The Experts below are selected from a list of 24288 Experts worldwide ranked by ideXlab platform
Caixia Yang - One of the best experts on this subject based on the ideXlab platform.
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Estimation of Lyapunov exponents from a time series for n-dimensional state space using Nonlinear Mapping
Nonlinear Dynamics, 2012Co-Authors: Caixia Yang, Christine Qiong Wu, Pei ZhangAbstract:It has been demonstrated that when estimating Lyapunov exponents using a time series, Nonlinear Mapping used for characterizing the evolution of the neighbors leads to more accurate negative exponents and is more robust to noise in the times series. However, the number of unknown elements of the matrices associated with Nonlinear Mapping increases significantly with the embedding dimensions of the state space where the dynamics is reconstructed. Such unknown coefficients are solved from a set of linear algebraic equations based on the least square-root fit method. Derivation of such linear equations and computer programming are tedious and error prone especially for the systems with high embedding dimensions. In this work, we develop a general form of the linear algebraic equations and the corresponding computer program in terms of arbitrary embedding dimensions. A stable robotic system with all negative Lyapunov exponents and the Lorenz system with positive, zero, and negative exponents are used to demonstrate the efficacy of the proposed method. The work can contribute significantly to estimating Lyapunov exponents for systems with large embedding dimensions.
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on stability analysis via lyapunov exponents calculated from a time series using Nonlinear Mapping a case study
Nonlinear Dynamics, 2010Co-Authors: Caixia YangAbstract:The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear Mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the Nonlinear approximation of the local neighborhood-to-neighborhood Mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear Mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear Mapping, (1) the accuracy of the negative exponents calculated using the Nonlinear Mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.
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On stability analysis via Lyapunov exponents calculated from a time series using Nonlinear Mapping—a case study
Nonlinear Dynamics, 2009Co-Authors: Caixia Yang, Qiong WuAbstract:The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear Mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the Nonlinear approximation of the local neighborhood-to-neighborhood Mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear Mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear Mapping, (1) the accuracy of the negative exponents calculated using the Nonlinear Mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.
Algirdas Mykolas Montvilas - One of the best experts on this subject based on the ideXlab platform.
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Features of Sequential Nonlinear Mapping
Informatica, 2003Co-Authors: Algirdas Mykolas MontvilasAbstract:In the paper the sequential Nonlinear Mapping has been investigated in order to reveal its features. The method was investigated by a plenty of experiments using various sorts of data. For illustrations there are presented results using “marginal” data: the first data gives the smallest Mapping error, and the other data gives the largest one. The sequential Nonlinear Mapping has been investigated according ability to differ the data groups (clustering) when at the beginning the number of groups is taken to be less than really exists. It was showed that the sequential Nonlinear Mapping differs the groups of data well even though the number of them is taken to be less by one than really exists. The experiments show that the factor for correction co-ordinates on the plane for the sequential Nonlinear Mapping can be taken in the range from 0.25 to 0.75. Mapping errors depend on both the sort of initial conditions and the nature of data.
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Sequential Nonlinear Mapping versus Simultaneous One
Informatica (lithuanian Academy of Sciences), 2002Co-Authors: Algirdas Mykolas MontvilasAbstract:In the paper two methods for data structure analysis and visualisation are presented: the simultaneous Nonlinear Mapping (Sammon, 1969) and the sequential one (Montvilas, 1995). These two methods were compared according ability to map the data on the plane, Mapping accuracy and a Mapping time. It was showed that the sequential Nonlinear Mapping has some bigger total Mapping error but needs considerable less calculation time than that of the simultaneous one. Examples are given.
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Pre-processing of Data for Nonlinear Mapping
Informatica (lithuanian Academy of Sciences), 2001Co-Authors: Algirdas Mykolas MontvilasAbstract:The sequential Nonlinear Mapping is suitable for sequential detection of states of dynamic systems (Montvilas, 1999a). In addition, it can indicate the undesirable states and even the damages of dynamic systems. The last is complicated when the damage is caused by a small changing of respective parameter describing the state. In the paper the problem of Nonlinear Mapping to be sensitive for the small changing of parameters and the problem related with dimensionality of parameters are solved by using a special pre-processing of data. Examples are given.
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On sequential Nonlinear Mapping for data structure analysis
Informatica (lithuanian Academy of Sciences), 1995Co-Authors: Algirdas Mykolas MontvilasAbstract:An algorithm for the sequential analysis of multivariate data structure is presented. The algorithm is based on the sequential Nonlinear Mapping of L-dimensional vectors from the L-hyperspace into a lower-dimensional (two-dimensional) vectors such that the inner structure of distances among the vectors is preserved. Expressions for the sequential Nonlinear Mapping are obtained. The Mapping error function is chosen. Theoretical minimum amount of the very beginning simultaneously mapped vectors is obtained.
S Hasselmann - One of the best experts on this subject based on the ideXlab platform.
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on the Nonlinear Mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion
Journal of Geophysical Research, 1991Co-Authors: K Hasselmann, S HasselmannAbstract:A new, closed Nonlinear integral transformation relation is derived describing the Mapping of a two-dimensional ocean wave spectrum into a synthetic aperture radar (SAR) image spectrum. The general integral relation is expanded in a power series with respect to orders of Nonlinearity and velocity bunching. The individual terms of the series can be readily computed using fast Fourier transforms. The convergence of the series is rapid. The series expansion is also useful in identifying the different contributions to the net imaging process, consisting of the real aperture radar (RAR) cross-section modulation, the Nonlinear motion (velocity bunching) effects, and their various interaction products. The lowest term of the expansion with respect to Nonlinearity order yields a simple quasi-linear approximate Mapping relation consisting of the standard linear SAR modulation expression multiplied by an additional Nonlinear Gaussian azimuthal cutoff factor. The cutoff scale is given by the rms azimuthal (velocity bunching) displacement. The same cutoff factor applies to all terms of the power series expansion. The Nonlinear Mapping relation is inverted using a standard first-guess wave spectrum as regularization term. This is needed to overcome the basic 180° Mapping ambiguity and the loss of information beyond the azimuthal cutoff. The inversion is solved numerically using an iteration technique based on the successive application of the explicit solution for the quasi-linear Mapping approximation, with interposed corrections invoking the full Nonlinear Mapping expression. A straightforward application of this technique, however, generally yields unrealistic discontinuities of the best fit wave spectrum in the transition region separating the low azimuthal wave number domain, in which useful SAR information is available and the wave spectrum is modified, from the high azimuthal wave number region beyond the azimuthal cutoff, where the first-guess wave spectrum is retained. This difficulty is overcome by applying a two-step inversion procedure. In the first step the energy level of the wave spectrum is adjusted, and the wave number plane rotated and rescaled, without altering the shape of the spectrum. Using the resulting globally fitted spectrum as the new first-guess input spectrum, the original inversion method is then applied without further constraints in a second step to obtain a final fine-scale optimized spectrum. The forward Mapping relation and inversion algorithms are illustrated for three Seasat cases representing different wave conditions corresponding to weakly, moderately, and strongly Nonlinear imaging conditions.
Qiong Wu - One of the best experts on this subject based on the ideXlab platform.
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On stability analysis via Lyapunov exponents calculated from a time series using Nonlinear Mapping—a case study
Nonlinear Dynamics, 2009Co-Authors: Caixia Yang, Qiong WuAbstract:The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear Mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the Nonlinear approximation of the local neighborhood-to-neighborhood Mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear Mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear Mapping, (1) the accuracy of the negative exponents calculated using the Nonlinear Mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.
Yitzhak Romem - One of the best experts on this subject based on the ideXlab platform.
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on pattern classification with sammon s Nonlinear Mapping an experimental study
Pattern Recognition, 1998Co-Authors: Boaz Lerner, Hugo Guterman, Mayer Aladjem, Itshak Dinsteint, Yitzhak RomemAbstract:Abstract Sammon's Mapping is conventionally used for exploratory data projection, and as such is usually inapplicable for classification. In this paper we apply a neural network (NN) implementation of Sammon's Mapping to classification by extracting an arbitrary number of projections. The projection map and classification accuracy of the Mapping are compared with those of the auto-associative NN (AANN), multilayer perceptron (MLP) and principal component (PC) feature extractor for chromosome data. We demonstrate that chromosome classification based on Sammon's (unsupervised) Mapping is superior to the classification based on the AANN and PC feature extractor and highly comparable with that based on the (supervised) MLP. c 1998 Pattern Recognition Society.
