Time series with tailored nonlinearities

It is demonstrated how to generate time series with tailored nonlinearities by inducing well-defined constraints on the Fourier phases. Correlations between the phase information of adjacent phases and (static and dynamic) measures of nonlinearities are established and their origin is explained. By applying a set of simple constraints on the phases of an originally linear and uncorrelated Gaussian time series, the observed scaling behavior of the intensity distribution of empirical time series can be reproduced. The power law character of the intensity distributions being typical for, e.g., turbulence and financial data can thus be explained in terms of phase correlations.

Figure 1

Gaussian random uncorrelated noise (black, filled circles). The colored points show time series with linear phase correlations among adjacent phases ($\mathrm{\Delta }=1$) with ${\sigma }_{\eta }=1.0$ and $d\varphi =0$ (blue, triangles) and $d\varphi =\pi$ (red, crosses).

Figure 2

Upper row: Nonlinear prediction error $\psi$ (upper left) and average connectivity $\kappa$ (upper right) versus phase correlation coefficient $c(\mathrm{\Delta }=1)$ for 900 time series with imposed phase correlations derived from Gaussian noise as input time series $g\left(t\right)$. Lower row: Same as upper row but with the Mrk 766 x-ray observation as input time series $g\left(t\right)$. The data from revolution 999 was binned with a bin size of 50 s leading to a time series with 1540 points. The insets show the corresponding results for 400 IAAFT surrogates.

Figure 3

Time series $g\left(t\right)$ that is obtained from white Gaussian noise by imposing a set of six different linear phase correlations.

Figure 4

Cumulative probability distribution $P\left(g\right)$ of the normalized positive (black) and negative [red (gray)] values of $g\left(t\right)$. The black dashed and red (gray) dotted lines show the respective distributions for the initial white Gaussian noise.

Figure 5

Cumulative probability distribution $P\left(v\right)$ of the normalized volatility $v$. The black dashed line shows the respective distributions for the initial white Gaussian noise.