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

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