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Abstract

his 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.