Observational data of natural systems, as measured in astrophysical, geophysical or physiological... more Observational data of natural systems, as measured in astrophysical, geophysical or physiological experiments are typically quite di erent from those obtained in laboratories. Due to the peculiarities with these data, well-known characteristics processes, such as periodicities or fractal dimension, often do not provide a suitable description. To study such data, we present here the use of measures of complexity, which are mainly basing on symbolic dynamics. We distinguish two types of such quantities: traditional measures (e.g. algorithmic complexity) which are measures of randomness and alternative measures (e.g. -complexity) which relate highest complexity to some critical points. It is important to note that there is no optimum measure of complexity. Its choice should depend on the context. Mostly, a combination of some such quantities is appropriate. Applying this concept to three examples in astrophysics, cardiology and cognitive psychology, we show that it can be helpful also in cases where other tools of data analysis fail.
Using a special technique of data analysis, we have found 34 grand minima of solar activity in a ... more Using a special technique of data analysis, we have found 34 grand minima of solar activity in a 7700-year-long tJ. 14 C record. The method used rests on a proper filtering of the tJ. 14 C record and the extrapolation of verifiable results for the later history back in time. Additionally, we have applied a method of nonlinear dynamics, the recurrence rate, to back up the results. Our findings are not contradictory to the record of grand minima by Eddy [1977J but constitute a considerable extension. Hence it has become possible to look closer at the validity of models. This way, we have tested the model of Barnes et al. [1980J. There are hints for that the grand minima might solely be driven by the 209-year period found in the tJ. 14 C record.
Using a special technique of data analysis, we have found out 34 grand minima of solar activity o... more Using a special technique of data analysis, we have found out 34 grand minima of solar activity obtained from a 7,700 years long 14 C record. The method used rests on a proper ltering of the 14 C record and the extrapolation of veri able results for the later history back in time. Additionally, we use a method of nonlinear dynamics, the recurrence rate, to back up the results. Our ndings are not contradictory to the record of solar maxima resp. minima by Eddy 5], but constitute a considerable extension. Hence, it has become possible to look closer at the validity of models. This way, we have tested several models for solar activity, esp. the model of Barnes et al. 1]. There are hints for that the grand minima might solely be driven by the 209 year period found in the 14 C record.
We have discussed some tools from nonlinear dynamics which may help to analyze transient phenomen... more We have discussed some tools from nonlinear dynamics which may help to analyze transient phenomena, such as solar bursts. The structure function known from turbulence theory is an appropriate method to nd out some scaling behavior of uctuations in time. More generally, the wavelet analysis, which is some generalization of the power spectrum, exhibits information on the location as well as the size of hidden characteristic features. Applying both techniques to microwave bursts, we have found some scaling properties that refer to the existence of hierarchic time structures. This is in good accordance with the electric circuit model for describing the are-particle energization process.
Rock-magnetic and palaeo-magnetic data from sediments of two Italian lakes provide a reconstructi... more Rock-magnetic and palaeo-magnetic data from sediments of two Italian lakes provide a reconstruction of the Earth' magnetic field in the last 100 kyr. Since the measurements contain also the climate signal, we apply the Independent Component Analysis in order to extract the signal of the Earth' magnetic field.
Charged dust grains in circumplanetary environments experience, beyond various de- terministic fo... more Charged dust grains in circumplanetary environments experience, beyond various de- terministic forces, also stochastic perturbations: E.g., fluctuations of the magnetic field, the charge of the grains etc. Here, we investigate the dynamics of a dust population in a circular orbit around the planet which is perturbed by a stochastic magnetic field B , modeled by an isotropi- cally Gaussian white noise. The resulting perturbation equations give rise to a modi- 2 fied diffusion of the inclinations and eccentricities x D [t +/- sin[2nt]/(2n)] (x - alias for eccentricity e and the inclination i, t - time). The diffusion coefficient is found to be D = [G]2/n, where the gyrofrequency and the orbital frequency are denoted by G, and n, respectively. This behavior has been checked by numerical experiments. We have chosen dust grains (1µm in radius) initially moving in circular orbits around a planet (Jupiter) and integrated numerically their trajectories over their typical lifetimes (100 ye...
Laser beam melt ablation - a contact-free machining process - oers several advantages compared to... more Laser beam melt ablation - a contact-free machining process - oers several advantages compared to conventional processing mechanisms: there exists no tool wear and even extremely hard or brittle materials can be processed. During ablation the workpiece is molten by a CO2-laser beam, this melt is then driven out by the impulse of a process gas. The idea behind laser ablation is rather simple, but it has a major limitation in practical applications: with increasing ablation rates surface quality of the workpiece processed declines rapidly. At high ablation rates, depending on the process parameters dierent periodic-like structures can be observed on the ablated surface. These structures show a dependence on the line energy, which has been identi ed as a fundamental control parameter. In dependence on this parameter several regimes with dierent behaviours of the process have been separated. These regimes are distinguishable as well in the surfaces obtained as in the signals gained by t...
. The structure of time series and letter sequences is investigated using the concepts of entropy... more . The structure of time series and letter sequences is investigated using the concepts of entropy and complexity. First conditional entropy and transinformation are introduced and several generalizations are discussed. Further several measures of complexity are introduced and discussed. ...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
We study the problem of simultaneous estimation of parameters and unobserved states from noisy da... more We study the problem of simultaneous estimation of parameters and unobserved states from noisy data of nonlinear time-continuous systems, including the case of additive stochastic forcing. We propose a solution by adapting the recently developed statistical method of unscented Kalman filtering to this problem. Due to its recursive and derivative-free structure, this method minimizes the cost function in a computationally efficient and robust way. It is found that parameters as well as unobserved components can be estimated with high accuracy, including confidence bands, from heavily noise-corrupted data.
ABSTRACT The application of chaos theory has become popular to understand the nature of various f... more ABSTRACT The application of chaos theory has become popular to understand the nature of various features of solar activity because most of them are far from regular. The usual approach, however, that is based on finding low-dimensional structures of the underlying processes seems to be successful only in a few exceptional cases, such as in rather coherent phenomena as coronal pulsations. It is important to note that most phenomena in solar radio emission are more complex. We present two kinds of techniques from nonlinear dynamics which can be useful to analyse such phenomena: Fragmentation processes observed in solar spike events are studied by means of symbolic dynamics methods. Different measures of complexity calculated from such observations reveal that there is some order in this fragmentation. Bursts are a typical transient phenomenon. To study energization processes causing impulsive microwave bursts, the wavelet analysis is applied. It exhibits structural differences of the pre- and post-impulsive phase in cases where the power spectra of both are not distinct.
We analyse the X-ray light curves of compact objects using linear and nonlinear time series analy... more We analyse the X-ray light curves of compact objects using linear and nonlinear time series analysis methods. A Power Density Spectrum (PDS) describes the overall second order properties of the observed data well. To look beyond we propose the nonlinear Q-statistic to detect an asymmetry of the time series. This allows us to find relevant time scales. This method even grants a subclassification of the known states of X-ray sources.
We study possible interrelations between the 300-year record of the yearly sunspot numbers and th... more We study possible interrelations between the 300-year record of the yearly sunspot numbers and the solar inertial motion (SIM) using the recently developed technique of synchronization analysis. Phase synchronization of the sunspot cycle and the SIM is found and statistically confirmed in three epochs (1734-1790, 1855-1875 and 1907-1960) of the whole period 1700-2000. These results give quantitative support to the hypothesis that there is a weak interaction between the solar activity and the SIM.
We analyze the variability in the X-ray lightcurves of the black hole candidate Cygnus X-1 by lin... more We analyze the variability in the X-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the over-all second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.
Observational data of natural systems, as measured in astrophysical, geophysical or physiological... more Observational data of natural systems, as measured in astrophysical, geophysical or physiological experiments are typically quite di erent from those obtained in laboratories. Due to the peculiarities with these data, well-known characteristics processes, such as periodicities or fractal dimension, often do not provide a suitable description. To study such data, we present here the use of measures of complexity, which are mainly basing on symbolic dynamics. We distinguish two types of such quantities: traditional measures (e.g. algorithmic complexity) which are measures of randomness and alternative measures (e.g. -complexity) which relate highest complexity to some critical points. It is important to note that there is no optimum measure of complexity. Its choice should depend on the context. Mostly, a combination of some such quantities is appropriate. Applying this concept to three examples in astrophysics, cardiology and cognitive psychology, we show that it can be helpful also in cases where other tools of data analysis fail.
Using a special technique of data analysis, we have found 34 grand minima of solar activity in a ... more Using a special technique of data analysis, we have found 34 grand minima of solar activity in a 7700-year-long tJ. 14 C record. The method used rests on a proper filtering of the tJ. 14 C record and the extrapolation of verifiable results for the later history back in time. Additionally, we have applied a method of nonlinear dynamics, the recurrence rate, to back up the results. Our findings are not contradictory to the record of grand minima by Eddy [1977J but constitute a considerable extension. Hence it has become possible to look closer at the validity of models. This way, we have tested the model of Barnes et al. [1980J. There are hints for that the grand minima might solely be driven by the 209-year period found in the tJ. 14 C record.
Using a special technique of data analysis, we have found out 34 grand minima of solar activity o... more Using a special technique of data analysis, we have found out 34 grand minima of solar activity obtained from a 7,700 years long 14 C record. The method used rests on a proper ltering of the 14 C record and the extrapolation of veri able results for the later history back in time. Additionally, we use a method of nonlinear dynamics, the recurrence rate, to back up the results. Our ndings are not contradictory to the record of solar maxima resp. minima by Eddy 5], but constitute a considerable extension. Hence, it has become possible to look closer at the validity of models. This way, we have tested several models for solar activity, esp. the model of Barnes et al. 1]. There are hints for that the grand minima might solely be driven by the 209 year period found in the 14 C record.
We have discussed some tools from nonlinear dynamics which may help to analyze transient phenomen... more We have discussed some tools from nonlinear dynamics which may help to analyze transient phenomena, such as solar bursts. The structure function known from turbulence theory is an appropriate method to nd out some scaling behavior of uctuations in time. More generally, the wavelet analysis, which is some generalization of the power spectrum, exhibits information on the location as well as the size of hidden characteristic features. Applying both techniques to microwave bursts, we have found some scaling properties that refer to the existence of hierarchic time structures. This is in good accordance with the electric circuit model for describing the are-particle energization process.
Rock-magnetic and palaeo-magnetic data from sediments of two Italian lakes provide a reconstructi... more Rock-magnetic and palaeo-magnetic data from sediments of two Italian lakes provide a reconstruction of the Earth' magnetic field in the last 100 kyr. Since the measurements contain also the climate signal, we apply the Independent Component Analysis in order to extract the signal of the Earth' magnetic field.
Charged dust grains in circumplanetary environments experience, beyond various de- terministic fo... more Charged dust grains in circumplanetary environments experience, beyond various de- terministic forces, also stochastic perturbations: E.g., fluctuations of the magnetic field, the charge of the grains etc. Here, we investigate the dynamics of a dust population in a circular orbit around the planet which is perturbed by a stochastic magnetic field B , modeled by an isotropi- cally Gaussian white noise. The resulting perturbation equations give rise to a modi- 2 fied diffusion of the inclinations and eccentricities x D [t +/- sin[2nt]/(2n)] (x - alias for eccentricity e and the inclination i, t - time). The diffusion coefficient is found to be D = [G]2/n, where the gyrofrequency and the orbital frequency are denoted by G, and n, respectively. This behavior has been checked by numerical experiments. We have chosen dust grains (1µm in radius) initially moving in circular orbits around a planet (Jupiter) and integrated numerically their trajectories over their typical lifetimes (100 ye...
Laser beam melt ablation - a contact-free machining process - oers several advantages compared to... more Laser beam melt ablation - a contact-free machining process - oers several advantages compared to conventional processing mechanisms: there exists no tool wear and even extremely hard or brittle materials can be processed. During ablation the workpiece is molten by a CO2-laser beam, this melt is then driven out by the impulse of a process gas. The idea behind laser ablation is rather simple, but it has a major limitation in practical applications: with increasing ablation rates surface quality of the workpiece processed declines rapidly. At high ablation rates, depending on the process parameters dierent periodic-like structures can be observed on the ablated surface. These structures show a dependence on the line energy, which has been identi ed as a fundamental control parameter. In dependence on this parameter several regimes with dierent behaviours of the process have been separated. These regimes are distinguishable as well in the surfaces obtained as in the signals gained by t...
. The structure of time series and letter sequences is investigated using the concepts of entropy... more . The structure of time series and letter sequences is investigated using the concepts of entropy and complexity. First conditional entropy and transinformation are introduced and several generalizations are discussed. Further several measures of complexity are introduced and discussed. ...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
We study the problem of simultaneous estimation of parameters and unobserved states from noisy da... more We study the problem of simultaneous estimation of parameters and unobserved states from noisy data of nonlinear time-continuous systems, including the case of additive stochastic forcing. We propose a solution by adapting the recently developed statistical method of unscented Kalman filtering to this problem. Due to its recursive and derivative-free structure, this method minimizes the cost function in a computationally efficient and robust way. It is found that parameters as well as unobserved components can be estimated with high accuracy, including confidence bands, from heavily noise-corrupted data.
ABSTRACT The application of chaos theory has become popular to understand the nature of various f... more ABSTRACT The application of chaos theory has become popular to understand the nature of various features of solar activity because most of them are far from regular. The usual approach, however, that is based on finding low-dimensional structures of the underlying processes seems to be successful only in a few exceptional cases, such as in rather coherent phenomena as coronal pulsations. It is important to note that most phenomena in solar radio emission are more complex. We present two kinds of techniques from nonlinear dynamics which can be useful to analyse such phenomena: Fragmentation processes observed in solar spike events are studied by means of symbolic dynamics methods. Different measures of complexity calculated from such observations reveal that there is some order in this fragmentation. Bursts are a typical transient phenomenon. To study energization processes causing impulsive microwave bursts, the wavelet analysis is applied. It exhibits structural differences of the pre- and post-impulsive phase in cases where the power spectra of both are not distinct.
We analyse the X-ray light curves of compact objects using linear and nonlinear time series analy... more We analyse the X-ray light curves of compact objects using linear and nonlinear time series analysis methods. A Power Density Spectrum (PDS) describes the overall second order properties of the observed data well. To look beyond we propose the nonlinear Q-statistic to detect an asymmetry of the time series. This allows us to find relevant time scales. This method even grants a subclassification of the known states of X-ray sources.
We study possible interrelations between the 300-year record of the yearly sunspot numbers and th... more We study possible interrelations between the 300-year record of the yearly sunspot numbers and the solar inertial motion (SIM) using the recently developed technique of synchronization analysis. Phase synchronization of the sunspot cycle and the SIM is found and statistically confirmed in three epochs (1734-1790, 1855-1875 and 1907-1960) of the whole period 1700-2000. These results give quantitative support to the hypothesis that there is a weak interaction between the solar activity and the SIM.
We analyze the variability in the X-ray lightcurves of the black hole candidate Cygnus X-1 by lin... more We analyze the variability in the X-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the over-all second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.
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Papers by Udo Schwarz