Differential significance spectra

Practical astronomical time series analysis occasionally comes along with target and comparison datasets that show coincident peaks in the DFT amplitude spectra. In this case, SIGSPEC provides a possibility to compute the probability that a peak in the target dataset is significant in spite of a given peak in the comparison dataset. Moreover, multiple target and/or comparison datasets may be handled the same way. The idea is to identify common (instrumental and/or environmental) effects and to distinguish them from periodicities exclusively found in a target dataset.

In the .ini file, there are three different keywords reserved for the specification of dataset types. Each expects one integer parameter representing the MultiFile index of the dataset under consideration.

1. The keyword target specifies a target dataset.
2. The keyword comp specifies a comparison dataset.
3. The keyword skip specifies a dataset to be ignored.

To enhance the convenience for the user, not all files need to be specified. The keyword deftype may be used to assign a default dataset type.

1. Use deftype target to assign the target attribute by default. If no deftype keyword is provided, this setting is activated.
2. Use deftype comp to assign the comp attribute by default.
3. Use deftype skip to assign the skip attribute by default.

Sampling profiles need to be computed for target datasets only. If the keyword profile is given in the .ini file, sampling profiles will only be generated for target datasets, and the file assign.log will also contain target datasets only.

To make datasets comparable even if their quality is different, the DFT spectra of the comparison datasets are scaled according to the power integral over the entire frequency range under consideration.

Instead of the observables $c_k$, $k = 0,1,...,K$ of a comparison dataset, the transformed quantities

$$c^\prime _k := \frac{P\left( x_l\right)}{P\left( c_k\right)} c_k$$ (17)

are used, where $x_l$, $l = 0,1,...,L$ denotes the observables of the target dataset under consideration and $P$ indicates the power integral of the quantity in parentheses. A DFT is calculated for each comparison dataset. There are two options to determine the resulting amplitude $A_T$ to be compared to the target amplitude $A$.

By default, the sig measures the probability of a peak generated by noise at the same variance as that of the given time series. In case of computing differential sigs, the normalisation has to be modified, since part of the power found in the target spectrum is assumed due to corresponding power in a comparison spectrum. To take this into account appropriately, a factor

$$\gamma := \frac{P\left( x_l\right)}{dP}$$ (18)

is introduced, where $dP$ is the power integral of the difference between the target data and the transformed comparison data. Correspondingly, the differential sig is a measure of the additional power with respect to the comparison dataset to be due to noise.

1. If the keyword diff:comp is set in the .ini file, a weighted arithmetic mean of the Fourier vectors, averaged over all comparison datasets is used to calculate $A_T$. The numbers of data points the comparison datasets consist of are used as weights. This option considers signal common among the comparison datasets only if the phases are aligned. Following the formalism by Reegen (2007), the cartesian representation of the differential sig evaluates to
   
   $$\mathrm{sig}\left( a_{\mathrm{ZM}},b_{\mathrm{ZM}}\left\vert... ...ght) = \gamma\frac{K\log\mathrm{e}}{\left< x^2\right>}\:\times$$
   
   
   
   
   $$\left\lbrace\left[\frac{\left( a_{\mathrm{ZM}}-a_{T\,\mathrm... ...,\mathrm{ZM}}\right)\sin\theta _0}{\alpha _0}\right] ^2\right.$$
   
   
   
   

   $$\left. + \left[\frac{\left( a_{\mathrm{ZM}}-a_{T\,\mathrm{ZM... ...ZM}}\right)\cos\theta _0}{\beta _0}\right] ^2\right\rbrace\: .$$ (19)

   
   

2. If the keyword diff:compalign is set in the .ini file, a weighted arithmetic mean of the DFT amplitudes, averaged over all comparison datasets is considered as $A_T$. The numbers of data points the comparison datasets consist of are used as weights. This option considers signal common among the comparison datasets also if they lag in phase. The differential sig is obtained through
   

   $$\mathrm{sig}\left( A\left\vert\right. A_T\right) = \gamma\fr... ...\sin ^2\left(\theta - \theta _0\right)}{\beta _0^2}\right]\: ,$$ (20)

   
   
   following the annotation introduced by Reegen (2007).

The default setting is diff:off, which switches off the computation of differential sigs.

Additional output is provided in the spectra (see p.), where columns 6 and 7 contain the DFT amplitudes and phases of the transformed comparison dataset, respectively.

Example. The sample project diffsig illustrates the analysis of target and comparison time series using differential significance spectra. There are nine time series input files available, indexed from 000038 through 000046. The file diffsig.ini contains the lines

mfstart 38
multifile -1

which forces SIGSPEC to start with the file 000038.diffsig.dat and compute all available datasets. In this case, SIGSPEC takes into account all files from 000038 to 000046. The two lines

deftype target
comp 38

in the file diffsig.ini define the file 000038.diffsig.dat as a comparison dataset and the rest as targets. Thus differential significance spectra are calculated for all time series from 000039 through 000045, with respect to 000038 as comparison data. The calculation of differential sigs is activated by the line

diff:compalign

in the file diffsig.ini, which produces differential sigs without respect to phase lags between comparison and target signals. The computations are made faster by the lines

ufreq 7
siglimit 0
iterations 1

The sampling of the input file 000038.diffsig.dat represents the V photometry of IC4996#89 (see Example SigSpecNative, p.), and the observable is a synthetically generated signal with unit amplitude at a frequency of 3.125 cycles per day, plus Gaussian noise with 5 units rms deviation. The corresponding significance spectrum, as obtained by typing

SigSpec 000038.diffsig

is displayed in the bottom panel of Fig.26. The five upper panels contain the differential significance spectra of the time series 000039 to 000046. These datasets contain 11649 points and are based on the sampling used in the project harmonics (p.). Gaussian noise with a standard deviation of 100 units is generated. Just as in case of the comparison data, a sinusoid at 3.125 cycles per day is synthesized, but the phase is not the same as for 000038.diffsig.dat. The amplitudes of this signal are 5 units for 000039, 6 units for 000040, 7 units for 000041, 8 units for 000042, 9 units for 000043, 10 units for 000044, 11 units for 000045, and 12 units for 000046. With increasing signal amplitude in the target data, the differential sig of the main peak consistently increases. In Fig.26 the datasets 000039 to 000046 are displayed from bottom to top.

Figure 26: Differential significance spectra for the sample project diffsig. Bottom: significance spectrum of comparison data, representing a sinusoidal signal at $3.125$ cycles per day (grey line), plus Gaussian noise. Top eight panels: Differential significance spectra for target time series representing the Gaussian noise plus a sinusoidal signal at $3.125$ cycles per day. Both the time-domain sampling and the signal phase differ from the comparison data. From bottom to top, the amplitude of this signal increases.