Fast Kalman Filtering and Forward–Backward Smoothing via a Low-Rank Perturbative Approach

References

• Anderson, B., and Moore, J. (1979), Optimal Filtering, Englewood Cliffs, NJ: Prentice Hall.
   Google Scholar
• Antoulas, A. (2005), Approximation of Large-Scale Dynamical Systems, Philadelphia, PA: SIAM.
   Google Scholar
• Banerjee, S., Gelfand, A.E., Finley, A.O., and Sang, H. (2008), “Gaussian Predictive Process Models for Large Spatial Data Sets,” Journal of the Royal Statistical Society, Series B, 70, 825–848.
   Google Scholar
• Bickel, J., and Levina, E. (2008), “Regularized Estimation of Large Covariance Matrices,” The Annals of Statistics, 36, 199–227.
   Web of Science ®Google Scholar
• Briggs, W.L., Henson, V.E., and McCormick, S.F. (2000), A Multigrid Tutorial (2nd ed.), Philadelphia, PA: SIAM.
   Google Scholar
• Brockwell, P., and Davis, R. (1991), Time Series: Theory and Methods, New York: Springer.
   Google Scholar
• Brown, E., Frank, L., Tang, D., Quirk, M., and Wilson, M. (1998), “A Statistical Paradigm for Neural Spike Train Decoding Applied to Position Prediction From Ensemble Firing Patterns of Rat Hippocampal Place Cells,” Journal of Neuroscience, 18, 7411–7425.
   PubMed Web of Science ®Google Scholar
• Brown, E., Nguyen, D., Frank, L., Wilson, M., and Solo, V. (2001), “An Analysis of Neural Receptive Field Plasticity by Point Process Adaptive Filtering,” PNAS, 98, 12261–12266.
   PubMed Web of Science ®Google Scholar
• Chan, R.H., and Ng, M.K. (1996), “Conjugate Gradient Methods for Toeplitz Systems,” SIAM Review, 38, 427–482.
   Web of Science ®Google Scholar
• Chandrasekar, J., Kim, I., Bernstein, D., and Ridley, A. (2008), “Cholesky-Based Reduced-Rank Square-Root Kalman Filtering,” in American Control Conference , Red Hook, NY: IEEE, pp.3987–3992.
   Google Scholar
• Cressie, N., and Johannesson, G. (2008), “Fixed Rank Kriging for Very Large Spatial Data Sets,” Journal of the Royal Statistical Society, Series B, 70, 209–226.
   Google Scholar
• Cressie, N., Shi, T., and Kang, E.L. (2010), “Fixed Rank Filtering for Spatio-Temporal Data,” Journal of Computational and Graphical Statistics, 19, 724–745.
   Web of Science ®Google Scholar
• Czanner, G., Eden, U., Wirth, S., Yanike, M., Suzuki, W., and Brown, E. (2008), “Analysis of Between-Trial and Within-Trial Neural Spiking Dynamics,” Journal of Neurophysiology, 99, 2672–2693.
   PubMed Web of Science ®Google Scholar
• Davis, T. (2006), Direct Methods for Sparse Linear Systems, Philadelphia, PA: SIAM.
   Google Scholar
• Dayan, P., and Abbott, L. (2001), Theoretical Neuroscience, Cambridge, MA: MIT Press.
   Google Scholar
• DiMatteo, I., Genovese, C., and Kass, R. (2001), “Bayesian Curve Fitting With Free-Knot Splines,” Biometrika, 88, 1055–1073.
   Web of Science ®Google Scholar
• El Karoui, N. (2008), “Operator Norm Consistent Estimation of Large-Dimensional Sparse Covariance Matrices,” The Annals of Statistics, 36, 2717–2756.
   Web of Science ®Google Scholar
• Evensen, G. (2009), Data Assimilation: The Ensemble Kalman Filter, New York: Springer.
   Google Scholar
• Fahrmeir, L., and Kaufmann, H. (1991), “On Kalman Filtering, Posterior Mode Estimation and Fisher Scoring in Dynamic Exponential Family Regression,” Metrika, 38, 37–60.
   Google Scholar
• Fahrmeir, L., and Tutz, G. (1994), Multivariate Statistical Modelling Based on Generalized Linear Models, New York: Springer.
   Google Scholar
• Fedorov, V. (1972), Theory of Optimal Experiments, New York: Academic Press.
   Google Scholar
• Frank, L., Eden, U., Solo, V., Wilson, M., and Brown, E. (2002), “Contrasting Patterns of Receptive Field Plasticity in the Hippocampus and the Entorhinal Cortex: An Adaptive Filtering Approach,” Journal of Neuroscience, 22, 3817–3830.
   PubMed Web of Science ®Google Scholar
• Freestone, D., Aram, P., Dewar, M., Scerri, K., Grayden, D., and Kadirkamanathan, V. (2011), “A Data-Driven Framework for Neural Field Modeling,” NeuroImage, 56, 1043–1058.
   PubMedGoogle Scholar
• Furrer, R., and Bengtsson, T. (2007), “Estimation of High-Dimensional Prior and Posterior Covariance Matrices in Kalman Filter Variants,” Journal of Multivariate Analysis, 98, 227–255.
   Web of Science ®Google Scholar
• Galka, A., Ozaki, T., Muhle, H., Stephani, U., and Siniatchkin, M. (2008), “A Data-Driven Model of the Generation of Human EEG Based on a Spatially Distributed Stochastic Wave Equation,” Cognitive Neurodynamics, 2, 101–113.
   PubMedGoogle Scholar
• Golub, G.H., and Van Loan, C.F. (1996), Matrix Computations, Baltimore, MD: The Johns Hopkins University Press.
   Google Scholar
• Green, P., and Silverman, B. (1994), Nonparametric Regression and Generalized Linear Models, Boca Raton, FL: CRC Press.
   Google Scholar
• Hastie, T., Tibshirani, R., and Friedman, J. (2001), The Elements of Statistical Learning, New York: Springer.
   Google Scholar
• Hines, M. (1984), “Efficient Computation of Branched Nerve Equations,” International Journal of Bio-Medical Computing, 15, 69–76.
   PubMedGoogle Scholar
• Huggins, J., and Paninski, L. (2012), “Optimal Experimental Design for Sampling Voltage on Dendritic Trees,” Journal of Computational Neuroscience, 32, 347–366.
   PubMedGoogle Scholar
• Isaacson, E., and Keller, H. (1994), Analysis of Numerical Methods, Mineola, NY: Dover Publications.
   Google Scholar
• Kaufman, C.G., Schervish, M.J., and Nychka, D.W. (2008), “Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets,” Journal of the American Statistical Association, 103, 1545–1555.
   Web of Science ®Google Scholar
• Khan, U.A., and Moura, J. M.F. (2008), “Distributing the Kalman Filter for Large-Scale Systems,” IEEE Transactions on Signal Processing, 56, 4919–4935.
   Web of Science ®Google Scholar
• Koch, C. (1999), Biophysics of Computation, New York: Oxford University Press.
   Google Scholar
• Lew, S., Wolters, C.H., Dierkes, T., Röer, C., and MacLeod, R.S. (2009), “Accuracy and Run-Time Comparison for Different Potential Approaches and Iterative Solvers in Finite Element Method Based EEG Source Analysis,” Applied Numerical Mathmatics, 59, 1970–1988.
   PubMed Web of Science ®Google Scholar
• Lewi, J., Butera, R., and Paninski, L. (2009), “Sequential Optimal Design of Neurophysiology Experiments,” Neural Computation, 21, 619–687.
   PubMed Web of Science ®Google Scholar
• Long, C.J., Purdon, R.L., Temereanca, S., Desai, N.U., Hämäläinen, M., and Brown, E.N. (2006), “Large Scale Kalman Filtering Solutions to the Electrophysiological Source Localization Problem—A MEG Case Study,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 4532–4535.
   PubMedGoogle Scholar
• Paninski, L. (2010), “Fast Kalman Filtering on Quasilinear Dendritic Trees,” Journal of Computational Neuroscience, 28, 211–228.
   PubMedGoogle Scholar
• Paninski, L., Ahmadian, Y., Ferreira, D., Koyama, S., Rahnama, K., Vidne, M., Vogelstein, J., and Wu, W. (2010), “A New Look at State-Space Models for Neural Data,” Journal of Computational Neuroscience, 29, 107–126.
   PubMed Web of Science ®Google Scholar
• Papandreou, G., and Yuille, A. (2010), “Gaussian Sampling by Local Perturbations,” in Proceedings of NIPS. , pp. 1858–1866.
   Google Scholar
• Park, T., and Casella, G. (2008), “The Bayesian Lasso,” Journal of the American Statistical Association, 103, 681–686.
   Web of Science ®Google Scholar
• Pnevmatikakis, E. A., and Paninski, L. (2012), “Fast Interior-Point Inference in High-Dimensional, Sparse, Penalized State-Space Models,” in Fifteenth International Conference on Artificial Intelligence and Statistics (AISTATS). Journal of Machine Learning Research, Volume 22, pp. 895–904.
   Google Scholar
• Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. (1992), Numerical Recipes in C, New York: Cambridge University Press.
   Google Scholar
• Rabiner, L. (1989), “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition,” Proceedings of the IEEE, 77, 257–286.
   Web of Science ®Google Scholar
• Rahnama Rad, K., and Paninski, L. (2010), “Efficient Estimation of Two-Dimensional Firing Rate Surfaces via Gaussian Process Methods,” Network, 21, 142–168.
   PubMedGoogle Scholar
• Rue, H., and Held, L. (2005), Gaussian Markov Random Fields: Theory and Applications, Boca Raton, FL: CRC Press.
   Google Scholar
• Rybicki, G., and Hummer, D. (1991), “An Accelerated Lambda Iteration Method for Multilevel Radiative Transfer. I-Non-overlapping Lines With Background Continuum,” Astronomy and Astrophysics, 245, 171–181.
   Google Scholar
• Sabino, J. (2007), “Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method,” . unpublished Ph.D. thesis, Rice University.
   Google Scholar
• Seeger, M., and Nickisch, H. (2011), “Large Scale Bayesian Inference and Experimental Design for Sparse Linear Models,” SIAM Journal of Imaging Sciences, 4, 166–199.
   Google Scholar
• Shumway, R., and Stoffer, D. (2006), Time Series Analysis and Its Applications, New York: Springer.
   Google Scholar
• Solo, V. (2004), “State Estimation From High-Dimensional Data,” ICASSP, 2, 685–688.
   Google Scholar
• Stroud, J.R., Muller, P., and Sanso, B. (2001), “Dynamic Models for Spatiotemporal Data,” Journal of the Royal Statistical Society, Series B, 63, 673–689.
   Google Scholar
• Treebushny, D., and Madsen, H. (2005), “On the Construction of a Reduced Rank Square-Root Kalman Filter for Efficient Uncertainty Propagation,” Future Generation Computer Systems, 21, 1047–1055.
   Google Scholar
• Verlaan, M. (1998), “Efficient Kalman Filtering Algorithms for Hydrodynamic Models,” . unpublished Ph.D. thesis, TU Delft.
   Google Scholar
• Wikle, C., and Cressie, N. (1999), “A Dimension-Reduced Approach to Space-Time Kalman Filtering,” . Biometrika, 86, 815–829.
   Google Scholar
• Wolters, C. (2007), “The Finite Element Method in EEG/MEG Source Analysis,” SIAM News, 40, 1–2.
   Google Scholar
• Wood, S.N. (2006), “Low-Rank Scale-Invariant Tensor Product Smooths for Generalized Additive Mixed Models,” Biometrics, 62, 1025–1036.
   PubMed Web of Science ®Google Scholar