Stochastic Nonlinear State-space Models Research Articles

Deep Neural Network-Embedded Stochastic Nonlinear State-Space Models and Their Applications to Process Monitoring.

Process complexities are characterized by strong nonlinearities, dynamics, and uncertainties. Monitoring such a complex process requires a high-quality model describing the corresponding nonlinear dynamic behavior. The proposed model is constructed using deep neural networks (DNNs) to represent the state transition and observation generation, both of which constitute a stochastic nonlinear state-space model. A new bidirectional recurrent neural network (RNN), creating a connection of the hidden layer between a forward RNN and a backward RNN, is proposed to generate the filtering estimation and the smoothing estimation of process states which further generate observations with DNN-based process models. The smoothing estimator and the process model are first learned offline with all collected samples. Then the filtering estimator is fine-tuned by the learned smoother and process models to achieve real-time monitoring since the filter state is estimated based on the past and the current observations. Two indices are designed based on the learned model for monitoring the process anomaly. The proposed process monitoring model can deal with complex nonlinearities, process dynamics, and process uncertainties, all of which can be very challenging for the existing methods, such as kernel mapping and stacked auto-encoder. Two case studies validate that the effectiveness of the proposed method outperforms the other comparative methods by at least 10% when using the averaged fault detection rate in the industrial experimental data.

An adaptive extended Kalman filtering approach to nonlinear dynamic gene regulatory networks via short gene expression time series

Gene regulation is one of the most amazing processes in which living cells are involved. Different genes may cooperate to produce a particular reaction if more than one genes are suppressed by a gene. The gene triggers a certain genetic disease, for instance, through inactivating the gene that is responsible for cancer, the disease produced by this gene may be controlled. Gene control and interaction might be defined by a gene regulatory network. DNA microarray technology has provided an efficient method for evaluating expression levels of thousands of genes in a single experiment. It enables to trace the levels of the expression of thousands of genes at once. Evaluating gene expression levels may be beneficial for medical diagnosis, treatment and drug design in different circumstances. With an intent to understand the effective biological data and to identify the connections between the individual genes, most of the current research efforts are focused on clustering. Recently, remodelling of the gene regulatory networks over the data on the time series has become an increasing field of interest. As a matter of fact, choosing a good model for the gene regulatory networks is required for a significant data analysis. Several gene expression experiments produce data on short time series only on a few points in time due to the high costs of the evaluation. Time series generally represent the dynamic responses to, some medicines or other treatment methods. As a consequence, modelling the time series of the gene expressions which is needed to pick the functional information over the data on time series has become an increasingly interesting field of study. Since gene regulatory networks form a naturally stochastic structure, the observed data on the time series may be modelled by using non-linear stochastic state space models and the parameters included in the model may be estimated with the Extended Kalman Filter (EKF) method. There have been many studies regarding this topic made in the literature. However, the fact that the model is nonlinear may cause some problems on the estimations of Kalman Filter (KF) method. For this reason, the researches on the adaptive methods in the EKF are going on with the aim of supporting the estimations. In this research, application of the developed model on the gene regulatory networks has been examined. With the aim of corroborating estimation method, it has been decided that the adaptive extended Kalman Filter (AEKF) was proper for being used and malaria gene expression has been applied for the set of data on the time series. A results have been compared with the results of the former research [1], and it has been understood that the estimation results obtained through the developed model were more preferable.

Estimation and identification in batch processes with particle filters

This paper deals with the problem of real-time estimation and identification in batch processes represented by stochastic nonlinear state-space models (SSMs). To address this, the two problems – estimation and identification – are formulated under a common Bayesian filtering framework. In contrast to a continuous process, a batch process is typically operated for a relatively shorter duration, which limits the number of measurements available in a campaign to effectively solve the filtering problem. To overcome the limitations of existing solutions, we propose a two-dimensional dual particle filtering algorithm to solve the nonlinear filtering problem. The proposed algorithm performs filtering both along the time and the batch directions. This allows to combine measurements from multiple short campaigns for improved estimation and identification results. The efficacy of the proposed method is demonstrated on two simulation examples.

Joint Parameter and State Estimation Based on Marginal Particle Filter and Particle Swarm Optimization

In this paper, a method for the dual estimation is proposed. This approach combines extended marginal particle filter (EMPF) with particle swarm optimization (PSO) for simultaneous estimation of state and parameter values in nonlinear stochastic state–space models. In the proposed method, the states are estimated by EMPF and the parameters are estimated by PSO. The performance of proposed algorithm is evaluated in two examples. Simulation results demonstrate the feasibility and efficiency of the proposed method.

Bayesian identification of non-linear state-space models: Part II- Error analysis

In the last two decades, several methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) have been proposed for Bayesian identification of stochastic non-linear state-space models (SSMs). It is well known that the performance of these simulation based identification methods depends on the numerical approximations used in their design. We propose the use of posterior Cramér-Rao lower bound (PCRLB) as a mean square error (MSE) bound. Using PCRLB, a systematic procedure is developed to analyse the estimates delivered by Bayesian identification methods in terms of bias, MSE, and efficiency. The efficacy and utility of the proposed approach is illustrated through a numerical example.

Bayesian identification of non-linear state-space models: Part I- Input design

We propose an algorithm for designing optimal inputs for on-line Bayesian identification of stochastic non-linear state-space models. The proposed method relies on minimization of the posterior Cramér Rao lower bound derived for the model parameters, with respect to the input sequence. To render the optimization problem computationally tractable, the inputs are parametrized as a multi-dimensional Markov chain in the input space. The proposed approach is illustrated through a simulation example.

Fault detection and isolation in stochastic non-linear state-space models using particle filters

A novel model-based algorithm for fault detection and isolation (FDI) in stochastic non-linear systems is proposed. The algorithm monitors changes in the process behavior and identifies a corresponding fault using a bank of particle filters running in parallel. The particle filters are used to generate a sequence of hidden states which are then used in a log-likelihood ratio to detect and isolate the faults. The approach is demonstrated through an implementation on two highly nonlinear case studies—a multi-unit chemical reactor system and a polyethylene reactor system. The effectiveness and the robustness of the proposed algorithm are illustrated by comparing the results with FDI techniques that use EKF and UKF state estimators instead of particle filters.

Nonparametric particle filtering approaches for identification and inference in nonlinear state-space dynamic systems

Most system identification approaches and statistical inference methods rely on the availability of the analytic knowledge of the probability distribution function of the system output variables. In the case of dynamic systems modelled by hidden Markov chains or stochastic nonlinear state-space models, these distributions as well as that of the state variables themselves, can be unknown or untractable. In that situation, the usual particle Monte Carlo filters for system identification or likelihood-based inference and model selection methods have to rely, whenever possible, on some hazardous approximations and are often at risk. This review shows how a recent nonparametric particle filtering approach can be efficiently used in that context, not only for consistent filtering of these systems but also to restore these statistical inference methods, allowing, for example, consistent particle estimation of Bayes factors or the generalisation of model parameter change detection sequential tests. Real-life applications of these particle approaches to a microbiological growth model are proposed as illustrations.

Sequential Monte Carlo methods for parameter estimation in nonlinear state-space models

Stochastic nonlinear state-space models (SSMs) are prototypical mathematical models in geoscience. Estimating unknown parameters in nonlinear SSMs is an important issue for environmental modeling. In this paper, we present two recently developed methods that are based on the sequential Monte Carlo (SMC) method for parameter estimation in nonlinear SSMs. The first method, which belongs to classical statistics, is the SMC-based maximum likelihood estimation. The second method, belonging to Bayesian statistics, is Particle Markov Chain Monte Carlo (PMCMC). With a low-dimensional nonlinear SSM, the implementations of the two methods are demonstrated. It is concluded that these SMC-based parameter estimation methods are applicable to environmental modeling and geoscience.

On Gaussian Optimal Smoothing of Non-Linear State Space Models

In this note we shall present a new Gaussian approximation based framework for approximate optimal smoothing of non-linear stochastic state space models. The approximation framework can be used for efficiently solving non-linear fixed-interval, fixed-point and fixed-lag optimal smoothing problems. We shall also numerically compare accuracies of approximations, which are based on Taylor series expansion, unscented transformation, central differences and Gauss-Hermite quadrature.

A comparison of simultaneous state and parameter estimation schemes for a continuous fermentor reactor

This article proposes a maximum likelihood algorithm for simultaneous estimation of state and parameter values in nonlinear stochastic state-space models. The proposed algorithm uses a combination of expectation maximization, nonlinear filtering and smoothing algorithms. The algorithm is tested with three popular techniques for filtering namely particle filter (PF), unscented Kalman filter (UKF) and extended Kalman filter (EKF). It is shown that the proposed algorithm when used in conjunction with UKF is computationally more efficient and provides better estimates. An online recursive algorithm based on nonlinear filtering theory is also derived and is shown to perform equally well with UKF and ensemble Kalman filter (EnKF) algorithms. A continuous fermentation reactor is used to illustrate the efficacy of batch and online versions of the proposed algorithms.

Parameter Estimation for Discrete-Time Nonlinear Systems Using EM

In this paper we consider parameter estimation of general stochastic nonlinear state-space models using the Maximum Likelihood method. This is accomplished via the employment of an Expectation Maximisation algorithm, where the essential components involve a particle smoother for the expectation step, and a gradient-based search for the maximisation step. The utility of this method is illustrated with several nonlinear and non-Gaussian examples.

Risk-sensitive filtering and smoothing via reference probability methods

We address the risk-sensitive filtering problem which is minimizing the expectation of the exponential of the squared estimation error multiplied by a risk-sensitive parameter. Such filtering can be more robust to plant and noise uncertainty than minimum error variance filtering. Although optimizing a differently formulated performance index to that of the so-called H/sub /spl infin// filtering, risk-sensitive filtering leads to a worst case deterministic noise estimation problem given from the differential game associated with H/sub /spl infin// filtering. We consider a class of discrete-time stochastic nonlinear state-space models. We present linear recursions in the information state and the result for the filtered estimate that minimizes the risk-sensitive cost index. We also present fixed-interval smoothing results for each of these signal models. In addition, a brief discussion is included on relations of the risk-sensitive estimation problem to minimum variance estimation and a worst case estimation problem in a deterministic noise scenario related to minimax dynamic games. The technique used in this paper is the so-called reference probability method which defines a new probability measure where the observations are independent and translates the problem to the new measure. The optimization problem is solved using simple estimation theory in the new measure, and the results are interpreted as solutions in the original measure.

Identification of Nonlinear Stochastic Grey Box Models: Theory, Implementation, and Experiences

Abstract Grey box identification refers to the practice of identifying dynamical systems in model structures exploiting partial prior information. If this leads to nonlinear stochastic state space models, there is a question as to whether the complexity of analysis and software implementation is justifiable in terms of improved control and/or prediction. The paper covers the following three issues. Firstly, the theoretical foundations for solving the problem are reviewed. Secondly, the implementation of a general purpose grey box identification software is reported. And thirdly, with help of two industrial applications, it is demonstrated that the complexity of the method can indeed pay off in terms of simplifying control (case study 1) or in terms of improved prediction and parameter accuracy (case study 2).