Online Bayesian learning in dynamic models: an illustrative introduction to particle methods

This paper investigates the use of linear dynamic models (LDMs) to improve classification of single-trial EEG signals. Existing dynamic classification of EEG uses discrete-state hidden Markov models (HMMs) based on piecewise-stationary assumption, which is inadequate for modeling the highly non-stationary dynamics underlying EEG. The continuous hidden states of LDMs could better describe this continuously changing characteristic of EEG, and thus improve the classification performance. We consider two examples of LDM: a simple local level model (LLM) and a time-varying autoregressive (TVAR) state-space model. AR parameters and band power are used as features. Parameter estimation of the LDMs is performed by using expectation-maximization (EM) algorithm. We also investigate different covariance modeling of Gaussian noises in LDMs for EEG classification. The experimental results on two-class motor-imagery classification show that both types of LDMs outperform the HMM baseline, with the best relative accuracy improvement of 14.8% by LLM with full covariance for Gaussian noises. It may due to that LDMs offer more flexibility in fitting the underlying dynamics of EEG.

In this paper we present an overview of the state of the art in Kalman filtering and dynamic Bayesian linear and nonlinear models. We present some of the basic results including the derivation of Kalman filtering equations as well as recent advances in Kalman filter models and their extensions including non‐Gaussian state‐space models. In so doing, we take a Bayesian perspective and discuss parameter learning in state‐space models which typically involves Markov chain Monte Carlo and sequential Monte Carlo methods. We present particle filtering and Bayesian particle learning techniques for state space models and discuss recent advances.This article is categorized under:Applications of Computational Statistics > Signal and Image Processing and CodingStatistical Models > Bayesian ModelsStatistical Models > Time Series ModelsStatistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC)

The particle filter is well known as a state estimation method for nonlinear and non-Gaussian system. However, particle filter has the inherent drawbacks such as samples less of diversity and low tracking accuracy. In this paper, a novel particle filter algorithm with the Markov Chain Monte Carlo (MCMC) and consensus strategy is proposed. The authors utilize MCMC sampling method to make the particles more diversification. And the algorithm is optimized by consensus strategy, which makes the state estimates of all network nodes converge to a more precise value. Simulation results show that compared to existing methods, the proposed algorithm has superior performance.

The existence of many different dynamic large eddy simulation (LES) methods leads to questions about the theoretical foundation of dynamic LES methods. It was shown recently that the use of stochastic analysis enables a theoretically well based systematic derivation of a realizable linear dynamic model (LDM) and a realizable nonlinear dynamic model (NDM). A priori and a posteriori analyses of turbulent channel flow are used here to study the characteristic properties of these dynamic models. The LDM and NDM are compared with other dynamic models: the non-stabilized and stabilized dynamic Smagorinsky model (DSM), which is used in many applications of LES, and Wang-Bergstrom's dynamic model (WBDM), which represents an extension of the DSM. The DSM and WBDM do not represent realizable models because they are not derived as consequences of a realizable stochastic process. The comparisons reported here show that the LDM and NDM are based on a dynamic model formulation that avoids shortcomings of existing concepts. The LDM and NDM account for backscatter, and they are computationally stable without any modification. The LDM and NDM represent the instantaneous small scale structure of turbulence very well. Compared to the DSM and WBDM, respectively, the LDM and NDM are computationally more efficient.

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

Data assimilation is the process of estimating the state of dynamic systems (linear or nonlinear, Gaussian or non-Gaussian) as accurately as possible from noisy observational data. Although the Three Dimensional Variational (3D-VAR) methods, Four Dimensional Variational (4D-VAR) methods and Ensemble Kalman filter (EnKF) methods are widely used and effective for linear and Gaussian dynamics, new methods of data assimilation are required for the general situation, that is, nonlinear non-Gaussian dynamics. General Bayesian recursive estimation theory is reviewed in this thesis. The Bayesian estimation approach provides a rather general and powerful framework for handling nonlinear, non-Gaussian, as well as linear, Gaussian estimation problems. Despite a general solution to the nonlinear estimation problem, there is no closed-form solution in the general case. Therefore, approximate techniques have to be employed. In this thesis, the sequential Monte Carlo (SMC) methods, commonly referred to as the particle filter, is presented to tackle non-linear, non-Gaussian estimation problems. In this thesis, we use the SMC methods only for the nonlinear state estimation problem, however, it can also be used for the nonlinear parameter estimation problem. In order to demonstrate the new methods in the general nonlinear non-Gaussian case, we compare Sequential Monte Carlo (SMC) methods with the Ensemble Kalman Filter (EnKF) by performing data assimilation in nonlinear and non-Gaussian dynamic systems. The models used in this study are referred to as state-space models. The Lorenz 1963 and 1966 models serve as test beds for examining the properties of these assimilation methods when used in highly nonlinear dynamics. The application of Sequential Monte Carlo methods to different fixed parameters in dynamic models is considered. Four different scenarios in the Lorenz 1063 [sic] model and three different scenarios in the Lorenz 1996 model are designed in this study for both the SMC methods and EnKF method with different filter siz

In this article, we develop a Bayesian framework for parameter estimation of a computationally expensive dynamic epidemic model using time series epidemic data. Specifically, we work with a model for A/H1N1 influenza, which is implemented as a deterministic computer simulator, taking as input the underlying epidemic parameters and calculating the corresponding time series of reported infections. To obtain Bayesian inference for the epidemic parameters, the simulator is embedded in the likelihood for the reported epidemic data. However, the simulator is computationally slow, making it impractical to use in Bayesian estimation where a large number of simulator runs is required. We propose an efficient approximation to the simulator using an emulator, a statistical model that combines a Gaussian process (GP) prior for the output function of the simulator with a dynamic linear model (DLM) for its evolution through time. This modeling framework is both flexible and tractable, resulting in efficient posterior inference through Markov chain Monte Carlo (MCMC). The proposed dynamic emulator is then used in a calibration procedure to obtain posterior inference for the parameters of the influenza epidemic.

Timely detection of localised COVID-19 surges is essential for targeting limited health resources, yet most routine surveillance algorithms ignore spatial dependence and many Bayesian spatio-temporal models are computationally demanding. Evidence on Hamiltonian Monte Carlo (HMC) performance for outbreak detection in low- and middle-income country (LMIC) settings remains limited. We applied a Bayesian spatio-temporal hidden Markov model (HMM) to South African COVID-19 hospital mortality, comparing data-augmented Markov chain Monte Carlo (MCMC) with dynamic HMC. We conducted a retrospective ecological time-series study of in-hospital COVID- 19 deaths across 52 districts over 28 months (March 2020-June 2022), using national hospital surveillance linked to district-level health-system and population indicators. Death counts were modelled with a Poisson log-linear specification incorporating a smooth temporal trend, cyclic seasonality, spatial smoothing, and selected covariates, offset by expected deaths from admissions and a case-fatality ratio. Outbreaks were represented by a two-state HMM with latent indicators integrated out analytically and estimated via dynamic HMC in Stan. Eight candidate models were ranked using marginal likelihoods; the preferred model was re-fitted with both samplers to compare runtime, ESS, and convergence. A spatial HMM with marginalised outbreak states was strongly favoured over nonoutbreak and threshold-based alternatives. Posterior outbreak probabilities reproduced the four recognised national waves while revealing marked district-level heterogeneity, with early intense outbreaks in Western Cape and Gauteng districts and later peaks inland. Outbreaks were short-lived (mean around three months), and residual spatial risks indicated persistent excess mortality in the Eastern Cape and Free State. Dynamic HMC and MCMC yielded similar outbreak probability surfaces; however, HMC produced substantially larger ESS 1 (approximately 5,000 versus 63) and near-ideal convergence, whereas MCMC showed poor mixing. ESS per second was similar, so HMC's extra computation yielded more informative samples. A Bayesian spatio-temporal HMM fitted with dynamic HMC delivers coherent, spatially resolved outbreak probabilities and captures short-lived district-level mortality surges within broader national waves. Despite greater computational intensity, dynamic HMC offers superior mixing and convergence and is preferable for routine surveillance when adequate computing resources are available. The framework is transferable to other routinely collected surveillance data in South Africa and similar LMIC settings.

In business, dynamic models often provide valuable insights into the complex interactions between variablesover time. But recent research contends that the lagged dependent variable specification is too problematic foruse in most situations. More specifically, if residuals autocorrelation is present in a dynamic equation wherelagged values of the dependent variable appear as regressors, Ordinary least squares (OLS) estimates are biasedand generally inconsistent. For this reason it is important to have available tests against autocorrelation,particularly when it is a dynamic model. The Breusch-Godfrey (BG) test is the most appropriate test in thepresence of stochastic regressors such as lagged values of the dependent variable for higher order autocorrelation,which is asymptotically equivalent to the Durbin-Watson h test for first order autocorrelation. But Durbin htest is not applicable for second or higher order autocorrelation. Moreover these existing tests are not suitable forone-sided higher order autoregressive schemes. Whenever the sign of the parameters are known of aneconometric model, usual two-sided tests are no longer valid. In this situation, we propose a distance-basedone-sided Lagrange Multiplier (DLM) test, a likelihood based test, to test one-sided alternative. Monte Carlosimulations are conducted to compare power properties of the proposed DLM test with the BG test. It is foundthat the DLM test shows substantially improved power than two-sided counterparts for most of the casesconsidered.

Parameterization of a process-based tree-growth model: Comparison of optimization, MCMC and Particle Filtering algorithms

Model based feature enhancement techniques are constructed from acoustic models for speech and noise, together with a model of how the speech and noise produce the noisy observations. Most techniques incorporate either Gaussian mixture models (GMM) or hidden Markov models (HMM). This paper explores using a switching linear dynamic model (LDM) for the clean speech. The linear dynamics of the model capture the smooth time evolution of speech. The switching states of the model capture the piecewise stationary characteristics of speech. However, incorporating a switching LDM causes the enhancement problem to become intractable. With a GMM or an HMM, the enhancement running time is proportional to the length of the utterance. The switching LDM causes the running time to become exponential in the length of the utterance. To overcome this drawback, the standard generalized pseudo-Bayesian technique is used to provide an approximate solution of the enhancement problem. We present preliminary results demonstrating that, even with relatively small model sizes, substantial word error rate improvement can be achieved.

Suppose that one wishes to sample from the density π(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution π(u|x) can be defined, giving the joint distribution π(x, u) = π(x)π(u|x). A MCMC scheme that samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches. The revolutionary algorithm of Swendsen and Wang is one such example. Besides reviewing the Swendsen-Wang algorithm and its generalizations, this article introduces a new auxiliary variable method called partial decoupling. Two applications in Bayesian image analysis are considered: a binary classification problem in which partial decoupling out performs Swendsen-Wang and single-site Metropolis methods, and a positron emission tomography (PET) reconstruction that uses the gray level prior of Geman and McClure. A generalized Swendsen–Wang algorithm is developed for this problem, which reduces the computing time to the point where MCMC is a viable method of posterior exploration.

Event Abstract Back to Event Statistical uncertainty and sensitivity analysis of intracellular signaling models - through approximate Bayesian computation and variance based global sensitivity analysis Olivia Eriksson1, 2*, Alexandra Jauhiainen1, 3, Sara Maad Sasane4, Anu G Nair5, Carolina Sartorius4 and Jeanette Hellgren Kotaleski2, 5, 6 1 Contributed equally to this work, Sweden 2 Stockholm University, Department of Numerical Analysis and Computer Science, Sweden 3 AstraZeneca AB R&D, Early Clinical Biometrics, Sweden 4 Lund University, Centre for Mathematical Sciences, Sweden 5 KTH Royal Institute of Technology, School of Computer Science and Communication, Sweden 6 Karolinska Institute, Department of Neuroscience, Sweden Computational dynamical models describing intracellular signaling pathways of nerve cells are increasing in size and complexity as more information is obtained from experiments. Models describing protein interactions within neurons can contain over hundred protein reactions [1, 2]. These models are built from knowledge about the interaction topology, inferred from e.g. gene knock-out experiments, as well as from experimental quantitative data describing the input-output relationship of the observed system. The quantitative data are often sparse as compared to the size of the system, and translating the experimental information into dynamical models often results in large uncertainties in the parameter (e.g reaction rates) values. This uncertainty is partly due to the sparseness in experimental time series data (practical unidentifiability), but it is also an intrinsic feature of the modeled system (structural unidentifiability) and could reflect the possibility for biological variation [3]; in many biological systems the same function can be achieved in partly different ways due to degeneracy and redundancy. Here we investigate the extent of this parameter uncertainty, in a model describing calcium (Ca)-dependent activation of Calmodulin (CaM), Protein phosphatase 2B (PP2B) and Ca/CaM-dependent protein kinase II (CamKII) [4], which is an intracellular pathway of importance to synaptic plasticity. We presume that the structure of the model is correct and then characterize the uncertainty in the parameters based on data. This is done by sampling the subspace of parameter values which has an equally good fit to data (corresponding to the “viable space” in [5]), through a statistical approach known as Approximate Bayesian Computation (ABC) with Markov Chain Monte Carlo (MCMC) [6]. ABC has been proposed for parameter estimation in system biology models previously, albeit in slightly different settings, e.g. [7, 8], and can be applied to a wide range of problems. In order to characterize the viable space efficiently with several experimental datasets, we employ multivariate probability distributions called copulas as a part of the ABC sampling. The result is a multivariate posterior distribution for the parameters, which represents how much they are constrained by the data (i.e. the uncertainty) as illustrated in Figure 1 (pairwise plots of viable space for four parameters indicating constraints and dependencies before and after calibration to data). The next step is to look into how the parameter uncertainty is translated to uncertainty of the predictions made from the model. We also investigate the role and importance of different parameters, with respect to different model outputs including the predictions, by means of global sensitivity analysis [9]. Figure 2 illustrates the sensitivity of one of the outputs calculated by a method based on the decomposition of the variance [10, 11, 12], showing different types of sensitivity measures. Finally, we explore how the sensitivity of different subsets of parameters change as different parts of the viable space are investigated, which is shown in Figure 3, where sensitivity profiles are clustered into groups with different sensitivity characteristics. Figure 1 Figure 2 Figure 3 Acknowledgements This research was supported by funding from the Swedish e-Science Research Centre and the Human Brain Project References [1] Gutierrez-Arenas O, Eriksson O, Hellgren Kotaleski J (2014) Segregation and Crosstalk of D1 Receptor-Mediated Activation of ERK in Striatal Medium Spiny Neurons upon Acute Administration of Psychostimulants. PLoS Comput Biol 10, e1003445. [2] Nair AG, Gutierrez-Arenas O, Eriksson O, Vincent P, and Hellgren Kotaleski J (2015) Sensing Positive versus Negative Reward Signals through Adenylyl Cyclase-Coupled GPCRs in Direct and Indirect Pathway Striatal Medium Spiny Neurons. J. Neuroscience 35, 14017-14030. [3] Marder E, Goeritz ML, Otopalik AG (2015) Robust circuit rhythms in small circuits arise from variable circuit components and mechanisms. Curr Opin Neurobiol 31, 156-163. [4] Nair AG, Gutierrez-Arenas O, Eriksson O, Jauhiainen A, Blackwell KT and Kotaleski, JH (2014). Modeling intracellular signaling underlying striatal function in health and disease. Progress in molecular biology and translational science, 123, 277 [5] Zamora-Sillero E, Hafner M, Ibig A, Stelling J, and Wagner, A. (2011). Efficient characterization of high-dimensional parameter spaces for systems biology. BMC systems biology 5, 142. [6] Marjoram P, Molitor J, Plagnol V, and Tavare S (2003) Markov chain Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. U.S.A. 100, 15324–15328. [7] Toni T, Welch D, Strelkowa N, Ipsen A, and Stumpf MP. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J R Soc Interface, 6, 187–202. [8] Liepe J, Kirk P, Filippi S, Toni T, Barnes CP, and Stumpf MP. A framework for parameter estimation and model selection from experimental data in systems biology using approximate Bayesian computation. Nat Protoc, 9, 439–456. [9] Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S Global sensitivity analysis: the primer. In Wiley 2008 New York, NY:Wiley. [10] Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Math. Comput. Simul. 55, 271–280. [11] Saltelli A (2002) Making best use of model evaluations to compute sensitivity indices. Comp Phys Comm 145, 280 – 297. [12] Halnes G, Ulfhielm E Ljunggren EE, Kotaleski JH and Rospars JP (2009) Modelling and sensitivity analysis of the reactions involving receptor, G-protein and effector in vertebrate olfactory receptor neurons. Journal of computational neuroscience 27, 471-491. Keywords: sensitivity analysis, Approximate Bayesian Computation, copulas, intracellular signalling, Dynamical Modeling, global sensitivity analysis, Markov chain Monte Carlo, variance decomposition Conference: Neuroinformatics 2016, Reading, United Kingdom, 3 Sep - 4 Sep, 2016. Presentation Type: Investigator presentations Topic: Computational neuroscience Citation: Eriksson O, Jauhiainen A, Maad Sasane S, Nair A, Sartorius C and Hellgren Kotaleski J (2016). Statistical uncertainty and sensitivity analysis of intracellular signaling models - through approximate Bayesian computation and variance based global sensitivity analysis. Front. Neuroinform. Conference Abstract: Neuroinformatics 2016. doi: 10.3389/conf.fninf.2016.20.00014 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 31 May 2016; Published Online: 18 Jul 2016. * Correspondence: PhD. Olivia Eriksson, Contributed equally to this work, Stockholm, Sweden, olivia@kth.se Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Google Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Google Scholar Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski PubMed Olivia Eriksson Alexandra Jauhiainen Sara Maad Sasane Anu G Nair Carolina Sartorius Jeanette Hellgren Kotaleski Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlo sampling

Research purpose: The current work introduces a novel method for time series discriminant analysis (DA). Proposing a version for the Bayes classifier employing Dynamic Linear Models, which we denote by BCDLM This article explores the application of DLMs and the Bayes Classifier in time series classification to promote application in sustainability across diverse sectors. Method: This paper presents some computer simulation studies in which we generate four different scenarios corresponding to time series observations from various Dynamic Linear Models (DLMs). In Discriminant Analysis, we investigated strategies for estimating variance in models and compared the performance of the BCDLM with other common classifiers. Such datasets are composed of real-time series (data from SONY AIBO Robot and spectrometry of coffee types) and pseudo-time series (data from Swedish leaves adapted for time series). We also point out that algorithm was used to determine training and test sets in real-world applications. Results: Considering the real-time series examined in this paper, The results obtained indicate that the parametric approach developed represents a promising alternative for this class of DA problems, with observations of time series in a situation that is quite difficult in practice when we have series with large sizes with respect to the number of observations in the classes, even though more thorough studies are required. Conclusions: It concludes that the BCDLM performed comparably to the results of the classifiers 1NN, RDA, NBND and NBK and superior to the methods LDA and QDA. This offers a powerful combination for time series classification, enabling accurate predictions and informed decision-making in areas such as energy consumption, waste management, and resource allocation.