Surrogate Testing by Estimating the Local Dispersions in Phase Space

2003 ◽  
Vol 13 (08) ◽  
pp. 2241-2251 ◽  
Author(s):  
Radhakrishnan Nagarajan
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Stationary Time Series ◽  
Data Sets ◽  
Local Dispersion ◽  
Dimensional Phase Space ◽  
Clustering Approach ◽  
Principal Directions ◽  
The One ◽  
Value Decomposition

In this report, a clustering approach is presented to detect dynamical nonlinearity in a stationary time series. The one-dimensional time series sampled from a dynamical system is mapped on to the m-dimensional phase space by the method of delays. The vectors in the phase space are partitioned by k-means clustering technique. The local trajectory matrix for the vectors in each of the clusters is determined. The eigenvalues of the local trajectory matrices represent the variation along the principal directions and are obtained by singular value decomposition (SVD). The product of the eigenvalues represents the generalized variance and is a measure of the local dispersion in the phase space. The sum of the local dispersions (gT) is used as the discriminant statistic to classify data sets obtained from deterministic and stochastic settings. The surrogates were generated by the Iterated Amplitude Adjusted Fourier Transform (IAAFT) technique.

Degrees of Freedom

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Prediction Method ◽  
Data Sets ◽  
Nonlinear Prediction ◽  
Orthogonal Direction ◽  
Empirical Prediction ◽  
Long Run ◽  
Dimensional Phase Space ◽  
Natural Analogues

How many degrees of freedom are evident in a physical process represented by f(s, t)? In some form questions about “degrees of freedom” (d.o.f.) are common in mathematics, physics, statistics, and geophysics. This would mean, for instance, in how many independent directions a weight suspended from the ceiling could move. Dofs are important for three reasons that will become apparent in the remaining chapters. First, dofs are critically important in understanding why natural analogues can (or cannot) be applied as a forecast method in a particular problem (Chapter 7). Secondly, understanding dofs leads to ideas about truncating data sets efficiently, which is very important for just about any empirical prediction method (Chapters 7 and 8). Lastly, the number of dofs retained is one aspect that has a bearing on how nonlinear prediction methods can be (Chapter 10). In view of Chapter 5 one might think that the total number of orthogonal directions required to reproduce a data set is the dof. However, this is impractical as the dimension would increase (to infinity) with ever denser and slightly imperfect observations. Rather we need a measure that takes into account the amount of variance represented by each orthogonal direction, because some directions are more important than others. This allows truncation in EOF space without lowering the “effective” dof very much. We here think schematically of the total atmospheric or oceanic variance about the mean state as being made up by N equal additive variance processes. N can be thought of as the dimension of a phase space in which the atmospheric state at one moment in time is a point. This point moves around over time in the N-dimensional phase space. The climatology is the origin of the phase space. The trajectory of a sequence of atmospheric states is thus a complicated Lissajous figure in N dimensions, where, importantly, the range of the excursions in each of the N dimensions is the same in the long run. The phase space is a hypersphere with an equal probability radius in all N directions.

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

2013 ◽  
Vol 23 (06) ◽  
pp. 1330019
Author(s):  
F. J. MOLERO ◽  
J. C. VAN DER MEER ◽  
S. FERRER ◽  
F. J. CÉSPEDES
Keyword(s):  
Phase Space ◽  
Elliptic Functions ◽  
Singular Values ◽  
Nonlinear Oscillators ◽  
Momentum Mapping ◽  
Dimensional Model ◽  
Integrable Hamiltonian Systems ◽  
One Dimensional ◽  
Dimensional Phase Space ◽  
The One

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

Forecasting Exchange Rates

2016 ◽  
pp. 1864-1883
Author(s):  
Ahmed Radhwan ◽  
Mahmoud Kamel ◽  
Mohammed Y. Dahab ◽  
AboulElla Hassanien
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Exchange Rates ◽  
Support Vector Regression ◽  
Chaos Theory ◽  
Financial Time Series ◽  
Optimization Technique ◽  
Support Vector ◽  
Data Sets ◽  
Financial Time

Accurate forecasting for future events constitutes a fascinating challenge for theoretical and for applied researches. Foreign Exchange market (FOREX) is selected in this research to represent an example of financial systems with a complex behavior. Forecasting a financial time series can be a very hard task due to the inherent uncertainty nature of these systems. It seems very difficult to tell whether a series is stochastic or deterministic chaotic or some combination of these states. More generally, the extent to which a non-linear deterministic process retains its properties when corrupted by noise is also unclear. The noise can affect a system in different ways even though the equations of the system remain deterministic. Since a single reliable statistical test for chaoticity is not available, combining multiple tests is a crucial aspect, especially when one is dealing with limited and noisy data sets like in economic and financial time series. In this research, the authors propose an improved model for forecasting exchange rates based on chaos theory that involves phase space reconstruction from the observed time series and the use of support vector regression (SVR) for forecasting.Given the exchange rates of a currency pair as scalar observations, observed time series is first analyzed to verify the existence of underlying nonlinear dynamics governing its evolution over time. Then, the time series is embedded into a higher dimensional phase space using embedding parameters.In the selection process to find the optimal embedding parameters,a novel method based on the Differential Evolution (DE) geneticalgorithm(as a global optimization technique) was applied. The authors have compared forecasting accuracy of the proposed model against the ordinary use of support vector regression. The experimental results demonstrate that the proposed method, which is based on chaos theory and genetic algorithm,is comparable with the existing approaches.

scholarly journals Local Functional Coefficient Autoregressive Model for Multistep Prediction of Chaotic Time Series

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Liyun Su ◽  
Chenlong Li
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Chaos Theory ◽  
Nearest Neighbor ◽  
Simulated Data ◽  
Real Data ◽  
Chaotic Time Series ◽  
Dimensional Phase Space ◽  
Local Functional ◽  
Functional Coefficient

A new methodology, which combines nonparametric method based on local functional coefficient autoregressive (LFAR) form with chaos theory and regional method, is proposed for multistep prediction of chaotic time series. The objective of this research study is to improve the performance of long-term forecasting of chaotic time series. To obtain the prediction values of chaotic time series, three steps are involved. Firstly, the original time series is reconstructed inm-dimensional phase space with a time delayτby using chaos theory. Secondly, select the nearest neighbor points by using local method in them-dimensional phase space. Thirdly, we use the nearest neighbor points to get a LFAR model. The proposed model’s parameters are selected by modified generalized cross validation (GCV) criterion. Both simulated data (Lorenz and Mackey-Glass systems) and real data (Sunspot time series) are used to illustrate the performance of the proposed methodology. By detailed investigation and comparing our results with published researches, we find that the LFAR model can effectively fit nonlinear characteristics of chaotic time series by using simple structure and has excellent performance for multistep forecasting.

scholarly journals Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Anna Krakovská ◽  
Kristína Mezeiová ◽  
Hana Budáčová
Keyword(s):  
Time Series ◽  
State Space ◽  
Time Window ◽  
Nearest Neighbour ◽  
Embedding Dimension ◽  
State Space Reconstruction ◽  
Dimensional Phase Space ◽  
Nearest Neighbours ◽  
Takens Theorem ◽  
The One

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

scholarly journals CLUSTERING TIME SERIES OF REPEATED SCAN DATA OF SANDY BEACHES

2019 ◽  
Vol XLII-2/W13 ◽  
pp. 1039-1046
Author(s):  
R. Lindenbergh ◽  
S. van der Kleij ◽  
M. Kuschnerus ◽  
S. Vos ◽  
S. de Vries
Keyword(s):  
Time Series ◽  
Laser Scanning ◽  
Sandy Beaches ◽  
Data Sets ◽  
Data Set ◽  
Spatially Correlated ◽  
Long Time ◽  
Clustering Approach ◽  
Scan Data ◽  
Year 2000

<p><strong>Abstract.</strong> Sandy beaches are highly dynamic areas affected by different natural and anthropogenic effects. Large changes, caused by a storm for example, are in general well-understood and easy to measure. Most times, only small changes, at the centimeter scale, are occurring, but these changes accumulate significantly over periods from weeks to months. Laser scanning is a suitable technique to measure such small signals, as it is able to obtain dense 3D terrain data at centimeter level in a time span of minutes. In this work we consider two repeated laser scan data sets of two different beaches in The Netherlands. The first data set is from around the year 2000 and consists of six consecutive yearly airborne laser scan data sets of a beach on Texel. The second data set is from 2017 and consists of 30 consecutive daily terrestrial scans of a beach near The Hague. So far, little work has been done on time series analysis of repeated scan data. To obtain a first grouping of morphologic processes, we propose to use a simple un-supervised clustering approach, k-means clustering, on de-leveled, cumulative point-wise time series. The results for both regions of interest, obtained using k=5 and k=10 clusters, indicate that such clustering gives a meaningful decomposition of the morphological laser scan data into clusters that exhibit similar change patterns. At the same time, we realize that the chosen approach is just a first step in a wide open topic of clustering spatially correlated long time series of morphological laser scan data as are now obtained by permanent laser scanning.</p>

ARIMA Algebra

2019 ◽  
pp. 11-47
Author(s):  
David McDowall ◽  
Richard McCleary ◽  
Bradley J. Bartos
Keyword(s):  
Time Series ◽  
White Noise ◽  
Moving Average ◽  
Nonstationary Process ◽  
Time Frame ◽  
Stationary Time Series ◽  
Underlying Process ◽  
The Past ◽  
Seasonal Model ◽  
The One

Chapter 2 introduces ARIMA algebra. With a few exceptions, this material mirrors the authors’ earlier work. The chapter begins with stationary time series processes – white noise, moving average (MA), and autoregressive (AR) processes – and moves predictably to non-stationary and multiplicative (seasonal) models. Stationarity implies that the time series process operated identically in the past as it does in the present and that it will continue to operate identically in the future. Without stationarity, the properties of the time series would vary with the time frame and no inferences about the underlying process would be possible. A seasonally nonstationary process drifts or trends in annual steps. The “best” seasonal model structure is the one that transforms the series to white noise with the fewest number of parameters.

scholarly journals An application of two techniques for the analysis of short, multivariate non-stationary time-series of Mauritanian trawl survey data

2005 ◽  
Vol 62 (3) ◽  
pp. 353-359 ◽  
Author(s):  
Karim Erzini ◽  
Cheikh A. O. Inejih ◽  
Kim A. Stobberup
Keyword(s):  
Time Series ◽  
Factor Analysis ◽  
Fishing Effort ◽  
Stationary Time Series ◽  
Dynamic Factor ◽  
Data Sets ◽  
Trawl Survey ◽  
Explanatory Variables ◽  
Nao Index ◽  
Complementary Techniques

Abstract Min/max autocorrelation factor analysis (MAFA) and dynamic factor analysis (DFA) are complementary techniques for analysing short (>15–25 y), non-stationary, multivariate data sets. We illustrate the two techniques using catch rate (cpue) time-series (1982–2001) for 17 species caught during trawl surveys off Mauritania, with the NAO index, an upwelling index, sea surface temperature, and an index of fishing effort as explanatory variables. Both techniques gave coherent results, the most important common trend being a decrease in cpue during the latter half of the time-series, and the next important being an increase during the first half. A DFA model with SST and UPW as explanatory variables and two common trends gave good fits to most of the cpue time-series.

Research on Chaotic Characteristics of Diary Return Water Time Series in Qingtongxia Irrigation Area

2014 ◽  
Vol 998-999 ◽  
pp. 1429-1434
Author(s):  
Xin Yu Zhao ◽  
Jin Long Gao ◽  
Peng Fei Gu ◽  
Pei Huang
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Time Series Analysis ◽  
White Noise ◽  
Correlation Dimension ◽  
Stationary Time Series ◽  
Irrigation Area ◽  
Research Results ◽  
Analysis Phase ◽  
Return Water

For the problems that the characteristics of diary return water time series in Ningxia Qingtongxia Irrigation area ,this issue studied it by using time series analysis, phase space reconstruction and saturation correlation dimension and other methods.The research results showed that diary return water time series in Qingtongxia irrigation area is a non-white noise stationary time series, and it has the chaotic characteristics, the saturation correlation dimension of it is 1.81, the embedding dimensionality of it is 9.

Sign in / Sign up

Export Citation Format

Share Document