We study the extraction of nonlinear data models in high-dimensional spaces with modified self-organizing maps. We present a general algorithm which maps low-dimensional lattices into high-dimensional data manifolds without violation of topology. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. Moreover we present a second algorithm for the extraction of generalized principal curves comprising disconnected and branching manifolds. The performance of the algorithm is demonstrated for both one- and two-dimensional principal manifolds and also for the case of sparse data sets. As an application we reveal cluster structures in a set of real world data from the domain of ecotoxicology.

Big Data analytics has attracted intense interest recently for its attempt to extract information, knowledge and wisdom from Big Data. In industry, with the development of sensor technology and Information & Communication Technologies (ICT), reams of high-dimensional, streaming, and nonlinear data are being collected and curated to support decision-making. The detection of faults in these data is an important application in eMaintenance solutions, as it can facilitate maintenance decision-making. Early discovery of system faults may ensure the reliability and safety of industrial systems and reduce the risk of unplanned breakdowns. Complexities in the data, including high dimensionality, fast-flowing data streams, and high nonlinearity, impose stringent challenges on fault detection applications. From the data modelling perspective, high dimensionality may cause the notorious “curse of dimensionality” and lead to deterioration in the accuracy of fault detection algorithms. Fast-flowing data streams require algorithms to give real-time or near real-time responses upon the arrival of new samples. High nonlinearity requires fault detection approaches to have sufficiently expressive power and to avoid overfitting or underfitting problems. Most existing fault detection approaches work in relatively low-dimensional spaces. Theoretical studies on high-dimensional fault detection mainly focus on detecting anomalies on subspace projections. However, these models are either arbitrary in selecting subspaces or computationally intensive. To meet the requirements of fast-flowing data streams, several strategies have been proposed to adapt existing models to an online mode to make them applicable in stream data mining. But few studies have simultaneously tackled the challenges associated with high dimensionality and data streams. Existing nonlinear fault detection approaches cannot provide satisfactory performance in terms of smoothness, effectiveness, robustness and interpretability. New approaches are needed to address this issue. This research develops an Angle-based Subspace Anomaly Detection (ABSAD) approach to fault detection in high-dimensional data. The efficacy of the approach is demonstrated in analytical studies and numerical illustrations. Based on the sliding window strategy, the approach is extended to an online mode to detect faults in high-dimensional data streams. Experiments on synthetic datasets show the online extension can adapt to the time-varying behaviour of the monitored system and, hence, is applicable to dynamic fault detection. To deal with highly nonlinear data, the research proposes an Adaptive Kernel Density-based (Adaptive-KD) anomaly detection approach. Numerical illustrations show the approach’s superiority in terms of smoothness, effectiveness and robustness.

In der nichtlinearen Datenreihenanalyse hat sich seit etwa 10 Jahren eine Monte-Carlo-Testmethode etabliert, die Theiler-surrogatmethode, mit Hilfe derer entschieden werden kann, ob eine Datenreihe nichtlinearen Ursprungs sei. Diese Methode wird kritisiert, modifiziert und verallgemeinert. Das, was Theiler untersuchen will braucht andere Surrogatmethoden, die hier konstruiert werden. Und das, was Theiler untersucht braucht gar keine Monte-Carlo-Methoden. Mit Hilfe des in der Arbeit eingeführten Begriffs des Phasensignals werden Testmöglichkeiten dargelegt und Beziehungen zwischen den nichtlinearen Eigenschaften der Zeitreihe und deren Phasenspektrum erforscht. Das Phasensignal wird aus dem Phasenspektrum der Zeitreihe hergeleitet und registriert außerordentliche Geschehnisse im Zeitbereich sowie Phasenkopplungen im Frequenzbereich. <br /> <br /> Die gewonnenen Erkenntnisse werden auf das Problem der Polbewegung angewendet. Die Hypothese einer nichtlinearen Beziehung zwischen der atmosphärischen Erregung und der Polbewegung wird untersucht. Eine nichtlineare Behandlung wird nicht für nötig gehalten. / In the nonlinear data analysis there is a popular Monte Carlo Test method due to Theiler (it was established about 10 years ago), the Theiler surrogate method, which tests whether a time series is of a linear origin. This method is being criticized, modified and generalized in this thesis. What Theiler wants to test, needs other surrogate methods, which are constructed here. And what Theiler really tests, does not need Monte Carlo methods. With the help of the concept of the phase signal, that is introduced here, other test options are possible. The phase signal helps also in investigating the relations between the nonlinear characteristics of the time series and their phase spectrum. The phase signal is derived from the phase spectrum of the time series and registers extraordinary events in the time domain as well as phase couplings in the frequency domain. <br /> <br /> These theoretical approches are applied to the problem of polar motion. The hypothesis of a nonlinear relationship between the atmospheric excitation and the pole movement is examined. A nonlinear treatment is not considered necessary.