Abstract={It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time series even in the case when the amplitudes of measurement noise and uncontaminated signal are of the same order of magnitude. Our approach is based on a recently developed method for a nonparametric reconstruction of Langevin processes. Minimizing a proper non-negative function, the procedure is able to correctly extract strong measurement noise and to estimate drift and diffusion coefficients in the Langevin equation describing the evolution of the original uncorrupted signal. As input, the algorithm uses only the two first conditional moments extracted directly from the stochastic series and is therefore suitable for a broad panoply of different signals. To demonstrate the power of the method, we apply the algorithm to synthetic as well as climatological measurement data, namely, the daily North Atlantic Oscillation index, shedding light on the discussion of the nature of its underlying physical processes.},

Doi={10.1103/PhysRevE.81.041125},

Keywords={Langevin}

}

...

...

@@ -250,7 +287,7 @@

}

@Manual{RCT2015,

Title={R: A Language and Environment for Statistical Computing},

Title={{R}: A Language and Environment for Statistical Computing},

Address={Vienna, Austria},

Author={{R Core Team}},

...

...

@@ -268,7 +305,6 @@

Pages={383--409},

Volume={433},

Abstract={We present a stochastic analysis of a data set consisting of 1.25 x 10^7 samples of the local velocity measured in the turbulent region of a round free jet. We ﬁnd evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker--Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker--Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker--Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker--Planck equation also allows the joint probability density of N increments on N different scales p(v1 , r1 ; . . . ; vN , rN ) to be determined.},

Doi={10.1017/S0022112001003597}

}

...

...

@@ -281,7 +317,6 @@

Pages={P04020},

Volume={2010},

Abstract={Many fluctuation phenomena, in physics and other fields, can be modeled by Fokker--Planck or stochastic differential equations whose coefficients, associated with drift and diffusion components, may be estimated directly from the observed time series. Its correct characterization is crucial to determine the system quantifiers. However, due to the finite sampling rates of real data, the empirical estimates may significantly differ from their true functional forms. In the literature, low-order corrections, or even no corrections, have been applied to the finite-time estimates. A frequent outcome consists of linear drift and quadratic diffusion coefficients. For this case, exact corrections have been recently found, from It{\^{o}}--Taylor expansions. Nevertheless, model validation constitutes a necessary step before determining and applying the appropriate corrections. Here, we exploit the consequences of the exact theoretical results obtained for the linear--quadratic model. In particular, we discuss whether the observed finite-time estimates are actually a manifestation of that model. The relevance of this analysis is put into evidence by its application to two contrasting real data examples in which finite-time linear drift and quadratic diffusion coefficients are observed. In one case the linear--quadratic model is readily rejected while in the other, although the model constitutes a very good approximation, low-order corrections are inappropriate. These examples give warning signs about the proper interpretation of finite-time analysis even in more general diffusion processes.},

Doi={10.1088/1742-5468/2010/04/P04020}

}

...

...

@@ -298,29 +333,43 @@

Issue={1}

}

@Article{Rinn2016a,

Title={The Langevin Approach: An R Package for Modeling Stochastic Processes},

Author={Rinn, Philip and Lind, Pedro and W\"achter, Matthias and Peinke, Joachim},

Journal={Journal of Open Research Software},

Year={2016},

Number={1},

Pages={e34},

Volume={4},

Doi={10.5334/jors.123}

}

@Book{Risken1996,

Title={The Fokker-Planck Equation},

Author={Hannes Risken},

Publisher={Springer},

Publisher={Springer-Verlag},

Year={1996},

Keywords={Fokker-Planck}

}

@TechReport{Sanderson2010,

Title={Armadillo: An Open Source \proglang{C++} Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments},

Title={Armadillo: An Open Source {C}++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments},