Signal-to-Noise Ratio and Lomb-Scargle Periodogram

As pointed out by Reegen (2007), the SIGSPEC method represents a tool for an iterative frequency analysis of a zero-mean corrected time series superior to signal-to-noise ratio estimation (Breger et al. 1993) and Lomb-Scargle periodogram (Lomb 1976; Scargle 1982). However, in some situations these alternative methods may be desired or even more reasonable. Namely the Lomb-Scargle periodogram represents the optimum statistical approach to the problem if the mean observable is meaningful rather than set zero arbitrarily. The relations between sig and signal-to-noise ratio or Lomb-Scargle periodogram, respectively, are introduced and discussed by Reegen (2007).

In order to meet a user's requirement of signal-to-noise ratio-based DFT analysis or Lomb-Scargle periodograms as well, the SIGSPEC software offers the option to perform an analysis relying on amplitude signal-to-noise ratios by providing the keyword DFT in the .ini file. If this keyword is specified, all SIGSPEC computations rely on the approximation of sig by the amplitude signal-to-noise ratio according to

$$\mathrm{sig}\left( A\right)\approx\frac{K\log\mathrm{e}}{4}\,\frac{A^2}{\left< x^2\right>}\: ,$$ (27)

where $K$ represents the number of time series data, $A$ denotes the Fourier amplitude, and $\left< x^2\right>$ refers to the variance of the observable.

Second, the keyword Lomb forces SIGSPEC to evaluate Lomb-Scargle periodograms rather than significance spectra. In this case, the sig is approximated by

$$\mathrm{sig}\left( A\right)\approx\frac{K\log\mathrm{e}}{4}\,\frac{P_{\mathrm{LS}}}{\left< x^2\right>}\: ,$$ (28)

where $P_{\mathrm{LS}}$ denotes the power level in terms of the Lomb-Scargle periodogram.

Figure 37: Significance spectrum of the V photometry of IC4996#89 (grey) and approximation by the signal-to-noise ratio of DFT amplitudes (black).

Figure 38: Significance spectrum of the V photometry of IC4996#89 (grey) and approximation by the Lomb-Scargle periodogram (black).

Example. In the sample projects DFT and L-S, the input time series represents the V photometry of IC4996#89.

The file DFT.ini contains a single entry

DFT

which forces SIGSPEC to rely on the signal-to-noise ratio of DFT amplitudes. The screen output is:

   1 freq 3.13205  sig 9.75026  rms 0.00449592  csig 9.75026
   2 freq 3.98473  sig 6.80132  rms 0.00422861  csig 6.80083
   3 freq 5.40684  sig 5.31609  rms 0.0040257  csig 5.30209
   4 freq 17.3677  sig 4.1816  rms 0.00388775  csig 4.14988

The file L-S.ini contains a single keyword

Lomb

and SIGSPEC uses the Lomb-Scargle periodogram rather than sig for all computations. The screen output is:

   1 freq 3.13205  sig 9.75026  rms 0.00449592  csig 9.75026
   2 freq 3.98472  sig 6.79398  rms 0.00422861  csig 6.7935
   3 freq 5.40684  sig 5.31451  rms 0.0040257  csig 5.30033
   4 freq 17.3677  sig 4.18161  rms 0.00388775  csig 4.14977

The significance spectrum of the input time series is compared to the approximations by DFT amplitude signal-to-noise ratio and Lomb-Scargle periodogram in Figs.37 and 38, respectively.

A comparison of the two outputs and the screen output of the corresponding sig-based application (Example SigSpecNative, p.) reveals slightly different signal components. Especially for the second component the frequency of which is close to an integer multiple of 1 cycle per day and therefore susceptible to alias, the results are different for all three methods. However, the frequencies, amplitudes and phases in the files result.dat are in good agreement and reflect the numerical uncertainties of the MultiSine fitting procedure only.