During my fellowship, together with my coauthors, I managed to provide many important results concerning the resampling methods for cyclostationary and almost cyclostationary processes. According to my knowledge I provided the first bootstrap consistency results for the continuous time processes that are not observed continuously and for data with jitter. Moreover, I provided two new block bootstrap approaches (the Generalized Seasonal Tapered Block Bootstrap and the Extension of Moving Block Bootstrap). I showed the applicability of the block bootstrap for frequency domain parameters in a case when the period length is growing in time. These results will allow for better analysis of periodic and almost periodic signals. For time and frequency parameters not only estimates are now available but the bootstrap pointwise and simultaneous confidence intervals. As a consequence new tests can be introduced e.g. to detect significant frequencies, to test periodicity of data, to compare different signals.
My results can be applied to any data that contain periodic or almost periodic structure. Such data appear often in economics, vibroacoustics, mechanics, signal processing, medicine, hydrology, climatology. Already in my papers with the help of specialists I provided the real data applications in economics and mechanics. In the first case I used data published by Eurostat, concerning monthly industrial production for Mining and quarrying; manufacturing; electricity, gas, steam and air conditioning supply in France, Germany and Italy (see Figure 1). I tested the presence of the so-called trading-day effect or the calendar effect. The second application presented in my paper concerns a vibratory signal. My aim was to detect the fault of the gearbox. My results can be applied to any rotating machine to diagnose it by checking if a new significant frequency appeared. In such case an expert can easily find the fault of machine and repair it quickly. It is especially important when one works with big machines (e.g. in mines). Stopping them for long repairs is expensive and proper fault detection on the early stage can reduce costs significantly. Finally, my results for data with period changing in time can be used for analysis of chirp signals. Chirp signals appear in audio signals (animal communication, echolocation), radar and sonar systems, astrophysics (gravitational waves radiated by coalescing binaries), mechanics and vibrations (e.g. car engines), medicine (EEG data – epileptic seizure) and seismography.
