Incomplete phase-space method to reveal time delay from scalar time series

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay τ∗ and embedding dimension p for phase space reconstruction. The value of τ∗ can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that τ∗ should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay τw=(p−1)τ∗, which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally τw and τ∗. In this paper, we suggest a simple method for estimating τ∗ and τw based on symbolic analysis and symbolic entropy. As in the C-C method, τ∗ is estimated as the first local optimal time delay and τw as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes

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Combining Snow Depth From FY-3C and In Situ Data Over the Tibetan Plateau Using a Nonlinear Analysis Method

Frontiers in Physics ◽

10.3389/fphy.2021.672288 ◽

2021 ◽

Vol 9 ◽

Author(s):

Aixia Feng ◽

Feng Gao ◽

Qiguang Wang ◽

Aiqing Feng ◽

Qiang Zhang ◽

...

Keyword(s):

Time Series ◽

Time Delay ◽

Phase Space ◽

Tibetan Plateau ◽

Satellite Data ◽

Snow Depth ◽

Hydrological Cycle ◽

The Tibetan Plateau ◽

Analysis Method

Snow cover over the Tibetan Plateau plays a vital role in the regional and global climate system because it affects not only the climate but also the hydrological cycle and ecosystem. However, high-quality snow data are hindered due to the sparsity of observation networks and complex terrain in the region. In this study, a nonlinear time series analysis method called phase space reconstruction was used to obtain the Tibetan Plateau snow depth by combining the FY-3C satellite data and in situ data for the period 2014–2017. This method features making a time delay reconstruction of a phase space to view the dynamics. Both of the grids and their nearby in situ snow depth time series were reconstructed with two appropriate parameters called time delay and embedding dimension. The values of the snow depth for grids were averaged over the in situ observations and retrieval of the satellite if their two parameters were the same. That implies that the two trajectories of the time series had the same evolution trend. Otherwise, the snow depth values for grids were averaged over the in situ observation. If there were no in situ sites within the grids, the retrieval of the satellite remained. The results show that the integrated Tibetan Plateau snow depth (ITPSD) had an average bias of –1.35 cm and 1.14 cm, standard deviation of the bias of 3.96 cm and 5.67 cm, and root mean square error of 4.18 cm and 5.79 cm compared with the in situ data and FY-3C satellite data, respectively. ITPSD expressed the issue that snow depth is usually overestimated in mountain regions by satellites. This is due to the introduction of more station observations using a dynamical statistical method to correct the biases in the satellite data.

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A fuzzy-set-based Reconstructed Phase Space method for identification of temporal patterns in complex time series

IEEE Transactions on Knowledge and Data Engineering ◽

10.1109/tkde.2005.68 ◽

2005 ◽

Vol 17 (5) ◽

pp. 601-613 ◽

Cited By ~ 23

Author(s):

X. Feng ◽

H. Huang

Keyword(s):

Time Series ◽

Phase Space ◽

Fuzzy Set ◽

Temporal Patterns ◽

Complex Time ◽

Reconstructed Phase Space ◽

Space Method

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Machine Learning of Time Series Using Time-Delay Embedding and Precision Annealing

Neural Computation ◽

10.1162/neco_a_01224 ◽

2019 ◽

Vol 31 (10) ◽

pp. 2004-2024 ◽

Cited By ~ 4

Author(s):

Alexander J. A. Ty ◽

Zheng Fang ◽

Rivver A. Gonzalez ◽

Paul. J. Rozdeba ◽

Henry D. I. Abarbanel

Keyword(s):

Machine Learning ◽

Time Series ◽

Time Delay ◽

Global Minimum ◽

Statistical Data ◽

Dimensional Space ◽

Network Models ◽

Multilayer Perceptrons ◽

Task Time ◽

Scalar Time Series

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. We borrow two techniques used in statistical data assimilation in order to accomplish this task: time-delay embedding to prepare our input data and precision annealing as a training method. The precision annealing approach identifies the global minimum of the action ([Formula: see text]). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series [Formula: see text] and, using methods of nonlinear time series analysis, show how to produce a [Formula: see text]-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed [Formula: see text] time series. In that [Formula: see text]-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on [Formula: see text]-dimensional input and producing [Formula: see text]-dimensional outputs.

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A Methodology for the Modeling of Rail Vehicles From Time Series Measurements Using Time-Delay Neural Networks

ASME 2007 Rail Transportation Division Fall Technical Conference ◽

10.1115/rtdf2007-46022 ◽

2007 ◽

Author(s):

John Zolock ◽

Robert Greif

Keyword(s):

Neural Networks ◽

Time Series ◽

Time Delay ◽

Phase Space ◽

Dynamical Systems ◽

Dynamic Systems ◽

Phase Space Reconstruction ◽

Series Data ◽

Modeling Methodology ◽

Forcing Function

The main goal of this research is to develop and demonstrate a general, efficient, mathematically and theoretically based methodology to model nonlinear forced vibrating mechanical systems from time series measurements. A system identification modeling methodology for forced dynamical systems is presented based on dynamic system theory and nonlinear time series analysis that employs phase space reconstruction (delay vector embedding) for modeling of dynamical systems from time series data using time-delay neural networks (TDNN). The first part of this work details the modeling methodology including background on dynamic systems, phase space reconstruction, and neural networks. In the second part of this work the methodology is evaluated based on its ability to model selected analytical lumped parameter forced vibrating dynamic systems including an example of a linear system predicting lumped mass displacement using a displacement forcing. function The work discusses the application to nonlinear systems, multi degree-of-freedom systems, and multi-input systems. The methodology is further evaluated on its ability to model an analytical passenger rail vehicle predicting vertical wheel/rail force using vertical rail profile as input. Studying the neural modeling methodology using an analytical systems shows the clearest observations from results which provide prospective users of this tool an understanding of the expectations and limitations of the modeling methodology.

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A Methodology for the Modeling of Forced Dynamical Systems From Time Series Measurements Using Time-Delay Neural Networks

Journal of Vibration and Acoustics ◽

10.1115/1.2981096 ◽

2008 ◽

Vol 131 (1) ◽

Cited By ~ 4

Author(s):

John Zolock ◽

Robert Greif

Keyword(s):

Neural Networks ◽

Time Series ◽

Time Delay ◽

Phase Space ◽

Dynamical Systems ◽

Dynamic Systems ◽

Phase Space Reconstruction ◽

Series Data ◽

Modeling Methodology ◽

Forcing Function

The main goal of this research was to develop and present a general, efficient, mathematical, and theoretical based methodology to model nonlinear forced-vibrating mechanical systems from time series measurements. A system identification modeling methodology for forced dynamical systems is presented based on a dynamic system theory and a nonlinear time series analysis that employ phase space reconstruction (delay vector embedding) in modeling dynamical systems from time series data using time-delay neural networks. The first part of this work details the modeling methodology, including background on dynamic systems, phase space reconstruction, and neural networks. In the second part of this work, the methodology is evaluated based on its ability to model selected analytical lumped-parameter forced-vibrating dynamic systems, including an example of a linear system predicting lumped mass displacement subjected to a displacement forcing function. The work discusses the application to nonlinear systems, multiple degree of freedom systems, and multiple input systems. The methodology is further evaluated on its ability to model an analytical passenger rail car predicting vertical wheel∕rail force using a measured vertical rail profile as the input function. Studying the neural modeling methodology using analytical systems shows the clearest observations from results, providing prospective users of this tool an understanding of the expectations and limitations of the modeling methodology.

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TIME SERIES ANALYSIS IN RECONSTRUCTED STATE SPACES

Stochastics and Dynamics ◽

10.1142/s0219493701000060 ◽

2001 ◽

Vol 01 (01) ◽

pp. 85-111 ◽

Cited By ~ 4

Author(s):

HOLGER KANTZ

Keyword(s):

Time Series ◽

Phase Space ◽

Chaotic Systems ◽

Time Series Data ◽

Series Data ◽

Nonstationary Processes ◽

State Spaces ◽

Phase Space Methods ◽

Vector Valued ◽

Scalar Time Series

Concepts for the analysis of observed scalar time series data in reconstructed vector valued phase spaces are reviewed. Originally exclusively designed for data from deterministic chaotic systems, phase space methods were recently extended to usage for nonlinear stochastic and for nonstationary processes.

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DETECTING CHAOS TIME SERIES VIA COMPLEX NETWORK FEATURE

Modern Physics Letters B ◽

10.1142/s0217984911027133 ◽

2011 ◽

Vol 25 (23) ◽

pp. 1889-1896 ◽

Cited By ~ 4

Author(s):

QIANG TANG ◽

JUNCHAN ZHAO ◽

TIESONG HU

Keyword(s):

Time Series ◽

Time Delay ◽

Phase Space ◽

Complex Network ◽

Stream Flow ◽

Yangtze River ◽

Random Series ◽

Coordinate Method ◽

Network Feature ◽

Periodic Time Series

In this paper, an effective method from time series to complex network via phase space reconstruction is introduced. We reconstruct the phase space from a time series by the time-delay coordinate method. Each state vector of phase space is regarded as a vertex of network and the connection is based on the distance of the vertices in phase space. The networks corresponding to various time series, the x component of the chaotic Rössler system, noisy periodic time series and random series, display different topology feature. So we can determine whether a time series is chaotic series by the topology feature of corresponding network. Finally, the daily stream-flow series of Yangtze River is investigated to validate the effective of our method.

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