Nonstationary Stochastic Systems Research Articles

Synthesis of Nonlinear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

The article is devoted to Bayes optimization problems of nonlinear observable stochastic systems (NLOStSs) based on wavelet canonical expansions (WLCEs). Input stochastic processes (StPs) and output StPs of considered nonlinearly StSs depend on random parameters and additive independent Gaussian noises. For stochastic synthesis we use a Bayes approach with the given loss function and minimum risk condition. WLCEs are formed by covariance function expansion coefficients of two-dimensional orthonormal basis of wavelet with a compact carrier. New results: (i) a common Bayes’ criteria synthesis algorithm for NLOStSs by WLCE is presented; (ii) partial synthesis algorithms for three of Bayes’ criteria (minimum mean square error, damage accumulation and probability of error exit outside the limits) are given; (iii) an approximate algorithm based on statistical linearization; (iv) three test examples. Applications: wavelet optimization and parameter calibration in complex measurement and control systems. Some generalizations are formulated.

Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

This article is devoted to analysis and optimization problems of stochastic systems based on wavelet canonical expansions. Basic new results: (i) for general Bayes criteria, a method of synthesized methodological support and a software tool for nonstationary normal (Gaussian) linear observable stochastic systems by Haar wavelet canonical expansions are presented; (ii) a method of synthesis of a linear optimal observable system for criterion of the maximal probability that a signal will not exceed a particular value in absolute magnitude is given. Applications: wavelet model building of essentially nonstationary stochastic processes and parameters calibration.

Software tools for analysis and synthesis of stochastic systems with high availability (XII)

The article proceeds the thematic cycle dedicated to analysis and synthesis for mean square criteria software tools for vector nonstationary (e.g. shock disturbances) linear stochastic systems with high availability (StSHA). Methodological support is based on Haar wavelet expansions (WLE) and wavelet canonical expansions (WLCE). Sect.1 is dedicated to problems statements. General basic algorithm is given in Sect. 2. Algorithms for StSHA with linear parameters and additive noises based on WLE and WLCE are described in Sect.3. Software tools in MATLAB are considered in Sect. 4. For typical shock disturbances bank of algorithms is developed. Complex of examples is given in Sect.5. Some generalizations are presented.

A Dynamic Systems Approach for Detecting and Localizing of Infarct-Related Artery in Acute Myocardial Infarction Using Compressed Paper-Based Electrocardiogram (ECG).

Timely evaluation and reperfusion have improved the myocardial salvage and the subsequent recovery rate of the patients hospitalized with acute myocardial infarction (MI). Long waiting time and time-consuming procedures of in-hospital diagnostic testing severely affect the timeliness. We present a Poincare pattern ensemble-based method with the consideration of multi-correlated non-stationary stochastic system dynamics to localize the infarct-related artery (IRA) in acute MI by fully harnessing information from paper-based Electrocardiogram (ECG). The vectorcardiogram (VCG) diagnostic features extracted from only 2.5-s long paper ECG recordings were used to hierarchically localize the IRA—not mere localization of the infarcted cardiac tissues—in acute MI. Paper ECG records and angiograms of 106 acute MI patients collected at the Heart Artery and Vein Center at Fresno California and the 12-lead ECG signals from the Physionet PTB online database were employed to validate the proposed approach. We reported the overall accuracies of 97.41% for healthy control (HC) vs. MI, 89.41 ± 9.89 for left and right culprit arteries vs. others, 88.2 ± 11.6 for left main arteries vs. right-coronary-ascending (RCA) and 93.67 ± 4.89 for left-anterior-descending (LAD) vs. left-circumflex (LCX). The IRA localization from paper ECG can be used to timely triage the patients with acute coronary syndromes to the percutaneous coronary intervention facilities.

On Problem of Stability in Probability for "Partial" Equilibrium Positions of Nonlinear Stochastic Systems

The stability problems naturally arise in applications either from the requirement of proper performance of a system or in assessing system capability. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: adaptive stabilization; spacecraft stabilization (especially stabilization by rotors); drift of the gyroscope axis; Lotka-Volterra ecological principle, e.t.c. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of stability. The article studies the problem of stability for nonlinear stochastic systems of differential equations Ito: stability with respect to a part of the variables in probability of partial zero equilibrium position. Initial perturbations of variables that do not define the given equilibrium position can be large (belonging to an arbitrary compact set) with respect to one part of the variables and arbitrary with respect to their other part. A conditions of stability of this type are obtained in the context of a stochastic analog of the Lyapunov functions method, which generalize a number of existing results. Example is given. The problem of unification of process of studying stability problems of stationary and non-stationary nonlinear stochastic systems of differential equations is also discussed.

Design of optimal state controller robust to external disturbance for one class of nonstationary stochastic systems

Consideration was given to the problem of optimal control of a stochastic dynamic system subject to uncontrollable stochastic disturbances. The system is defined over a finite time interval, and its diffusion coefficient depends linearly on the state vector, control signal, and external disturbance. The feedback controller was assumed to be nonstationary, and linear in the state vector. A law of optimal control robust to the external disturbance was established.

New Approach to Noncausal Identification of Nonstationary Stochastic FIR Systems Subject to Both Smooth and Abrupt Parameter Changes

In this technical note, we consider the problem of finite-interval parameter smoothing for a class of nonstationary linear stochastic systems subject to both smooth and abrupt parameter changes. The proposed parallel estimation scheme combines the estimates yielded by several exponentially weighted basis function algorithms. The resulting smoother automatically adjusts its smoothing bandwidth to the type and rate of nonstationarity of the identified system. It also allows one to account for the distribution of the measurement noise.

Locally-adaptive Kalman smoothing approach to identification of nonstationary stochastic systems

The problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data, is considered. It is shown how several competing Kalman-type parameter smoothers, differing in smoothness constraints and memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. It also allows one to account for parameter jumps and for the distribution of measurement noise.

Non-stationary forward flux sampling

We present a method, Non-Stationary Forward Flux Sampling, that allows efficient simulation of rare events in both stationary and non-stationary stochastic systems. The method uses stochastic branching and pruning to achieve uniform sampling of trajectories in phase space and time, leading to accurate estimates for time-dependent switching propensities and time-dependent phase space probability densities. It is suitable for equilibrium or non-equilibrium systems, in or out of stationary state, including non-Markovian or externally driven systems. We demonstrate the validity of the technique by applying it to a one-dimensional barrier crossing problem that can be solved exactly, and show its usefulness by applying it to the time-dependent switching of a genetic toggle switch.

Dynamic Bayesian modelling of non-stationary stochastic systems using constrained least square estimation and gradient descent optimisation

A dynamic Bayesian network (DBN) is a statistical tool particularly for representing stochastic casual systems using probability and graph theories. The most important procedure in constructing a DBN is selecting the best parameter vector given as conditional probability distribution through a proper learning algorithm. This study presents a novel parameter learning methodology for Markov chain (MC) and hidden Markov model (HMM) DBN using the constrained least square method and the gradient descent optimisation, respectively. The former is employed for satisfying the probability axiom in an MC model and the latter is applied to derive adjustment rules for HMM parameters. The authors primitively assume that an observation probability vector is necessarily predefined prior to applying of the proposed learning algorithm for both models. Simulation experiment is achieved to test their learning algorithm for modelling non-stationary stochastic systems. The authors additionally provide qualitative comparative study with recently addressed learning methodologies of DBN models.

First Passage Problems for Nonstationary Discrete-Time Stochastic Control Systems

This paper concerns first passage problems for nonstationary nonlinear discrete-time stochastic control systems. The state and control spaces are general Borel spaces, the transition probabilities are nonstationary, the costs/rewards are time-varying and may be unbounded. The optimal control problem is to minimize the expected discounted costs incurred until the first passage time to some target set, in which the discount factors are allowed to be both time- and state-dependent. Under reasonably mild conditions we establish the so-called first passage optimality equations, and prove that the optimal cost/reward functions satisfy the optimality equations. Furthermore, from the optimality equations we show the existence of optimal Markov policies.

Locally Adaptive Cooperative Kalman Smoothing and Its Application to Identification of Nonstationary Stochastic Systems

One of the central problems of the stochastic approximation theory is the proper adjustment of the smoothing algorithm to the unknown, and possibly time-varying, rate and mode of variation of the estimated signals/parameters. In this paper we propose a novel locally adaptive parallel estimation scheme which can be used to solve the problem of fixed-interval Kalman smoothing in the presence of model uncertainty. The proposed solution is based on the idea of cooperative smoothing-the Bayesian extension of the leave-one-out cross-validation approach to model selection. Within this approach the smoothed estimates are evaluated as a convex combination of the estimates provided by several competing smoothers. We derive computationally attractive algorithms allowing for cooperative Kalman smoothing and show how the proposed approach can be applied to identification of nonstationary stochastic systems.

On noncausal weighted least squares identification of nonstationary stochastic systems

In this paper, we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted (windowed) least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. We optimize the window shape for a certain class of parameter variations and we derive computationally attractive recursive smoothing algorithms for such an optimized case.

Nonstationary discrete-time deterministic and stochastic control systems: Bounded and unbounded cases

This paper is about nonstationary nonlinear discrete-time deterministic and stochastic control systems with Borel state and control spaces, with either bounded or unbounded costs. The control problem is to minimize an infinite-horizon total cost performance index. Using dynamic programming arguments we show that, under suitable assumptions, the optimal cost functions satisfy optimality equations, which in turn give a procedure to find optimal control policies. We also prove the convergence of value iteration (or successive approximations) functions. Several examples illustrate our results under different sets of assumptions.

On noncausal identification of nonstationary stochastic systems

AbstractIn this paper we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. It also allows one to account for the distribution of measurement noise, and in particular – to cope with heavy-tailed disturbances, such as Laplacian noise, or light-tailed disturbances, such as uniform noise.

Nonstationary discrete-time deterministic and stochastic control systems with infinite horizon

This article is about nonstationary nonlinear discrete-time deterministic and stochastic control systems with Borel state and control spaces, possibly noncompact control constraint sets, and unbounded costs. The control problem is to minimise an infinite-horizon total cost performance index. Using dynamic programming arguments we show that, under suitable assumptions, the optimal cost functions satisfy optimality equations, which in turn give a procedure to find optimal control policies.

An investigation of setup instability in non-stationary stochastic inventory systems

In stochastic inventory systems unfolding uncertainties in demand lead to the revision of earlier replenishment plans which in turn results in an instability or so-called system nervousness. In this paper, we provide the grounds for measuring system nervousness in non-stationary demand environments, and gauge the stability and the cost performances of ( R, S) and ( s, S) inventory policies. Our results reveal that, both the stability and the cost performance of inventory policies are affected by the demand pattern as well as the cost parameters, and the ( R, S) policy has the potential to replace the cost-optimal ( s, S) policy for systems with limited flexibility.

Data-based mechanistic modeling, forecasting, and control

This article briefly reviews the main aspects of the generic data based mechanistic (DBM) approach to modeling stochastic dynamic systems and shown how it is being applied to the analysis, forecasting, and control of environmental and agricultural systems. The advantages of this inductive approach to modeling lie in its wide range of applicability. It can be used to model linear, nonstationary, and nonlinear stochastic systems, and its exploitation of recursive estimation means that the modeling results are useful for both online and offline applications. To demonstrate the practical utility of the various methodological tools that underpin the DBM approach, the article also outlines several typical, practical examples in the area of environmental and agricultural systems analysis, where DBM models have formed the basis for simulation model reduction, control system design, and forecasting.

Gauge theory of self-similar system

On the basis of a dilatation invariant Lagrangian, equations governing probability density and gauge potential of the non-stationary self-similar stochastic system are determined. It is shown that an automodel regime is realized at small time interval determined by the Tsallis’ parameter q>1. An exponential decay occurs at large time where the dilatation parameter and the partial scale tend to constant values.

Recursive Estimation and Modelling of Nonstationary and Nonlinear Time-Series

This paper presents a unified approach to nonlinear and nonstationary time-series analysis for a fairly wide class of linear time variable parameter (TVP) or nonlinear systems. The method theory exploits recursive filtering and fixed interval smoothing algorithms to derive TVP linear model approximations to the nonlinear or nonstationary stochastic system, on the basis of data obtained from the system during planned experiments or passive monitoring exercises. This TVP model includes the State Dependent type of Model (SDM) as a special case, and two particular SDM forms, due to Priestley and Young, are discussed in detail. The paper concludes with three practical examples: the first based on the modelling of data from a simulated nonlinear growth equation ; the second concerned with the adaptive forecasting and smoothing of the Box-Jenkins Airline Passenger data; and the third providing a critical appraisal of state dependent modelling applied to the famous Sunspot time-series.