Stochastic State Space Research Articles

Parameter identification and uncertainty quantification in stochastic state space models and its application to texture analysis

In this paper, a computational framework, which enables efficient and robust parameter identification, as well as uncertainty quantification in state space models based on Itô stochastic processes, is presented. For optimization, a Maximum Likelihood approach based on the system's corresponding Fokker-Planck equation is followed. Gradient information is included by means of an adjoint approach, which is based on the Lagrangian of the optimization problem. To quantify the uncertainty of the Maximum-A-Posteriori estimates of the model parameters, a Bayesian inference approach based on Markov Chain Monte Carlo simulations, as well as profile likelihoods are implemented and compared in terms of runtime and accuracy. The framework is applied to experimental electron backscatter diffraction data of a fatigued metal film, where the aim is to develop a model, which consistently and physically meaningfully captures the metal's microstructural changes that are caused by external loading.

Research on dynamic nonlinear input prediction of fault diagnosis based on fractional differential operator equation in high-speed train control system.

In order to control the nonlinear high-speed train with high robustness, the fractional order control of nonlinear switching systems is studied. The fractional order controller is designed for a class of nonlinear switching systems by the fractional order backstepping method. In this paper, a simple and effective online updating scheme of model coefficients is proposed by using the flexibility of the model predictive control algorithm and its wide range of model accommodation. A stochastic discrete nonlinear state space model describing the mechanical behavior of a single particle in a high-speed train is constructed, and the maximum likelihood estimation of the parameters of a high-speed train is transformed into an optimization problem with great expectations. Finally, numerical comparison experiments of motion characters of two high-speed trains are given. The results show the effectiveness of the proposed identification method.

Stability Analysis of the Bat Algorithm Described as a Stochastic Discrete-Time State-Space System

The main problem with the soft-computing algorithms is a determination of their parameters. The tuning rules are very general and need experiments during a trial and error method. The equations describing the bat algorithm have the form of difference equations, and the algorithm can be treated as a stochastic discrete-time system. The behaviour of this system depends on its dynamic and preservation stability conditions. The paper presents the stability analysis of the bat algorithm described as a stochastic discrete-time state-space system. The observability and controllability analyses were made in order to verify the correctness of the model describing the dynamic of BA. Sufficient conditions for stability are derived based on the Lyapunov stability theory. They indicate the recommended areas of the location of the parameters. The analysis of the position of eigenvalues of the state matrix shows how the different values of parameters affect the behaviour of the algorithm. They indicate the recommended area of the location of the parameters. Simulation results confirm the theory-based analysis.

Kalman filter-based centralized controller design for non-square multi-input multi-output processes

In this paper, a simple and effective centralized controller is presented to control non-square multi-input multi-output industrial processes with heavy interactions and significant time-delays. The control objective is to achieve a good load disturbance rejection and set point tracking. The controller and process model equations are represented in a stochastic state-space. Also, the state and measurement equations respectively use general random walk model and finite impulse response model of the plant. To design the controller, a quadratic cost function including the control error vector as well as the input vector and its changes is introduced. A Kalman filter algorithm is used to solve the discrete algebraic Riccati equation and to estimate the state vector of the controller recursively. This novel use of Kalman filter combines the observer and controller in a block. No square down/up, pairing and pseudo-inversing, is required for the proposed control structure. This leads to the simplicity of controller design. To evaluate the performance of the proposed control strategy, it is applied to a simulation example and an industrial case study with heavy interactions and significant time delays, an atmospheric distillation column with 7 inputs and 4 outputs. Simulation results show that the proposed controller has a good performance in both load disturbance rejection and setpoint tracking.

Stochastic Thermodynamics: A Dynamical Systems Approach

In this paper, we develop an energy-based, large-scale dynamical system model driven by Markov diffusion processes to present a unified framework for statistical thermodynamics predicated on a stochastic dynamical systems formalism. Specifically, using a stochastic state space formulation, we develop a nonlinear stochastic compartmental dynamical system model characterized by energy conservation laws that is consistent with statistical thermodynamic principles. In particular, we show that the difference between the average supplied system energy and the average stored system energy for our stochastic thermodynamic model is a martingale with respect to the system filtration. In addition, we show that the average stored system energy is equal to the mean energy that can be extracted from the system and the mean energy that can be delivered to the system in order to transfer it from a zero energy level to an arbitrary nonempty subset in the state space over a finite stopping time.

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.

Accurate State Estimation in Continuous-Discrete Stochastic State-Space Systems With Nonlinear or Nondifferentiable Observations

This paper presents a novel method of nonlinear Kalman filtering, which unifies the best features of the accurate continuous-discrete extended and cubature Kalman filters. More precisely, the time updates in the discussed state estimator are done as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All this allows accurate predictions of the state mean and covariance matrix to be combined with accurate measurement updates. Moreover, the new filter is particularly effective for continuous-discrete stochastic systems with nonlinear and/or nondifferentiable observations. The efficiency of this mixed-type method is shown in comparison to the performance of the original accurate continuous-discrete extended and cubature Kalman filters in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, with sufficiently long sampling times.

An Approach for Resiliency Quantification of Large Scale Systems

We quantify the resiliency of large scale systems upon changes encountered beyond the normal system behavior. Formal definitions for resiliency and change are provided together with general steps for resiliency quantification and a set of resiliency metrics that can be used to quantify the effects of changes. A formalization of the approach is also shown in the form of a set of four algorithms that can be applied when large scale systems are modeled through stochastic analytic state space models (monolithic models or interacting sub-models). In particular, in the case of interacting submodels, since resiliency quantification involves understanding the transient behavior of the system, fixed-point variables evolve with time leading to non-homogenous Markov chains. At the best of our knowledge, this is the first paper facing this problem in a general way. The proposed approach is applied to an Infrastructure-as-a-Service (IaaS) Cloud use case. Specifically, we assess the impact of changes in demand and available capacity on the Cloud resiliency and we show that the approach proposed in this paper can scale for a real sized Cloud without significantly compromising the accuracy.

State Estimation of Macromotion Positioning Tables Based on Switching Kalman Filter

In this brief, a switching Kalman filtering (SKF) method is proposed for the state estimation of macromotion positioning tables disturbed by random noises. In this method, the macromotion positioning tables working in noisy environment are described by the so-called stochastic sandwich models with a dead zone. In this scheme, a nonsmooth stochastic state space model is constructed first by introducing several embedded switch functions. These functions are used to describe the effect of a dead zone. Then, an SKF is developed based on the obtained nonsmooth stochastic state space model. The operating states of this filter can be switched automatically among different operating zones according to the change of system operating conditions. Moreover, the convergence of proposed switching filtering method is discussed. Then, the proposed SKF method is implemented to an X–Y macromotion positioning table for state estimation in a noisy case.

Change Point Detection and Estimation of the Two-Sided Jumps of Asset Returns Using a Modified Kalman Filter

In the first part of the paper, the positive and negative jumps of NASDAQ daily (log-) returns and three of its stocks are estimated based on the methodology presented by Theodosiadou et al. 2016, where jumps are assumed to be hidden random variables. For that reason, the use of stochastic state space models in discrete time is adopted. The daily return is expressed as the difference between the two-sided jumps under noise inclusion, and the recursive Kalman filter algorithm is used in order to estimate them. Since the estimated jumps have to be non-negative, the associated pdf truncation method, according to the non-negativity constraints, is applied. In order to overcome the resulting underestimation of the empirical time series, a scaling procedure follows the stage of truncation. In the second part of the paper, a nonparametric change point analysis concerning the (variance–) covariance is applied to the NASDAQ return time series, as well as to the estimated bivariate jump time series derived after the scaling procedure and to each jump component separately. A similar change point analysis is applied to the three other stocks of the NASDAQ index.

Stochastic wave finite element method in uncertain elastic media through the second order perturbation

In this work, the stochastic wave finite element (SWFE) method for uncertain media through the second-order perturbation is formulated. A parametric approach for uncertainties is considered and combined to the finite element technique. The stochastic state space formulation is detailed in this work. The originality of this paper is the study of the second-order perturbation. The sensitivity and the precision of the SWFE approach are treated through the second-order perturbation introduced in the structural parameters. The question of the statistics of the propagation constants and the wave modes is considered. Comparisons with analytical results and Monte Carlo simulations are performed.

Introduction of a new continuous time and state space stochastic process in condition prediction

Periodic condition assessments of pavements together with condition predictions are the basis for investment decisions in every pavement management system (PMS). Typical approaches include surveys of distress types every 3–6 years with analysis rating and calculation of condition indices for road safety and/or structural health. Furthermore, advanced PMS prediction models allow a comparison of maintenance alternatives and an optimisation of investment strategies. This paper presents an overview of current survey and rating approaches in Germany, Switzerland and Austria, together with an impact analysis of different methods, utilised deterministic performance functions and condition threshold (trigger) values for all major distress types. The core of this paper is a comparison of common deterministic condition prediction models with discrete stochastic approaches and prediction models based on advanced regression techniques mainly from scientific literature and an innovative stochastic continuous time and continuous state space process (HOFFMANN – Process). All prediction models are applied to real-world data from condition surveys in Austria and the long-term pavement performance Database (USA) at single-section and network level. The paper provides evidence why deterministic prediction approaches are leading to substantial bias in condition distribution and remaining service life as they do not account for the stochastic nature of pavements. Classic Markov-chain approaches do not account for censoring of survey data and neglect changes in transition probabilities with increasing age. Applying common bivariate and multiple regression techniques may also lead to certain bias due to collinearity effects and specification bias. The paper provides mathematical evidence on ways to avoid these shortcomings based on the presented innovative stochastic process leading to a higher reliability in condition assessment, rating and accuracy of condition predictions. The aspects of censoring, distress-specific assignment and optimisation of treatments with this new HOFFMANN-process will be covered in forthcoming papers.

System Entropy Measurement of Stochastic Partial Differential Systems

System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential dynamic equations in the spatio-temporal domain. No efficient method currently exists that can calculate the system entropy of stochastic partial differential systems (SPDSs) in consideration of the effects of intrinsic random fluctuation and compartment diffusion. In this study, a novel indirect measurement method is proposed for calculating of system entropy of SPDSs using a Hamilton–Jacobi integral inequality (HJII)-constrained optimization method. In other words, we solve a nonlinear HJII-constrained optimization problem for measuring the system entropy of nonlinear stochastic partial differential systems (NSPDSs). To simplify the system entropy measurement of NSPDSs, the global linearization technique and finite difference scheme were employed to approximate the nonlinear stochastic spatial state space system. This allows the nonlinear HJII-constrained optimization problem for the system entropy measurement to be transformed to an equivalent linear matrix inequalities (LMIs)-constrained optimization problem, which can be easily solved using the MATLAB LMI-toolbox (MATLAB R2014a, version 8.3). Finally, several examples are presented to illustrate the system entropy measurement of SPDSs.

Natural Resource Management for Nonlinear Stochastic Biotic-Abiotic Ecosystems: Robust Reference Tracking Control Strategy Using Limited Set of Controllers

The management of natural resources has become an important subject because of the increasing population. To meet human requirements for resources such as food and water, many control strategies have been proposed to improve biomass production and nutrient supply. However, implementing these strategies may be limited by the expense of large numbers of control devices and by the effects of stochastic perturbations. In this study, a robust reference tracking control strategy for the natural resource management of nonlinear stochastic biotic-abiotic ecosystems is proposed, using a limited set of controllers, to regulate the systematic dynamics achieving a desired reference trajectory under the influence of intrinsic fluctuations and environmental disturbances. To simplify the design procedure and make the robust reference tracking control strategy more feasible, we propose a fuzzy stochastic partial differrential equations system to represent the ecosystem, itself approximated by a fuzzy stochastic spatial state space model, based on a finite difference scheme. This allows replacement of a complex Hamilton-Jacobi integral inequality by an equivalent set of local linear matrix inequalities which can be easily solved. We verify the efficiency of the proposed approach using a nonlinear stochastic biomass-nutrient control example and compare the robust tracking control performance for different arrangements of control devices. Managers may select appropriate tracking control schemes based on this comparison. The robust reference tracking control strategy can be applied not only to agricultural systems but also to biophysical systems in ecological conservation, ecosystem restoration or engineering.

Dynamic Forecasting and Control Algorithms of Glaucoma Progression for Clinician Decision Support

In managing chronic diseases such as glaucoma, the timing of periodic examinations is crucial, as it may significantly impact patients’ outcomes. We address the question of when to monitor a glaucoma patient by integrating a dynamic, stochastic state space system model of disease evolution with novel optimization approaches to predict the likelihood of progression at any future time. Information about each patient’s disease state is learned sequentially through a series of noisy medical tests. This information is used to determine the best time to next test based on each patient’s individual disease trajectory as well as population information. We develop closed-form solutions and study structural properties of our algorithm. While some have proposed that fixed-interval monitoring can be improved upon, our methodology validates a sophisticated model-based approach to doing so. Based on data from two large-scale, 10+ years clinical trials, we show that our methods significantly outperform fixed-interval schedules and age-based threshold policies by achieving greater accuracy of identifying progression with fewer examinations. Although this work is motivated by our collaboration with glaucoma specialists, the methodology developed is applicable to a variety of chronic diseases.

State-Space Approach to Structural Representation of Perturbed Pitch Period Sequences in Voice Signals

The aim of this study was to propose a state space-based approach to model perturbed pitch period sequences (PPSs), extracted from real sustained vowels, combining the principal features of disturbed real PPSs with structural analysis of stochastic time series and state space methods. The PPSs were obtained from a database composed of 53 healthy subjects. State space models were developed taking into account different structures and complexity levels. PPS features such as trend, cycle, and irregular structures were considered. Model parameters were calculated using optimization procedures. For each PPS, state estimates were obtained combining the developed models and diffuse initialization with filtering and smoothing methods. Statistical tests were applied to objectively evaluate the performance of this method. Statistical tests demonstrated that the proposed approach correctly represented more than the 75% of the database with a significance value of 0.05. In the analysis, structural estimates suitably characterized the dynamics of the PPSs. Trend estimates proved to properly represent slow long-term dynamics, whereas cycle estimates captured short-term autoregressive dependencies. The present study demonstrated that the proposed approach is suitable for representing and analyzing real perturbed PPSs, also allowing to extract further information related to the phonation process.

Robust maximum likelihood estimation for stochastic state space model with observation outliers

The objective of this paper is to develop a robust maximum likelihood estimation (MLE) for the stochastic state space model via the expectation maximisation algorithm to cope with observation outliers. Two types of outliers and their influence are studied in this paper: namely,the additive outlier (AO) and innovative outlier (IO). Due to the sensitivity of the MLE to AO and IO, we propose two techniques for robustifying the MLE: the weighted maximum likelihood estimation (WMLE) and the trimmed maximum likelihood estimation (TMLE). The WMLE is easy to implement with weights estimated from the data; however, it is still sensitive to IO and a patch of AO outliers. On the other hand, the TMLE is reduced to a combinatorial optimisation problem and hard to implement but it is efficient to both types of outliers presented here. To overcome the difficulty, we apply the parallel randomised algorithm that has a low computational cost. A Monte Carlo simulation result shows the efficiency of the proposed algorithms.

A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction

—Time series models provide a powerful tool to extract nonstationary features from measured data. In this article, a statistical framework based upon a dynamic harmonic regression model for examining modal behavior is provided. In this model, temporal patterns in measured data are modeled within a stochastic state space setting. Estimates of the states or time-varying parameters are then obtained using an optimal estimation method based on the Kalman filter. Techniques to estimate future values of the unobserved signal are also analyzed. The widely applicable technique is illustrated on both simulated and measured data. Factors that affect the performance of the method are discussed, including the effects of non-linear trends, data quality, and sampling design. Connections with other modal identification methods are also investigated.

Motion artifact reduction in PPG signals from face: face tracking & stochastic state space modeling approach.

The Photoplethymography(PPG) is generally measured on a finger or an ear using contact sensors. The recent several studies using non-contact sensor such as CCD camera and web-cam to measure PPG have been introduced. However the motion artifact issue is also emerging in non-contact camera sensing similar to contact-type one because it is sensitive to artifacts generated by subject's head and body motion. In this paper, the two sequential approaches for a motion artifact reduction algorithm are presented; the one is a face tracking method that detects and tracks the skin region of face which is containing PPG signals, the other is the reduction method of motion artifact due to various head & face movement such as roll, yaw, pitch, translation and scale. Results of the proposed KF are compared to these of the FIR band pass filter(BPF).

Comparison of the performance of particle filter algorithms applied to tracking of a disease epidemic

We present general methodology for sequential inference in nonlinear stochastic state-space models to simultaneously estimate dynamic states and fixed parameters. We show that basic particle filters may fail due to degeneracy in fixed parameter estimation and suggest the use of a kernel density approximation to the filtered distribution of the fixed parameters to allow the fixed parameters to regenerate. In addition, we show that “seemingly” uninformative uniform priors on fixed parameters can affect posterior inferences and suggest the use of priors bounded only by the support of the parameter. We show the negative impact of using multinomial resampling and suggest the use of either stratified or residual resampling within the particle filter. As a motivating example, we use a model for tracking and prediction of a disease outbreak via a syndromic surveillance system. Finally, we use this improved particle filtering methodology to relax prior assumptions on model parameters yet still provide reasonable estimates for model parameters and disease states.