In an attempt to find parameters of a time series which are absolutely robust with respect to non... more In an attempt to find parameters of a time series which are absolutely robust with respect to nonlinear distortion, we introduce a function called the entropy profile which measures in some sense the distance between the given process and white noise. This concept combines a clear definition and a simple algorithm, which apply to arbitrary stationary time series, with an informative graphical representation similar to the Fourier spectrum. For sequences derived from onedimensional maps, the entropy profile indicates periodic and almost periodic behavior and the presence of Markov partitions.
The possibility of state prediction in deterministic chaotic systems, which are described by 1-D ... more The possibility of state prediction in deterministic chaotic systems, which are described by 1-D maps, is discussed in the light o f information theory. A quantity h(l) is defined which represents the production of uncertainty on a future state by the chaotic dynamics (intrinsic noise) after / time steps have passed. h(l) is related to the Lyapunov characteristic exponent. Moreover, the influence of the measuring process (overlappings o f mapped boxes o f state space partition) and external noise on the state predictability are investigated quantitatively.
. This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing c... more . This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing called ranking. It is of special relevance in nonlinear time seriesanalysis and may cause several computational advantages. The second subject isthe definition and estimation of a generalized mutual information which is useful toanalyse statistical dependences in scalar or multivariate time series. A fast algorithmfor
ABSTRACT . We investigate the cardiorespiratory system of a newborn piglet during REM and NON--RE... more ABSTRACT . We investigate the cardiorespiratory system of a newborn piglet during REM and NON--REM sleep as well as general anesthesia, hypoxia, and cholinergic blockade. The coordinated behavior of heart rate fluctuation and respiratory movement reflects essential capabilities of the autonomic coordination. A corresponding multivariate data analysis was done by means of several nonlinear methods: generalized mutual information, redundancy and surrogate data, window pattern entropy, and computation of phase relations. Some of them are applied for the first time in this context. 1 Introduction The stable operation of organisms is based on the complex coordination between several physiological subsystems. In the present paper we use different nonlinear techniques of multivariate time series analysis to address corresponding interactions within the autonomic nervous system (ANS). Relevant nonlinear properties of the heart rate dynamics are confirmed which may improve concepts of medical treatment...
. This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing c... more . This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing called ranking. It is of special relevance in nonlinear time seriesanalysis and may cause several computational advantages. The second subject isthe definition and estimation of a generalized mutual information which is useful toanalyse statistical dependences in scalar or multivariate time series. A fast algorithmfor
We propose a new method to analyse the spatio-temporal statistical dependencies in multivariate t... more We propose a new method to analyse the spatio-temporal statistical dependencies in multivariate time series. When applying this universal method to magnetoencephalogram (MEG) data, we can visualize the spatio-temporal information flow in the human cortex on a certain level of coarse graining.
Nonuniform separation of orbits initially close to each other is measured by several quantities w... more Nonuniform separation of orbits initially close to each other is measured by several quantities which are derived from the statistics ofgrowth rates ofsmall perturbations. Using these measuresof nonuniformity, a Belousov-Zhabotinsky map (BZ map), the logistic map, and the tent map are compared. The extremely nonuniform BZ map shows a remarkable response to external noise: the state predictability can be improved by an increase in noise power.
We introduce complexity parameters for time series based on comparison of neighboring values. The... more We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
ABSTRACT We propose two methods to measure all (linear and nonlinear) statistical dependences in ... more ABSTRACT We propose two methods to measure all (linear and nonlinear) statistical dependences in a stationary time series. Presuming ergodicity, the measures can be obtained from efficient numerical algorithms. KEY WORDS: Entropy; time series analysis; mutual information; contingency; chaotic map; quadratic map. 1 Introduction The measurement of statistical dependences is one of the fundamental problems in time series analysis. For example, given a finite data sequence, there is reason to look for a predictor if there are statistical dependences between past and future states. In data compressing coding system statistical dependences between letters are used to reduce the Bit rate (see e. g. [2, 13, 22]). There are several quantities and algorithms to measure statistical dependences. All of them have their advantages and limitations. In section 2 we give a short review of some well--known "classical" methods which facilitates an evaluation of our procedures. In section 3 we propose our first m...
The European Physical Journal Special Topics, 2013
ABSTRACT We introduce a quantity called LE-statistic. It is an easily computable functional of or... more ABSTRACT We introduce a quantity called LE-statistic. It is an easily computable functional of ordinal data with versatile applications. We demonstrate its usefulness as a statistic in a nonparametric independence test of paired samples, and as a complexity measure of a scalar time series. For chaotic orbits of one-dimensional dynamical systems it is related to the Lyapunov characteristic exponent.
For piecewise monotone interval maps, we show that the Kolmogorov Sinai entropy can be obtained f... more For piecewise monotone interval maps, we show that the Kolmogorov Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.
The maintenance of balance while sitting or standing requires a control mechanism which can maint... more The maintenance of balance while sitting or standing requires a control mechanism which can maintain upright posture as well as adapt quickly and exibly to changes in the environment. Some sort of dynamical control must link visual, auditory, vestibular, and proprioceptive perceptual input to the motoric responses required to activate appropriate muscle groups in order to maintain balance. This dynamical control mechanism needs to use perceptual input to predict the future state of posture with respect to the environment if adaptive balance is to be maintained under changing conditions. These constraints suggest that a purely stochastic random-walk postural control system is unlikely, although others have been unable to reject a linear stochastic model for postural control of quiet standing.
In an attempt to find parameters of a time series which are absolutely robust with respect to non... more In an attempt to find parameters of a time series which are absolutely robust with respect to nonlinear distortion, we introduce a function called the entropy profile which measures in some sense the distance between the given process and white noise. This concept combines a clear definition and a simple algorithm, which apply to arbitrary stationary time series, with an informative graphical representation similar to the Fourier spectrum. For sequences derived from onedimensional maps, the entropy profile indicates periodic and almost periodic behavior and the presence of Markov partitions.
The possibility of state prediction in deterministic chaotic systems, which are described by 1-D ... more The possibility of state prediction in deterministic chaotic systems, which are described by 1-D maps, is discussed in the light o f information theory. A quantity h(l) is defined which represents the production of uncertainty on a future state by the chaotic dynamics (intrinsic noise) after / time steps have passed. h(l) is related to the Lyapunov characteristic exponent. Moreover, the influence of the measuring process (overlappings o f mapped boxes o f state space partition) and external noise on the state predictability are investigated quantitatively.
. This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing c... more . This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing called ranking. It is of special relevance in nonlinear time seriesanalysis and may cause several computational advantages. The second subject isthe definition and estimation of a generalized mutual information which is useful toanalyse statistical dependences in scalar or multivariate time series. A fast algorithmfor
ABSTRACT . We investigate the cardiorespiratory system of a newborn piglet during REM and NON--RE... more ABSTRACT . We investigate the cardiorespiratory system of a newborn piglet during REM and NON--REM sleep as well as general anesthesia, hypoxia, and cholinergic blockade. The coordinated behavior of heart rate fluctuation and respiratory movement reflects essential capabilities of the autonomic coordination. A corresponding multivariate data analysis was done by means of several nonlinear methods: generalized mutual information, redundancy and surrogate data, window pattern entropy, and computation of phase relations. Some of them are applied for the first time in this context. 1 Introduction The stable operation of organisms is based on the complex coordination between several physiological subsystems. In the present paper we use different nonlinear techniques of multivariate time series analysis to address corresponding interactions within the autonomic nervous system (ANS). Relevant nonlinear properties of the heart rate dynamics are confirmed which may improve concepts of medical treatment...
. This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing c... more . This chapter is concerned with two subjects. The first one is a method ofsignal preprocessing called ranking. It is of special relevance in nonlinear time seriesanalysis and may cause several computational advantages. The second subject isthe definition and estimation of a generalized mutual information which is useful toanalyse statistical dependences in scalar or multivariate time series. A fast algorithmfor
We propose a new method to analyse the spatio-temporal statistical dependencies in multivariate t... more We propose a new method to analyse the spatio-temporal statistical dependencies in multivariate time series. When applying this universal method to magnetoencephalogram (MEG) data, we can visualize the spatio-temporal information flow in the human cortex on a certain level of coarse graining.
Nonuniform separation of orbits initially close to each other is measured by several quantities w... more Nonuniform separation of orbits initially close to each other is measured by several quantities which are derived from the statistics ofgrowth rates ofsmall perturbations. Using these measuresof nonuniformity, a Belousov-Zhabotinsky map (BZ map), the logistic map, and the tent map are compared. The extremely nonuniform BZ map shows a remarkable response to external noise: the state predictability can be improved by an increase in noise power.
We introduce complexity parameters for time series based on comparison of neighboring values. The... more We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.
ABSTRACT We propose two methods to measure all (linear and nonlinear) statistical dependences in ... more ABSTRACT We propose two methods to measure all (linear and nonlinear) statistical dependences in a stationary time series. Presuming ergodicity, the measures can be obtained from efficient numerical algorithms. KEY WORDS: Entropy; time series analysis; mutual information; contingency; chaotic map; quadratic map. 1 Introduction The measurement of statistical dependences is one of the fundamental problems in time series analysis. For example, given a finite data sequence, there is reason to look for a predictor if there are statistical dependences between past and future states. In data compressing coding system statistical dependences between letters are used to reduce the Bit rate (see e. g. [2, 13, 22]). There are several quantities and algorithms to measure statistical dependences. All of them have their advantages and limitations. In section 2 we give a short review of some well--known "classical" methods which facilitates an evaluation of our procedures. In section 3 we propose our first m...
The European Physical Journal Special Topics, 2013
ABSTRACT We introduce a quantity called LE-statistic. It is an easily computable functional of or... more ABSTRACT We introduce a quantity called LE-statistic. It is an easily computable functional of ordinal data with versatile applications. We demonstrate its usefulness as a statistic in a nonparametric independence test of paired samples, and as a complexity measure of a scalar time series. For chaotic orbits of one-dimensional dynamical systems it is related to the Lyapunov characteristic exponent.
For piecewise monotone interval maps, we show that the Kolmogorov Sinai entropy can be obtained f... more For piecewise monotone interval maps, we show that the Kolmogorov Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.
The maintenance of balance while sitting or standing requires a control mechanism which can maint... more The maintenance of balance while sitting or standing requires a control mechanism which can maintain upright posture as well as adapt quickly and exibly to changes in the environment. Some sort of dynamical control must link visual, auditory, vestibular, and proprioceptive perceptual input to the motoric responses required to activate appropriate muscle groups in order to maintain balance. This dynamical control mechanism needs to use perceptual input to predict the future state of posture with respect to the environment if adaptive balance is to be maintained under changing conditions. These constraints suggest that a purely stochastic random-walk postural control system is unlikely, although others have been unable to reject a linear stochastic model for postural control of quiet standing.
Das Buch führt in die Physik der
nichtlinearen dissipativen (dynamischen) Systeme ein.
Trotz des... more Das Buch führt in die Physik der nichtlinearen dissipativen (dynamischen) Systeme ein. Trotz des deterministischen Charakters der (nichtlinearen) Bewegungsgleichungen kann die Zustandsentwicklung hier oftmals nur sehr schlecht vorhergesagt werden. Im Buch werden allgemeine theoretische Konzepte zur Beschreibung dieser Systeme vorgestellt und mit vielen konkreten Beispielen illustriert. Insbesondere ist dies ein parametrisch erregtes Pendel. Das Buch richtet sich vor allem an Studenten der Physik oder anderer naturwissenschaftlicher Studiengänge höherer Semester.
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nichtlinearen dissipativen (dynamischen) Systeme ein.
Trotz des deterministischen Charakters der
(nichtlinearen) Bewegungsgleichungen
kann die Zustandsentwicklung hier oftmals nur sehr
schlecht vorhergesagt werden.
Im Buch werden allgemeine theoretische Konzepte zur Beschreibung dieser Systeme vorgestellt und mit vielen konkreten Beispielen illustriert.
Insbesondere ist dies ein parametrisch erregtes Pendel.
Das Buch richtet sich vor allem an Studenten der Physik
oder anderer naturwissenschaftlicher Studiengänge höherer Semester.