Change of the nature of a test when surrogate data are applied

Surrogate data is a well-known method in nonlinear time series analysis and it has been widely used in testing nonlinearity. Fourier transform-based surrogates are artificially generated time series which share the linear properties of the observed series. They can be used for the generation of critical values for test statistics. In this paper we will show that the variance of these critical values may be of the same order as the variance of the test statistic itself. This changes the nature of the test because the test rejects if the test statistic divided by the critical value exceeds 1. An example is a test for normality that checks higher-order empirical cumulants. We will show that such a test is transformed to a test on (circular) stationarity.

Figure 1

Plot of the histogram of the test statistic $T_3$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 2

Plot of the histogram of the test statistic $T_4$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 3

Plot of the histogram of the test statistic $T_8$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 4

Plot of the histogram of the test statistic $T_9$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 5

Plot of the histogram of the test statistic $T_{10}$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 6

Plot of the histogram of the test statistic $T_{11}$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).

Figure 7

Plot of the histogram of the test statistic $T_{12}$ (upper plot) and plot of the histogram of $95\%$ quantiles based on surrogate data (lower plot).