No abstract

No abstract

Ragwitz and Kantz [1] propose a correction to a method for the reconstruction of Fokker-Planck equations from time series data. In [2,3,4, 5] a method was presented which directly applied the mathematical definitions of the drift D (1) and diffusion term D (2) [6] for an estimate from time series. Here different moments of conditional probability densities (pdf) for finite step sizes ∆ in the limit ∆ → 0 have to be estimated. Ragwitz and Kantz state that previous results have not been checked and that indispensable finite time step ∆ correction have to be employed for reliable estimates of D (2) . We want to add the following comments.Ragwitz and Kantz base their investigation on an estimate of the finite time conditional probability in terms of a Gaussian, eq. (7) of their paper. There is, however, no reason that for finite ∆ the conditional pdf is Gaussian. The exact expressions for the conditional moments up to the order ∆ 2 can be unambigously derived from the Fokker-Planck equation [7]:For (x − x 0 ) 2 |x 0 the ansatz (7) of [1] neglects the last two terms which are important for processes involving multiplicative noise. However, intermittency of turbulence is related to a multiplicatice process. This remark especially applies to the wind data presented in [1]. The validation of their method based on a Langevin (equation (9)) only works since purely additive noise is considered.For the investigation of turbulence Ragwitz and Kantz claim to obtain remarkable correction, as shown in their Fig. 6. Our approach in [5] is based on an estimate of the diffusion term using the limit ∆ → 0 yielding a dependency which can be approximated by a low order polynomial. In order to improve on this estimate the coefficients of this polynomial have been varied in such a way that the solution of the corresponding Fokker-Plank equation for finite values of ∆ yields an accurate representation of the measured one. In other words, in a second step, we have performed a parametric estimation of the diffusion term. In Figure 1 we present a case where a large correction to the ∆ → 0 estimation of D (2) had to be introduced (usually corrections are of the order of some The range of ∆ which can be taken for the estimate of D (2) must be choosen carefully in order to ensure that the Markovian property holds, see [5]. Since for each estimated value of D (2) a finite number of data point is used, a statistical error can be estimated for D (2) (cf. Fig. 1). These errors naturally increase considerably for large values of x (compare Fig. 6 [1] and Fig 13 [5].) To conclude, a deeper understanding of finite time correlations are of interest. The correct terms of higher orders in ∆ [7] may be used to improve on the estimation of drift and diffusion terms. Up to now the best way for estimating diffusion coefficients is to combine a nonparametric estimate for ∆ → 0 with a functional ansatz, i.e. a suitable polynomial ansatz. Refining the estimates of the coefficients by parametric methods as for instance described by [8] leads to improved resu...

Around the world wind energy is starting to become a major energy provider in electricity markets, as well as participating in ancillary services markets to help maintain grid stability. The reliability of system operations and smooth integration of wind energy into electricity markets has been strongly supported by years of improvement in weather and wind power forecasting systems. Deterministic forecasts are still predominant in utility practice although truly optimal decisions and risk hedging are only possible with the adoption of uncertainty forecasts. One of the main barriers for the industrial adoption of uncertainty forecasts is the lack of understanding of its information content (e.g., its physical and statistical modeling) and standardization of uncertainty forecast products, which frequently leads to mistrust towards uncertainty forecasts and their applicability in practice. This paper aims at improving this understanding by establishing a common terminology and reviewing the methods to determine, estimate, and communicate the uncertainty in weather and wind power forecasts. This conceptual analysis of the state of the art highlights that: (i) end-users should start to look at the forecast's properties in order to map different uncertainty representations to specific wind energy-related user requirements; (ii) a multidisciplinary team is required to foster the integration of stochastic methods in the industry sector. A set of recommendations for standardization and improved training of operators are provided along with examples of best practices.

This letter reports on a new method of analysing experimentally gained time series with respect to different types of noise involved, namely, we show that it is possible to differentiate between dynamical and measurement noise. This method does not depend on previous knowledge of model equations. For the complicated case of a chaotic dynamics spoiled at the same time by dynamical and measurement noise, we even show how to extract from data the magnitude of both types of noise. As a further result, we present a new criterion to verify the correct embedding for chaotic dynamics with dynamical noise.