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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - ResMet (Resampling methods for nonstationary stochastic processes)

Teaser

Summary

My project was dedicated to applications of resampling methods in statistical inference for the processes with periodic and almost periodic structures. The mentioned processes in the literature are called cyclostationary (CS) and almost cyclostationary (ACS). Studies of CS/ACS processes started by Gladyshev in 1961. Since then the subject has developed fast and found applications in many branches of vibroacoustics, mechanics, signal analysis and climatology. The wide range of possible applications resulted in the thousands of papers that were published through the last 60 years. Unfortunately, the analysis of CS/ACS processes is very difficult and many problems are still unsolved. My objective was to provide tools for practitioners that will help them:
â€¢ to deal with short samples from periodic data, to construct confidence intervals for time and frequency characteristics of CS and ACS processes;
â€¢ to test significance of particular frequencies and hence be sure that the mean function was correctly removed from the data;
â€¢ to analyze data in the presence of jitter effect i.e., in the situation when time instances in which the considered process is observed are disturbed;
â€¢ to analyze data randomly sampled i.e., data for which times between the consecutive observations are not always equally spaced;
â€¢ to deal with data with non-zero mean function;
â€¢ to analyze data for which period is changing in time;
â€¢ to choose suitable for the considered problem bootstrap technique.
The obtained results can be applied in any setting, where data with periodic or almost periodic structure appear e.g., in economics, vibroacoustics, mechanics, signal processing, medicine, hydrology, climatology.
My additional objective was to make French habilitation, which will allow me to get professor position, supervise PhD students and create my own research group.

Work performed

During the fellowship I considered the following research problems:
1. applicability of the bootstrap resampling techniques for non-uniformly spaced data and data with jitter effect;
2. applicability of the subsampling method for nonstationary time series with non-zero mean function;
3. construction of tests allowing to check periodicity of the covariance function and to detect significance of the Fourier frequencies;
4. validity of the various bootstrap approaches for periodic characteristics of periodically correlated time series;
5. applicability of resampling techniques for the coefficients of the mean and the autocovariance functions of the time series with growing period;
6. modification of the existing bootstrap techniques to improve their properties;
7. review of periodic autoregressive moving average methods based on Fourier representation of periodic coefficients;
8. pointing out the optimal approach for specific estimation problems. Practitioner need to know which method in a given problem should be used;

Main achieved results:
a. I showed validity of the Moving Block Bootstrap for non-uniformly spaced data and data with jitter effect. I applied our results to the real vibratory signal. To prepare mentioned application I worked with the specialists from the LASPI laboratory in Roanne;
b. I proposed the conditions under which the subsampling method can be applied to nonstationary time series with non-zero mean function;
c. I used the method from point b. to construct tests allowing for detection of the significant Fourier frequencies in the mean function of periodically correlated processes and for investigating periodicity of the covariance function of the considered data. With the help of the specialist from the Cracow University of Economics I applied these results to Eurostat data, concerning monthly industrial production for mining and quarrying; manufacturing; electricity, gas, steam and air conditioning supply in three countries;
d. I proposed the extension of the Moving Block Bootstrap method, which is applicable for all time and frequency domain characteristics of periodically correlated time series. In particular it can be used for periodic characteristics like seasonal means and seasonal variances that are often considered in economics applications;
e. I showed applicability of the Generalized Seasonal Block Bootstrap for the coefficients of the mean and the autocovariance functions of the time series with growing period;
f. I introduced the Generalized Seasonal Tapered Block Bootstrap method, which allows to reduce some negative effects that appear during construction of the bootstrap sample while standard bootstrap approaches are used;
g. I provided a review of periodic autoregressive moving average methods based on Fourier representation of periodic coefficients and discussed existing open problems;
h. I performed a large simulation study to compare different bootstrap approaches and to find method for the block length choice;
i. I obtained French habilitation.

Final results

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 â€“ epileptic seizure) and seismography.