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  • 1.
    Umenberger, Jack
    et al.
    Uppsala University, Uppsala, Sweden.
    Schön, Thomas B.
    Uppsala University, Uppsala, Sweden.
    Lindsten, Fredrik
    Uppsala University, Uppsala, Sweden.
    Bayesian identification of state-space models via adaptive thermostats2019In: Proceedings of the 58th IEEE Conference on Decision and Control (CDC), 2019Conference paper (Refereed)
  • 2.
    Vaicenavicius, Juozas
    et al.
    Uppsala University, Uppsala, Sweden; Veoneer Inc..
    Widmann, David
    Uppsala University, Uppsala, Sweden.
    Andersson, Carl
    Uppsala University, Uppsala, Sweden.
    Lindsten, Fredrik
    Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Science & Engineering.
    Roll, Jacob
    Veoneer Inc..
    Schön, Thomas B.
    Uppsala University, Uppsala, Sweden.
    Evaluating model calibration in classification2019In: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, 2019, Vol. 89Conference paper (Refereed)
    Abstract [en]

    Probabilistic classifiers output a probability distribution on target classes rather than just a class prediction. Besides providing a clear separation of prediction and decision making, the main advantage of probabilistic models is their ability to represent uncertainty about predictions. In safetycritical applications, it is pivotal for a model to possess an adequate sense of uncertainty, which for probabilistic classifiers translates into outputting probability distributions that are consistent with the empirical frequencies observed from realized outcomes. A classifier with such a property is called calibrated. In this work, we develop a general theoretical calibration evaluation framework grounded in probability theory, and point out subtleties present in model calibration evaluation that lead to refined interpretations of existing evaluation techniques. Lastly, we propose new ways to quantify and visualize miscalibration in probabilistic classification, including novel multidimensional reliability diagrams.

  • 3.
    Andersson Naesseth, Christian
    et al.
    Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Science & Engineering.
    Lindsten, Fredrik
    Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Science & Engineering.
    Schon, Thomas B.
    Uppsala Univ, Sweden.
    High-Dimensional Filtering Using Nested Sequential Monte Carlo2019In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 67, no 16, p. 4177-4188Article in journal (Refereed)
    Abstract [en]

    Sequential Monte Carlo (SMC) methods comprise one of the most successful approaches to approximate Bayesian filtering. However, SMC without a good proposal distribution can perform poorly, in particular in high dimensions. We propose nested sequential Monte Carlo, a methodology that generalizes the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correctSMCalgorithm. This way, we can compute an "exact approximation" of, e. g., the locally optimal proposal, and extend the class of models forwhichwe can perform efficient inference using SMC. We showimproved accuracy over other state-of-the-art methods on several spatio-temporal state-space models.

  • 4.
    Jacob, Pierre E.
    et al.
    Harvard Univ, MA 02138 USA.
    Lindsten, Fredrik
    Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Science & Engineering.
    Schön, Thomas B.
    Uppsala Univ, Sweden.
    Smoothing With Couplings of Conditional Particle Filters2019In: Journal of the American Statistical Association, ISSN 0162-1459, E-ISSN 1537-274XArticle in journal (Refereed)
    Abstract [en]

    In state-space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and CI can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains, with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly informative observations, and for a realistic Lotka-Volterra model with an intractable transition density. Supplementary materials for this article are available online.

  • 5.
    Calafat, Francisco M.
    et al.
    Natl Oceanog Ctr, Joseph Proudman Bldg,6 Brownlow St, Liverpool L3 5DA, Merseyside, England.
    Wahl, Thomas
    Univ Cent Florida, Natl Ctr Integrated Coastal Res, 12800 Pegasus Dr,Suite 211, Orlando, FL 32816 USA;Univ Cent Florida, Dept Civil Environm & Construct Engn, USA.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Williams, Joanne
    Natl Oceanog Ctr, England.
    Frajka-Williams, Eleanor
    Univ Southampton, Ocean & Earth Sci, European Way, England.
    Coherent modulation of the sea-level annual cycle in the United States by Atlantic Rossby waves2018In: Nature Communications, ISSN 2041-1723, E-ISSN 2041-1723, Vol. 9, article id 2571Article in journal (Refereed)
    Abstract [en]

    Changes in the sea-level annual cycle (SLAC) can have profound impacts on coastal areas, including increased flooding risk and ecosystem alteration, yet little is known about the magnitude and drivers of such changes. Here we show, using novel Bayesian methods, that there are significant decadal fluctuations in the amplitude of the SLAC along the United States Gulf and Southeast coasts, including an extreme event in 2008-2009 that is likely (probability = 68%) unprecedented in the tide-gauge record. Such fluctuations are coherent along the coast but decoupled from deep-ocean changes. Through the use of numerical and analytical ocean models, we show that the primary driver of these fluctuations involves incident Rossby waves that generate fast western-boundary waves. These Rossby waves project onto the basin-wide upper mid-ocean transport (top 1000 m) leading to a link with the SLAC, wherein larger SLAC amplitudes coincide with enhanced transport variability.

  • 6.
    Wigren, Anna
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Murray, Lawrence
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Improving the particle filter in high dimensions using conjugate artificial process noise2018In: 18th IFAC Symposium on System IdentificationSYSID 2018 Proceedings, Elsevier, 2018, Vol. 51, p. 670-675Conference paper (Refereed)
    Abstract [en]

    The particle filter is one of the most successful methods for state inference and identification of general non-linear and non-Gaussian models. However, standard particle filters suffer from degeneracy of the particle weights, in particular for high-dimensional problems. We propose a method for improving the performance of the particle filter for certain challenging state space models, with implications for high-dimensional inference. First we approximate the model by adding artificial process noise in an additional state update, then we design a proposal that combines the standard and the locally optimal proposal. This results in a bias-variance trade-off, where adding more noise reduces the variance of the estimate but increases the model bias. The performance of the proposed method is empirically evaluated on a linear-Gaussian state space model and on the non-linear Lorenz'96 model. For both models we observe a significant improvement in performance over the standard particle filter.

  • 7.
    Svensson, Andreas
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Learning nonlinear state-space models using smooth particle-filter-based likelihood approximations2018In: 18th IFAC Symposium on System IdentificationSYSID 2018 Proceedings, Elsevier, 2018, p. 652-657Conference paper (Refereed)
    Abstract [en]

    When classical particle filtering algorithms are used for maximum likelihood parameter estimation in nonlinear state-space models, a key challenge is that estimates of the likelihood function and its derivatives are inherently noisy. The key idea in this paper is to run a particle filter based on a current parameter estimate, but then use the output from this particle filter to re-evaluate the likelihood function approximation also for other parameter values. This results in a (local) deterministic approximation of the likelihood and any standard optimization routine can be applied to find the maximum of this approximation. By iterating this procedure we eventually arrive at a final parameter estimate.

  • 8.
    Svensson, Andreas
    et al.
    Uppsala universitet, Reglerteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Reglerteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Learning of state-space models with highly informative observations: A tempered sequential Monte Carlo solution2018In: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 104, p. 915-928Article in journal (Refereed)
    Abstract [en]

    Probabilistic (or Bayesian) modeling and learning offers interesting possibilities for systematic representation of uncertainty using probability theory. However, probabilistic learning often leads to computationally challenging problems. Some problems of this type that were previously intractable can now be solved on standard personal computers thanks to recent advances in Monte Carlo methods. In particular, for learning of unknown parameters in nonlinear state-space models, methods based on the particle filter (a Monte Carlo method) have proven very useful. A notoriously challenging problem, however, still occurs when the observations in the state-space model are highly informative, i.e. when there is very little or no measurement noise present, relative to the amount of process noise. The particle filter will then struggle in estimating one of the basic components for probabilistic learning, namely the likelihood p(datalparameters). To this end we suggest an algorithm which initially assumes that there is substantial amount of artificial measurement noise present. The variance of this noise is sequentially decreased in an adaptive fashion such that we, in the end, recover the original problem or possibly a very close approximation of it. The main component in our algorithm is a sequential Monte Carlo (SMC) sampler, which gives our proposed method a clear resemblance to the SMC2 method. Another natural link is also made to the ideas underlying the approximate Bayesian computation (ABC). We illustrate it with numerical examples, and in particular show promising results for a challenging Wiener-Hammerstein benchmark problem.

  • 9.
    Schön, Thomas B.
    et al.
    Uppsala universitet, Reglerteknik, Sweden.
    Svensson, Andreas
    Uppsala universitet, Reglerteknik, Sweden.
    Murray, Lawrence
    Uppsala universitet, Reglerteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo2018In: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 104, p. 866-883Article in journal (Refereed)
    Abstract [en]

    Probabilistic modeling provides the capability to represent and manipulate uncertainty in data, models, predictions and decisions. We are concerned with the problem of learning probabilistic models of dynamical systems from measured data. Specifically, we consider learning of probabilistic nonlinear state-space models. There is no closed-form solution available for this problem, implying that we are forced to use approximations. In this tutorial we will provide a self-contained introduction to one of the state-of-the-art methods the particle Metropolis-Hastings algorithm which has proven to offer a practical approximation. This is a Monte Carlo based method, where the particle filter is used to guide a Markov chain Monte Carlo method through the parameter space. One of the key merits of the particle Metropolis-Hastings algorithm is that it is guaranteed to converge to the "true solution" under mild assumptions, despite being based on a particle filter with only a finite number of particles. We will also provide a motivating numerical example illustrating the method using a modeling language tailored for sequential Monte Carlo methods. The intention of modeling languages of this kind is to open up the power of sophisticated Monte Carlo methods including particle Metropolis-Hastings to a large group of users without requiring them to know all the underlying mathematical details.

  • 10.
    Jacob, Pierre
    et al.
    Harvard University, USA.
    Lindsten, Fredrik
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Retracted article: Smoothing with Couplings of Conditional Particle Filters2018In: Journal of the American Statistical Association, ISSN 0162-1459, E-ISSN 1537-274XArticle in journal (Refereed)
    Abstract [en]

    In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and confidence intervals can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains, with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly-informative observations, and for a realistic Lotka-Volterra model with an intractable transition density.

  • 11.
    Risuleo, Riccardo S.
    et al.
    Royal Institute of Technology, Stockholm, Sweden.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering. Uppsala University, Uppsala, Sweden.
    Hjalmarsson, Håkan
    KTH - Royal Institute of Technology, Stockholm, Sweden.
    Semi-parametric kernel-based identification of Wiener systems2018In: Proceedings of the 57th IEEE Conference on Decision and Control (CDC), IEEE, 2018Conference paper (Refereed)
    Abstract [en]

    We present a technique for kernel-based identification of Wiener systems. We model the impulse response of the linear block with a Gaussian process. The static nonlinearity is modeled with a combination of basis functions. The coefficients of the static nonlinearity are estimated, together with the hyperparameters of the covariance function of the Gaussian process model, using an iterative algorithm based on the expectation-maximization method combined with elliptical slice sampling to sample from the posterior distribution of the impulse response given the data. The same sampling method is then used to find the posterior-mean estimate of the impulse response. We test the proposed algorithm on a benchmark of randomly-generated Wiener systems.

  • 12.
    Singh, S. S.
    et al.
    Univ Cambridge, Dept Engn, Trumpington St, Cambridge CB2 1PZ, England.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Moulines, E.
    Ecole Polytech, Ctr Math Appl, Route Saclay, F-91128 Palaiseau, France.
    Blocking strategies and stability of particle Gibbs samplers2017In: Biometrika, ISSN 0006-3444, E-ISSN 1464-3510, Vol. 104, no 4, p. 953-969Article in journal (Refereed)
    Abstract [en]

    Sampling from the posterior probability distribution of the latent states of a hidden Markov model is nontrivial even in the context of Markov chain Monte Carlo. To address this, Andrieu et al. (2010) proposed a way of using a particle filter to construct a Markov kernel that leaves the posterior distribution invariant. Recent theoretical results have established the uniform ergodicity of this Markov kernel and shown that the mixing rate does not deteriorate provided the number of particles grows at least linearly with the number of latent states. However, this gives rise to a cost per application of the kernel that is quadratic in the number of latent states, which can be prohibitive for long observation sequences. Using blocking strategies, we devise samplers that have a stable mixing rate for a cost per iteration that is linear in the number of latent states and which are easily parallelizable.

  • 13.
    Lindsten, Fredrik
    et al.
    Uppsala University, Sweden.
    Johansen, A. M.
    University of Warwick, England.
    Andersson Naesseth, Christian
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Kirkpatrick, B.
    Intrepid Net Comp, MT USA.
    Schön, T. B.
    Uppsala University, Sweden.
    Aston, J. A. D.
    University of Cambridge, England.
    Bouchard-Cote, A.
    University of British Columbia, Canada.
    Divide-and-Conquer With Sequential Monte Carlo2017In: Journal of Computational And Graphical Statistics, ISSN 1061-8600, E-ISSN 1537-2715, Vol. 26, no 2, p. 445-458Article in journal (Refereed)
    Abstract [en]

    We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved subproblems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed divide-and-conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging subproblems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem. Supplementary materials including proofs and additional numerical results are available online.

  • 14.
    Rainforth, Tom
    et al.
    The University of Oxford, Oxford, United Kingdom.
    Andersson Naesseth, Christian
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Lindsten, Fredrik
    Uppsala University, Uppsala, Sweden.
    Paige, Brooks
    The University of Oxford, Oxford, United Kingdom.
    Meent, Jan-Willem van de
    The University of Oxford, Oxford, United Kingdom.
    Doucet, Arnaud
    The University of Oxford, Oxford, United Kingdom.
    Wood, Frank
    The University of Oxford, Oxford, United Kingdom.
    Interacting Particle Markov Chain Monte Carlo2016In: Proceedings of the 33rd International Conference on Machine Learning, 2016, Vol. 48Conference paper (Refereed)
    Abstract [en]

    We introduce interacting particle Markov chain Monte Carlo (iPMCMC), a PMCMC method based on an interacting pool of standard and conditional sequential Monte Carlo samplers. Like related methods, iPMCMC is a Markov chain Monte Carlo sampler on an extended space. We present empirical results that show significant improvements in mixing rates relative to both noninteracting PMCMC samplers and a single PMCMC sampler with an equivalent memory and computational budget. An additional advantage of the iPMCMC method is that it is suitable for distributed and multi-core architectures.

  • 15.
    Lindsten, Fredrik
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Bunch, Pete
    Department of Engineering, University of Cambridge, Cambridge, UK.
    Särkkä, Simo
    Department of Electrical Engineering and Automation, Aalto University, Aalto, Finland.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Godsill, Simon J.
    Department of Engineering, University of Cambridge, Cambridge, UK.
    Rao–Blackwellized particle smoothers for conditionally linear Gaussian models2016In: IEEE Journal on Selected Topics in Signal Processing, ISSN 1932-4553, E-ISSN 1941-0484, Vol. 10, no 2, p. 353-365Article in journal (Refereed)
    Abstract [en]

    Sequential Monte Carlo (SMC) methods, such as the particle filter, are by now one of the standard computational techniques for addressing the filtering problem in general state-space models. However, many applications require post-processing of data offline. In such scenarios the smoothing problem-in which all the available data is used to compute state estimates-is of central interest. We consider the smoothing problem for a class of conditionally linear Gaussian models. We present a forward-backward-type Rao-Blackwellized particle smoother (RBPS) that is able to exploit the tractable substructure present in these models. Akin to the well known Rao-Blackwellized particle filter, the proposed RBPS marginalizes out a conditionally tractable subset of state variables, effectively making use of SMC only for the “intractable part” of the model. Compared to existing RBPS, two key features of the proposed method are: 1) it does not require structural approximations of the model, and 2) the aforementioned marginalization is done both in the forward direction and in the backward direction.

  • 16.
    Wågberg, Johan
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Lindsten, Fredrik
    Department of Engineering, University of Cambridge, Cambridge, United Kingdom.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Uppsala.
    Bayesian nonparametric identification of piecewise affine ARX systems2015In: 17th IFAC Symposium on System IdentificationSYSID 2015 Proceedings / [ed] Yanlong Zhao, Elsevier, 2015, Vol. 48, p. 709-714Conference paper (Refereed)
    Abstract [en]

    We introduce a Bayesian nonparametric approach to identification of piecewise affine ARX systems. The clustering properties of the Dirichlet process are used to construct a prior over the partition of the regressor space as well as the parameters of each local model. This enables us to probabilistically reason about and to identify the number of modes, the partition of the regressor space, and the linear dynamics of each local model from data. By appropriate choices of base measure and likelihood function, we give explicit expressions for how to perform both inference and prediction. Simulations and experiments on real data from a pick and place machine are used to illustrate the capabilities of the new approach.

  • 17.
    Andersson Naesseth, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Lindsten, Fredrik
    The University of Cambridge, Cambridge, United Kingdom.
    Schön, Thomas
    Uppsala University, Uppsala, Sweden.
    Nested Sequential Monte Carlo Methods2015In: Proceedings of The 32nd International Conference on Machine Learning / [ed] Francis Bach, David Blei, Journal of Machine Learning Research (Online) , 2015, Vol. 37, p. 1292-1301Conference paper (Refereed)
    Abstract [en]

    We propose nested sequential Monte Carlo (NSMC), a methodology to sample from sequences of probability distributions, even where the random variables are high-dimensional. NSMC generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. Furthermore, NSMC can in itself be used to produce such properly weighted samples. Consequently, one NSMC sampler can be used to construct an efficient high-dimensional proposal distribution for another NSMC sampler, and this nesting of the algorithm can be done to an arbitrary degree. This allows us to consider complex and high-dimensional models using SMC. We show results that motivate the efficacy of our approach on several filtering problems with dimensions in the order of 100 to 1 000.

  • 18.
    Bunch, Pete
    et al.
    University of Cambridge, UK.
    Lindsten, Fredrik
    University of Cambridge, UK.
    Singh, Sumeetpal S.
    University of Cambridge, UK.
    Particle Gibbs with refreshed backward simulation2015In: Proceedings of the 40th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Institute of Electrical and Electronics Engineers (IEEE), 2015Conference paper (Refereed)
    Abstract [en]

    The particle Gibbs algorithm can be used for Bayesian parameter estimation in Markovian state space models. Sometimes the resulting Markov chains mix slowly when the component particle filter suffers from degeneracy. This effect can be somewhat alleviated using backward simulation. In this paper we show how a simple modification to this scheme, which we refer to as refreshed backward simulation, can further improve the mixing. This works by sampling new state values simultaneously with the corresponding ancestor indexes. Although the necessary conditional distributions cannot be sampled directly, we provide suitable Markov kernels which target them. The efficacy of this new scheme is demonstrated with a simulation example.

  • 19.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Dept. of Information Technology, Uppsala University, Uppsala, Sweden.
    Particle Metropolis-Hastings using gradient and Hessian information2015In: Statistics and computing, ISSN 0960-3174, E-ISSN 1573-1375, Vol. 25, no 1, p. 81-92Article in journal (Other academic)
    Abstract [en]

    Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining MCMC and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles.

    We propose two alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.

  • 20.
    Riabiz, Marina
    et al.
    Signal Processing and Communications Laboratory, Engineering Department, University of Cambridge, UK.
    Lindsten, Fredrik
    Signal Processing and Communications Laboratory, Engineering Department, University of Cambridge, UK.
    Godsill, Simon J.
    Signal Processing and Communications Laboratory, Engineering Department, University of Cambridge, UK.
    Pseudo-Marginal MCMC for Parameter Estimation in Alpha-Stable Distributions2015In: Proceedings of the 17th IFAC Symposium on System Identification (SYSID), Elsevier, 2015, Vol. 48, p. 472-477Conference paper (Refereed)
    Abstract [en]

    The α-stable distribution is very useful for modelling data with extreme values and skewed behaviour. The distribution is governed by two key parameters, tail thickness and skewness, in addition to scale and location. Inferring these parameters is difficult due to the lack of a closed form expression of the probability density. We develop a Bayesian method, based on the pseudo-marginal MCMC approach, that requires only unbiased estimates of the intractable likelihood. To compute these estimates we build an adaptive importance sampler for a latentvariable- representation of the α-stable density. This representation has previously been used in the literature for conditional MCMC sampling of the parameters, and we compare our method with this approach.

  • 21.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Schön, Thomas Bo
    Department of Information Technology, Uppsala University.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Quasi-Newton particle Metropolis-Hastings2015In: Proceedings of the 17th IFAC Symposium on System Identification., Elsevier, 2015, Vol. 48 Issue 28, p. 981-986Conference paper (Refereed)
    Abstract [en]

    Particle Metropolis-Hastings enables Bayesian parameter inference in general nonlinear state space models (SSMs). However, in many implementations a random walk proposal is used and this can result in poor mixing if not tuned correctly using tedious pilot runs. Therefore, we consider a new proposal inspired by quasi-Newton algorithms that may achieve similar (or better) mixing with less tuning. An advantage compared to other Hessian based proposals, is that it only requires estimates of the gradient of the log-posterior. A possible application is parameter inference in the challenging class of SSMs with intractable likelihoods.We exemplify this application and the benefits of the new proposal by modelling log-returns offuture contracts on coffee by a stochastic volatility model with alpha-stable observations.

  • 22.
    Özkan, Emre
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    University of Cambridge, England.
    Fritsche, Carsten
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Recursive Maximum Likelihood Identification of Jump Markov Nonlinear Systems2015In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 63, no 3, p. 754-765Article in journal (Refereed)
    Abstract [en]

    We present an online method for joint state and parameter estimation in jump Markov non-linear systems (JMNLS). State inference is enabled via the use of particle filters which makes the method applicable to a wide range of non-linear models. To exploit the inherent structure of JMNLS, we design a Rao-Blackwellized particle filter (RBPF) where the discrete mode is marginalized out analytically. This results in an efficient implementation of the algorithm and reduces the estimation error variance. The proposed RBPF is then used to compute, recursively in time, smoothed estimates of complete data sufficient statistics. Together with the online expectation maximization algorithm, this enables recursive identification of unknown model parameters including the transition probability matrix. The method is also applicable to online identification of jump Markov linear systems(JMLS). The performance of the method is illustrated in simulations and on a localization problem in wireless networks using real data.

  • 23.
    Lacoste-Julien, Simon
    et al.
    INRIA - Sierra Project-Team École Normale Supérieure, Paris, France.
    Lindsten, Fredrik
    University of Cambridge, Cambridge, England, United Kingdom.
    Bach, Francis
    INRIA - Sierra Project-Team École Normale Supérieure, Paris, France.
    Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering2015In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics, 2015Conference paper (Refereed)
  • 24.
    Schön, Thomas Bo
    et al.
    Department of Information Technology, Uppsala University.
    Lindsten, Fredrik
    Department of Engineering, University of Cambridge, UK.
    Dahlin, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Wågberg, Johan
    Department of Information Technology, Uppsala University.
    Andersson Naesseth, Christian
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Svensson, Andreas
    Department of Information Technology, Uppsala University.
    Dai, Liang
    Department of Information Technology, Uppsala University.
    Sequential Monte Carlo Methods for System Identification2015In: Proceedings of the 17th IFAC Symposium on System Identification., Elsevier, 2015, Vol. 48, p. 775-786Conference paper (Refereed)
    Abstract [en]

    One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more than two decades ago), provide numerical solutions to the nonlinear state estimation problems arising in SSMs. When combined with additional identification techniques, these algorithms provide solid solutions to the nonlinear system identification problem. We describe two general strategies for creating such combinations and discuss why SMC is a natural tool for implementing these strategies.

  • 25.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering. University of Cambridge, England.
    Douc, Randal
    Telecom SudParis, France.
    Moulines, Eric
    Telecom ParisTech, France.
    Uniform Ergodicity of the Particle Gibbs Sampler2015In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 42, no 3, p. 775-797Article in journal (Refereed)
    Abstract [en]

    The particle Gibbs sampler is a systematic way of using a particle filter within Markov chain Monte Carlo. This results in an off-the-shelf Markov kernel on the space of state trajectories, which can be used to simulate from the full joint smoothing distribution for a state space model in a Markov chain Monte Carlo scheme. We show that the particle Gibbs Markov kernel is uniformly ergodic under rather general assumptions, which we will carefully review and discuss. In particular, we provide an explicit rate of convergence, which reveals that (i) for fixed number of data points, the convergence rate can be made arbitrarily good by increasing the number of particles and (ii) under general mixing assumptions, the convergence rate can be kept constant by increasing the number of particles superlinearly with the number of observations. We illustrate the applicability of our result by studying in detail a common stochastic volatility model with a non-compact state space.

  • 26.
    Andersson Naesseth, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Capacity estimation of two-dimensional channels using Sequential Monte Carlo2014In: 2014 IEEE Information Theory Workshop, 2014, p. 431-435Conference paper (Refereed)
    Abstract [en]

    We derive a new Sequential-Monte-Carlo-based algorithm to estimate the capacity of two-dimensional channel models. The focus is on computing the noiseless capacity of the 2-D (1, ∞) run-length limited constrained channel, but the underlying idea is generally applicable. The proposed algorithm is profiled against a state-of-the-art method, yielding more than an order of magnitude improvement in estimation accuracy for a given computation time.

  • 27.
    Frigola, Roger
    et al.
    Department of Engineering, University of Cambridge, UK.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas B.
    Deptartment of Information Technology, Uppsala University, Sweden.
    Rasmussen, Carl E.
    Department of Engineering, University of Cambridge, UK.
    Identification of Gaussian process state-space models with particle stochastic approximation EM2014In: Proceedings of the 19th IFAC World Congress, Elsevier, 2014, Vol. 47Conference paper (Refereed)
    Abstract [en]

    Gaussian process state-space models (GP-SSMs) are a very exible family of models of nonlinear dynamical systems. They comprise a Bayesian nonparametric representation of the dynamics of the system and additional (hyper-)parameters governing the properties of this nonparametric representation. The Bayesian formalism enables systematic reasoning about the uncertainty in the system dynamics. We present an approach to maximum likelihood identification of the parameters in GP-SSMs, while retaining the full nonparametric description of the dynamics. The method is based on a stochastic approximation version of the EM algorithm that employs recent developments in particle Markov chain Monte Carlo for efficient identification.

  • 28.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Particle filter-based Gaussian process optimisation for parameter inference2014In: Proceedings of the 19th IFAC World Congress, 2014 / [ed] Edward Boje and Xiaohua Xia, 2014, p. 8675-8680Conference paper (Refereed)
    Abstract [en]

    We propose a novel method for maximum-likelihood-based parameter inference in nonlinear and/or non-Gaussian state space models. The method is an iterative procedure with three steps. At each iteration a particle filter is used to estimate the value of the log-likelihood function at the current parameter iterate. Using these log-likelihood estimates, a surrogate objective function is created by utilizing a Gaussian process model. Finally, we use a heuristic procedure to obtain a revised parameter iterate, providing an automatic trade-off between exploration and exploitation of the surrogate model. The method is profiled on two state space models with good performance both considering accuracy and computational cost.

  • 29.
    Gunnarsson, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology. Ericsson Research, Linköping, Sweden.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Carlsson, N.
    Ericsson Research, Linköping, Sweden; Volvo Cars, Gothanburg, Sweden.
    Particle filtering for network-based positioning terrestrial radio networks2014In: Data Fusion & Target Tracking 2014: Algorithms and Applications (DF&TT 2014), IET Conference on, Institution of Engineering and Technology , 2014, Vol. 2014, no 629 CPConference paper (Refereed)
    Abstract [en]

    There is strong interest in positioing in wireless networks, partly to support end user service needs, but also to support network management with network-based network information. The focus in this paper is on the latter, while using measurements that are readily available in wireless networks. We show how thesignal direction of departure and inter-distance between the base station and the mobile terminal can be estimated, and how particle filters and smoothers can be used to post-process the measurements. The methods are evaluated in a live 3GPP LTE network with promising results inlcuding position error medians of less than 100 m.

  • 30.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jordan, Michael I.
    University of Calif Berkeley, CA 94720 USA; University of Calif Berkeley, CA 94720 USA.
    Schon, Thomas B.
    Uppsala University, Sweden.
    Particle Gibbs with Ancestor Sampling2014In: Journal of machine learning research, ISSN 1532-4435, E-ISSN 1533-7928, Vol. 15, p. 2145-2184Article in journal (Refereed)
    Abstract [en]

    Particle Markov chain Monte Carlo (PMCMC) is a systematic way of combining the two main tools used for Monte Carlo statistical inference: sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). We present a new PMCMC algorithm that we refer to as particle Gibbs with ancestor sampling (PGAS). PGAS provides the data analyst with an off-the-shelf class of Markov kernels that can be used to simulate, for instance, the typically high-dimensional and highly autocorrelated state trajectory in a state-space model. The ancestor sampling procedure enables fast mixing of the PGAS kernel even when using seemingly few particles in the underlying SMC sampler. This is important as it can significantly reduce the computational burden that is typically associated with using SMC. PGAS is conceptually similar to the existing PG with backward simulation (PGBS) procedure. Instead of using separate forward and backward sweeps as in PGBS, however, we achieve the same effect in a single forward sweep. This makes PGAS well suited for addressing inference problems not only in state-space models, but also in models with more complex dependencies, such as non-Markovian, Bayesian nonparametric, and general probabilistic graphical models.

  • 31.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology. Uppsala University.
    Second-Order Particle MCMC for Bayesian Parameter Inference2014In: Proceedings of the 19th IFAC World Congress, 2014, p. 8656-8661Conference paper (Refereed)
    Abstract [en]

    We propose an improved proposal distribution in the Particle Metropolis-Hastings (PMH) algorithm for Bayesian parameter inference in nonlinear state space models. This proposal incorporates second-order information about the parameter posterior distribution, which can be extracted from the particle filter already used within the PMH algorithm. The added information makes the proposal scale-invariant, simpler to tune and can possibly also shorten the burn-in phase. The proposed algorithm has a computational cost which is proportional to the number of particles, i.e. the same as the original marginal PMH algorithm. Finally, we provide two numerical examples that illustrates some of the possible benefits of adding the second-order information.

  • 32.
    Andersson Naesseth, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    University of Cambridge, Cambridge, UK.
    Schön, Thomas
    Uppsala University, Uppsala, Sweden.
    Sequential Monte Carlo for Graphical Models2014In: Advances in Neural Information Processing Systems, 2014, p. 1862-1870Conference paper (Refereed)
    Abstract [en]

    We propose a new framework for how to use sequential Monte Carlo (SMC) algorithms for inference in probabilistic graphical models (PGM). Via a sequential decomposition of the PGM we find a sequence of auxiliary distributions defined on a monotonically increasing sequence of probability spaces. By targeting these auxiliary distributions using SMC we are able to approximate the full joint distribution defined by the PGM. One of the key merits of the SMC sampler is that it provides an unbiased estimate of the partition function of the model. We also show how it can be used within a particle Markov chain Monte Carlo framework in order to construct high-dimensional block-sampling algorithms for general PGMs.

  • 33.
    Taghavi, Ehsan
    et al.
    School of Computational Science and Engineering, McMaster University.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Svensson, Lennart
    Division of Signals and Systems, Chalmers University.
    Schön, Thomas B.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Adaptive stopping for fast particle smoothing2013In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), IEEE , 2013, p. 6293-6297Conference paper (Refereed)
    Abstract [en]

    Particle smoothing is useful for offline state inference and parameter learning in nonlinear/non-Gaussian state-space models. However, many particle smoothers, such as the popular forward filter/backward simulator (FFBS), are plagued by a quadratic computational complexity in the number of particles. One approach to tackle this issue is to use rejection-sampling-based FFBS (RS-FFBS), which asymptotically reaches linear complexity. In practice, however, the constants can be quite large and the actual gain in computational time limited. In this contribution, we develop a hybrid method, governed by an adaptive stopping rule, in order to exploit the benefits, but avoid the drawbacks, of RS-FFBS. The resulting particle smoother is shown in a simulation study to be considerably more computationally efficient than both FFBS and RS-FFBS.

  • 34.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    An Efficient Stochastic Approximation EM Algorithm using Conditional Particle Filters2013In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing, IEEE conference proceedings, 2013, p. 6274-6278Conference paper (Refereed)
    Abstract [en]

    I present a novel method for maximum likelihood parameter estimation in nonlinear/non-Gaussian state-space models. It is an expectation maximization (EM) like method, which uses sequential Monte Carlo (SMC) for the intermediate state inference problem. Contrary to existing SMC-based EM algorithms, however, it makes efficient use of the simulated particles through the use of particle Markov chain Monte Carlo (PMCMC) theory. More precisely, the proposed method combines the efficient conditional particle filter with ancestor sampling (CPF-AS) with the stochastic approximation EM (SAEM) algorithm. This results in a procedure which does not rely on asymptotics in the number of particles for convergence, meaning that the method is very computationally competitive. Indeed, the method is evaluated in a simulation study, using a small number of particles with promising results.

  • 35.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas B.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Backward simulation methods for Monte Carlo statistical inference2013In: Foundations and Trends in Machine Learning, ISSN 1935-8237, Vol. 6, no 1, p. 1-143Article in journal (Refereed)
    Abstract [en]

    Monte Carlo methods, in particular those based on Markov chains and on interacting particle systems, are by now tools that are routinely used in machine learning. These methods have had a profound impact on statistical inference in a wide range of application areas where probabilistic models are used. Moreover, there are many algorithms in machine learning which are based on the idea of processing the data sequentially, first in the forward direction and then in the backward direction. In this tutorial we will review a branch of Monte Carlo methods based on the forward-backward idea, referred to as backward simulators. These methods are useful for learning and inference in probabilistic models containing latent stochastic processes. The theory and practice of backward simulation algorithms have undergone a significant development in recent years and the algorithms keep finding new applications. The foundation for these methods is sequential Monte Carlo (SMC). SMC-based backward simulators are capable of addressing smoothing problems in sequential latent variable models, such as general, nonlinear/non-Gaussian state-space models (SSMs). However, we will also clearly show that the underlying backward simulation idea is by no means restricted to SSMs. Furthermore, backward simulation plays an important role in recent developments of Markov chain Monte Carlo (MCMC) methods. Particle MCMC is a systematic way of using SMC within MCMC. In this framework, backward simulation gives us a way to significantly improve the performance of the samplers. We review and discuss several related backward-simulation-based methods for state inference as well as learning of static parameters, both using a frequentistic and a Bayesian approach.

  • 36.
    Frigola, Roger
    et al.
    Department of Engineering, University of Cambridge, UK.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas B.
    Deptartment of Information Technology, Uppsala University, Sweden.
    Rasmussen, Carl E.
    Department of Engineering, University of Cambridge, UK.
    Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC2013In: Advances in Neural Information Processing Systems 26 / [ed] C.J.C. Burges, L. Bottou, M. Welling, Z. Ghahramani and K.Q. Weinberger, 2013Conference paper (Refereed)
    Abstract [en]

    State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once an approximation of the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. We make use of sparse Gaussian process models to greatly reduce the computational complexity of the approach.

  • 37.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jordan, Michael I.
    University of Calif Berkeley, CA USA .
    Bayesian semiparametric Wiener system identification2013In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 49, no 7, p. 2053-2063Article in journal (Refereed)
    Abstract [en]

    We present a novel method for Wiener system identification. The method relies on a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process model for the static nonlinearity. We avoid making strong assumptions, such as monotonicity, on the nonlinear mapping. Stochastic disturbances, entering both as measurement noise and as process noise, are handled in a systematic manner. The nonparametric nature of the Gaussian process allows us to handle a wide range of nonlinearities without making problem-specific parameterizations. We also consider sparsity-promoting priors, based on generalized hyperbolic distributions, to automatically infer the order of the underlying dynamical system. We derive an inference algorithm based on an efficient particle Markov chain Monte Carlo method, referred to as particle Gibbs with ancestor sampling. The method is profiled on two challenging identification problems with good results. Blind Wiener system identification is handled as a special case.

  • 38.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Particle filters and Markov chains for learning of dynamical systems2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) methods provide computational tools for systematic inference and learning in complex dynamical systems, such as nonlinear and non-Gaussian state-space models. This thesis builds upon several methodological advances within these classes of Monte Carlo methods.Particular emphasis is placed on the combination of SMC and MCMC in so called particle MCMC algorithms. These algorithms rely on SMC for generating samples from the often highly autocorrelated state-trajectory. A specific particle MCMC algorithm, referred to as particle Gibbs with ancestor sampling (PGAS), is suggested. By making use of backward sampling ideas, albeit implemented in a forward-only fashion, PGAS enjoys good mixing even when using seemingly few particles in the underlying SMC sampler. This results in a computationally competitive particle MCMC algorithm. As illustrated in this thesis, PGAS is a useful tool for both Bayesian and frequentistic parameter inference as well as for state smoothing. The PGAS sampler is successfully applied to the classical problem of Wiener system identification, and it is also used for inference in the challenging class of non-Markovian latent variable models.Many nonlinear models encountered in practice contain some tractable substructure. As a second problem considered in this thesis, we develop Monte Carlo methods capable of exploiting such substructures to obtain more accurate estimators than what is provided otherwise. For the filtering problem, this can be done by using the well known Rao-Blackwellized particle filter (RBPF). The RBPF is analysed in terms of asymptotic variance, resulting in an expression for the performance gain offered by Rao-Blackwellization. Furthermore, a Rao-Blackwellized particle smoother is derived, capable of addressing the smoothing problem in so called mixed linear/nonlinear state-space models. The idea of Rao-Blackwellization is also used to develop an online algorithm for Bayesian parameter inference in nonlinear state-space models with affine parameter dependencies.

    List of papers
    1. Backward simulation methods for Monte Carlo statistical inference
    Open this publication in new window or tab >>Backward simulation methods for Monte Carlo statistical inference
    2013 (English)In: Foundations and Trends in Machine Learning, ISSN 1935-8237, Vol. 6, no 1, p. 1-143Article in journal (Refereed) Published
    Abstract [en]

    Monte Carlo methods, in particular those based on Markov chains and on interacting particle systems, are by now tools that are routinely used in machine learning. These methods have had a profound impact on statistical inference in a wide range of application areas where probabilistic models are used. Moreover, there are many algorithms in machine learning which are based on the idea of processing the data sequentially, first in the forward direction and then in the backward direction. In this tutorial we will review a branch of Monte Carlo methods based on the forward-backward idea, referred to as backward simulators. These methods are useful for learning and inference in probabilistic models containing latent stochastic processes. The theory and practice of backward simulation algorithms have undergone a significant development in recent years and the algorithms keep finding new applications. The foundation for these methods is sequential Monte Carlo (SMC). SMC-based backward simulators are capable of addressing smoothing problems in sequential latent variable models, such as general, nonlinear/non-Gaussian state-space models (SSMs). However, we will also clearly show that the underlying backward simulation idea is by no means restricted to SSMs. Furthermore, backward simulation plays an important role in recent developments of Markov chain Monte Carlo (MCMC) methods. Particle MCMC is a systematic way of using SMC within MCMC. In this framework, backward simulation gives us a way to significantly improve the performance of the samplers. We review and discuss several related backward-simulation-based methods for state inference as well as learning of static parameters, both using a frequentistic and a Bayesian approach.

    Keywords
    Bayesian learning, Markov chain Monte Carlo, Nonlinear signal processing, Particle smoothing, Sequential Monte Carlo
    National Category
    Control Engineering Probability Theory and Statistics
    Identifiers
    urn:nbn:se:liu:diva-98294 (URN)10.1561/2200000045 (DOI)
    Projects
    CNDMCADICS
    Funder
    Swedish Research Council
    Available from: 2013-10-07 Created: 2013-10-07 Last updated: 2013-10-08
    2. Ancestor Sampling for Particle Gibbs
    Open this publication in new window or tab >>Ancestor Sampling for Particle Gibbs
    2012 (English)In: Proceedings of the 26th Conference on Neural Information Processing Systems, 2012Conference paper, Published paper (Refereed)
    Abstract [en]

    We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error. Several application examples are discussed, including Rao-Blackwellized particle smoothing and inference in degenerate state-space models.

    Keywords
    Particle Gibbs, Sampling
    National Category
    Probability Theory and Statistics Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-88610 (URN)9781627480031 (ISBN)
    Conference
    26th Conference on Neural Information Processing Systems, Lake Tahoe, NV, USA, 3-6 December, 2012
    Projects
    CADICSCNDM
    Funder
    Swedish Research Council
    Available from: 2013-02-13 Created: 2013-02-13 Last updated: 2013-10-08
    3. Bayesian semiparametric Wiener system identification
    Open this publication in new window or tab >>Bayesian semiparametric Wiener system identification
    2013 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 49, no 7, p. 2053-2063Article in journal (Refereed) Published
    Abstract [en]

    We present a novel method for Wiener system identification. The method relies on a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process model for the static nonlinearity. We avoid making strong assumptions, such as monotonicity, on the nonlinear mapping. Stochastic disturbances, entering both as measurement noise and as process noise, are handled in a systematic manner. The nonparametric nature of the Gaussian process allows us to handle a wide range of nonlinearities without making problem-specific parameterizations. We also consider sparsity-promoting priors, based on generalized hyperbolic distributions, to automatically infer the order of the underlying dynamical system. We derive an inference algorithm based on an efficient particle Markov chain Monte Carlo method, referred to as particle Gibbs with ancestor sampling. The method is profiled on two challenging identification problems with good results. Blind Wiener system identification is handled as a special case.

    Place, publisher, year, edition, pages
    Elsevier, 2013
    Keywords
    System identification, Wiener, Block-oriented models, Gaussian process, Semiparametric, Particle filter, Ancestor sampling, Particle Markov chain Monte Carlo
    National Category
    Engineering and Technology
    Identifiers
    urn:nbn:se:liu:diva-95954 (URN)10.1016/j.automatica.2013.03.021 (DOI)000321233900011 ()
    Note

    Funding Agencies|project Calibrating Nonlinear Dynamical Models|621-2010-5876|Swedish Research Council||CADICS||Linnaeus Center||

    Available from: 2013-08-19 Created: 2013-08-12 Last updated: 2017-12-06
    4. An Efficient Stochastic Approximation EM Algorithm using Conditional Particle Filters
    Open this publication in new window or tab >>An Efficient Stochastic Approximation EM Algorithm using Conditional Particle Filters
    2013 (English)In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing, IEEE conference proceedings, 2013, p. 6274-6278Conference paper, Published paper (Refereed)
    Abstract [en]

    I present a novel method for maximum likelihood parameter estimation in nonlinear/non-Gaussian state-space models. It is an expectation maximization (EM) like method, which uses sequential Monte Carlo (SMC) for the intermediate state inference problem. Contrary to existing SMC-based EM algorithms, however, it makes efficient use of the simulated particles through the use of particle Markov chain Monte Carlo (PMCMC) theory. More precisely, the proposed method combines the efficient conditional particle filter with ancestor sampling (CPF-AS) with the stochastic approximation EM (SAEM) algorithm. This results in a procedure which does not rely on asymptotics in the number of particles for convergence, meaning that the method is very computationally competitive. Indeed, the method is evaluated in a simulation study, using a small number of particles with promising results.

    Place, publisher, year, edition, pages
    IEEE conference proceedings, 2013
    Keywords
    Maximum likelihood, Stochastic approximation, Conditional particle filter
    National Category
    Control Engineering Signal Processing
    Identifiers
    urn:nbn:se:liu:diva-93459 (URN)10.1109/ICASSP.2013.6638872 (DOI)000329611506087 ()
    Conference
    38th International Conference on Acoustics, Speech, and Signal Processing, Vancouver, Canada, 26-31 May, 2013
    Projects
    CNDMCADICS
    Funder
    Swedish Research Council, 621-2010-5876
    Available from: 2013-06-03 Created: 2013-06-03 Last updated: 2014-02-20
    5. Rao-Blackwellized Particle Smoothers for Mixed Linear/Nonlinear State-Space Models
    Open this publication in new window or tab >>Rao-Blackwellized Particle Smoothers for Mixed Linear/Nonlinear State-Space Models
    2013 (English)In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing, IEEE conference proceedings, 2013, p. 6288-6292Conference paper, Published paper (Refereed)
    Abstract [en]

    We consider the smoothing problem for a class of conditionally linear Gaussian state-space (CLGSS) models, referred to as mixed linear/nonlinear models. In contrast to the better studied hierarchical CLGSS models, these allow for an intricate cross dependence between the linear and the nonlinear parts of the state vector. We derive a Rao-Blackwellized particle smoother (RBPS) for this model class by exploiting its tractable substructure. The smoother is of the forward filtering/backward simulation type. A key feature of the proposed method is that, unlike existing RBPS for this model class, the linear part of the state vector is marginalized out in both the forward direction and in the backward direction.

    Place, publisher, year, edition, pages
    IEEE conference proceedings, 2013
    Keywords
    Rao-Blackwellization, Particle smoothing, Backward simulation, Sequential Monte Carlo
    National Category
    Signal Processing Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-93460 (URN)10.1109/ICASSP.2013.6638875 (DOI)000329611506090 ()
    Conference
    38th International Conference on Acoustics, Speech, and Signal Processing, Vancouver, Canada, 26-31 May, 2013
    Projects
    CNDMCADICS
    Funder
    Swedish Research Council, 621-2010-5876
    Available from: 2013-06-03 Created: 2013-06-03 Last updated: 2014-02-20Bibliographically approved
    6. A non-degenerate Rao-Blackwellised particle filter for estimating static parameters in dynamical models
    Open this publication in new window or tab >>A non-degenerate Rao-Blackwellised particle filter for estimating static parameters in dynamical models
    2012 (English)In: Proceedings of the 16th IFAC Symposium on System Identification, 2012Conference paper, Oral presentation only (Refereed)
    Abstract [en]

    The particle filter (PF) has emerged as a powerful tool for solving nonlinear and/or non-Gaussian filtering problems. When some of the states enter the model linearly, this can be exploited by using particles only for the "nonlinear" states and employing conditional Kalman filters for the "linear" states; this leads to the Rao-Blackwellised particle filter (RBPF). However, it is well known that the PF fails when the state of the model contains some static parameter. This is true also for the RBPF, even if the static states are marginalised analytically by a Kalman filter. The reason is that the posterior density of the static states is computed conditioned on the nonlinear particle trajectories, which are bound to degenerate over time. To circumvent this problem, we propose a method for targeting the posterior parameter density, conditioned on just the current nonlinear state. This results in an RBPF-like method, capable of recursive identification of nonlinear dynamical models with affine parameter dependencies.

    Keywords
    Particle Filtering/Monte Carlo Methods; Nonlinear System Identification; Recursive Identification
    National Category
    Electrical Engineering, Electronic Engineering, Information Engineering
    Identifiers
    urn:nbn:se:liu:diva-81271 (URN)10.3182/20120711-3-BE-2027.00184 (DOI)
    Conference
    16th IFAC Symposium on System Identification, Brussels, Belgium, July 11-13, 2012.
    Projects
    CADICSCNDM
    Funder
    Swedish Research Council
    Available from: 2012-09-10 Created: 2012-09-10 Last updated: 2013-10-08
    7. An Explicit Variance Reduction Expression for the Rao-Blackwellised Particle Filter
    Open this publication in new window or tab >>An Explicit Variance Reduction Expression for the Rao-Blackwellised Particle Filter
    2011 (English)In: Proceedings of the 18th IFAC World Congress, 2011, p. 11979-11984Conference paper, Published paper (Refereed)
    Abstract [en]

    Particle filters (PFs) have shown to be very potent tools for state estimation in nonlinear and/or non-Gaussian state-space models. For certain models, containing a conditionally tractable substructure (typically conditionally linear Gaussian or with finite support), it is possible to exploit this structure in order to obtain more accurate estimates. This has become known as Rao-Blackwellised particle filtering (RBPF). However, since the RBPF is typically more computationally demanding than the standard PF per particle, it is not always beneficial to resort to Rao-Blackwellisation. For the same computational effort, a standard PF with an increased number of particles, which would also increase the accuracy, could be used instead. In this paper, we have analysed the asymptotic variance of the RBPF and provide an explicit expression for the obtained variance reduction. This expression could be used to make an efficient discrimination of when to apply Rao-Blackwellisation, and when not to.

    Keywords
    Particle filtering, Monte-Carlo methods, Rao-Blackwellised particle filter, Marginalised particle filter, Rao-Blackwellisation, Variance reduction
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-81259 (URN)10.3182/20110828-6-IT-1002.02920 (DOI)978-3-902661-93-7 (ISBN)
    Conference
    18th IFAC World Congress, Milano, Italy 28 August-2 September, 2011
    Projects
    CADICSCNDM
    Funder
    Swedish Research Council
    Available from: 2012-09-10 Created: 2012-09-10 Last updated: 2013-10-08
  • 39.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Particle Metropolis Hastings using Langevin Dynamics2013In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing, IEEE conference proceedings, 2013, p. 6308-6312Conference paper (Refereed)
    Abstract [en]

    Particle Markov Chain Monte Carlo (PMCMC) samplers allow for routine inference of parameters and states in challenging nonlinear problems. A common choice for the parameter proposal is a simple random walk sampler, which can scale poorly with the number of parameters.

    In this paper, we propose to use log-likelihood gradients, i.e. the score, in the construction of the proposal, akin to the Langevin Monte Carlo method, but adapted to the PMCMC framework. This can be thought of as a way to guide a random walk proposal by using drift terms that are proportional to the score function. The method is successfully applied to a stochastic volatility model and the drift term exhibits intuitive behaviour.

  • 40.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Bunch, Pete
    University of Cambridge, United Kingdom.
    Godsill, Simon J.
    University of Cambridge, United Kingdom.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Rao-Blackwellized Particle Smoothers for Mixed Linear/Nonlinear State-Space Models2013In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing, IEEE conference proceedings, 2013, p. 6288-6292Conference paper (Refereed)
    Abstract [en]

    We consider the smoothing problem for a class of conditionally linear Gaussian state-space (CLGSS) models, referred to as mixed linear/nonlinear models. In contrast to the better studied hierarchical CLGSS models, these allow for an intricate cross dependence between the linear and the nonlinear parts of the state vector. We derive a Rao-Blackwellized particle smoother (RBPS) for this model class by exploiting its tractable substructure. The smoother is of the forward filtering/backward simulation type. A key feature of the proposed method is that, unlike existing RBPS for this model class, the linear part of the state vector is marginalized out in both the forward direction and in the backward direction.

  • 41.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas B.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Svensson, Lennart
    Division of Signals and Systems, Chalmers University.
    A non-degenerate Rao-Blackwellised particle filter for estimating static parameters in dynamical models2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012Conference paper (Refereed)
    Abstract [en]

    The particle filter (PF) has emerged as a powerful tool for solving nonlinear and/or non-Gaussian filtering problems. When some of the states enter the model linearly, this can be exploited by using particles only for the "nonlinear" states and employing conditional Kalman filters for the "linear" states; this leads to the Rao-Blackwellised particle filter (RBPF). However, it is well known that the PF fails when the state of the model contains some static parameter. This is true also for the RBPF, even if the static states are marginalised analytically by a Kalman filter. The reason is that the posterior density of the static states is computed conditioned on the nonlinear particle trajectories, which are bound to degenerate over time. To circumvent this problem, we propose a method for targeting the posterior parameter density, conditioned on just the current nonlinear state. This results in an RBPF-like method, capable of recursive identification of nonlinear dynamical models with affine parameter dependencies.

  • 42.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas B.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jordan, Michael I.
    University of California, Berkeley, USA.
    A Semiparametric Bayesian Approach to Wiener System Identification2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 1137-1142Conference paper (Refereed)
    Abstract [en]

    We consider a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process (GP) model for the static nonlinearity. The GP model is a flexible model that can describe different types of nonlinearities while avoiding making strong assumptions such as monotonicity. We derive an inferential method based on recent advances in Monte Carlo statistical methods, known as Particle Markov Chain Monte Carlo (PMCMC). The idea underlying PMCMC is to use a particle filter (PF) to generate a sample state trajectory in a Markov chain Monte Carlo sampler. We use a recently proposed PMCMC sampler, denoted particle Gibbs with backward simulation, which has been shown to be efficient even when we use very few particles in the PF. The resulting method is used in a simulation study to identify two different Wiener systems with non-invertible nonlinearities.

  • 43.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jordan, Michael I.
    University of California, Berkeley.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ancestor Sampling for Particle Gibbs2012In: Proceedings of the 26th Conference on Neural Information Processing Systems, 2012Conference paper (Refereed)
    Abstract [en]

    We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error. Several application examples are discussed, including Rao-Blackwellized particle smoothing and inference in degenerate state-space models.

  • 44.
    Wills, Adrian
    et al.
    University of Newcastle, Australia.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ninness, Brett
    University of Newcastle, Australia.
    Estimation of Linear Systems using a Gibbs Sampler2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 203-208Conference paper (Refereed)
    Abstract [en]

    This paper considers a Bayesian approach to linear system identification. One motivation is the advantage of the minimum mean square error of the associated conditional mean estimate. A further motivation is the error quantifications afforded by the posterior density which are not reliant on asymptotic in data length derivations. To compute these posterior quantities, this paper derives and illustrates a Gibbs sampling approach, which is a randomized algorithm in the family of Markov chain Monte Carlo methods. We provide details on a numerically robust implementation of the Gibbs sampler. In a numerical example, the proposed method is illustrated to give good convergence properties without requiring any user tuning.

  • 45.
    Dahlin, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindsten, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas Bo
    Dept. of Information Technology, Uppsala University, Uppsala, Sweden.
    Wills, Adrian George
    School of EECS, University of Newcastle, Australia .
    Hierarchical Bayesian approaches for robust inference in ARX models2012In: Proceedings from the 16th IFAC Symposium on System Identification, 2012 / [ed] Michel Kinnaert, International Federation of Automatic Control , 2012, Vol. 16 Part 1, p. 131-136Conference paper (Refereed)
    Abstract [en]

    Gaussian innovations are the typical choice in most ARX models but using other distributions such as the Student's t could be useful. We demonstrate that this choice of distribution for the innovations provides an increased robustness to data anomalies, such as outliers and missing observations. We consider these models in a Bayesian setting and perform inference using numerical procedures based on Markov Chain Monte Carlo methods. These models include automatic order determination by two alternative methods, based on a parametric model order and a sparseness prior, respectively. The methods and the advantage of our choice of innovations are illustrated in three numerical studies using both simulated data and real EEG data.

  • 46.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On the Use of Backward Simulation in the Particle Gibbs Sampler2012In: Proceedings of the 37th IEEE International Conference on Acoustics, Speech, and Signal Processing, IEEE , 2012, p. 3845-3848Conference paper (Refereed)
    Abstract [en]

    The particle Gibbs (PG) sampler was introduced in [Andrieu et al. (2010)] as a way to incorporate a particle filter (PF) in a Markov chain Monte Carlo (MCMC) sampler. The resulting method was shown to be an efficient tool for joint Bayesian parameter and state inference in nonlinear, non-Gaussian state-space models. However, the mixing of the PG kernel can be very poor when there is severe degeneracy in the PF. Hence, the success of the PG sampler heavily relies on the, often unrealistic, assumption that we can implement a PF without suffering from any considerate degeneracy. However, as pointed out by Whiteley in the discussion following [Andrieu et al. (2010)], the mixing can be improved by adding a backward simulation step to the PG sampler. Here, we investigate this further, derive an explicit PG sampler with backward simulation (denoted PG-BSi) and show that this indeed is a valid MCMC method. Furthermore, we show in a numerical example that backward simulation can lead to a considerable increase in performance over the standard PG sampler.

  • 47.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jordan, Michael I.
    University of California, Berkeley, USA.
    A Semiparametric Bayesian Approach to Wiener System Identification2011Report (Other academic)
    Abstract [en]

    We consider a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process (GP) model for the static nonlinearity. The GP model is a flexible model that can describe different types of nonlinearities while avoiding making strong assumptions such as monotonicity. We derive an inferential method based on recent advances in Monte Carlo statistical methods, known as Particle Markov Chain Monte Carlo (PMCMC). The idea underlying PMCMC is to use a particle filter (PF) to generate a sample state trajectory in a Markov chain Monte Carlo sampler. We use a recently proposed PMCMC sampler, denoted particle Gibbs with backward simulation, which has been shown to be efficient even when we use very few particles in the PF. The resulting method is used in a simulation study to identify two different Wiener systems with non-invertible nonlinearities.

  • 48.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Olsson, Jimmy
    Lund University, Sweden.
    An Explicit Variance Reduction Expression for the Rao-Blackwellised Particle Filter2011In: Proceedings of the 18th IFAC World Congress, 2011, p. 11979-11984Conference paper (Refereed)
    Abstract [en]

    Particle filters (PFs) have shown to be very potent tools for state estimation in nonlinear and/or non-Gaussian state-space models. For certain models, containing a conditionally tractable substructure (typically conditionally linear Gaussian or with finite support), it is possible to exploit this structure in order to obtain more accurate estimates. This has become known as Rao-Blackwellised particle filtering (RBPF). However, since the RBPF is typically more computationally demanding than the standard PF per particle, it is not always beneficial to resort to Rao-Blackwellisation. For the same computational effort, a standard PF with an increased number of particles, which would also increase the accuracy, could be used instead. In this paper, we have analysed the asymptotic variance of the RBPF and provide an explicit expression for the obtained variance reduction. This expression could be used to make an efficient discrimination of when to apply Rao-Blackwellisation, and when not to.

  • 49.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Clustering using Sum-of-Norms Regularization: With Application to Particle Filter Output Computation2011Report (Other academic)
    Abstract [en]

    We present a novel clustering method, formulated as a convex optimization problem. The method is based on over-parameterization and uses a sum-of-norms (SON) regularization to control the trade-off between the model fit and the number of clusters. Hence, the number of clusters can be automatically adapted to best describe the data, and need not to be specified a priori. We apply SON clustering to cluster the particles in a particle filter, an application where the number of clusters is often unknown and time varying, making SON clustering an attractive alternative.

  • 50.
    Lindsten, Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Clustering using Sum-of-Norms Regularization: With Application to Particle Filter Output Computation2011In: Proceedings of the 2011 IEEE Statistical Signal Processing Workshop, 2011, p. 201-204Conference paper (Refereed)
    Abstract [en]

    We present a novel clustering method, formulated as a convex optimization problem. The method is based on over-parameterization and uses a sum-of-norms (SON) regularization to control the trade-off between the model fit and the number of clusters. Hence, the number of clusters can be automatically adapted to best describe the data, and need not to be specified a priori. We apply SON clustering to cluster the particles in a particle filter, an application where the number of clusters is often unknown and time varying, making SON clustering an attractive alternative.

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