Dynamic System Parameter Estimation Research Articles

Least Squares Estimation of Multifactor Uncertain Differential Equations with Applications to the Stock Market

Multifactor uncertain differential equations are powerful tools for studying dynamic systems under multi-source noise. A key challenge in this study is how to accurately estimate unknown parameters based on the framework of uncertainty theory in multi-source noise environments. To address this core problem, this paper innovatively proposes a least-squares estimation method. The essence of this method lies in constructing statistical invariants with a symmetric uncertainty distribution based on observational data and determining specific parameters by minimizing the distance between the population distribution and the empirical distribution of the statistical invariant. Additionally, two numerical examples are provided to help readers better understand the practical operation and effectiveness of this method. In addition, we also provide a case study of JD.com’s stock prices to illustrate the advantages of the method proposed in this paper, which not only provides a new idea and method for addressing the problem of dynamic system parameter estimation but also provides a new perspective and tool for research and application in related fields.

Digital Fault Locator for Double End Fed Transmission Lines

The paper presents a digital fault locator by dynamic system parameter estimation for a double end fed transmission line. The method uses about 1/6 cycle of recorded fault data and doesn‟t require filtering of dc offset and high-frequency components. The system differential equations are based on a lumped parameter line model, Thevenin equivalents at both ends of the line and an unknown fault resistance. The accuracy is demonstrated by a representative set of tests results obtained with computer simulation

Robust Bayesian state and parameter estimation framework for stochastic dynamical systems with combined time-varying and time-invariant parameters

We consider the joint state and parameter estimation for dynamical systems having both time-varying and time-invariant parameters. For systems with time-invariant parameters, it has previously been demonstrated that although nonlinear filters may be used for concurrent state and parameter estimation, employing the Markov Chain Monte Carlo (MCMC) algorithm for estimating time-invariant parameters with nested nonlinear filters for state estimation provides more reliable estimates. Following established methods for parameter estimation using filters, we augment the state vector to include the original system states in addition to the subset of the parameters that are known to vary in time. Conventionally, both time-varying and time-invariant parameters would be appended in the state vector, and thus for the purpose of estimation, both would free to vary in time. However, allowing parameters that are known to be time-invariant to change in time introduces non-physical dynamics into the original system. To avoid inducing artificial dynamics, the time-invariant parameters are estimated using MCMC. Furthermore, by estimating the time-invariant parameters by MCMC, the dimensionality of the augmented state is smaller and the nonlinearity in the ensuing state space model will tend to be weaker compared to the conventional approach. The time-varying parameters are perturbed by random noise processes, and the amplitude of the artificial noise is estimated alongside the time-invariant system parameters using MCMC. Estimating the noise strength in this procedure circumvents the need for manual tuning. We illustrate the above-described approach for a simple dynamical system in which some model parameters are time-varying, while the remaining parameters are time-invariant. We adopt the extended Kalman filter (EKF) for state estimation and the estimation of the time-varying system parameters, but reserve the task of estimating time-invariant parameters, including the model noise strength and the artificial noise strength associated with the time-varying parameter to the MCMC algorithm.

Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises the functional tensor-train approximation of the inverse Rosenblatt transport recently developed by Dolgov et al. (Stat Comput 30:603–625, 2020) to a wide class of high-dimensional non-negative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared tensor-train decomposition which preserves the monotonicity. More crucially, we integrate the proposed order-preserving functional tensor-train transport into a nested variable transformation framework inspired by the layered structure of deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficiency of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.

Maximum likelihood-based robust state estimation over a horizon length during measurement outliers

In this paper, a novel approach towards horizon-based maximum likelihood (ML) state estimator is proposed that makes the state estimation process more robust against unmodeled and unstructured noise and disturbances in the state-space models. State space models provide a powerful way to perform state and parameter estimation for dynamical systems. However, if the measurements are contaminated by outliers and disturbances with no known models, the estimation process is highly biased. A ML-based approach is used to find a batch solution for the filtering problem. Based on ML solution, a robust algorithm is proposed that seamlessly estimates dynamic state in the presence of zero or non-zero mean measurement outliers. The proposed algorithm that dictates this switching uses an explicit outlier detection mechanism that enables its seamless working. Simulations have been carried out for a moving target tracking application as an example to demonstrate the resilience of the proposed method against zero-mean and non-zero mean measurement outliers in comparison to state-of-the-art methods.

Bayesian inference for dynamical systems.

Bayesian inference is a common method for conducting parameter estimation for dynamical systems. Despite the prevalent use of Bayesian inference for performing parameter estimation for dynamical systems, there is a need for a formalized and detailed methodology. This paper presents a comprehensive methodology for dynamical system parameter estimation using Bayesian inference and it covers utilizing different distributions, Markov Chain Monte Carlo (MCMC) sampling, obtaining credible intervals for parameters, and prediction intervals for solutions. A logistic growth example is given to illustrate the methodology.

A Novel and Computationally Efficient Joint Unscented Kalman Filtering Scheme for Parameter Estimation of a Class of Nonlinear Systems

Unscented Kalman filter (UKF) is one type of the sigma point Kalman filters and it is based on unscented transformation. UKF is used for parameter estimation of various dynamic systems and for such purpose either joint UKF (JUKF) or dual UKF (DUKF) schemes are considered. JUKF is based on estimating states and parameters together by using only one filter. For DUKF, states and parameters are decoupled and two separate filters are considered. In this paper, a modification to standard JUKF is proposed for parameter estimation which is based on decoupling parameter vector and updating parameter estimates by considering the error transformation between measurements and transformed sigma points during measurement update into the parameter errors. A linear transformation is proposed for such a purpose. Thus, the computational complexity of the standard JUKF is reduced significantly since parameters are decoupled from the state vector while the convergence of parameter estimate(s) is guaranteed. The new modified JUKF scheme is promising to be used for the parameter estimation of dynamic systems for which a linear transformation between measurement and parameter errors can be obtained. The effectiveness of this new scheme is proven by applying it to two nonlinear dynamic systems.

Model selection and parameter estimation of dynamical systems using a novel variant of approximate Bayesian computation

Abstract Model selection is a challenging problem that is of importance in many branches of the sciences and engineering, particularly in structural dynamics. By definition, it is intended to select the most likely model among a set of competing models that best matches the dynamic behaviour of a real structure and better predicts the measured data. The Bayesian approach which is based essentially on the evaluation of a likelihood function is one of the most popular approach to deal with model selection and parameter estimation issues. However, in some circumstances, the likelihood function is either intractable or not available even in a closed form. To overcome this issue, the likelihood-free or approximate Bayesian computation (ABC) algorithm has been introduced in the literature, which relaxes the need for an explicit likelihood function to measure the level of agreement between model predictions and measurements. However, ABC algorithms suffer from a low acceptance rate of samples which is actually a common problem with the traditional Bayesian methods. To overcome this shortcoming and alleviate the computational burden, a new variant of the ABC algorithm based on an ellipsoidal Nested Sampling (NS) technique is introduced in this paper; it has been called ABC-NS. Through this paper, it will be shown how the new algorithm is a promising alternative to deal with parameter estimation and model selection issues. It promises drastic speedups and provides a good approximation of the posterior distributions. To demonstrate its robust computational efficiency, four illustrative examples are given. Firstly, the efficiency of the algorithm is demonstrated to deal with parameter estimation. Secondly, two examples based on simulated and real data are given to demonstrate the efficiency of the algorithm to deal with model selection in structural dynamics.

Performance of objective functions and optimisation procedures for parameter estimation in system biology models

Mathematical modelling of signalling pathways aids experimental investigation in system and synthetic biology. Ever increasing data availability prompts the development of large dynamic models with numerous parameters. In this paper, we investigate how the number of unknown parameters affects the convergence of three frequently used optimisation algorithms and four objective functions. We compare objective functions that use data-driven normalisation of the simulations with those that use scaling factors. The data-driven normalisation of the simulation approach implies that simulations are normalised in the same way as the data, making both directly comparable. The scaling factor approach, which is commonly used for parameter estimation in dynamic systems, introduces scaling factors that multiply the simulations to convert them to the scale of the data. Here we show that the scaling factor approach increases, compared to data-driven normalisation of the simulations, the degree of practical non-identifiability, defined as the number of directions in the parameter space, along which parameters are not identifiable. Further, the results indicate that data-driven normalisation of the simulations greatly improve the speed of convergence of all tested algorithms when the overall number of unknown parameters is relatively large (74 parameters in our test problems). Data-driven normalisation of the simulations also markedly improve the performance of the non-gradient-based algorithm tested even when the number of unknown parameters is relatively small (10 parameters in our test problems). As the models and the unknown parameters increase in size, the data-driven normalisation of the simulation approach can be the preferred option, because it does not aggravate non-identifiability and allows for obtaining parameter estimates in a reasonable amount of time.

Estimating the parameters of dynamical systems from Big Data using Sequential Monte Carlo samplers

In this paper the authors present a method which facilitates computationally efficient parameter estimation of dynamical systems from a continuously growing set of measurement data. It is shown that the proposed method, which utilises Sequential Monte Carlo samplers, is guaranteed to be fully parallelisable (in contrast to Markov chain Monte Carlo methods) and can be applied to a wide variety of scenarios within structural dynamics. Its ability to allow convergence of one’s parameter estimates, as more data is analysed, sets it apart from other sequential methods (such as the particle filter).

Using Modified Intelligent Experimental Design in Parameter Estimation of Chaotic Systems

Computational modeling plays an important role in prediction and optimization of real systems and processes. Models usually have some parameters which should be set up to the proper value. Therefore, parameter estimation is known as an important part of the modeling and system identification. It usually refers to the process of using sampled data to estimate the optimum values of parameters. The accuracy of model can be increased by adjusting its parameters to the optimum value which need a richer dataset. One simple solution for having a richer dataset is increasing the amount of data, but that can be costly and time consuming. When using data from animals or people, it is especially important to have a proper plan. There are several available methods for parameter estimation in dynamical systems; however there are some basic differences in chaotic systems due to their sensitivity to initial condition (butterfly effect). Accordingly, in this paper, a new cost function which is proper for chaotic systems is applied to the chaotic one-dimensional map. Then the efficiency of a newly introduced intelligent method experimental design in extracting proper data is investigated. The results show the success of the proposed method.

Parameter estimation for dynamical systems with discrete events and logical operations.

Ordinary differential equation (ODE) models are frequently used to describe the dynamic behaviour of biochemical processes. Such ODE models are often extended by events to describe the effect of fast latent processes on the process dynamics. To exploit the predictive power of ODE models, their parameters have to be inferred from experimental data. For models without events, gradient based optimization schemes perform well for parameter estimation, when sensitivity equations are used for gradient computation. Yet, sensitivity equations for models with parameter- and state-dependent events and event-triggered observations are not supported by existing toolboxes. In this manuscript, we describe the sensitivity equations for differential equation models with events and demonstrate how to estimate parameters from event-resolved data using event-triggered observations in parameter estimation. We consider a model for GFP expression after transfection and a model for spiking neurons and demonstrate that we can improve computational efficiency and robustness of parameter estimation by using sensitivity equations for systems with events. Moreover, we demonstrate that, by using event-outputs, it is possible to consider event-resolved data, such as time-to-event data, for parameter estimation with ODE models. By providing a user-friendly, modular implementation in the toolbox AMICI, the developed methods are made publicly available and can be integrated in other systems biology toolboxes. We implement the methods in the open-source toolbox Advanced MATLAB Interface for CVODES and IDAS (AMICI, https://github.com/ICB-DCM/AMICI ). jan.hasenauer@helmholtz-muenchen.de. Supplementary data are available at Bioinformatics online.

Consistency of maximum likelihood estimation for some dynamical systems

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.

Regularized EM algorithm for sparse parameter estimation in nonlinear dynamic systems with application to gene regulatory network inference

Parameter estimation in dynamic systems finds applications in various disciplines, including system biology. The well-known expectation-maximization (EM) algorithm is a popular method and has been widely used to solve system identification and parameter estimation problems. However, the conventional EM algorithm cannot exploit the sparsity. On the other hand, in gene regulatory network inference problems, the parameters to be estimated often exhibit sparse structure. In this paper, a regularized expectation-maximization (rEM) algorithm for sparse parameter estimation in nonlinear dynamic systems is proposed that is based on the maximum a posteriori (MAP) estimation and can incorporate the sparse prior. The expectation step involves the forward Gaussian approximation filtering and the backward Gaussian approximation smoothing. The maximization step employs a re-weighted iterative thresholding method. The proposed algorithm is then applied to gene regulatory network inference. Results based on both synthetic and real data show the effectiveness of the proposed algorithm.

Diagnosis of inhomogeneous insulation degradation in electric cables by distributed shunt conductance estimation

For the diagnosis of inhomogeneous insulation degradation in electric cables, the estimation of distributed shunt conductance is studied in this paper. Gradual growth of the shunt conductance is a consequence of degradation of the dielectric properties of the insulator. The proposed estimation method is based on voltage and current measurements at a single end of the cable. After the linearization of the bilinear term of the telegrapher's equations through a perturbation approach, the Kalman filter is applied to transform the problem of dynamic system parameter estimation to a simple linear regression problem. Results of numerical simulations are presented to demonstrate the feasibility of the proposed method. In particular, it is shown that the weak sensitivity of the available measurements to the shunt conductance can be compensated by long time data samples.

Parameter estimation in time-triggered and event-triggered model-based control of uncertain systems

In this article on-line parameter estimation of dynamical systems is addressed in the context of model-based networked control systems (MB-NCSs). Stability conditions that are robust to parameter uncertainties and lack of feedback for extended intervals of time are presented. The updated model is used to control the real system the next time feedback information is unavailable. Additionally, new estimation models are proposed that offer better convergence properties than typical state-space parameter estimation methods; common assumptions such as availability of persistently exciting inputs and estimation of only a canonical form of the system are relaxed. The implementation of upgraded models in MB-NCSs results in better usage of the network by allowing longer intervals without the need for measurement updates.

Diagnosis of inhomogeneous insulation degradation in electric cables by distributed shunt conductance estimation

For the diagnosis of inhomogeneous insulation degradation in electrical cables, the estimation of the distributed shunt conductance is studied in this paper. Gradual growth of the shunt conductance is a consequence of degradation of the dielectric properties of the insulator. The proposed estimation method is based on voltage and current measurements at a single end of the cable. After the linearization of the bilinear term of the telegrapher's equations through a perturbation approach, the Kalman filter is applied to transform the problem of dynamic system parameter estimation to a simple linear regression problem. Results of numerical simulations are presented to demonstrate the feasibility of the proposed method. In particular, it is shown that the weak sensitivity of the available measurements to the shunt conductance can be compensated by long time data samples.

PSO with Adaptive Mutation and Inertia Weight and Its Application in Parameter Estimation of Dynamic Systems

An important problem in engineering is the unknown parameters estimation in nonlinear systems. In this paper, a novel adaptive particle swarm optimization (APSO) method is proposed to solve this problem. This work considers two new aspects, namely an adaptive mutation mechanism and a dynamic inertia weight into the conventional particle swarm optimization (PSO) method. These mechanisms are employed to enhance global search ability and to increase accuracy. First, three well-known benchmark functions namely Griewank, Rosenbrock and Rastrigrin are utilized to test the ability of a search algorithm for identifying the global optimum. The performance of the proposed APSO is compared with advanced algorithms such as a nonlinearly decreasing weight PSO (NDWPSO) and a real-coded genetic algorithm (GA), in terms of parameter accuracy and convergence speed. It is confirmed that the proposed APSO is more successful than other aforementioned algorithms. Finally, the feasibility of this algorithm is demonstrated through estimating the parameters of two kinds of highly nonlinear systems as the case studies.

Ｕｎｓｃｅｎｔｅｄ Ｋａｌｍａｎ Ｆｉｌｔｅｒを用いた力学系の状態およびパラメータ推定

Ｕｎｓｃｅｎｔｅｄ Ｋａｌｍａｎ Ｆｉｌｔｅｒを用いた力学系の状態およびパラメータ推定

Оценка параметров зашумленных динамических систем

Оценка параметров зашумленных динамических систем