Die vorliegende Arbeit befaßt sich mit instationären, uni- oder multivariaten Zeitreihen, die bei der Beobachtung komplexer nichtlinearer dynamischer Systeme entstehen und sich der Modellierung durch ein globales Modell entziehen. In vielen natürlichen oder gesellschaftlichen Prozessen kann man jedoch wiederkehrende Phänomene beobachten, die von deren Rhythmen beeinflußt sind; ebenso lassen sich in technischen Prozessen beispielsweise aufgrund einer bedarfsorientierten Steuerung wiederholte, aber nicht periodische Verhaltensweisen ausmachen. Für solche Systeme und Zeitreihen wird deshalb vorgeschlagen, eine partielle Modellierung durch mehrere lokale Modelle vorzunehmen, die wiederkehrende Phänomene in Form zeitlich begrenzter Muster beschreiben. Um den Unwägbarkeiten dieser und sich anschließender Aufgabenstellungen Rechnung zu tragen, werden in dieser Arbeit durchgehend unscharfe Ansätze zur Modellierung von Mustern und ihrer Weiterverarbeitung gewählt und ausgearbeitet. Die Aufgabenstellung der Erkennung von Mustern in fortlaufenden Zeitreihen wird dahingehend verallgemeinert, daß unvollständige, sich noch in Entwicklung befindliche Musterinstanzen erkannt werden können. Basierend auf ebendieser frühzeitigen Erkennung kann der Verlauf der Zeitreihe -- und damit das weitere Systemverhalten -- lokal prognostiziert werden. Auf Besonderheiten und Schwierigkeiten, die sich aus der neuartigen Aufgabe der Online-Erkennung von Mustern ergeben, wird jeweils vermittels geeigneter Beispiele eingegangen, ebenso die praktische Verwendbarkeit des musterbasierten Vorhersageprinzips anhand realer Daten dokumentiert. / This dissertation focuses on non-stationary multivariate time series stemming from the observation of complex nonlinear dynamical systems. While one global model for such systems and time series may not always be feasible, we may observe recurring phenomena (patterns) in some of these time series. These phenomena might, for example, be caused by the rhythms of natural or societal processes, or a demand-oriented control of technical processes. For such systems and time series a partial modelling by means of multiple local models is being proposed. To cope with the intrinsic uncertainties of this task, fuzzy methods and models are being used throughout this work. Means are introduced for modelling and recognition of patterns in multivariate time series. Based on a novel method for the early recognition of incomplete patterns in streaming time series, a short-time prediction becomes feasible. Peculiarities and intrinsic difficulties of an online recognition of incomplete patterns are being discussed with the help of suitable examples. The usability of the pattern-based prediction approach is being demonstrated by means of real-world data.

Nowadays remote sensing is a well-established method and technique of providing data. The current development shows the availability of systems with very high geometric resolution for the monitoring of vegetation. At the same time, however, the value of temporally high-resolution data is underestimated, particularly in applications focusing on the detection of short-term changes. These can be natural processes like natural disasters as well as changes caused by anthropogenic interventions. These include economic activities such as forestry, agriculture or mining but also processes which are intended to convert previously used areas into natural or near-natural surfaces. The Königsbrücker Heide is a former military training site located about 30 km north of the Saxon state capitol Dresden. After the withdrawal of the Soviet forces in 1992 and after nearly 100 years of military use this site was declared as nature reserve in 1996. The management of the whole protection area is implemented in three different management zone. Based on MODIS-NDVI time series between 2000 and 2016 different developments are apparent in the nature development zone and the zone of controlled succession. Nevertheless, the analyses also show that short-term changes, so called breaks in the vegetation development cannot be described using linear trend models. The complete understanding of vegetation trends is only given if discontinuities in vegetation development are considered. Structural breaks in the NDVI time series can be found simultaneously in the whole study area. Hence it can be assumed that these breaks have a more natural character, caused for example by climatic conditions like temperature or precipitation. Otherwise, especially in the zone of controlled succession structural breaks can be detected which cannot be traced back to natural conditions. Final analyses of the spatial distribution of breakpoints as well as their frequency depending on the respective protection zone allow a detailed view to vegetation development in the Köonigsbrüucker Heide.

In 1989 about 1.5 million soldiers were stationed in Germany. With the political changes in the early 1990s a substantial decline of the staff occurred on currently 200,000 employees in the armed forces and less than 60,000 soldiers of foreign forces. These processes entailed conversions of large areas not longer used for military purposes, especially in the new federal states in the eastern part of Germany. One of these conversion areas is the former military training area Königsbrück in Saxony. For the analysis of vegetation and its development over time, the Normalized Difference Vegetation Index (NDVI) has established as one of the most important indicators. In this context, the questions arise whether MODIS NDVI products are suitable to determine conversion processes on former military territories like military training areas and what development processes occurred in the 'Königsbrücker Heide' in the past 15 years. First, a decomposition of each series in its trend component, seasonality and the remaining residuals is performed. For the trend component different regression models are tested. Statistical analysis of these trends can reveal different developments, for example in nature development zones (without human impact) and zones of controlled succession. The presented work ow is intended to show the opportunity to support a high temporal resolution monitoring of conversion areas such as former military training areas.

The value of remote sensing data is particularly evident where an areal monitoring is needed to provide information on the earth's surface development. The use of temporal high resolution time series data allows for detecting short-term changes. In Kogi State in Nigeria different vegetation types can be found. As the major population in this region is living in rural communities with crop farming the existing vegetation is slowly being altered. The expansion of agricultural land causes loss of natural vegetation, especially in the regions close to the rivers which are suitable for crop production. With regard to these facts, two questions can be dealt with covering different aspects of the development of vegetation in the Kogi state, the determination and evaluation of the general development of the vegetation in the study area (trend estimation) and analyses on a short-term behavior of vegetation conditions, which can provide information about seasonal effects in vegetation development. For this purpose, the GIMMS-NDVI data set, provided by the NOAA, provides information on the normalized difference vegetation index (NDVI) in a geometric resolution of approx. 8 km. The temporal resolution of 15 days allows the already described analyses. For the presented analysis data for the period 1981-2012 (31 years) were used. The implemented work flow mainly applies methods of time series analysis. The results show that in addition to the classical seasonal development, artefacts of different vegetation periods (several NDVI maxima) can be found in the data. The trend component of the time series shows a consistently positive development in the entire study area considering the full investigation period of 31 years. However, the results also show that this development has not been continuous and a simple linear modeling of the NDVI increase is only possible to a limited extent. For this reason, the trend modeling was extended by procedures for detecting structural breaks in the time series.

Die Analyse von Proxy-Zeitreihen aus Paläoklimaarchiven wie zum Beispiel Baumringen, Seesedimenten, Tropfsteinen und Eisbohrkernen mittels gefensterter Rekurrenznetzwerkanalyse ermöglicht die Identifizierung und Charakterisierung dynamischer Anomalien in der Klimavariabilität der Vergangenheit. Das Ziel der vorliegenden Arbeit ist die Entwicklung einer zuverlässigeren Routine zur gefensterten Rekurrenznetzwerkanalyse. Aufbauend auf dem bestehenden methodischen Rahmen werden die Bereiche der Phasenraumrekonstruktion und des Signifikanztests als verbesserungsfähig identifiziert. Deshalb werden verschiedene Methoden zur Rekonstruktion des Phasenraums aus unregelmäßig abgetasteten, verrauschten Daten verglichen. Außerdem wird ein allgemeiner flächenweiser Signifikanztest eingeführt, der, basierend auf einem ausgewählten Nullmodell, Korrelationen in den Analyseergebnissen numerisch abschätzt, um damit das Problem hoher Raten an falsch positiv signifikanten Ergebnissen zu adressieren. Im zweiten Teil der Arbeit wird die entwickelte Methodik genutzt, um die nichtlineare Variabilität des Klimas der Vergangenheit in Nord- und Südamerika zu untersuchen, indem vier reale Zeitreihen verschiedener Proxys studiert werden. Außerdem werden Proxy-System-Modelle genutzt, um auf die Frage der Eignung von Daten verschiedener Paläoklimaarchive zur Charakterisierung der Klimavariabilität mittels gefensterter Rekurrenznetzwerkanalyse einzugehen. Mit der Arbeit wird der Einsatz nichtlinearer Methoden zur Analyse von Paläoklima-Zeitreihen vorangebracht, das Potential und die Grenzen der gefensterten Rekurrenznetzwerkanalyse aufgezeigt und zukünftige relevante Fragestellungen, die die erhaltenen Ergebnisse und Schlussfolgerungen komplementieren können, identifiziert. / Studying palaeoclimate proxy data from archives such as tree rings, lake sediments, speleothems, and ice cores using windowed recurrence network analysis offers the possibility to characterise dynamical anomalies in past climate variability. This thesis aims at developing a more reliable framework of windowed recurrence network analysis by comparing different phase space reconstruction approaches for non-uniformly sampled noisy data and by tackling the problem of increased numbers of false positive significant points when correlations within the analysis results can not be neglected. For this, different phase space reconstruction approaches are systematically compared and a generalised areawise significance test which implements a numerical estimation of the correlations within the analysis results is introduced. In particular, the test can be used to identify patches of possibly false positive significant points. The developed analysis framework is applied to detect and characterise dynamical anomalies in past climate variability in North and South America by studying four real-world palaeoclimatic time series from different archives. Furthermore, the question whether palaeoclimate proxy time series from different archives are equally well suited for tracking past climate dynamics with windowed recurrence network analysis is approached by using the framework of proxy system modelling. This thesis promotes the use of non-linear methods for analysing palaeoclimate proxy time series, provides a detailed assessment of potentials and limitations of windowed recurrence network analysis and identifies future research directions that can complement the obtained results and conclusions.

The forecasting of time series data is an integral component for management, planning, and decision making. Following the Big Data trend, large amounts of time series data are available in many application domains. The highly dynamic and often noisy character of these domains in combination with the logistic problems of collecting data from a large number of data sources, imposes new requirements on the forecasting process. A constantly increasing number of time series has to be forecasted, preferably with low latency AND high accuracy. This is almost impossible, when keeping the traditional focus on creating one forecast model for each individual time series. In addition, often used forecasting approaches like ARIMA need complete historical data to train forecast models and fail if time series are intermittent. A method that addresses all these new requirements is the cross-sectional forecasting approach. It utilizes available data from many time series of the same domain in one single model, thus, missing values can be compensated and accurate forecast results can be calculated quickly. However, this approach is limited by a rigid training data selection and existing forecasting methods show that adaptability of the model to the data increases the forecast accuracy. Therefore, in this paper we present CSAR a model that extends the cross-sectional paradigm by adding more flexibility and allowing fine grained adaptations to the analyzed data. In this way, we achieve an increased forecast accuracy and thus a wider applicability.

Das Klimasystem der Erde besteht aus zahlreichen interagierenden Teilsystemen, die sich über verschiedene Zeitskalen hinweg verändern, was zu einer äußerst komplizierten räumlich-zeitlichen Klimavariabilität führt. Das Verständnis von Prozessen, die auf verschiedenen räumlichen und zeitlichen Skalen ablaufen, ist ein entscheidender Aspekt bei der numerischen Wettervorhersage. Die Variabilität des Klimas, ein sich selbst konstituierendes System, scheint in Mustern auf großen Skalen organisiert zu sein. Die Verwendung von Klimanetzwerken hat sich als erfolgreicher Ansatz für die Erkennung der räumlichen Ausbreitung dieser großräumigen Muster in der Variabilität des Klimasystems erwiesen. In dieser Arbeit wird mit Hilfe von Klimanetzwerken gezeigt, dass die Klimavariabilität nicht nur auf größeren Skalen (Asiatischer Sommermonsun, El Niño/Southern Oscillation), sondern auch auf kleineren Skalen, z.B. auf Wetterzeitskalen, in Mustern organisiert ist. Dies findet Anwendung bei der Erkennung einzelner tropischer Wirbelstürme, bei der Charakterisierung binärer Wirbelsturm-Interaktionen, die zu einer vollständigen Verschmelzung führen, und bei der Untersuchung der intrasaisonalen und interannuellen Variabilität des Asiatischen Sommermonsuns. Schließlich wird die Anwendbarkeit von Klimanetzwerken zur Analyse von Vorhersagefehlern demonstriert, was für die Verbesserung von Vorhersagen von immenser Bedeutung ist. Da korrelierte Fehler durch vorhersagbare Beziehungen zwischen Fehlern verschiedener Regionen aufgrund von zugrunde liegenden systematischen oder zufälligen Prozessen auftreten können, wird gezeigt, dass Fehler-Netzwerke helfen können, die räumlich kohärenten Strukturen von Vorhersagefehlern zu untersuchen. Die Analyse der Fehler-Netzwerk-Topologie von Klimavariablen liefert ein erstes Verständnis der vorherrschenden Fehlerquelle und veranschaulicht das Potenzial von Klimanetzwerken als vielversprechendes Diagnoseinstrument zur Untersuchung von Fehlerkorrelationen. / The Earth’s climate system consists of numerous interacting subsystems varying over a multitude of time scales giving rise to highly complicated spatio-temporal climate variability. Understanding processes occurring at different scales, both spatial and temporal, has been a very crucial problem in numerical weather prediction. The variability of climate, a self-constituting system, appears to be organized in patterns on large scales. The climate networks approach has been very successful in detecting the spatial propagation of these large scale patterns of variability in the climate system. In this thesis, it is demonstrated using climate network approach that climate variability is organized in patterns not only at larger scales (Asian Summer Monsoon, El Niño-Southern Oscillation) but also at shorter scales, e.g., weather time scales. This finds application in detecting individual tropical cyclones, characterizing binary cyclone interaction leading to a complete merger, and studying the intraseasonal and interannual variability of the Asian Summer Monsoon. Finally, the applicability of the climate network framework to understand forecast error properties is demonstrated, which is crucial for improvement of forecasts. As correlated errors can arise due to the presence of a predictable relationship between errors of different regions because of some underlying systematic or random process, it is shown that error networks can help to analyze the spatially coherent structures of forecast errors. The analysis of the error network topology of a climate variable provides a preliminary understanding of the dominant source of error, which shows the potential of climate networks as a very promising diagnostic tool to study error correlations.

Diese Dissertation beantwortet verschiedene politikrelevante ökonomische Fragen in den Bereichen Handelspolitik, Geldpolitik, sowie Rohstoffmärkte und Energieökonomik mit Hilfe von strukturellen Vektorautoregressionsmodellen (SVAR). SVARs stellen eine effektive Möglichkeit dar, die Beziehungen zwischen verschiedenen makroökonomischen und/oder Finanzmarkt-Variablen zu modellieren und werden verwendet, um die dynamischen kausalen Effekte von ökonomischen Schocks zu schätzen. Für jede ökonomische Fragestellung wird eine Identifikationsstrategie angewandt, die auf die betrachteten Daten und ihre statistischen Eigenschaften sowie die zugrundeliegenden Annahmen über ökonomische Mechanismen zwischen den betrachteten Zeitreihen zugeschnitten ist. Im Einzelnen besteht diese Dissertation aus vier Kapiteln. In den ersten beiden Kapiteln werden die Auswirkungen von Handelspolitik auf Finanzmärkte und auf die Makroökonomie geschätzt. Das dritte Kapitel liefert einen methodischen Beitrag zur SVAR-Literatur, der in einer Anwendung zu den Effekten von Geldpolitik dargestellt wird. Das letzte Kapitel verlässt die Felder der Handels- und Geldpolitik und wendet sich Rohstoffmärkten und der Energiewirtschaft zu, stützt sich dabei aber ebenfalls auf Zeitreihenmethoden. Es analysiert die Rolle von Metallen in der Energiewende. / This dissertation answers various policy relevant economic questions in the fields of trade policy, monetary policy, and commodity markets and energy economics using structural vector autoregression (SVAR) models. SVARs constitute a parsimonious way to model the relations between different macroeconomic and/or financial variables and they are used to estimate the dynamic causal effects of economic shocks. For each economic question, this dissertation applies an identification strategy that is tailored to the relevant data and its statistical properties as well as the underlying assumptions about economic mechanisms among the regarded time series. Specifically, this dissertation consists of four chapters. The first two chapters estimate the effects of trade policy on financial markets and on the macroeconomy. The third chapter makes a methodological contribution to the SVAR literature in an application to monetary policy shocks. The final chapter moves away from trade and monetary policy to commodity markets and energy economics but also relies on time series methods. It analyzes the role of metals for the clean energy transition.

Time series forecasting is challenging as sophisticated forecast models are computationally expensive to build. Recent research has addressed the integration of forecasting inside a DBMS. One main benefit is that models can be created once and then repeatedly used to answer forecast queries. Often forecast queries are submitted on higher aggregation levels, e. g., forecasts of sales over all locations. To answer such a forecast query, we have two possibilities. First, we can aggregate all base time series (sales in Austria, sales in Belgium...) and create only one model for the aggregate time series. Second, we can create models for all base time series and aggregate the base forecast values. The second possibility might lead to a higher accuracy but it is usually too expensive due to a high number of base time series. However, we actually do not need all base models to achieve a high accuracy, a sample of base models is enough. With this approach, we still achieve a better accuracy than an aggregate model, very similar to using all models, but we need less models to create and maintain in the database. We further improve this approach if new actual values of the base time series arrive at different points in time. With each new actual value we can refine the aggregate forecast and eventually converge towards the real actual value. Our experimental evaluation using several real-world data sets, shows a high accuracy of our approaches and a fast convergence towards the optimal value with increasing sample sizes and increasing number of actual values respectively.

This thesis deals with data-driven stochastic models of daily temperature time series recorded at weather stations. These univariate time series are long-range correlated, i.e. their autocorrelation functions possess a power-law decay. In addition, their marginal distributions violate Gaussianity and their response functions are nonlinear, calling for nonlinear models. We present two methods for inferring nonlinear long-range correlated stochastic models of single-trajectory data and use them to reconstruct models of daily mean temperature data recorded at Potsdam Telegrafenberg, Germany. The first method employs fractional filtering using the estimated Hurst exponent of the time series. We render the time series short-range correlated with the first-order difference approximation of the Grünwald-Letnikov fractional derivative, the inverse of the fractional integration operation used in ARFIMA processes. Subsequently, we reconstruct a Markovian model of the fractionally differenced time series. The second inference method is ‘fractional Onsager-Machlup optimization’ (fOMo), a maximum likelihood framework apt to infer nonlinear force and diffusion terms of overdamped stochastic differential equations driven by arbitrarily correlated Gaussian noise, in particular fractional Gaussian noise. The optimization corresponds to the minimization of a stochastic action as studied in statistical field theory. The optimal drift and diffusion terms then render a given time series the most probable path of the model. Both inference methods show excellent results for temperature time series. They are applicable to other stationary, monofractal time series and thus may prove beneficial in biophysics, e.g. active matter dynamics and anomalous diffusion, neurophysics and finance. Finally, we employ stochastic temperature models reconstructed via the fractional filtering method for predictions. A forecast of the first frost date at Potsdam Telegrafenberg using the mean first-passage time of model trajectories and the zero degree temperature line shows small predictive power. The second application extends the stochastic temperature model to include an external forcing by a meteorological index time series that is associated to long-lived circulation patterns in the atmosphere. A causal analysis of Arctic Oscillation (AO) and North-Atlantic Oscillation indices and European extreme temperatures reveals the largest influence of the AO index on daily extreme winter temperatures in southern Scandinavia. We therefore reconstruct a nonlinear long-range correlated stochastic model of daily maximum and minimum winter temperatures recorded at Visby Flygplats, Sweden, with external driving by the AO index. Binary temperature forecasts show predictive power for up to 35 (30) days lead time for daily maximum (minimum) temperatures. An AR(1) model possesses predictive power for only 10 (5) days lead time for daily maximum (minimum) temperature, proving the potential of nonlinear long-range correlated models for predictions.:1 Introduction 1.1 Long-Range Correlations in Geophysical Time Series 1.2 Stochastic Modeling of Geophysical Time Series 1.3 Structure of the Thesis 2 Preliminaries 2.1 Time Series and Stochastic Processes 2.1.1 Stochastic Processes 2.1.2 Basic Concepts of Time Series Analysis 2.1.3 Classification of Stochastic Processes 2.1.4 Inference of Stochastic Processes 2.2 Markov Processes 2.2.1 Fokker-Planck Equation 2.2.2 Langevin Equation 2.2.3 Stochastic Integration 2.2.4 Correspondence of Langevin Equation and Fokker-Planck Equation 2.2.5 Numerical Solution of Langevin Equation 2.2.6 Path Integral Formulation 2.2.7 Discrete-Time Processes 2.3 Long-Range Correlated Processes 2.3.1 Self-Similarity and Long-Range Correlations 2.3.2 Fractional Calculus 2.3.3 Fractional Brownian Motion and Fractional Gaussian Noise 2.3.4 Stochastic Differential Equations driven by fGn 2.3.5 Numerical Solution of SDE driven by fGn 2.3.6 ARFIMA Processes 2.4 Estimation of the Hurst parameter 2.4.1 Estimation Methods 2.4.2 Detrended Fluctuation Analysis 2.5 Discussion of Previous Approaches to Modeling LRC Data 2.5.1 Generalized Langevin Equation 2.5.2 Modified Discrete Langevin Equation 2.5.3 Atmospheric Response Functions 3 Inference via Fractional Differencing 3.1 Surface Temperature Time Series 3.2 Fractional Differencing of Time Series 3.2.1 Removing Long-Range Correlations 3.2.2 Memory Selection 3.2.3 Testing for Markovianity 3.3 Finite-Time Kramers-Moyal Analysis 3.3.1 Kernel-Based Regression of Kramers-Moyal Moments 3.3.2 The Adjoint Fokker-Planck Equation 3.3.3 Numerical Procedure 3.3.4 Inferred Drift and Diffusion Terms 3.3.5 Model Data Generation 3.3.6 Results for Temperature Anomalies 3.4 Discrete-Time Langevin Equation 3.4.1 Estimation of Force and Diffusion Terms 3.4.2 Model Data Generation 3.4.3 Nonlinear Toy Model 3.4.4 Application to Temperature Data 3.4.5 Results for Temperature Anomalies 3.5 Discussion 4 Inference via Fractional Onsager-Machlup Optimization 4.1 Derivation of the Maximum Likelihood Estimator 4.2 Analytical Approaches 4.2.1 Force Estimation for Fixed Diffusion 4.2.2 Diffusion Estimation for Fixed Drift 4.2.3 Fractional Ornstein-Uhlenbeck Process 4.2.4 Superposition of Noise Processes 4.3 Numerical Procedure 4.4 Toy Model with Double-Well Potential 4.4.1 Comparison with Markovian Estimate 4.4.2 Finite-Size Error Scaling 4.5 Application to Temperature Data 4.5.1 Consistency of Inferred Drift and Diffusion 4.5.2 Comparison of Synthetic Data and Temperature 4.5.3 Residual Noise 4.6 Discussion 5 Predictions with Long-Range Correlated Models 5.1 First Frost Date 5.1.1 Forecast Ensemble and Forecast Error 5.1.2 Numerical Details 5.1.3 Results 5.2 Causal Analysis of Meteorological Indices and European Extreme Temperatures 5.2.1 Measures for Causal Influence 5.2.2 Causal Analysis Results 5.2.3 Causal Analysis for Visby Flygplats, Sweden 5.3 Forecasting Winter Temperature Extremes at Visby Flygplats, Sweden 5.3.1 Model Inference and Forecast 5.3.2 Root-Mean-Square Error Analysis 5.3.3 Binary Forecasts of Temperature Extremes 5.4 Discussion 6 Conclusion and Outlook 6.1 Inference of Nonlinear LRC Models 6.2 Predictions with LRC models 6.3 Further Research Directions 6.3.1 Method Extensions 6.3.2 Meteorological Applications 6.3.3 Data Interpolation 6.3.4 Anomalous Diffusion and Active Matter Dynamics Bibliography / Diese Arbeit befasst sich mit datengetriebenen stochastischen Modellen von Tagestemperatur-Zeitreihen, die von Wetterstationen aufgezeichnet wurden. Diese univariaten Zeitreihen sind langreichweitig korreliert, d.h. ihre Autokorrelationsfunktionen fallen gemäß eines Potenzgesetzes ab. Darüber hinaus sind ihre Randverteilungen nicht-Gaußsch und ihre Antwortfunktionen nichtlinear, was nichtlineare Modelle erforderlich macht. Wir stellen zwei Methoden zur Rekonstruktion nichtlinearer, langreichweitig korrelierter stochastischer Modelle von Einzeltrajektorien vor und verwenden sie zur Rekonstruktion von Modellen aus Tagesmitteltemperaturdaten, die an der Wetterstation Potsdam Telegrafenberg, Deutschland, aufgezeichnet wurden. Die erste Methode verwendet eine fraktionale Filterung unter Verwendung des geschätzten Hurst-Exponenten der Zeitreihe. Dabei werden die langreichweitigen Korrelationen der Zeitreihe mit der Differenzenapproximation erster Ordnung der fraktionalen Grünwald-Letnikov-Ableitung, der inversen Operation der in ARFIMA-Prozessen verwendeten fraktionalen Integration, entfert. Anschließend rekonstruieren wir ein Markov-Modell der fraktional differenzierten, nun kurzreichweitig korrelierten Zeitreihe. Die zweite Inferenzmethode ist die ‘fractional Onsager-Machlup optimization’ (fOMo), ein Maximum-Likelihood-Schätzer, der nichtlineare Kraft- und Diffusionsterme von überdämpften stochastischen Differentialgleichungen rekonstruiert, die von beliebig korreliertem Gaußschen Rauschen, insbesondere fraktionalem Gaußschen Rauschen, angetrieben werden. Die Optimierung entspricht der Minimierung einer stochastischen Wirkung, wie sie in der statistischen Feldtheorie untersucht wird. Die optimalen Drift- und Diffusionsterme machen die gegebene Zeitreihe dann zum wahrscheinlichsten Pfad des Modells. Beide Inferenzmethoden zeigen exzellente Ergebnisse für Temperaturzeitreihen. Sie sind auf weitere stationäre, monofraktale Zeitreihen anwendbar und können daher in der Biophysik, z. B. der Dynamik aktiver Materie und anomaler Diffusion, in der Neurophysik und im Finanzwesen nützlich sein. Schließlich verwenden wir stochastische Temperatur-Modelle, die mit Hilfe der Methode der fraktionalen Filterung rekonstruiert wurden, für Vorhersagen. Eine Vorhersage des ersten Frosttages im Herbst mit Temperaturdaten der Wetterstation Potsdam Telegrafenberg unter Verwendung der mittleren Erstauftreffszeit von Modelltrajektorien und der Null-Grad-Temperaturlinie zeigt nur geringe Vorhersagekraft. Die zweite Anwendung erweitert das stochastische Temperaturmodell um einen zusätzlichen Antrieb durch eine meteorologische Indexzeitreihe, welche langlebige Zirkulationsmuster in der Atmosphäre charakterisiert. Eine Kausalsanalyse des Einflusses der Indizes der Arktischen Oszillation und der Nordatlantischen Oszillation auf Extremtemperaturen in Europa zeigt den größten Einfluss des Arktischen-Oszillations-Index auf die täglichen Maximal- und Minimaltemperaturen im Winter in Südskandinavien. Darauf aufbauend rekonstruieren wir ein nichtlineares, langreichweitig korreliertes stochastisches Modell der Tagesmaximal- und -minimaltemperaturen im Winter der Wetterstation Visby Flygplats in Schweden mit zusätzlichem Antrieb durch den Arktischen Oszillationsindex. Binäre Vorhersagen des Modells besitzen einen Vorhersagehorizont von bis zu 35 (30) Tagen für Tages-Maximal-(Minimal-)Temperaturen. Binäre Vorhersagen mithilfe eines AR(1)-Modells besitzen einen Vorhersagehorizont von nur 10 (5) Tagen für tägliche Maximal-(Minimal-)Temperaturen. Dies beweist das Potenzial nichtlinearer, langreichweitig korrelierter Modelle für Vorhersagen.:1 Introduction 1.1 Long-Range Correlations in Geophysical Time Series 1.2 Stochastic Modeling of Geophysical Time Series 1.3 Structure of the Thesis 2 Preliminaries 2.1 Time Series and Stochastic Processes 2.1.1 Stochastic Processes 2.1.2 Basic Concepts of Time Series Analysis 2.1.3 Classification of Stochastic Processes 2.1.4 Inference of Stochastic Processes 2.2 Markov Processes 2.2.1 Fokker-Planck Equation 2.2.2 Langevin Equation 2.2.3 Stochastic Integration 2.2.4 Correspondence of Langevin Equation and Fokker-Planck Equation 2.2.5 Numerical Solution of Langevin Equation 2.2.6 Path Integral Formulation 2.2.7 Discrete-Time Processes 2.3 Long-Range Correlated Processes 2.3.1 Self-Similarity and Long-Range Correlations 2.3.2 Fractional Calculus 2.3.3 Fractional Brownian Motion and Fractional Gaussian Noise 2.3.4 Stochastic Differential Equations driven by fGn 2.3.5 Numerical Solution of SDE driven by fGn 2.3.6 ARFIMA Processes 2.4 Estimation of the Hurst parameter 2.4.1 Estimation Methods 2.4.2 Detrended Fluctuation Analysis 2.5 Discussion of Previous Approaches to Modeling LRC Data 2.5.1 Generalized Langevin Equation 2.5.2 Modified Discrete Langevin Equation 2.5.3 Atmospheric Response Functions 3 Inference via Fractional Differencing 3.1 Surface Temperature Time Series 3.2 Fractional Differencing of Time Series 3.2.1 Removing Long-Range Correlations 3.2.2 Memory Selection 3.2.3 Testing for Markovianity 3.3 Finite-Time Kramers-Moyal Analysis 3.3.1 Kernel-Based Regression of Kramers-Moyal Moments 3.3.2 The Adjoint Fokker-Planck Equation 3.3.3 Numerical Procedure 3.3.4 Inferred Drift and Diffusion Terms 3.3.5 Model Data Generation 3.3.6 Results for Temperature Anomalies 3.4 Discrete-Time Langevin Equation 3.4.1 Estimation of Force and Diffusion Terms 3.4.2 Model Data Generation 3.4.3 Nonlinear Toy Model 3.4.4 Application to Temperature Data 3.4.5 Results for Temperature Anomalies 3.5 Discussion 4 Inference via Fractional Onsager-Machlup Optimization 4.1 Derivation of the Maximum Likelihood Estimator 4.2 Analytical Approaches 4.2.1 Force Estimation for Fixed Diffusion 4.2.2 Diffusion Estimation for Fixed Drift 4.2.3 Fractional Ornstein-Uhlenbeck Process 4.2.4 Superposition of Noise Processes 4.3 Numerical Procedure 4.4 Toy Model with Double-Well Potential 4.4.1 Comparison with Markovian Estimate 4.4.2 Finite-Size Error Scaling 4.5 Application to Temperature Data 4.5.1 Consistency of Inferred Drift and Diffusion 4.5.2 Comparison of Synthetic Data and Temperature 4.5.3 Residual Noise 4.6 Discussion 5 Predictions with Long-Range Correlated Models 5.1 First Frost Date 5.1.1 Forecast Ensemble and Forecast Error 5.1.2 Numerical Details 5.1.3 Results 5.2 Causal Analysis of Meteorological Indices and European Extreme Temperatures 5.2.1 Measures for Causal Influence 5.2.2 Causal Analysis Results 5.2.3 Causal Analysis for Visby Flygplats, Sweden 5.3 Forecasting Winter Temperature Extremes at Visby Flygplats, Sweden 5.3.1 Model Inference and Forecast 5.3.2 Root-Mean-Square Error Analysis 5.3.3 Binary Forecasts of Temperature Extremes 5.4 Discussion 6 Conclusion and Outlook 6.1 Inference of Nonlinear LRC Models 6.2 Predictions with LRC models 6.3 Further Research Directions 6.3.1 Method Extensions 6.3.2 Meteorological Applications 6.3.3 Data Interpolation 6.3.4 Anomalous Diffusion and Active Matter Dynamics Bibliography