), §2.4, who credit Carpenter et al.; Arulampalam et al. (2002), §V-A, credit Kitagawa (1996).
C. Chahbazian, K. Dahia, N. Merlinge, B. Winter-Bonnet, A. Blanc, and C. Musso. Recursive posterior Cramér-Rao lower bound on Lie groups. Automatica, 160:111422, 2024. Proposition 2 extends the recursion of Tichavský, Muravchik, and Nehorai (1998) to unimodular matrix Lie groups, and §5, eqs. (68) to (72), gives its matrices for Gaussian noise entering by the group exponential. Free: PDF, HAL.
Z. Chen, H. Biggie, N. Ahmed, S. Julier, and C. Heckman. Kalman filter auto-tuning through enforcing chi-squared normalized error distributions with Bayesian optimization. arXiv:2306.07225,
A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1):1–38, 1977. Free: PDF, course copy.
P. Ding. On the conditional distribution of the multivariate t distribution. The American Statistician, 70(3):293–295, 2016. Free: arXiv.
R. Douc, O. Cappé, and E. Moulines. Comparison of resampling schemes for particle filtering. arXiv:cs/0507025, 2005. Free: arXiv.
D. C. Dowson and B. V. Landau. The Fréchet distance between multivariate normal distributions. Journal of Multivariate Analysis, 12(3):450–455, 1982. Not read. Described in Fréchet distance and Wasserstein metric.
A. Doucet, N. de Freitas, K. Murphy, and S. Russell. Rao-Blackwellised particle filtering for dynamic Bayesian networks. In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI), pages 176–183, 2000. Free: PDF.
J. Duník, M. Soták, M. Veselý, O. Straka, and W. Hawkinson. Design of Rao–Blackwellized point-mass filter with application in terrain aided navigation. IEEE Transactions on Aerospace and Electronic Systems, 55(1):251–272, 2019. Not read. Described in Pocoma et al. (2023), §II-D and §III, which cite it for the correction and prediction.
W. Fischer and K. Meier-Hellstern. The Markov-modulated Poisson process (MMPP) cookbook. Performance Evaluation, 18(2):149–171, 1993. Not read. The likelihood is as given in S. Mews, J.-O. Koslik, and R. Langrock, How to build your latent Markov model, arXiv:2406.19157, §3.1.
Á. F. García-Fernández, L. Svensson, M. R. Morelande, and S. Särkkä. Posterior linearization filter: principles and implementation using sigma points. IEEE Transactions on Signal Processing, 63(20):5561–5573, 2015. Algorithm 1, the iterated posterior linearization filter; §IV-C on earlier iterated sigma-point filters. Free: author's version.
A. Gelb, editor. Applied Optimal Estimation. MIT Press, 1974. Not read. The Gauss-Markov process is described in Ornstein-Uhlenbeck process; no free source read describes the linearized Kalman filter.
S. J. Godsill, A. Doucet, and M. West. Monte Carlo smoothing for nonlinear time series. Journal of the American Statistical Association, 99(465):156–168, 2004. Backward simulation for particles, which decisions.md [smoothing] leaves out until a process model exposes its transition density.
G. H. Golub and C. F. Van Loan. Matrix Computations. Fourth edition, Johns Hopkins University Press, 2013. Not read. The comments link QR decomposition, Cholesky decomposition, and Square root of a matrix.
N. J. Gordon, D. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F – Radar and Signal Processing, 140(2):107–113,
W. K. Grassmann, M. I. Taksar, and D. P. Heyman. Regenerative analysis and steady state distributions for Markov chains. Operations Research, 33(5):1107–1116, 1985. Not read. The GTH algorithm, as the code implements it (Gaussian elimination of the states from the last, the diagonal never read), is described in Q. Bu and Y. Q. Zhao, GTH algorithm, censored Markov chains, and RG-factorization in block-form, arXiv:2604.14347, §2.
N. Hashimzade, O. Kirsanov, T. Kirsanova, and J. Maih. On Bayesian filtering for Markov regime switching models. arXiv:2402.08051, 2024. Free: arXiv.
R. E. Helmick, W. D. Blair, and S. A. Hoffman. Fixed-interval smoothing for Markovian switching systems. IEEE Transactions on Information Theory, 41(6):1845–1855, 1995. The two-filter IMM smoother, which decisions.md [smoothing] rejects, since it needs backward-time dynamics.
C. Hertzberg, R. Wagner, U. Frese, and L. Schröder. Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds. Information Fusion, 14(1):57–77, 2013. Free: arXiv.
N. J. Higham. The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications, 26(4):1179–1193, 2005. The method of Eigen's matrix exponential, which ContinuousTimeMarkovChain::transition() calls.
J. D. Hol, T. B. Schön, and F. Gustafsson. On resampling algorithms for particle filters. In IEEE Nonlinear Statistical Signal Processing Workshop, pages 79–82, 2006. Free: PDF.
M. F. Huber, T. Bailey, H. Durrant-Whyte, and U. D. Hanebeck. On entropy approximation for Gaussian mixture random vectors. In IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI), pages 181–188, 2008. The upper bound on a mixture's entropy, the components' weighted entropies plus the entropy of the weights, which bounds the lower end of kullbackLeiblerCollapseBounds().
L. Isserlis. On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12(1–2):134–139, 1918. Free: PDF.
K. Ito and K. Xiong. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control, 45(5):910–927, 2000. The Gauss-Hermite filter, which GaussHermite implements as Särkkä (2013) states it.
S. J. Julier. The spherical simplex unscented transformation. In Proceedings of the American Control Conference, volume 3, pages 2430–2434, 2003. The spherical simplex set, which SphericalSimplex builds by the construction of Julier and Uhlmann (2004).
S. J. Julier and J. K. Uhlmann. Unscented filtering and nonlinear estimation. Proceedings of the IEEE, 92(3):401–422, 2004. Not read. The transform, its weights, and the Kalman filter built on it are as given in Särkkä (2013), §5.5, §5.6, and §6.5; no free source read describes the spherical simplex construction.
P. G. Kaminski, A. E. Bryson, and S. F. Schmidt. Discrete square root filtering: a survey of current techniques. IEEE Transactions on Automatic Control, 16(6):727–736, 1971. The square-root covariance filter as a triangularization of stacked factors, in prediction and in the update. Not read. Described in K. S. Tracy, A square-root Kalman filter using only QR decompositions, arXiv:2208.06452, §3, which forms the innovation factor and the posterior separately rather than as one array.
R. E. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1):35–45, 1960. Free: PDF, course copy.
T. Ke, P. Agrawal, Y. Zhang, W. Zhen, C. X. Guo, T. Sharp, and R. C. Dutoit. PC-SRIF: Preconditioned Cholesky-based square root information filter for vision-aided inertial navigation. arXiv:2409.11372, 2024. Free: arXiv.
C.-J. Kim. Dynamic linear models with Markov-switching. Journal of Econometrics, 60(1–2):1–22,
G. Kitagawa. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5(1):1–25, 1996. Not read. Stratified resampling is described in Douc et al. (2005), §2.3, and the smoother in Särkkä (2013), §11.1.
A. Kong, J. S. Liu, and W. H. Wong. Sequential imputations and Bayesian missing data problems. Journal of the American Statistical Association, 89(425):278–288, 1994. Not read. Described in Bergman (1999), §6.4.1, equation (6.28), and Arulampalam et al. (2002), §V-A.
S. Kullback and R. A. Leibler. On information and sufficiency. Annals of Mathematical Statistics, 22(1):79–86, 1951. Free: open access.
T. Lefebvre, H. Bruyninckx, and J. De Schutter. Comment on "A new method for the nonlinear transformation of means and covariances in filters and estimators". IEEE Transactions on Automatic Control, 47(8):1406–1409, 2002. Free: PDF, course copy.
W. J. Lentz. Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics, 15(3):668–671, 1976. Not read. Described in Lentz's algorithm.
J. Lin. Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37(1):145–151, 1991. Not read. Described in Jensen-Shannon divergence.
J. S. Liu and R. Chen. Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association, 93(443):1032–1044, 1998. Not read. Residual resampling is described in Douc et al. (2005), §2.2, and resampling on the effective sample size in Arulampalam et al. (2002), §V-A.
F. L. Markley. Attitude error representations for Kalman filtering. Journal of Guidance, Control, and Dynamics, 26(2):311–317, 2003. The text checked is the NASA preprint of 2002 (NTRS 20020060647), whose reset section says the reset leaves the covariance unchanged. Free: NASA preprint.
M. W. Mueller, M. Hehn, and R. D'Andrea. Covariance correction step for Kalman filtering with an attitude. Journal of Guidance, Control, and Dynamics, 40(9):2301–2306, 2017. The text checked is the authors' preprint. Free: preprint.
L. M. Murray, A. Lee, and P. E. Jacob. Parallel resampling in the particle filter. Journal of Computational and Graphical Statistics, 25(3):789–805, 2016. Free: arXiv.
C. Musso, N. Oudjane, and F. Le Gland. Improving regularised particle filters. In A. Doucet, N. de Freitas, and N. Gordon, editors, Sequential Monte Carlo Methods in Practice, pages 247–271. Springer, 2001. Not read. Described in Arulampalam et al. (2002), §V-B3, with an Epanechnikov kernel where the library uses a Gaussian one.
A. G. O. Mutambara. Decentralized Estimation and Control for Multisensor Systems. CRC Press,
N. Nadarajah. Retrodiction for Multitarget Tracking. PhD thesis, McMaster University, 2009. Chapter 8 derives and states the backward IMM smoother. Free: PDF.
N. Nadarajah, R. Tharmarasa, M. McDonald, and T. Kirubarajan. IMM forward filtering and backward smoothing for maneuvering target tracking. IEEE Transactions on Aerospace and Electronic Systems, 48(3):2673–2678, 2012. The journal version of Nadarajah (2009), chapter 8; the code follows the thesis, which is the text checked. Not read. The comment that cites it follows Nadarajah (2009), which this pass did not recheck.
S. A. Pasha, B.-N. Vo, H. D. Tuan, and W.-K. Ma. A Gaussian mixture PHD filter for jump Markov system models. IEEE Transactions on Aerospace and Electronic Systems, 45(3):919–936, 2009. Free: repository.
X. Pennec. Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 25(1):127–154, 2006. Not read. The iteration is as given in Hertzberg et al. (2013), §3.4. Free copy: CORE, not yet read.
E. Pocoma, L. Storrer, F. Quitin, L. Vandendorpe, P. De Doncker, and F. Horlin. Direct tracking of a wireless transmitter based on Rao-Blackwellized point mass filter. IEEE Transactions on Vehicular Technology, 2023. §II-C gives the factorization into the block and the rest, §II-D and Algorithm 1 the filter, equations (17) and (18) the correction for a measurement of the block alone, and §III-D the prediction between grids. Free: PDF, author's copy.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes: The Art of Scientific Computing, third edition. Cambridge University Press, 2007. Not read. The series and continued fraction are described in Incomplete gamma function; no free source read gives the switch between them or rtsafe's safeguard.
L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257–286, 1989. Free: PDF, course copy.
A. S. Rahmathullah, L. Svensson, and D. Svensson. Merging-based forward-backward smoothing on Gaussian mixtures. In Proceedings of the 17th International Conference on Information Fusion,
H. E. Rauch, F. Tung, and C. T. Striebel. Maximum likelihood estimates of linear dynamic systems. AIAA Journal, 3(8):1445–1450, 1965. Not read. Described in Särkkä (2013), §8.2 and §10.1.
A. R. Runnalls. Kullback-Leibler approach to Gaussian mixture reduction. IEEE Transactions on Aerospace and Electronic Systems, 43(3):989–999, 2007. Free: preprint.
S. Särkkä, P. Bunch, and S. J. Godsill. A backward-simulation based Rao-Blackwellized particle smoother for conditionally linear Gaussian models. In 16th IFAC Symposium on System Identification (SYSID), 2012. The Rao-Blackwellized smoother that decisions.md [smoothing] defers. Free: author's PDF.
S. Särkkä. Bayesian Filtering and Smoothing. Cambridge University Press, 2013. Free: author's PDF.
S. F. Schmidt. Application of state-space methods to navigation problems. In C. T. Leondes, editor, Advances in Control Systems, volume 3, pages 293–340. Academic Press, 1966. The consider, or Schmidt-Kalman, update.
T. Schön, F. Gustafsson, and P.-J. Nordlund. Marginalized particle filters for mixed linear/nonlinear state-space models. IEEE Transactions on Signal Processing, 53(7):2279–2289, 2005. Free: preprint.
B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and Hall, 1986. Not read. Described in Multivariate kernel density estimation, with a diagonal bandwidth where the library takes the data's covariance.
V. Šmídl and M. Gašperin. Rao-Blackwellized point mass filter for reliable state estimation. In Proceedings of the 16th International Conference on Information Fusion, pages 312–318, 2013. Not read. Described in Pocoma et al. (2023), §II-D.
J. Solà. Quaternion kinematics for the error-state Kalman filter. Technical report, arXiv:1711.02508, 2017. §6.3, the ESKF reset, eqs. (284) to (288) and (294). Free: arXiv.
J. Solà, J. Deray, and D. Atchuthan. A micro Lie theory for state estimation in robotics. Technical report IRI-TR-18-01, arXiv:1812.01537, 2018 (version 9, 2021). Appendix B, the right Jacobian of SO(3), eq. (143). Free: arXiv.
R. van der Merwe and E. A. Wan. The square-root unscented Kalman filter for state and parameter-estimation. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), volume 6, pages 3461–3464, 2001. Sigma points spread from the covariance's factor, and the predicted factor by triangularizing their weighted deviations with the noise's factor; the code departs in the update, which triangularizes the joint array rather than applying their rank-one downdates, and takes the measurement's statistical linearization. Not read. No free source read describes the square-root unscented Kalman filter.
N. Vasconcelos and A. Lippman. Learning mixture hierarchies. In Advances in Neural Information Processing Systems 11 (NIPS 1998), pages 606–612. MIT Press, 1999. Free: PDF.
A. J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13(2):260–269, 1967. Not read. Described in Rabiner (1989), §III-B.
B. P. Welford. Note on a method for calculating corrected sums of squares and products. Technometrics, 4(3):419–420, 1962. Not read. Described in [Algorithms for calculating variance](https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Welford's_online_algorithm).
I. J. Thompson and A. R. Barnett. Coulomb and Bessel functions of complex arguments and order. Journal of Computational Physics, 64(2):490–509, 1986. Not read. Described in Lentz's algorithm.
P. Tichavský, C. H. Muravchik, and A. Nehorai. Posterior Cramér-Rao bounds for discrete-time nonlinear filtering. IEEE Transactions on Signal Processing, 46(5):1386–1396, 1998. Proposition 1, eq. (21), is the recursion on the information matrix, and eqs. (34) to (36) its matrices for additive Gaussian noise; Section IV treats a singular process noise. Free: PDF, SEDICI.
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