Time series columns containing statistical weights

Furthermore, one or more columns with statistical weights may be specified using the keyword col:weights. The keyword accepts two arguments: the first is the column index, the second is a floating-point value, say $p_n$ for the $n$th weights column. Given $N$ weights columns indexed according to $n = 1,\, ...,\, N$, the total weight for the $m$th row is evaluated using the weights $w_{nm}$ in the individual columns according to

$$\Gamma _m := \prod _{n=1}^{N}\gamma _{nm}^{p_n}\: .$$ (4)

Weights need not be normalised, this is performed by SIGSPEC.

Time, observable, and weights columns have to consist of floating-point numbers only. SIGSPEC checks these columns before starting the computations. If a non-numeric entry is found in one of the special columns, the program terminates with an error message.

Caution: SIGSPEC does not support the exponential annotation (e.g. 4.234E03 or 1.0385e-03)!

Example. The sample project weights contains a dataset with statistical weights in column 3 the squares of which are used by SIGSPEC, as specified by the .ini file entry

col:weights 3 2

The input time series weights.dat represents the sampling of IC4996#89 (V), and the magnitudes were synthesized by

1. a sinusoid with frequency 4.68573 cycles per day, amplitude 17.27 mmag,
2. Gaussian noise with 25 mmag rms deviation only for the measurements between HJD 2452524 and HJD 2452525,
3. Gaussian noise with 2.5 mmag rms deviation for all other nights.

Figure 3: Light curve for the sample project weights.

The resulting light curve is displayed in Fig.3. Fig.4 compares the frequency domain output (a closeup for frequencies below 10 cycles per day) with and without employing the weights. Without weights, the peak at 4.7 cycles per day visible, but not the most significant one. Moreover, there is no signal that reaches the sig threshold of 5.

   1 freq 5.68136  sig 3.75547  rms 0.088716  csig 3.75547

Figure 4: Grey: Fourier spectra for the sample project weights. Left: significance spectra. Right: DFT amplitudes. Top: spectra of the unweighted time series. Bottom: spectra employing statistical weights. The significant components are indicated by dots with dashed drop lines (file weights/result.dat). The default sig threshold of 5 is represented by a horizontal dashed line in the left panels.

Column 3 in the time series input file weights.dat contains zeroes for the measurements between HJD2452524 and HJD2452525 and values of 1 for the rest. Consequently, in this example, the exponent 2 assigned to the keyword col:weights in the file weights.ini does not affect the weighting: the result would be the same if, e.g.,

col:weights 3 1

were given instead of

col:weights 3 2

Employing the weights column, SIGSPEC provides the component at 4.7 cycles per day as the only significant signal:

   1 freq 4.67968  sig 20.395  rms 0.029129  csig 20.395
   2 freq 30.5489  sig 4.47468  rms 0.0252866  csig 4.47468