Sequential Markov Chain Monte Carlo Research Articles

Supply Curves in Electricity Markets: A Framework for Dynamic Modeling and Monte Carlo Forecasting

We introduce a new time series model for supply curves in a short-term electricity market. The model accounts for the contribution of different resources to the aggregate supply curve, as well as the impact of external factors on the price and amount of supplies provided by each of those resources. We equip the proposed model with a unified Monte Carlo methodology for tracking the latent variables, forecasting, and hyperparameter estimation. Specifically, we present a sequential Markov chain Monte Carlo (S-MCMC) algorithm for tracking the latent variables of the model, which in turn enables forecasting day-ahead supply curves. We present two stochastic variants of the expectation-maximization (EM) algorithm for estimating the hyperparameters of the proposed model. Both variants of EM employ S-MCMC in their expectation steps. We apply the proposed framework to the Turkish electricity market and show its performance on a real dataset from that market.

Uncertainty quantification in reservoirs with faults using a sequential approach

Reservoir simulation is critically important for optimally managing petroleum reservoirs. Often, many of the parameters of the model are unknown and cannot be measured directly. These parameters must then be inferred from production data at the wells. This is an inverse problem which can be formulated within a Bayesian framework to integrate prior knowledge with observational data. Markov Chain Monte Carlo (MCMC) methods are commonly used to solve Bayesian inverse problems by generating a set of samples which can be used to characterize the posterior distribution. In this work, we present a novel MCMC algorithm which uses a sequential transition kernel designed to exploit the redundancy which is often present in time series data from reservoirs. This method can be used to efficiently generate samples from the Bayesian posterior for time-dependent models. While this method is general and could be useful for many different models. We consider a Bayesian inverse problem in which we wish to infer fault transmissibilities from measurements of pressure at wells using a two-phase flow model. We demonstrate how the sequential MCMC algorithm presented here can be more efficient than a standard Metropolis-Hastings MCMC approach for this inverse problem. We use integrated autocorrelation times along with mean-squared jump distances to determine the performance of each method for the inverse problem.

Sequential Inference Methods for Non-Homogeneous Poisson Processes With State-Space Prior

The non-homogeneous Poisson process provides a generalised framework for the modelling of random point data by allowing the intensity of point generation to vary across its domain of interest (time or space). The use of non-homogeneous Poisson processes have arisen in many areas of signal processing and machine learning, but application is still largely limited by its intractable likelihood function and the lack of computationally efficient inference schemes, although some methods do exist for the batch data case. In this paper, we propose for the first time a sequential framework for intensity inference which combines the non-homogeneous model of Poisson data with continuous-time state-space models for their time-varying intensity. This approach enables us to design efficient online inference schemes, for which we propose a novel sequential Markov chain Monte Carlo (SMCMC) algorithm, as is demanded by many applications where point data arrive sequentially and decisions need to be made with low latency. The proposed approach is compared with competing methods on synthetic datasets and tested with high-frequency financial order book data, showing in general improved performance and better computational efficiency than the main batch-based competitor algorithm, and better performance than a simple baseline kernel estimation scheme.

Limit theorems for sequential MCMC methods

AbstractBoth sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$ -inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

Invertible Particle-Flow-Based Sequential MCMC With Extension to Gaussian Mixture Noise Models

Sequential state estimation in non-linear and non-Gaussian state spaces has a wide range of applications in statistics and signal processing. One of the most effective non-linear filtering approaches, particle filtering, suffers from weight degeneracy in high-dimensional filtering scenarios. Several avenues have been pursued to address high-dimensionality. Among these, particle flow particle filters construct effective proposal distributions by using invertible flow to migrate particles continuously from the prior distribution to the posterior, and sequential Markov chain Monte Carlo (SMCMC) methods use a Metropolis-Hastings (MH) accept-reject approach to improve filtering performance. In this paper, we propose to combine the strengths of invertible particle flow and SMCMC by constructing a composite Metropolis-Hastings (MH) kernel within the SMCMC framework using invertible particle flow. In addition, we propose a Gaussian mixture model (GMM)-based particle flow algorithm to construct effective MH kernels for multi-modal distributions. Simulation results show that for high-dimensional state estimation example problems the proposed kernels significantly increase the acceptance rate with minimal additional computational overhead and improve estimation accuracy compared with state-of-the-art filtering algorithms.

A Sequential MCMC Model for Reliability Evaluation of Offshore Wind Farms Considering Severe Weather Conditions

The offshore wind farms (OWF) are susceptible to the severe weather, which can cause the increase of component failure rate and has significant influence on the maintenance process and the reliability of wind farms. This paper proposes a sequential Markov chain Monte Carlo (MCMC) model for reliability evaluation of the OWF considering the impact of severe offshore weather. First, main factors affecting the wind turbine (WT) failure rate are analyzed. Second, a time-varying analytical model for the WT failure rate affected by wind speed and lightning is established. Three types of WT failure rates are considered: the failure rate under normal weather, that under strong wind, and that affected by lightning. Moreover, a time-varying analytical model for the repair time of main components of OWF is established by considering the influence of severe weather on offshore transportation time and maintenance efficiency after component failure. The MCMC model takes into account the temporal correlation of the weather and the repair process of failed component in the reliability evaluation. The model enables simultaneous simulation of the weather intensity and component state. For each system state generated by the MCMC model, a breadth-first search (BFS) method is applied to analyze the connectivity of the WTs and the sink node. Finally, the output of the wind farm is determined based on the wind speed data at this state. The expected energy not supply (EENS) and the generation ratio availability (GRA) indices of the OWF are evaluated to demonstrate the effectiveness of the proposed models. Further, the effects of other factors such as the enhanced protection for WT, the use of helicopter, and the weather characteristics of the OWF location on the reliability of OWF are discussed.

Gradient-Based Sequential Markov Chain Monte Carlo for Multitarget Tracking With Correlated Measurements

Measurements in wireless sensor networks (WSNs) are often correlated both in space and in time. This paper focuses on tracking multiple targets in WSNs by taking into consideration these measurement correlations. A sequential Markov Chain Monte Carlo (SMCMC) approach is proposed in which a Metropolis within Gibbs refinement step and a likelihood gradient proposal are introduced. This SMCMC filter is applied to case studies with cellular network received signal strength data in which the shadowing component correlations in space and time are estimated. The efficiency of the SMCMC approach compared to particle filtering, as well as the gradient proposal compared to a basic prior proposal, are demonstrated through numerical simulations. The accuracy improvement with the gradient-based SMCMC is above $90\%$ when using a low number of particles. Thanks to its sequential nature, the proposed approach can be applied to various WSN applications, including traffic mobility monitoring and prediction.

Embarrassingly parallel sequential Markov-chain Monte Carlo for large sets of time series

Bayesian computation crucially relies on Markov chain Monte Carlo (MCMC) algorithms. In the case of massive data sets, running the Metropolis-Hastings sampler to draw from the posterior distribution becomes prohibitive due to the large number of likelihood terms that need to be calculated at each iteration. In order to perform Bayesian inference for a large set of time series, we consider an algorithm that combines “divide and conquer” ideas previously used to design MCMC algorithms for big data with a sequential MCMC strategy. The performance of the method is illustrated using a large set of financial data.

Accelerating Markov Chain Monte Carlo with Active Subspaces

The Markov chain Monte Carlo (MCMC) method is the computational workhorse for Bayesian inverse problems. However, MCMC struggles in high-dimensional parameter spaces, since its iterates must sequentially explore the high-dimensional space. This struggle is compounded in physical applications when the nonlinear forward model is computationally expensive. One approach to accelerate MCMC is to reduce the dimension of the state space. Active subspaces are part of an emerging set of tools for subspace-based dimension reduction. An active subspace in a given inverse problem indicates a separation between a low-dimensional subspace that is informed by the data and its orthogonal complement that is constrained by the prior. With this information, one can run the sequential MCMC on the active variables while sampling independently according to the prior on the inactive variables. However, this approach to increase efficiency may introduce bias. We provide a bound on the Hellinger distance between the true posterior and its active subspace-exploiting approximation. And we demonstrate the active subspace-accelerated MCMC on two computational examples: (i) a two-dimensional parameter space with a quadratic forward model and one-dimensional active subspace and (ii) a 100-dimensional parameter space with a PDE-based forward model and a two-dimensional active subspace.

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems. In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion and Hamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms.

Electrocardiogram Signal Modeling With Adaptive Parameter Estimation Using Sequential Bayesian Methods

The automatic classification of electrocardiogram (ECG) signals is of great clinical significance in eliminating the strenuous process of manually annotating ECG recordings. Although statistical models describing ECG signal dynamics currently exist, they depend considerably on a priori information and user-specified model parameters. Also, ECG beat morphologies, which vary greatly across different individuals and disease states, cannot easily be described by a single representation. In this paper, we propose sequential Bayesian based methods to effectively model and adaptively select parameters of ECG signals. We first consider an adaptive framework based on a sequential Bayesian tracking method that adaptively selects the best cardiac parameters by minimizing the estimation error and does not require early-stage processing to obtain prior signal information. We then present ECG modeling techniques using the interacting multiple model (IMM) and sequential Markov chain Monte Carlo (SMCMC) methods combined with simultaneous model selection. Both these methods can adaptively choose between different representations to model various ECG beat morphologies without requiring prior ECG information. The performance of the proposed algorithms is demonstrated using real ECG data. Finally, we develop a Bayesian maximum-likelihood based classifier to classify different types of cardiac arrhythmias using which, correction classification rates of 90% and 98% are obtained, when considering features obtained from the estimated model parameters of the adaptive framework, and both the IMM and SMCMC methods, respectively.

Markov Chain Monte Carlo Detectors for Channels With Intersymbol Interference

In this paper, we propose novel low-complexity soft-in soft-out (SISO) equalizers using the Markov chain Monte Carlo (MCMC) technique. We develop a bitwise MCMC equalizer (b-MCMC) that adopts a Gibbs sampler to update one bit at a time, as well as a group-wise MCMC (g-MCMC) equalizer where multiple symbols are updated simultaneously. The g-MCMC equalizer is shown to outperform both the b-MCMC and the linear minimum mean square error (MMSE) equalizer significantly for channels with severe amplitude distortion. Direct application of MCMC to channel equalization requires sequential processing which leads to long processing delay. We develop a parallel processing algorithm that reduces the processing delay by orders of magnitude. Numerical results show that both the sequential and parallel processing MCMC equalizers perform similarly well and achieve a performance that is only slightly worse than the optimum maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> (MAP) equalizer. The MAP equalizer, on the other hand, has a complexity that grows exponentially with the size of the memory of the channel, while the complexity of the proposed MCMC equalizers grows linearly.