Particle filtering methods are widely applied in sequential state estimation within nonlinear non-Gaussian state space model. However, the traditional particle filtering methods suffer the weight degeneracy in the high-dimensional state space model. Currently, there are many methods to improve the performance of particle filtering in high-dimensional state space model. Among these, the more advanced method is to construct the Sequential Makov chian Monte Carlo (SMCMC) framework by implementing the Composite Metropolis-Hasting (MH) Kernel. In this paper, we proposed to discrete the Zig-Zag Sampler and apply the Zig-Zag Sampler in the refinement stage of the Composite MH Kernel within the SMCMC framework which is implemented the invertible particle flow in the joint draw stage. We evaluate the performance of proposed method through numerical experiments of the challenging complex high-dimensional filtering examples. Nemurical experiments show that in high-dimensional state estimation examples, the proposed method improves estimation accuracy and increases the acceptance ratio compared with state-of-the-art filtering methods.
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Stochastic epidemic models provide an interpretable probabilistic description of the spread of a disease through a population. Yet, fitting these models when the epidemic process is only partially observed is a notoriously difficult task due to the intractability of the likelihood for many classical models. To remedy this issue, this article introduces a novel data-augmented MCMC algorithm for fast and exact Bayesian inference for the stochastic SIR model given discretely observed infection incidence counts. In a Metropolis-Hastings step, new event times of the latent data are jointly proposed from a surrogate process that closely resembles the SIR, and from which we can efficiently generate epidemics compatible with the observed data. The proposed DA-MCMC algorithm is fast and, since the latent data are generated from a faithful approximation of the target model, a large portion thereof can be updated per iteration without prohibitively lowering the acceptance rate. We find that the method explores the high-dimensional latent space efficiently and scales to outbreaks with hundreds of thousands of individuals, and we show that the Markov chain underlying the algorithm is uniformly ergodic. We validate its performance via thorough simulation experiments and a case study on the 2013-2015 Ebola outbreak in Western Africa.
We propose a novel coupled rejection-sampling method for sampling from couplings of arbitrary distributions. The method relies on accepting or rejecting coupled samples coming from dominating marginals. Contrary to existing acceptance-rejection methods, the variance of the execution time of the proposed method is limited and stays finite as the two target marginals approach each other in the sense of the total variation norm. In the important special case of coupling multivariate Gaussians with different means and covariances, we derive positive lower bounds for the resulting coupling probability of our algorithm, and we then show how the coupling method can be optimised using convex optimisation. Finally, we show how we can modify the coupled-rejection method to propose from coupled ensemble of proposals, so as to asymptotically recover a maximal coupling. We then apply the method to derive a novel parallel coupled particle filter resampling algorithm, and show how it can be used to speed up unbiased MCMC methods based on couplings.
Many real-world problems require one to estimate parameters of interest, in a Bayesian framework, from data that are collected sequentially in time. Conventional methods for sampling from posterior distributions, such as {Markov Chain Monte Carlo} can not efficiently address such problems as they do not take advantage of the data's sequential structure. To this end, sequential methods which seek to update the posterior distribution whenever a new collection of data become available are often used to solve these types of problems. Two popular choices of sequential method are the Ensemble Kalman filter (EnKF) and the sequential Monte Carlo sampler (SMCS). While EnKF only computes a Gaussian approximation of the posterior distribution, SMCS can draw samples directly from the posterior. Its performance, however, depends critically upon the kernels that are used. In this work, we present a method that constructs the kernels of SMCS using an EnKF formulation, and we demonstrate the performance of the method with numerical examples.
We propose a novel multibody dynamics simulation framework that can efficiently deal with large-dimensionality and complementarity multi-contact conditions. Typical contact simulation approaches perform contact impulse-level fixed-point iteration (IL-FPI), which has high time-complexity from large-size matrix inversion and multiplication, as well as susceptibility to ill-conditioned contact situations. To circumvent this, we propose a novel framework based on velocity-level fixed-point iteration (VL-FPI), which, by utilizing a certain surrogate dynamics and contact nodalization (with virtual nodes), can achieve not only inter-contact decoupling but also their inter-axes decoupling (i.e., contact diagonalization). This then enables us to one-shot/parallel-solve the contact problem during each VL-FPI iteration-loop, while the surrogate dynamics structure allows us to circumvent large-size/dense matrix inversion/multiplication, thereby, significantly speeding up the simulation time with improved convergence property. We theoretically show that the solution of our framework is consistent with that of the original problem and, further, elucidate mathematical conditions for the convergence of our proposed solver. Performance and properties of our proposed simulation framework are also demonstrated and experimentally-validated for various large-dimensional/multi-contact scenarios including deformable objects.
Data collected from wearable devices and smartphones can shed light on an individual's patterns of behavior and circadian routine. Phone use can be modeled as alternating between the state of active use and the state of being idle. Markov chains and alternating recurrent event models are commonly used to model state transitions in cases such as these, and the incorporation of random effects can be used to introduce time-of-day effects. While state labels can be derived prior to modeling dynamics, this approach omits informative regression covariates that can influence state memberships. We instead propose a recurrent event proportional hazards (PH) regression to model the transitions between latent states. We propose an Expectation-Maximization (EM) algorithm for imputing latent state labels and estimating regression parameters. We show that our E-step simplifies to the hidden Markov model (HMM) forward-backward algorithm, allowing us to recover a HMM in addition to PH models. We derive asymptotic distributions for our model parameter estimates and compare our approach against competing methods through simulation as well as in a digital phenotyping study that followed smartphone use in a cohort of adolescents with mood disorders.
The recent developments in the machine learning domain have enabled the development of complex multivariate probabilistic forecasting models. Therefore, it is pivotal to have a precise evaluation method to gauge the performance and predictability power of these complex methods. To do so, several evaluation metrics have been proposed in the past (such as Energy Score, Dawid-Sebastiani score, variogram score), however, they cannot reliably measure the performance of a probabilistic forecaster. Recently, CRPS-sum has gained a lot of prominence as a reliable metric for multivariate probabilistic forecasting. This paper presents a systematic evaluation of CRPS-sum to understand its discrimination ability. We show that the statistical properties of target data affect the discrimination ability of CRPS-Sum. Furthermore, we highlight that CRPS-Sum calculation overlooks the performance of the model on each dimension. These flaws can lead us to an incorrect assessment of model performance. Finally, with experiments on the real-world dataset, we demonstrate that the shortcomings of CRPS-Sum provide a misleading indication of the probabilistic forecasting performance method. We show that it is easily possible to have a better CRPS-Sum for a dummy model, which looks like random noise, in comparison to the state-of-the-art method.
Hamiltonian Monte Carlo (HMC) is a powerful Markov Chain Monte Carlo (MCMC) method for sampling from complex high-dimensional continuous distributions. However, in many situations it is necessary or desirable to combine HMC with other Metropolis-Hastings (MH) samplers. The common HMC-within-Gibbs strategy implies a trade-off between long HMC trajectories and more frequent other MH updates. Addressing this trade-off has been the focus of several recent works. In this paper we propose Metropolis Augmented Hamiltonian Monte Carlo (MAHMC), an HMC variant that allows MH updates within HMC and eliminates this trade-off. Experiments on two representative examples demonstrate MAHMC's efficiency and ease of use when compared with within-Gibbs alternatives.
A method of Sequential Log-Convex Programming (SLCP) is constructed that exploits the log-convex structure present in many engineering design problems. The mathematical structure of Geometric Programming (GP) is combined with the ability of Sequential Quadratic Program (SQP) to accommodate a wide range of objective and constraint functions, resulting in a practical algorithm that can be adopted with little to no modification of existing design practices. Three test problems are considered to demonstrate the SLCP algorithm, comparing it with SQP and the modified Logspace Sequential Quadratic Programming (LSQP). In these cases, SLCP shows up to a 77% reduction in number of iterations compared to SQP, and an 11% reduction compared to LSQP. The airfoil analysis code XFOIL is integrated into one of the case studies to show how SLCP can be used to evolve the fidelity of design problems that have initially been modeled as GP compatible. Finally, a methodology for design based on GP and SLCP is briefly discussed.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
This paper presents a safety-aware learning framework that employs an adaptive model learning method together with barrier certificates for systems with possibly nonstationary agent dynamics. To extract the dynamic structure of the model, we use a sparse optimization technique, and the resulting model will be used in combination with control barrier certificates which constrain feedback controllers only when safety is about to be violated. Under some mild assumptions, solutions to the constrained feedback-controller optimization are guaranteed to be globally optimal, and the monotonic improvement of a feedback controller is thus ensured. In addition, we reformulate the (action-)value function approximation to make any kernel-based nonlinear function estimation method applicable. We then employ a state-of-the-art kernel adaptive filtering technique for the (action-)value function approximation. The resulting framework is verified experimentally on a brushbot, whose dynamics is unknown and highly complex.
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