The development of time-frequency analysis techniques allow astronomers to successfully deal with the non-stationary time series that originate from unstable physical mechanisms. We applied a recently developed time-frequency analysis method, the Hilbert-Huang transform (HHT), to two non-stationary phenomena: the superorbital modulation in the high-mass X-ray binary SMC X-1 and the quasi-periodic oscillation (QPO) of the AGN RE J1034+396. From the analysis of SMC X-1, we obtained a Hilbert spectrum that shows more detailed information in both the time and frequency domains. Then, a phase- resolved analysis of both the spectra and the orbital profiles was presented. From the spectral analysis, we noticed that the iron line production is dominated by different regions of this binary system in different superorbital phases. Furthermore, a pre-eclipse dip lying at orbital phase ~ 0.6 - 0.85 was discovered dur- ing the superorbital transition state. We further applied the HHT to analyze the QPO of RE J1034+396. From the Hilbert spectrum and the O - C analysis results, we suggest that it is better to divide the evolu- tion of the QPO into three epochs according to their different periodicities. The correlations between the QPO periods and corresponding uxes were also different in these three epochs. The change in periodicity and the relationships could be interpreted as the change in oscillation mode based on the diskoseismology model.

Real-life time series characteristic data has significant amount of non-stationary components, especially periodic components in nature. Extracting such components has required many ad-hoc techniques with external parameters set by users in a case-by-case manner. In this study, we used Empirical Mode Decomposition Method from Hilbert-Huang Transform to extract them in a systematic manner with least number of ad-hoc parameters set by users. After the periodic components are removed, the remaining time-series data can be analyzed with traditional methods such as ARIMA model. Then we suggest a different way of setting control chart limits for characteristic data with periodic components in addition to ARIMA components.

Real-life time series characteristic data has significant amount of non-stationary components, especially periodic components in nature. Extracting such components has required many ad-hoc techniques with external parameters set by users in case-by-case m

We present the Hilbert-Huang transform (HHT) analysis on the quasi-periodic modulation of SMC X-1. SMC X-1, consistingof a neutron star and a massive companion, exhibits superorbital modulation with a period varying between ~40 d and ~65 d.We applied the HHT on the light curve observed by the All-Sky Monitor onboard Rossi X-ray Timing Explorer (RXTE) to obtainthe instantaneous frequency of the superorbital modulation of SMC X-1. The resultant Hilbert spectrum is consistent with thedynamic power spectrum while it shows more detailed information in both the time and frequency domains. According to theinstantaneous frequency, we found a correlation between the superorbital period and the modulation amplitude. Combiningthe spectral observation made by the Proportional Counter Array onboard RXTE and the superorbital phase derived in theHHT, we performed a superorbital phase-resolved spectral analysis of SMC X-1. An analysis of the spectral parameters versusthe orbital phase for different superorbital states revealed that the diversity of nH has an orbital dependence. Furthermore,we obtained the variation in the eclipse profiles by folding the All Sky Monitor light curve with orbital period for differentsuperorbital states. A dip feature, similar to the pre-eclipse dip of Her X-1, can be observed only in the superorbital ascendingand descending states, while the width is anti-correlated with the X-ray flux.

We investigate the sunspot area data spanning from solar cycles 1 (March 1755) to 23 (December 2010) in time domain. For this purpose, we employ the Hilbert transform analysis method, which is used in the field of information theory. One of the most important advantages of this method is that it enables the simultaneous study of associations between the amplitude and the phase in various timescales. In this pilot study, we adopt the alternating sunspot area as a function of time, known as Bracewell transformation. We first calculate the instantaneous amplitude and the instantaneous phase. As a result, we confirm a ~22-year periodic behavior in the instantaneous amplitude. We also find that a behavior of the instantaneous amplitude with longer periodicities than the ~22-year periodicity can also be seen, though it is not as straightforward as the obvious ~22-year periodic behavior revealed by the method currently proposed. In addition to these, we note that the phase difference apparently correlates with the instantaneous amplitude. On the other hand, however, we cannot see any obvious association of the instantaneous frequency and the instantaneous amplitude. We conclude by briefly discussing the current status of development of an algorithm for the solar activity forecast based on the method presented, as this work is a part of that larger project.