Effects of data windows on the methods of surrogate data

To generate surrogate data in nonlinear time series analysis, the Fourier transform is generally used. In the calculation of the Fourier transform, the time series is assumed to be periodic. Because such an assumption does not always hold true, the estimation accuracy of the Fourier transformed data and thus the power spectra is reduced. Due to such an estimation error, it is also possible that the surrogate test will lead to a false conclusion; for example, that a linear time series is nonlinear. In this paper, we experimentally evaluated the effects of data windows from the viewpoint of false rejections with several types of surrogate data. Our results indicate that if the data length becomes shorter, the false rejections by the data windows are reduced to a greater extent. However, if the data length is sufficient, the use of data windows is not a viable option. In the worst possible case wherein the linear memory of the original data is very long as in the nonstationary case, the critical length of the data for which the data windows were effective was approximately 1000.

Figure 1

Shapes of the data windows.

Figure 2

Characteristics of the ensemble variance ${\sigma }_{\left\{x\left(t\right)\right\}}^2$ of the AR model $x\left(t\right)$ by varying the coefficient $a$ of the AR model.

Figure 3

Results of the false rejection rates with the FT surrogate data.

Figure 4

Results of the false rejection rates with the FS surrogate data.

Figure 5

Results of the false rejection rates with the IAAFT surrogate data.