Wind Energy

132 D. Kleinhans and R. Friedrich
D (1) E(x=0.5)
0.5
0
−0.5
−1
−1.5
10 100 1000
N
10000 100000
Fig. 23.1. Estimated drift function D (1)
E (x =0.5) of synthetic Ornstein-Uhlenbeck
process as function of N. Dashed region marks the symmetric error-bars corresponding
to (23.7)
E[D (1) (x=0.5)]
40
35
30
25
20
15
10
5
0
1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01
τ
Fig. 23.2. Error of drift estimate as function of time increment τ (triangles).
A divergent behaviour for τ → 0 is evident. The solid line represents the best
fit f(τ) =0.0045/ √ τ
�
EN D (1)
E (x0,τ)
�
�
�
�
� 2 D
=
τ
(2)
E (x0,τ)
�
D
−
N
(1)
E (x0,τ)
�2 . (23.7)
N
In sum the statistical error in the estimated drift function is proportional
to (Nτ) −1/2 . Figures 23.1 and 23.2 illustrate the divergent behaviour of the
estimated drift coefficient D (1)
E in the cases of few data-points and small time
increments considering as example data from an Ornstein-Uhlenbeck process.
23.5 Conclusion
In conclusion there is simple method to estimate the dynamical drift and
diffusion functions from measured data. This method can be used to describe