Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

Deep learning has been used to train neural networks to estimate the Remaining Useful Life (RUL) of a machine given sensor signals from that machine. This has resulted in some accurate state-of-the-art RUL estimation models. Here we present a novel way to construct a generative model to estimate the RUL and quantify its uncertainty. It takes the form of a Linear Gaussian State Space Model (LGSSM) and is trained using the Kalman Filter; hence, uncertainty is quantified using this LGSSM instead of by other popular methods like Monte Carlo Dropout or Deep Ensembles. This means we can train by directly using the marginal log-likelihood loss and don’t require multiple samples to represent the uncertainty. However, this method is limited to using Gaussian distributions to quantify uncertainty. We avoid needing to use non-linear variants of the filter by processing the sensors using a neural network to represent noncausal sensor sequences as a “control variable” in the LGSSM. This noncausal representation is shown to be important for achieving state-of-the-art performance. The model is tested on a turbofan engine and dust filter dataset. The code can be found on GitHub. 1 1 https://github.com/StarMarco/Kalman_DVAE • Deep Neural Networks encode time window of sensor signals. • These are used as control/forcing variables of a Linear Gaussian State Space Model (LGSSM). • The “observables" of the LGSSM are the Remaining Useful Life (RUL) values of the machine. • The Kalman filtering algorithm is used to estimate the marginal log-likelihood of these RUL estimates. • After training the LGSSM can be evaluated over time to estimate the RUL trajectories.

The paper presents a broad general review of the state space approach to time series analysis. It begins with an introduction to the linear Gaussian state space model. Applications to problems in practical time series analysis are considered. The state space approach is briefly compared with the Box–Jenkins approach. The Kalman filter and smoother and the simulation smoother are described. Missing observations, forecasting and initialisation are considered. A representation of a multivariate series as a univariate series is displayed. The construction and maximisation of the likelihood function are discussed. An application to real data is presented. The treatment is extended to non-Gaussian and nonlinear state space models. A simulation technique based on importance sampling is described for analysing these models. The use of antithetic variables in the simulation is considered. Bayesian analysis of the models is developed based on an extension of the importance sampling technique. Classical and Bayesian methods are applied to a real time series.

Time variant reliability model updating in instrumented dynamical systems based on Girsanov’s transformation

Particle filter-based Gaussian process optimisation for parameter inference

Application of a Linear Gaussian State Space Model and the Kalman Filter in Estimating Expectations of the Natural Logarithmic Monthly Nav Returns of the Open – End Funds of Major Investment Banks.

Sequential Monte Carlo (SMC) methods (also known as particle filter) provide a way to solve the state estimation problem in nonlinear non-Gaussian state space models (SSM) through numerical approximation. Particle smoothing is one retrospective state estimation method based on particle filtering. In this paper, we propose a new particle smoother. The basic idea is easy and leads to a forward-backward procedure, where the Metropolis-Hastings algorithm is used to resample the filtering particles. The goodness of the new scheme is assessed using a nonlinear SSM. It is concluded that this new particle smoother is suitable for state estimation in complicated dynamical systems.

Nudging is a popular algorithmic strategy in numerical filtering to deal with the problem of inference in high-dimensional dynamical systems. We demonstrate in this paper that general nudging techniques can also tackle another crucial statistical problem in filtering, namely the misspecification of the transition kernel. Specifically, we rely on the formulation of nudging as a general operation increasing the likelihood and prove analytically that, when applied carefully, nudging techniques implicitly define state-space models that have higher marginal likelihoods for a given (fixed) sequence of observations. This provides a theoretical justification of nudging techniques as data-informed algorithmic modifications of state-space models to obtain robust models under misspecified dynamics. To demonstrate the use of nudging, we provide numerical experiments on linear Gaussian state-space models and a stochastic Lorenz 63 model with misspecified dynamics and show that nudging offers a robust filtering strategy for these cases.

In time-series analysis, state-space models (SSMs) have been widely used to estimate the conditional probability distributions of hidden variables and parameter values, as well as to understand structures that can generate the data. For example, the Kalman filter is used to analytically calculate the conditional probability distribution on linear SSMs in terms of minimizing the variance, and several extensions, such as the unscented Kalman filter and particle filter, have been applied to calculate the approximate distribution on nonlinear SSMs. Recently, the approximate Bayesian computation (ABC) has been applied to such time-series filtering to handle intractable likelihoods; however, it remains problematic with respect to consistently achieving a reduction of the estimation bias, evaluating the validity of the models, and dealing with replicated observations. To address these problems, in this paper, we propose a novel method combined with the kernel ABC to perform filtering, parameter estimation, and the model evaluation in SSMs. Simulation studies show that the proposed method produces inference that is comparable to other ABC methods, with the advantage of not requiring a careful calibration of the ABC threshold. In addition, we evaluate the performance of the model-selection capability using true and competitive models on synthetic data from nonlinear SSMs. Finally, we apply the proposed method to real data in rat circadian oscillations, and demonstrated the usefulness in practical situations.

Nonlinear state-space models are ubiquitous in modeling real-world dynamical systems. Sequential Monte Carlo (SMC) techniques, also known as particle methods, are a well-known class of parameter estimation methods for this general class of state-space models. Existing SMC-based techniques rely on excessive sampling of the parameter space, which makes their computation intractable for large systems or tall data sets. Bayesian optimization techniques have been used for fast inference in state-space models with intractable likelihoods. These techniques aim to find the maximum of the likelihood function by sequential sampling of the parameter space through a single SMC approximator. Various SMC approximators with different fidelities and computational costs are often available for sample-based likelihood approximation. In this paper, we propose a multi-fidelity Bayesian optimization algorithm for the inference of general nonlinear state-space models (MFBO-SSM), which enables simultaneous sequential selection of parameters and approximators. The accuracy and speed of the algorithm are demonstrated by numerical experiments using synthetic gene expression data from a gene regulatory network model and real data from the VIX stock price index.

Nonlinear mixed-effects state space models with applications to HIV dynamics

This paper proposes a nonlinear state space model (SSM) to estimate the health degradation and predict the remaining useful life(RUL) of industry asset. Expectation Maximization (EM) algorithm and particle filtering (PF) are introduced to estimate SSM parameters. A case study is utilized to predict RUL of industry asset.

The problem of sequential detection of spontaneous changes in equations for unobserved state variables and observations of linear and nonlinear multivariate state-space models is considered. We consider the case of new additive terms and the case of changing coefficients of these equations. The proposed nonparametric method uses the idea of the “moving window” of observations. In Theorem 4.1 the exponential rate of convergence to zero for type 1 error probability (a wrong decision about a change) is proved. In Theorem 4.2 we consider type 2 error probability and the normalized (by the volume N of the moving window) delay time in change-point detection. Asymptotic optimality of the proposed method is proved in Theorem 5.1. Experimental study includes Monte Carlo examples of sequentially detected changes in coefficients of linear and nonlinear state-space models.

Recent massive increase of the computational power has allowed to rebirth of Monte Carlo integration and its application of Bayesian filtering, or particle filters. Particle filters evaluate a posterior probability distribution of the state variable based on observations in Monte Carlo simulation using so-called importance sampling. However, the filter performance is deteriorated by degeneracy phenomena in the importance weights. Recognizing the similarities and the difference of the processes between the particle filters and evolution strategies, an evolutionary computation approaches, a novel filter called the evolution strategies based particle filter (ESP) has been proposed to circumvent this difficulty and to improve the performance. Here, the ESP filter is applied to fault detection of nonlinear stochastic state space models. Its applicability is exemplified by numerical simulation studies

The restricted mean time in favor (RMT-IF) of treatment is a nonparametric effect size for complex life history data. It is defined as the net average time the treated spend in a more favorable state than the untreated over a prespecified time window. It generalizes the familiar restricted mean survival time (RMST) from the two-state life-death model to account for intermediate stages in disease progression. The overall estimand can be additively decomposed into stage-wise effects, with the standard RMST as a component. Alternate expressions of the overall and stage-wise estimands as integrals of the marginal survival functions for a sequence of landmark transitioning events allow them to be easily estimated by plug-in Kaplan-Meier estimators. The dynamic profile of the estimated treatment effects as a function of follow-up time can be visualized using a multilayer, cone-shaped "bouquet plot." Simulation studies under realistic settings show that the RMT-IF meaningfully and accurately quantifies the treatment effect and outperforms traditional tests on time to the first event in statistical efficiency thanks to its fuller utilization of patient data. The new methods are illustrated on a colon cancer trial with relapse and death as outcomes and a cardiovascular trial with recurrent hospitalizations and death as outcomes. The R-package rmt implements the proposed methodology and is publicly available from the Comprehensive R Archive Network (CRAN).

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