Synthetic time series

To illustrate how the 3-step procedure in STARS works, two time series of length 60, each consisting of three different regimes, were generated using the random number generator. The characteristics of the regimes are shown in the table below, and the data are presented in Fig. 1.

First, the test for regime shifts in the mean was performed with the following parameters: the target significance level p = 0.1 , the cut-off length l = 20, and the Huber's parameter h = 1. Looking at the chart in Fig. 1, it is not clear whether the time series contain regime shifts in the mean, red noise, or both. Therefore, it was decided to estimate and filter the red noise using the IP4 method with the subsample size of 12. The results of this part of the analysis are presented in Fig. 2 for time series x and in Fig. 3 for y.

After the stepwise trends are removed from x and y, the residuals are placed in worksheet "ResM", which becomes active if the "Variance" box was checked. If not, switch to the worksheet prior to testing for regime shifts in the variance. In our example, this second step of the analysis was performed with the same parameters as for the mean, i.e. p = 0.1 and l = 20, and the results are presented in Fig. 4 for x and in Fig. 5 for y.

For the third, final step of the analysis make sure that worksheet "NormV", where the normalized values of the residuals are placed, is active. The test for shifts in the correlation coefficient was performed with the same parameters, p = 0.1 and l = 20; however, it is not necessary that these parameters should be the same for each step. Depending on the situation, they can be adjusted separately for shifts in the mean, variance, and the correlation coefficient. Figure 6 shows three distinctive regimes in the correlation coefficient between x and y, along with their confidence intervals. The correlation coefficients were adjusted for small sample size using the Olkin-Pratt formula (see Zimmerman et al., 2003). The significance level for the difference in the correlation coefficients for the adjacent regimes were also calculated.