Udo Schwarz

Measures of complexity in signal analysis

Observational data of natural systems, as measured in astrophysical, geophysical or physiological experiments are typically quite different 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. e‐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.

The unscented Kalman filter : a powerful tool for data analysis

We present a derivation of the unscented Kalman filter (UKF) as an approximation to the optimal Bayesian filter equations. The potentials of the UKF are then demonstrated for the problem of simultaneous estimation of states and parameters from noise corrupted data of nonlinear dynamical systems. The UKF even works for the chaotic Chua system which includes nondifferentiable terms.

Entropy, Complexity, Predictability, and Data Analysis of Time Series and Letter Sequences

The structure of time series and letter sequences is investigated using the concepts of entropy and complexity. First, conditional entropy and mutual information are introduced and several generalizations are discussed. Further, several measures of complexity are introduced and discussed. The capability of these concepts to describe the structure of time series and letter sequences generated by nonlinear maps, data series from meteorology, astrophysics, cardiology, cognitive psychology, and finance is investigated. The relation between the complexity and the predictability of information strings is discussed. The relation between local order and the predictability of time series is investigated.

IS THE SOLAR ACTIVITY CYCLE SYNCHRONIZED WITH THE SOLAR INERTIAL MOTION

The 300 year record of the yearly sunspot numbers and numerically generated trajectory of the solar inertial motion (SIM) were subjects of a synchronization analysis. Phase synchronization of the sunspot cycle and a fast component of the SIM have been found and confirmed with statistical significance in three epochs (1727–1757, 1802–1832 and 1863–1922) of the entire 1700–1997 record. This result can be considered as a quantitative support for the hypothesis that there is a weak interaction of gravity and solar activity.

Linear and nonlinear characterization of surfaces from a laser beam melt ablation process

We apply linear and nonlinear methods to study the properties of surfaces generated by a laser beam melt ablation process. As a result we present a characterization and ordering of the surfaces depending on the adjusted process parameters. Our findings give some insight into the performance of two widely applied multifractal analysis methods—the detrended fluctuation analysis and the wavelet transform modulus maxima method—on short real world data.

Comparison of different methods for the evaluation of treatment effects from the sleep EEG of patients with major depression.

In healthy subjects, sleep has a typical structure of three to five cyclic transitions between different sleep states. In major depression, this regular pattern is often destroyed but can be reestablished during successful treatment. The differences between healthy and abnormal sleep are generally assessed in a time-consuming process, which consists of determining the nightly variations of the sleep states (the hypnogram) based on visual inspection of the electroencephalogram (EEG), electrooculogram, and electromyogram. In this study, three different methods of sleep EEG analysis (spectrum, outlier, and recurrence analysis) have been examined with regard to their ability to extract information about treatment effects in patients with major depression. Our data suggest that improved sleep patterns during treatment with antidepressant medication can be identified with an appropriate analysis of the EEG. By comparing different methods, we have found that many treatment effects identified by spectrum analysis can be reproduced by the much simpler technique of outlier analysis. Finally, the cyclic structure of sleep and its modification by antidepressant treatment is best illustrated by a non-linear approach, the so-called recurrence method.

Analysis of solar spike events by means of symbolic dynamics methods

Using quantities of symbolic dynamics, such as mutual information, Shannon information and algorithmic complexity, we have searched for interrelations of spikes emitted simultaneously at different frequencies during the impulsive phase of a flare event. As the spikes are related to the flare energy release and are interpreted as emissions originating at different sites having different magnetic field strengths, any relation in frequency is interpretated as a relation in space. This approach is appropriate to characterize such spatio-temporal patterns, whereas the popular estimate of fractal dimensions can be applied to low-dimensional systems only

ANALYSIS OF HIGH-RESOLUTION MICROELECTRODE EEG RECORDINGS IN AN ANIMAL MODEL OF SPONTANEOUS LIMBIC SEIZURES

We perform a systematic data analysis on high resolution (0.5–12 kHz) multiarray microelectrode recordings from an animal model of spontaneous limbic epilepsy, to investigate the role of high frequency oscillations and the occurrence of early precursors for seizures. Results of spectral analysis confirm the importance of very high frequency oscillations (even greater than 600 Hz) in normal (healthy) and abnormal (epileptic) hippocampus. Furthermore, we show that the measures of Recurrence Quantification Analysis (RQA) and Recurrence Time Statistics (RTS) are successful in indicating, rather uniquely, the onset of ictal state and the occurrence of some warnings/precursors during the pre-ictal state, in contrast to the linear measures investigated.

Predicting thermal displacements in modular tool systems

In the last decade, there has been an increasing interest in compensating thermally induced errors to improve the manufacturing accuracy of modular tool systems. These modular tool systems are interfaces between spindle and workpiece and consist of several complicatedly formed parts. Their thermal behavior is dominated by nonlinearities, delay and hysteresis effects even in tools with simpler geometry and it is difficult to describe it theoretically. Due to the dominant nonlinear nature of this behavior the so far used linear regression between the temperatures and the displacements is insufficient. Therefore, in this study we test the hypothesis whether we can reliably predict such thermal displacements via nonlinear temperature-displacement regression functions. These functions are estimated first from learning measurements using the alternating conditional expectation (ACE) algorithm and then tested on independent data sets. First, we analyze data that were generated by a finite element spindle model. We find that our approach is a powerful tool to describe the relation between temperatures and displacements for simulated data. Next, we analyze the temperature-displacement relationship in a silent real experimental setup, where the tool system is thermally forced. Again, the ACE algorithm is powerful to estimate the deformation with high precision. The corresponding errors obtained by using the nonlinear regression approach are 10-fold lower in comparison to multiple linear regression analysis. Finally, we investigate the thermal behavior of a modular tool system in a working milling machine and again get promising results. The thermally induced errors can be estimated with 1-2 microm accuracy using this nonlinear regression analysis. Therefore, this approach seems to be very useful for the development of new modular tool systems.

MODELING THERMAL DISPLACEMENTS IN MODULAR TOOL SYSTEMS

There is an important interest in compensating thermally induced errors of modular tool systems to improve the manufacturing accuracy. In this paper, we test the hypothesis whether we can predict s...