The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Matthew B. Kennel - One of the best experts on this subject based on the ideXlab platform.
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Globally enumerating unstable periodic orbits for Observed data using symbolic dynamics.
Chaos (Woodbury N.Y.), 2007Co-Authors: Michael Buhl, Matthew B. KennelAbstract:The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an Observed Time Series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from Observed Time Series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.
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Statistically relaxing to generating partitions for Observed Time-Series data.
Physical Review E, 2005Co-Authors: Michael Buhl, Matthew B. KennelAbstract:We introduce a relaxation algorithm to estimate approximations to generating partitions for Observed dynamical Time Series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies. We construct an energylike functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to Observed data. As each Observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.
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Estimating good discrete partitions from Observed data: symbolic false nearest neighbors.
Physical review letters, 2003Co-Authors: Matthew B. Kennel, Michael BuhlAbstract:A symbolic analysis of Observed Time Series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the Observed Time Series and is sensible even in the presence of noise when no truly generating partition is possible.
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Estimating Good Discrete Partitions from Observed Data: Symbolic False Nearest Neighbors
AIP Conference Proceedings, 2003Co-Authors: Matthew B. Kennel, Michael BuhlAbstract:A symbolic analysis of Observed Time Series data typically requires making a discrete partition of a continuous state space containing observations of the dynamics. A particular kind of partition, called “generating”, preserves all dynamical information of a deterministic map in the symbolic representation, but such partitions are not obvious beyond one dimension, and existing methods to find them require significant knowledge of the dynamical evolution operator or the spectrum of unstable periodic orbits. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition: avoiding topological degeneracies. It requires only the Observed Time Series and is sensible even in the presence of noise when no truly generating partition is possible. Because of its resemblance to a geometrical statistic frequently used for reconstructing valid Time‐delay embeddings, we call the algorithm “symbolic false nearest neighbors”.
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Statistical Test for Dynamical Nonstationarity in Observed Time-Series Data
arXiv: Chaotic Dynamics, 1995Co-Authors: Matthew B. KennelAbstract:Information in the Time distribution of points in a state space reconstructed from Observed data yields a test for ``nonstationarity''. Framed in terms of a statistical hypothesis test, this numerical algorithm can discern whether some underlying slow changes in parameters have taken place. The method examines a fundamental object in nonlinear dynamics, the geometry of orbits in state space, with corrections to overcome difficulties in real dynamical data which cause naive statistics to fail.
Michael Buhl - One of the best experts on this subject based on the ideXlab platform.
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Globally enumerating unstable periodic orbits for Observed data using symbolic dynamics.
Chaos (Woodbury N.Y.), 2007Co-Authors: Michael Buhl, Matthew B. KennelAbstract:The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an Observed Time Series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from Observed Time Series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.
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Statistically relaxing to generating partitions for Observed Time-Series data.
Physical Review E, 2005Co-Authors: Michael Buhl, Matthew B. KennelAbstract:We introduce a relaxation algorithm to estimate approximations to generating partitions for Observed dynamical Time Series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies. We construct an energylike functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to Observed data. As each Observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.
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Estimating good discrete partitions from Observed data: symbolic false nearest neighbors.
Physical review letters, 2003Co-Authors: Matthew B. Kennel, Michael BuhlAbstract:A symbolic analysis of Observed Time Series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the Observed Time Series and is sensible even in the presence of noise when no truly generating partition is possible.
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Estimating Good Discrete Partitions from Observed Data: Symbolic False Nearest Neighbors
AIP Conference Proceedings, 2003Co-Authors: Matthew B. Kennel, Michael BuhlAbstract:A symbolic analysis of Observed Time Series data typically requires making a discrete partition of a continuous state space containing observations of the dynamics. A particular kind of partition, called “generating”, preserves all dynamical information of a deterministic map in the symbolic representation, but such partitions are not obvious beyond one dimension, and existing methods to find them require significant knowledge of the dynamical evolution operator or the spectrum of unstable periodic orbits. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition: avoiding topological degeneracies. It requires only the Observed Time Series and is sensible even in the presence of noise when no truly generating partition is possible. Because of its resemblance to a geometrical statistic frequently used for reconstructing valid Time‐delay embeddings, we call the algorithm “symbolic false nearest neighbors”.
Rene Garello - One of the best experts on this subject based on the ideXlab platform.
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Time Series nonlinearity modeling: a Giannakis formula type approach
Signal Processing, 2003Co-Authors: Jean-marc Le Caillec, Rene GarelloAbstract:In this paper, we propose a method for identifying the coefficients of a simplified Second Order Volterra Model (SOVM) driven by a normal i.i.d. white noise. The interest of estimating the coefficients of such a model is to easily model nonlinear Time Series by identifying a linear spectrum and a nonlinear spectrum. In fact, the nonlinear spectrum is the spectrum of output data of a quadratic system (squarer) driven by a normal i.i.d. white noise while the linear spectrum is the output data spectrum of a linear system driven by the same noise. Consequently, by estimating the linear and nonlinear spectrum components, the proposed algorithm locates (in the Fourier domain) and quantifies the nonlinear artifacts in an Observed Time Series, this Observed Time Series being the output of a nonlinear system and the input data of this system not being available. The method for estimating the model coefficients is quite simple and is based on the ratio of products of Higher Order Cumulants. For this reason, the method of identification is close to Giannakis' formula which identifies the coefficients of a linear system driven by a non symmetric noise and also uses the ratio of cumulants. In this paper, we also address the question of order selection of both parts of the simplified SOVM (i.e. the linear and quadratic parts) based on hypothesis testing, the order of each part interfering strongly in the final results. Finally, we propose a method for verifying that the higher order statistics (HOS) of the Observed Time Series are matched with the HOS derived from the estimated coefficients, thus proving that the Time Series is well modeled by the estimated nonlinear parametric model.
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Time Series nonlinearity modeling : a Giannakis formula type approach
Signal Processing, 2003Co-Authors: Jean-marc Le Caillec, Rene GarelloAbstract:In this paper, we propose a method for identifying the coefficients of a simplified Second Order Volterra Model (SOVM) driven by a normal i.i.d. white noise. The interest of estimating the coefficients of such a model is to easily model nonlinear Time Series by identifying a linear spectrum and a nonlinear spectrum. In fact, the nonlinear spectrum is the spectrum of output data of a quadratic system (squarer) driven by a normal i.i.d. white noise while the linear spectrum is the output data spectrum of a linear system driven by the same noise. Consequently, by estimating the linear and nonlinear spectrum components, the proposed algorithm locates (in the Fourier domain) and quantifies the nonlinear artifacts in an Observed Time Series, this Observed Time Series being the output of a nonlinear system and the input data of this system not being available. The method for estimating the model coefficients is quite simple and is based on the ratio of products of Higher Order Cumulants. For this reason, the method of identification is close to Giannakis' formula which identifies the coefficients of a linear system driven by a non symmetric noise and also uses the ratio of cumulants.
Devin Kilminster - One of the best experts on this subject based on the ideXlab platform.
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Estimating a generating partition from Observed Time Series: symbolic shadowing.
Physical review. E Statistical nonlinear and soft matter physics, 2004Co-Authors: Yoshito Hirata, Kevin Judd, Devin KilminsterAbstract:We propose a deterministic algorithm for approximating a generating partition from a Time Series using tessellations. Using data generated by Hénon and Ikeda maps, we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order, and provide good estimates of the metric and topological entropies. The algorithm gives useful results even with a short noisy Time Series.
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Estimating a generating partition from Observed Time Series: Symbolic shadowing
Physical Review E, 2004Co-Authors: Yoshito Hirata, Kevin Judd, Devin KilminsterAbstract:We propose a deterministic algorithm for approximating a generating partition from a Time Series using tessellations. Using data generated by H\'enon and Ikeda maps, we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order, and provide good estimates of the metric and topological entropies. The algorithm gives useful results even with a short noisy Time Series.
Christiaan Heij - One of the best experts on this subject based on the ideXlab platform.
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Positivity conditions for stochastic state space modelling of Time Series
Econometric Reviews, 1992Co-Authors: Christiaan Heij, Teun Kloek, Andre LucasAbstract:This short paper clarifies some aspects of the balancing method for state space modelling of Observed Time Series. This method may fail to satisfy the so-called positive real condition for stochastic processes. We illustrate this by theoretical spectral analysis and also by simulating univariate ARMA (1,1) models.
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A modified canonical correlation approach to approximate state space modelling
[1991] Proceedings of the 30th IEEE Conference on Decision and Control, 1Co-Authors: Christiaan Heij, Berend RoordaAbstract:The authors describe a procedure for modeling Observed Time Series by means of a linear system. A system is characterized by its behavior, i.e., the set of all Time Series compatible with the system laws. The objective is to find a simple system that contains a Time Series that is close to the Observed one. The dynamical relations between the variables are modeled in two steps. First, an approximate state trajectory is constructed, and then an approximate linear system is determined on the basis of the Observed Time Series and its state trajectory. The structure of the resulting system, i.e., the decomposition of variables into inputs and outputs and the number of state variables, is not specified a priori, but is chosen on basis of the data. The authors consider some alternative ways to construct a state trajectory, using canonical correlation and singular value analysis. The resulting procedures are illustrated by some simulations. >
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On identifiability of finite dimensional linear systems
Proceedings of the 27th IEEE Conference on Decision and Control, 1Co-Authors: Christiaan HeijAbstract:The problem of system identification on the basis of an Observed Time Series is considered for the class of linear, Time-invariant, complete systems. A concept of system identifiability is defined. A procedure for exact modeling of multivariable finite Time Series is introduced which has the desirable properties of monotonicity and convergence. The class of controllable systems is identifiable by this procedure. >
