7 Conclusions

Time series analysis is a part of standard mathematical statistics. In time series analysis, we formulated the three stage period analysis (TSPA) capable of detecting any arbitrary periodic function in weighted data (Sect. 3). The free and "other'' model parameter error estimates were then presented (Sect. 4), and the determination of data distribution in time (Sect. 5). In statistics, the idea that the TSPA represents $\chi^2$-tests performed for m independent frequencies allowed significance estimates. The Kolmogorov-Smirnov test supplied further modelling information. The linear correlations between the phase residuals $\bar{\phi}'$ and $\bar{\phi}$ were introduced for the spurious periodicity identification that is crucial, e.g. in multiperiodicity studies (Breger 1997). In general, all three stages of TSPA may sometimes be unnecessary, e.g. the analysis of some data might begin directly from the GSch.

One of our main goals was to provide the precise formulation of all methods applied in Paper I. Several examples of the TSPA analysis of real data were therefore presented in Sect. 6. Two subsets of the normalized photometry of V 1794 Cyg were studied in Sects. 6.1 and 6.2. The TSPA and complementary methods performed perfectly in the first case. The second case illustrated problematic results that are not at all rare for real data with any time series analysis method. Our methodology in analysing all 93 subsets of normalized photometry of V 1794 Cyg in Paper I was summarized in Sect. 6.3. A higher order model was illustrated in Sect. 6.4, where the fifth order TSPA was applied to the V light curve of the cepheid variable BL Her. Even higher order models may be necessary, e.g. a TSPA with $K\!>\!5$ could be used to model the sharp primary and secondary minima interrupting the phases of constant brightness in eclipsing binaries, if the data phase coverage is adequate. The case of "pure noise'' was studied in Sect. 6.5, where the first order TSPA was applied to the V light curve of the nonvariable star SAO50205.

We shall now compare our method to the popular Lomb-Scargle periodogram, i.e. the advantages and limitations of the TSPA with respect to the innumerable other available time series analysis methods are not reviewed in detail. Except for the $\Theta_{\mathrm{grid}}$ scaling (Eq. 7), our GSch with $K\!=\!1$ and unity weights equals the Lomb-Scargle periodogram (Scargle 1982: "... fitting sine waves...''). In this case the RSch would give the same results, if combined to a high overfilling factor in the GSch, e.g. OFAC in Press & Teukolsky (1988). But the Lomb-Scargle periodogram analysis fails when the correct model is $K\!\geq\!2$(e.g. with the $\chi^2$-criterion), while the TSPA with any K utilizes this error information. The Lomb-Scargle periodogram has been compared to several other time series analysis methods, e.g. by Schwarzenberg-Czerny (1996) when introducing a method that substituted a Fourier series model of Kharmonics with an equivalent orthogonal polynomial series. It was also pointed out that the analysis of variance test statistic (ANOVA) combined to and applied in this orthogonal polynomial method is uniformly the most powerful test statistic. Schwarzenberg-Czerny (1996) emphasized that unlike ANOVA, many frequently applied test statistics are normalized by dividing with the empirical variance of the data, which causes unreliable significance estimates. We conclude that the $\chi^2_0$ significance estimates of the TSPA model are reliable, if only if the accuracy of the data is precisely known. But even when the unknown accuracy of individual measurements prevents significance estimates, cases like the analysis of the 93 subsets of V 1794 Cyg normalized photometry confirm that our method detects real periods and identifies spurious ones (Table 1). Furthermore, the bootstrap model parameter error estimates are always reliable, because these estimates are based on the modelling residuals that reflect the accuracy of the data, be it known or unknown. In conclusion, the TSPA and complementary methods, being relatively easy to apply and interpret, are efficient in searching for periodicity in large weighted time series (e.g. Paper I).

Acknowledgements

The work was partly supported by the EC Human Capital and Mobility (Networks) project "Late type stars: activity, magnetism, turbulence'' No. ERBCHRXCT940483. We wish to thank the (anonymous) referee for valuable comments on the manuscript.