IMPROVED PARAMETER ESTIMATION FROM NOISY TIME SERIES FOR NONLINEAR DYNAMICAL SYSTEMS

2007 ◽  
Vol 17 (05) ◽  
pp. 1741-1752 ◽  
Author(s):  
TOMOMICHI NAKAMURA ◽  
YOSHITO HIRATA ◽  
KEVIN JUDD ◽  
DEVIN KILMINSTER ◽  
MICHAEL SMALL
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Dynamical Systems ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Parameter Estimates ◽  
Nonlinear Dynamical ◽  
Observational Noise

In this paper we consider the problem of estimating the parameters of a nonlinear dynamical system given a finite time series of observations that are contaminated by observational noise. The least squares method is a standard method for parameter estimation, but for nonlinear dynamical systems it is well known that the least squares method can result in biased estimates, especially when the noise is significant relative to the nonlinearity. In this paper, it is demonstrated that by combining nonlinear noise reduction and least squares parameter fitting it is possible to obtain more accurate parameter estimates.

MODELING NONLINEAR TIME SERIES USING IMPROVED LEAST SQUARES METHOD

2006 ◽  
Vol 16 (02) ◽  
pp. 445-464 ◽  
Author(s):  
TOMOMICHI NAKAMURA ◽  
MICHAEL SMALL
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Dynamical System ◽  
Information Criteria ◽  
Nonlinear Dynamical ◽  
Numerical Studies ◽  
Correct Model ◽  
Building Models ◽  
Observational Noise

We improve the least squares (LS) method for building models of a nonlinear dynamical system given finite time series which are contaminated by observational noise. When the noise level is low, the LS method gives good estimates for the parameters, however, the models selected as the best by information criteria often tend to be over-parameterized or even degenerate. We observe that the correct model is not selected as the best model despite belonging to the chosen model class. To overcome this, we propose a simple but very effective idea to use the LS method more appropriately. We apply the method for model selection. Numerical studies indicate that the method can be used to apply information criteria more effectively, and generally avoid over-fitting and model degeneracy.

NONLINEAR DYNAMICAL SYSTEM IDENTIFICATION FROM UNCERTAIN AND INDIRECT MEASUREMENTS

2004 ◽  
Vol 14 (06) ◽  
pp. 1905-1933 ◽  
Author(s):  
HENNING U. VOSS ◽  
JENS TIMMER ◽  
JÜRGEN KURTHS
Keyword(s):  
Parameter Estimation ◽  
Measurement Errors ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Recursive Estimation ◽  
Cost Functions ◽  
Parameter Estimates ◽  
Nonlinear Dynamical ◽  
Estimation Techniques ◽  
The Cost

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

Joint Maximum a Posteriori Smoother for State and Parameter Estimation in Nonlinear Dynamical Systems*

2012 ◽  
Vol 45 (16) ◽  
pp. 900-905 ◽  
Author(s):  
Dimas A. Dutra ◽  
Bruno O.S. Teixeira ◽  
Luis A. Aguirre
Keyword(s):  
Parameter Estimation ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
Nonlinear Dynamical

scholarly journals Sparse identification of nonlinear dynamical systems via reweighted ℓ1-regularized least squares

2021 ◽  
Vol 376 ◽  
pp. 113620 ◽  
Author(s):  
Alexandre Cortiella ◽  
Kwang-Chun Park ◽  
Alireza Doostan
Keyword(s):  
Dynamical Systems ◽  
Least Squares ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Regularized Least Squares

scholarly journals Phase Synchronization Is the Amplified Result by the Hilbert Transform

2015 ◽  
Vol 2015 ◽  
pp. 1-3 ◽  
Author(s):  
Ming-Chi Lu ◽  
Hsing-Chung Ho ◽  
Chen-An Chan ◽  
Chia-Ju Liu ◽  
Jiann-Shing Lih ◽  
...  
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Hilbert Transform ◽  
Phase Synchronization ◽  
Nonlinear Dynamical Systems ◽  
The State ◽  
Nonlinear Dynamical ◽  
Large Correlation ◽  
Current Usage ◽  
The Hilbert Transform

We investigate the interplay between phase synchronization and amplitude synchronization in nonlinear dynamical systems. It is numerically found that phase synchronization intends to be established earlier than amplitude synchronization. Nevertheless, amplitude synchronization (or the state with large correlation between the amplitudes) is crucial for the maintenance of a high correlation between two time series. A breakdown of high correlation in amplitudes will lead to a desynchronization of two time series. It is shown that these unique features are caused essentially by the Hilbert transform. This leads to a deep concern and criticism on the current usage of phase synchronization.

scholarly journals Modeling of information channel by using of pseudorandom signals of nonlinear dynamical system

2020 ◽  
Vol 22 (4) ◽  
pp. 79-87
Author(s):  
I. K. Nasyrov ◽  
V. V. Andreev
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Detection Algorithm ◽  
Main Step ◽  
Operating Characteristics ◽  
Nonlinear Dynamical ◽  
Pseudorandom Signals ◽  
Chaotic Signals ◽  
Correlation Properties

Pseudorandom signals of nonlinear dynamical systems are studied and the possibility of their application in information systems analyzed. Continuous and discrete dynamical systems are considered: Lorenz System, Bernoulli and Henon maps. Since the parameters of dynamical systems (DS) are included in the equations linearly, the principal possibility of the state linear control of a nonlinear DS is shown. The correlation properties comparative analysis of these DSs signals is carried out.. Analysis of correlation characteristics has shown that the use of chaotic signals in communication and radar systems can significantly increase their resolution over the range and taking into account the specific properties of chaotic signals, it allows them to be hidden. The representation of nonlinear dynamical systems equations in the form of stochastic differential equations allowed us to obtain an expression for the likelihood functional, with the help of which many problems of optimal signal reception are solved. It is shown that the main step in processing the received message, which provides the maximum likelihood functionals, is to calculate the correlation integrals between the components and the systems under consideration. This made it possible to base the detection algorithm on the correlation reception between signal components. A correlation detection receiver was synthesized and the operating characteristics of the receiver were found.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950030 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Aboozar Ghaffari ◽  
Sajad Jafari ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dictionary Learning ◽  
Nonlinear Dynamical Systems ◽  
Sparse Recovery ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
Bifurcation Points ◽  
Long Time

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

A New Method for Equilibria and Stability Analysis of Nonlinear Dynamical Systems

2014 ◽  
Vol 534 ◽  
pp. 131-136
Author(s):  
Long Cao ◽  
Yi Hua Cao
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
Vehicle Model ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Linearized System ◽  
Continuation Algorithm

A novel method based on numerical continuation algorithm for equilibria and stability analysis of nonlinear dynamical system is introduced and applied to an aircraft vehicle model. Dynamical systems are usually modeled with differential equations, while their equilibria and stability analysis are pure algebraic problems. The newly-proposed method in this paper provides a way to solve the equilibrium equation and the eigenvalues of the locally linearized system simultaneously, which avoids QR iterations and can save much time.

Sign in / Sign up

Export Citation Format

Share Document