A New Cost Function for Parameter Estimation of Chaotic Systems Using Return Maps as Fingerprints

2014 ◽  
Vol 24 (10) ◽  
pp. 1450134 ◽  
Author(s):  
Sajad Jafari ◽  
Julien C. Sprott ◽  
Viet-Thanh Pham ◽  
S. Mohammad Reza Hashemi Golpayegani ◽  
Amir Homayoun Jafari
Keyword(s):  
Parameter Estimation ◽  
Cost Function ◽  
Chaotic Systems ◽  
Time Series Data ◽  
Initial Conditions ◽  
Model Systems ◽  
Series Data ◽  
Return Maps ◽  
Active Interest ◽  
The Time Domain

Estimating parameters of a model system using observed chaotic scalar time series data is a topic of active interest. To estimate these parameters requires a suitable similarity indicator between the observed and model systems. Many works have considered a similarity measure in the time domain, which has limitations because of sensitive dependence on initial conditions. On the other hand, there are features of chaotic systems that are not sensitive to initial conditions such as the topology of the strange attractor. We have used this feature to propose a new cost function for parameter estimation of chaotic models, and we show its efficacy for several simple chaotic systems.

Causation Entropy Identifies Sparsity Structure for Parameter Estimation of Dynamic Systems

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Pileun Kim ◽  
Jonathan Rogers ◽  
Jie Sun ◽  
Erik Bollt
Keyword(s):  
Parameter Estimation ◽  
Time Series Data ◽  
Multivariate Time Series ◽  
Measurement Data ◽  
Relative Magnitude ◽  
Initial Guess ◽  
Series Data ◽  
System Parameters ◽  
Estimation Problems ◽  
System States

Parameter estimation is an important topic in the field of system identification. This paper explores the role of a new information theory measure of data dependency in parameter estimation problems. Causation entropy is a recently proposed information-theoretic measure of influence between components of multivariate time series data. Because causation entropy measures the influence of one dataset upon another, it is naturally related to the parameters of a dynamical system. In this paper, it is shown that by numerically estimating causation entropy from the outputs of a dynamic system, it is possible to uncover the internal parametric structure of the system and thus establish the relative magnitude of system parameters. In the simple case of linear systems subject to Gaussian uncertainty, it is first shown that causation entropy can be represented in closed form as the logarithm of a rational function of system parameters. For more general systems, a causation entropy estimator is proposed, which allows causation entropy to be numerically estimated from measurement data. Results are provided for discrete linear and nonlinear systems, thus showing that numerical estimates of causation entropy can be used to identify the dependencies between system states directly from output data. Causation entropy estimates can therefore be used to inform parameter estimation by reducing the size of the parameter set or to generate a more accurate initial guess for subsequent parameter optimization.

scholarly journals ESTIMATION OF AUTOREGRESSIVE ROOTS NEAR UNITY USING PANEL DATA

2000 ◽  
Vol 16 (6) ◽  
pp. 927-997 ◽  
Author(s):  
Hyungsik R. Moon ◽  
Peter C.B. Phillips
Keyword(s):  
Time Series ◽  
Panel Data ◽  
Time Series Data ◽  
Initial Conditions ◽  
Series Data ◽  
Test Statistics ◽  
Rate Condition ◽  
Panel Models ◽  
Stochastic Trends ◽  
Efficient Extraction

Time series data are often well modeled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localizing parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localizing parameter are asymptotically biased. Some techniques are developed to overcome this difficulty, and consistent estimators of c in the region c < 0 are developed for panel models with deterministic and stochastic trends. A limit distribution theory is also established, and test statistics are constructed for exploring interesting hypotheses, such as the equivalence of local to unity parameters across subgroups of the population. The methods are applied to the empirically important problem of the efficient extraction of deterministic trends. They are also shown to deliver consistent estimates of distancing parameters in nonstationary panel models where the initial conditions are in the distant past. In the development of the asymptotic theory this paper makes use of both sequential and joint limit approaches. An important limitation in the operation of the joint asymptotics that is sometimes needed in our development is the rate condition n/T → 0. So the results in the paper are likely to be most relevant in panels where T is large and n is moderately large.

Global optimization for parameter estimation of differential-algebraic systems

2009 ◽  
Vol 63 (3) ◽  
Author(s):  
Michal Čižniar ◽  
Marián Podmajerský ◽  
Tomáš Hirmajer ◽  
Miroslav Fikar ◽  
Abderrazak Latifi
Keyword(s):  
Global Optimization ◽  
Parameter Estimation ◽  
Time Series Data ◽  
Optimization Method ◽  
Differential Algebraic Equations ◽  
Series Data ◽  
Global Optimality ◽  
Algebraic Equations ◽  
Estimation Of Parameters ◽  
Orthogonal Collocation

AbstractThe estimation of parameters in semi-empirical models is essential in numerous areas of engineering and applied science. In many cases, these models are described by a set of ordinary-differential equations or by a set of differential-algebraic equations. Due to the presence of non-convexities of functions participating in these equations, current gradient-based optimization methods can guarantee only locally optimal solutions. This deficiency can have a marked impact on the operation of chemical processes from the economical, environmental and safety points of view and it thus motivates the development of global optimization algorithms. This paper presents a global optimization method which guarantees ɛ-convergence to the global solution. The approach consists in the transformation of the dynamic optimization problem into a nonlinear programming problem (NLP) using the method of orthogonal collocation on finite elements. Rigorous convex underestimators of the nonconvex NLP problem are employed within the spatial branch-and-bound method and solved to global optimality. The proposed method was applied to two example problems dealing with parameter estimation from time series data.

scholarly journals Testing for Unit Roots in Time Series Data

1989 ◽  
Vol 5 (2) ◽  
pp. 256-271 ◽  
Author(s):  
Sastry G. Pantula
Keyword(s):  
Likelihood Ratio ◽  
Time Series Data ◽  
Initial Conditions ◽  
Asymptotic Properties ◽  
Unit Roots ◽  
Sequential Testing ◽  
Series Data ◽  
Stochastic Difference Equation ◽  
Asymptotic Results ◽  
Testing Procedures

Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 and the initial conditions (Y−p+1,…, Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.

scholarly journals Real age prediction from the transcriptome with RAPToR

2021 ◽  
Author(s):  
Romain Bulteau ◽  
Mirko Francesconi
Keyword(s):  
Gene Expression ◽  
Large Scale ◽  
Time Series Data ◽  
Expression Profiles ◽  
Computational Method ◽  
Model Systems ◽  
Developmental Expression ◽  
Model Organisms ◽  
Series Data ◽  
Developmental Variation

AbstractGenome-wide gene expression profiling is a powerful tool for exploratory analyses, providing a high dimensional picture of the state of a biological system. However, uncontrolled variation among samples can obscure and confound the effect of variables of interest. Uncontrolled developmental variation is often a major source of unknown expression variation in developmental systems. Existing methods to sort samples from transcriptomes require many samples to infer developmental trajectories and only provide a relative pseudo-time.Here we present RAPToR (Real Age Prediction from Transcriptome staging on Reference), a simple computational method to estimate the absolute developmental age of even a single sample from its gene expression with up to minutes precision. We achieve this by staging samples on high-resolution reference developmental expression profiles we build from existing time series data. We implemented RAPToR for the most common animal model systems: nematode, fruit fly, zebrafish, and mouse, and demonstrate application for non-model organisms. We show how developmental variation discovered by RAPToR can be exploited to increase power to detect differential expression and to untangle the signal of perturbations of interest even when it is completely confounded with development. We anticipate our RAPToR post-profiling staging strategy will be especially useful in large scale single organism profiling because it eliminates the need for synchronization or for a tedious and potentially difficult step of accurate staging before profiling.

scholarly journals White Noise Tests on the LSTM Model Trained with Double Pendulum

2021 ◽  
Vol 2137 (1) ◽  
pp. 012032
Author(s):  
Xisen Wang
Keyword(s):  
Time Series ◽  
White Noise ◽  
Time Series Data ◽  
Initial Conditions ◽  
Correlation Coefficients ◽  
Series Data ◽  
Double Pendulum ◽  
Auto Correlation ◽  
State Dependent ◽  
Auto Regressive

Abstract This paper describes the intrinsic qualities of a simple double pendulum (DP), with a visual representation, a rigorous deduction of the Lagrangian equation, and a concrete factor analysis. LSTM model was utilized to simulate the double pendulum’s periodic and chaotic behaviors and evaluates the effectiveness of the model. The auto-correlation coefficients was calculated. Meanwhile, Box-Pierce test and Ljung-Box tests for various state-dependent time series were conducted to give various initial conditions to explore the DP system’s random characteristics. The research results are as follows: 1) Chaos did not lead to direct randomness; 2) seasonality could coexist with chaos; 3) the highly auto-regressive nature of DP’s time series data are found. Therefore, it can be concluded that the chaos in a double pendulum has particular patterns (such as the positive relationship with the likelihood of being a random white noise series) that could be further explored.

Revisiting the Holt-Winters' Additive Method for Better Forecasting

2019 ◽  
Vol 15 (2) ◽  
pp. 43-57
Author(s):  
Seng Hansun ◽  
Vincent Charles ◽  
Christiana Rini Indrati ◽  
Subanar
Keyword(s):  
Time Series ◽  
Average Method ◽  
Data Science ◽  
Time Series Data ◽  
Initial Conditions ◽  
Moving Average ◽  
Series Data ◽  
Initial Values ◽  
Additive Method ◽  
Weighted Moving Average

Time series are one of the most common data types encountered by data scientists and, in the context of today's exponentially increasing data, learning how to best model them to derive meaningful insights is an important skill in the Big Data and Data Science toolbox. As a result, many researchers have dedicated their efforts to developing time series analysis methods to predict future values based on previously observed values. One of the well-known methods is the Holt-Winters' seasonal method, which is commonly used to capture the seasonality effect in time series data. In this study, the authors aim to build upon the Holt-Winters' additive method by introducing new formulas for finding the initial values. Obtaining more accurate estimations of the initial values could result in a better forecasting result. The authors use the basic principle found in the weighted moving average method to assign more weight to the most recent data and combine it with the original initial conditions found in the Holt-Winters' additive method. Based on the experiment performed, the authors conclude that the new formulas for finding the initial values in the Holt-Winters' additive method could give a better forecasting when compared to the traditional Holt-Winters' additive method and the weighted moving average method in terms of the accuracy level.

scholarly journals Linear Hybrid Deterministic Dynamic Modeling for Time-to-Event Processes: State and Parameter Estimations

2016 ◽  
Vol 5 (6) ◽  
pp. 32
Author(s):  
E. A. Appiah ◽  
G. S. Ladde
Keyword(s):  
Parameter Estimation ◽  
Discrete Time ◽  
Chemical Engineering ◽  
Time Series Data ◽  
Survival Function ◽  
Series Data ◽  
Dynamic Processes ◽  
Time To Event ◽  
Deterministic Dynamic ◽  
Special Cases

In this work, we initiate an innovative alternative modeling approach for time-to-event dynamic processes. The proposed approach is composed of the following basic components: (1) development of continuous-time state of dynamic process, (2) introduction of discrete-time dynamic intervention process, (3) formulation of continuous and discrete-time interconnected dynamic system, (4) utilizing Euler-type discretized schemes, and (5) introduction of conceptual and computational state and parameter estimation procedures. The presented approach is motivated by state and parameter estimation of time-to-event processes in biological, chemical, engineering, epidemiological, medical, military, multiple-markets and social dynamic processes under the influence of discrete-time intervention processes. The role and scope of our approach is exhibited by presenting several well-known hazard/risk rate and survival function estimates as special cases. Moreover, conceptual algorithms are illustrated by time-series data sets under the influence of intervention processes.

Conclusions

2019 ◽  
pp. 213-218
Author(s):  
Timothy G. Barraclough
Keyword(s):  
Time Series Data ◽  
Scale Up ◽  
Current Data ◽  
Model Systems ◽  
Series Data ◽  
Develop Model ◽  
Genome Data ◽  
Vital Component ◽  
Future Work

This final chapter summarizes conclusions from the book and highlights a few general areas for future work. The species model for the structure of diversity is found to be useful and largely supported by current data, but is open to future tests against explicit alternative models. It is also a vital component for understanding and predicting contemporary evolution in the diverse systems that all organisms live in. The common evolutionary framework for microbial and multicellular life is highlighted, while drawing attention to current gaps in understanding for each type of organism. Future work needs to scale up to develop model systems of diverse assemblages and clades, including time-series data ranging from contemporary to geological scales. The imminent avalanche of genome data for thousands of individuals sampled within and between species is identified as a key challenge and opportunity. Finally, this chapter repeats the challenge that evolutionary biologists should embrace diversity and need to attempt to predict evolution in diverse systems, in order to deliver solutions of benefit to society.

Interpolated Mapping System Identification From Time Series Data

1993 ◽  
Author(s):  
Faruk H. Bursal ◽  
Benson H. Tongue
Keyword(s):  
Time Series ◽  
System Identification ◽  
Time Series Data ◽  
Initial Conditions ◽  
Series Data ◽  
Time Interval ◽  
Identification Algorithm ◽  
Regular Grid ◽  
Time Step ◽  
Data Set

Abstract In this paper, a system identification algorithm based on Interpolated Mapping (IM) that was introduced in a previous paper is generalized to the case of data stemming from arbitrary time series. The motivation for the new algorithm is the need to identify nonlinear dynamics in continuous time from discrete-time data. This approach has great generality and is applicable to problems arising in many areas of science and engineering. In the original formulation, a map defined on a regular grid in the state space of a dynamical system was assumed to be given. For the formulation to become practically viable, however, the requirement of initial conditions being taken from such a regular grid needs to be dropped. In particular, one would like to use time series data, where the time interval between samples is identified with the mapping time step T. This paper is concerned with the resulting complications. Various options for extending the formulation are examined, and a choice is made in favor of a pre-processing algorithm for estimating the FS map based on local fits to the data set. The suggested algorithm also has smoothing properties that are desirable from the standpoint of noise reduction.

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