## 3. Data analysisSeveral methods are generally used to study the time-dependence of a periodic signal. A very popular example from variable star research is the so-called `O-C diagram' in which deviations between the observed and the predicted zero-phases of the signal are depicted over time. This method requires a sufficient signal-to-noise ratio, and good phase coverage because the signal must be `visible'. Another often-used method is to sub-divide the time series, and analyzing each sub-sample separately through periodogram analysis (Scargle 1982; Horne & Baliunas 1986). Using this method, Baliunas et al. (1985) and Donahue (1993) examined the year-to-year variations of the rotation periods of several solar-like stars (stellar butterfly diagrams). A more sophisticated method is the collection of Fourier transforms of subsamples formed by a window propagating over the entire series. This constitutes the Gabor transform. However, one disadvantage of the Gabor transform is that the number of oscillations inside the window depends on the frequency analyzed. Thus, the relative frequency resolution will vary over the frequency interval studied. The wavelet transform uses a window whose width is a function of the actual frequency. Several types of wavelets can be used. However, if the signal is sinusoidal, the wavelet should also be chosen to be sinusoidal. Thus, the Morlet wavelet (Grossmann & Morlet 1984) represents a sinusoidal oscillation contained within a Gaussian envelope. Then the wavelet transform can be written as where is the complex conjugate of where . The parameter Therefore, the relative frequency resolution is uniquely determined
by To evaluate the time-frequency behaviour we plot a wavelet map as the square of WT() (the `power') with dependence on time and frequency as a first step. A visual inspection of such a plot shows regions of enhanced power which, in case of a positive detection of the SDR, will follow systematic trends in the form of one or more `mountain ranges' or, even in a more complicated manner, as a series of isolated peaks as the peak frequency changes over the course of the activity cycle. Next, cuts are made along the frequency axis at discrete values of time, in order to find the absolute maximum power and also the relative maxima of power at each epoch and their corresponding frequencies. In the third step, these frequencies are plotted over time. This plot will represent a stellar butterfly diagram, i.e. the time evolution of rotation periods that are defined by the rotation experienced by plage regions causing modulation of the emission core of the solar disk-integrated Ca II K line. As in the case of Fourier analysis of unevenly sampled data, artifacts may result which can lead to misinterpretation of the wavelet map. Szatmáry, Vinkó & Gál (1994: SVG94) investigated the properties of the wavelet map in the cases of the following signal characteristics: i) varying amplitude, period, and zero-phase; ii) strong deviations of the signal from a sine curve; iii) double-mode oscillations; iv) the influence of noise; and, v) gaps in the time series. Based on their analysis, we assume the following expected characteristics might be found in wavelet pictures of our time series: i) Fig. 2 shows that strong variations in amplitude, period,
and phase exist, even on timescales close to the rotation period. A
variable amplitude will lead to a variation of power with time,
whereas jumps in zero-phase can disrupt a `mountain range' into
isolated peaks. A further loss of power occurs because the original
data are unevenly sampled, and while filling in the time series with
zeroes causes it to ii) Deviations from a sine curve will have little influence because strong deviations such as those in the light curve of eclipsing binaries cannot be expected. iii) Double-mode oscillations can arise from two ARs at different latitudes. The behaviour of the wavelet spectrum will depend on the time resolution because the superposition of two signals leads to a complicated but regular pattern in the time series. If the time-resolution is fine enough, the pattern will be visible in the wavelet map; otherwise the wavelet picture is no more informative than the Fourier spectrum, i.e., there will be two parallel mountain ranges in the wavelet map. If the two modes are close together in frequency; a beat phenomenon is expected. In this case, the wavelet map looks like amplitude modulation. Mode switching can, in principle, be localized in time which cannot be examined using Fourier analysis. iv) The influence of low amplitude noise on wavelet analysis is similar to its effect on Fourier analysis. However, in the case of strong noise, the Fourier spectrum is a better tool simply because the maximum number of data points is included in a window that has the maximum length possible. v) Gaps in the data are quite usual for stellar observations. Under certain circumstances they can produce side-lobes in the wavelet map, similar to `aliases' seen in Fourier analysis (Horne & Baliunas 1986). In this particular investigation, large (seasonal) gaps are not present; the characteristic gap size in this time series is only on the order of a few days. © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |