Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series
TL;DR: An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems, applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.
Abstract: An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems. The method is applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.
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TL;DR: A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.
2,942 citations
Cites methods from "Nonlinear forecasting as a way of d..."
...The amount of computation for the Wales method [38] (based on [36]) is also greater, although it is comparable to the present approach....
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...22 [36] G....
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TL;DR: A new method, based on nonlinear state space reconstruction, that can distinguish causality from correlation is introduced, and extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm).
Abstract: Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.
1,591 citations
Cites background or methods from "Nonlinear forecasting as a way of d..."
...Such statedependent behavior is a defining hallmark of complex nonlinear systems (3, 4), and nonlinearity is ubiquitous in nature (3–11)....
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...Note that CCM is related to the general notion of cross prediction (3, 25) but with important differences....
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...In more detail, CCM looks for the signature of X in Y’s time series by seeing if there is a correspondence between the “library” of points in the attractor manifold built from Y, MY, and points in the X manifold, MX; these two manifolds are constructed from laggedcoordinates of the time series variables Y and X respectively (3, 19, 24) (movies S1 and S2)....
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...This enables us to estimate states across manifolds using Y to estimate the state of X and vice-versa using nearest neighbors (3)....
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...This means that each variable can identify the state of the other (3, 19, 20, 24, 25) (e....
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TL;DR: It is shown that the dynamics of the reference (weight) vectors during the input-driven adaptation procedure are determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter and resemble the dynamicsof Brownian particles moving in a potential determined by a data point density.
Abstract: A neural network algorithm based on a soft-max adaptation rule is presented. This algorithm exhibits good performance in reaching the optimum minimization of a cost function for vector quantization data compression. The soft-max rule employed is an extension of the standard K-means clustering procedure and takes into account a neighborhood ranking of the reference (weight) vectors. It is shown that the dynamics of the reference (weight) vectors during the input-driven adaptation procedure are determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter and resemble the dynamics of Brownian particles moving in a potential determined by the data point density. The network is used to represent the attractor of the Mackey-Glass equation and to predict the Mackey-Glass time series, with additional local linear mappings for generating output values. The results obtained for the time-series prediction compare favorably with the results achieved by backpropagation and radial basis function networks. >
1,504 citations
TL;DR: In this paper, the authors describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos and present a variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation.
Abstract: We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.
1,381 citations
TL;DR: A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters.
Abstract: Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from this http URL . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.
1,356 citations
References
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TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.
4,619 citations
TL;DR: In this article, a measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable, and the relation of this measure to fractal dimension and information-theoretic entropy is discussed.
Abstract: A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
4,323 citations
TL;DR: An error estimate is presented for this forecasting technique for chaotic data, and its effectiveness is demonstrated by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow.
Abstract: We present a forecasting technique for chaotic data. After embedding a time series in a state space using delay coordinates, we ``learn'' the induced nonlinear mapping using local approximation. This allows us to make short-term predictions of the future behavior of a time series, using information based only on past values. We present an error estimate for this technique, and demonstrate its effectiveness by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow.
1,836 citations
TL;DR: Numerical techniques are presented for constructing nonlinear predictive models directly from time series data and scaling laws are developed which describe the data requirements for reliable predictions.
1,376 citations