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Generalized Cauchy difference iterative forecasting model for wind speed based on fractal time series

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Abstract

The local irregularity and global correlation of wind speed can be described by the fractal dimension D and the Hurst parameter H. However, the existing mathematical models imply a linear relationship between the fractal dimension D and the Hurst parameter H, which is not adequate to describe the complete characteristics of the wind speed. To overcome the descriptive limitations of the \(D-H\) linearity assumption, in this paper, we introduce novel model, based on the generalized Cauchy (GC) process, for simulation and forecasting of wind speed. In the model, the fractal dimension D and Hurst parameter H can be combined arbitrarily to describe the local irregularity and global correlation of the wind speed. Furthermore, the GC process is taken as the disturbance fluctuation term in the forecasting model, and a difference iteration form is obtained as difference equation and incremental distribution. In such model, the maximum forecasting range of the wind speed is determined by the maximum Lyapunov exponent. To illustrate the properties of the model and its performance, simulating and forecasting of the actual wind speeds in two regions are reported.

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Abbreviations

GC:

Generalized Cauchy

LSTM:

Long short-term memory neural network

LRD:

Long-range dependence

fBm:

Fractional Brownian motion

fGn:

Fractional Gaussian noise

ACF:

Auto-correlation function

SOA:

Score of accuracy

\(x\left( t \right) \) :

Stationary Gaussian process

\({R_{XX}}\left( {\cdot } \right) \) :

Auto-correlation function

D :

Fractal dimension

H :

Hurst parameter

\(\chi \) :

Maximum Lyapunov exponent

\(\upsilon \) :

Maximum forecasting step

\({B_H}\left( t \right) \) :

Fractional Brownian motion

\(B\left( t \right) \) :

Brownian motion

\(GC\left( t \right) \) :

Generalized Cauchy process

\(\varDelta GC\left( t \right) \) :

Increment of Generalized Cauchy process

\(\mu \) :

Drift coefficient

\(\delta \) :

Fluctuation coefficient

\(X\left( t \right) \) :

Wind speed fluctuation state at time t

\(X\left( {t{\mathrm{+ }}1} \right) \) :

Wind speed fluctuation state at time \(t+1\)

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Authors and Affiliations

  1. Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

    He Liu & Wanqing Song

  2. Politecnico di Milano, Via La Masa 34/3, 20156, Milan, Italy

    Enrico Zio

  3. CRC, PSL Research University, Sophia Antipolis, France

    Enrico Zio

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  1. He Liu
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Appendix: the calculation process of the Lyapunov exponent

Appendix: the calculation process of the Lyapunov exponent

The chaotic nature of the fractal time series can be judged by the maximum Lyapunov exponent, and the calculation process by the small data method is as follows [33, 35]:

Step 1: Defining the time series \(\left\{ {x\left( i \right) ,i = 1,2, \cdots ,n} \right\} \) and calculate the delay time \(\varepsilon \) by the ACF method.

Step 2: Calculating embedding dimension m by Cao algorithm.

Step 3: Reconstructing the phase space.

$$\begin{aligned}&X\left( i \right) = {\left[ {x\left( i \right) ,x\left( {i + \varepsilon } \right) , \cdots x\left( {i + \left( {m - 1} \right) \varepsilon } \right) } \right] ^T}\, \, \, \, \nonumber \\&\quad \left( i = 1,2, \cdots ,M\right) , \end{aligned}$$
(44)

where \(M = n - \left( {m - 1} \right) \varepsilon \).

Step 4: Calculating the maximum Lyapunov exponent \(\chi \).

$$\begin{aligned} \chi =\frac{{\ln {d_j}\left( i \right) - \ln {C_j}}}{{i\varDelta t}}, \end{aligned}$$
(45)

where \({C_j} = {d_j}\left( 0 \right) \) and \({d_j}\left( i \right) \) are the distances between each reference point \(X\left( j \right) \) and the neighboring point \(X\left( {{\hat{j}}} \right) \) after the i-th discrete time step:

$$\begin{aligned}&{d_j}\left( i \right) {\mathrm{= min}}\left\| {X\left( {j + i} \right) - X\left( {{\hat{j}} + i} \right) } \right\| ,\nonumber \\&\qquad \qquad i = 1,2, \cdots ,min\left( {M - j,M - {\hat{j}}} \right) . \end{aligned}$$
(46)

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Liu, H., Song, W. & Zio, E. Generalized Cauchy difference iterative forecasting model for wind speed based on fractal time series. Nonlinear Dyn 103, 759–773 (2021). https://doi.org/10.1007/s11071-020-06150-z

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