I’m curious about units on the Power and Semi-amplitude axes of these plots. Attached are results for the delta Scuti star ZZ Mic for just under one month of data from TESS.
Searching the Internet for the definitions of power and semi-amplitude spectra, the usual description is that the power is the square of the amplitude (not semi-amplitude). Looking at the units on the attached plots, that is clearly not the case.
Can anyone explain the situation?
Roy
I’ve experimented and searched further. I ran the same analysis with the same data in Peranso. Here is a screenshot of the result.
Roy
I meant to include the statement that Peranso does not appear to give the option of plotting semi-amplitude against frequency in the DC DFT - plotting Theta seems to be the only possibility.
Roy
Has anyone had a chance to look at the issue raised in the above posts?
Roy
Hi Roy
Apologies again for the delay. Chipping away at a new VStar release since early last year (with Max) has kept me occupied, among other things lately.
I also wanted to think and read a bit more before responding.
I’ll start with what Grant Foster says in Analyzing Light Curves. The following excepts are from Section 7, Fourier Periodogram:
The power, by which we judge the statistical quality of the fit between our model and the data, is based on the sum of squared residuals with the best-fit parameters.
The DCDFT is based on a least-squares fit of the data to three trial functions, a siane function at the trial frequency, a cosine function at the trial frequency, and a constant. Hence it’s based on a model of the form
xj = β0 + β1 cos(2πνt) + β2 sin(2πνt) + εj, (7.1)
and computes the power level to represent the statistical significance of the fit. Other Fourier methods (like the DFT and FFT) are based on simpler computations which reduce to just about the same thing when the time sampling is regular, i.e., when the data are evenly spaced in time.
As an aside, there are of course more sin & cos terms in a Fourier model with multiple harmonics.
The example power spectrum periodogram plots he gives show Power on the y-axis.
In relation to the semi-amplitude spectrum plots, Grant has this to say:
But with Fourier periodograms we can compute other quantities to express the response at a given trial frequency. In particular, almost all Fourier analysis programs compute the amplitude (actually the semi-amplitude) as the (semi-)amplitude of the best-fit sinusoid at the given trial frequency. It’s important information; after all, if we decide we’ve detected a statistically significant periodicity in our data by matching it to a sinusoid, we’d like to know how big the sinusoid is!
…
The amplitude spectrum is similar to the power spectrum (i.e., periodogram); it shows a single dominant peak at the true signal frequency (1 cycle/day). Its height indicates the (semi-)amplitude of the signal…Still, power and amplitude are related; the power level is usually proportional to the square of the amplitude. In some cases this is exactly true and in almost all cases it’s a very good approximation.
Notice the text I have highlighted in bold above: the power level is usually proportional to the square of the amplitude. So, the power level is not equal to the square of the amplitude, but is proportional to it.
The example plots you show seem (at first glance) to be consistent with this revised definition (of proportionality) and with the comment about the background level of the amplitude spectrum mentioned next but I have
The background level of the amplitude spectrum indicates the typical (semi-) amplitude due to the noise. This gives us a good idea how large a signal has to be in order to be detected by the Fourier periodogram; its amplitude has to be big enough to rise above the noise in order to be detected.
Yet there are cases in which the power level is not approximately proportional to the square of the amplitude, especially when the time sampling of the data is highly irregular. In such cases it’s the power level which truly indicates significance, so it’s the power spectrum (periodogram) rather than the amplitude spectrum which enables us to assess whether or not periodic behavior exists in our data. Furthermore, for scientific analysis it’s customary to compute and plot power spectra, this is what your colleagues will be expecting. All in all scientific practice calls for the computation of power spectra, although amplitude spectra can be useful and informative.
I don’t yet know what the denotation of Theta in Peranso is but I have not searched extensively. Others who use Peranso may already know more. I did however find the following, in this paper:
Figures 3.1 through 3.7 show screenshots that depict the process of using Peranso. In the figures, “Theta” is used by Peranso to represent the power in a given period value.
Given the y-axis range in the Peranso and VStar screenshots, they appear to represent the same value or at least, the same range for this example, as you say, irrespective of name.
David
Hi David,
Thank you for your detailed reply, which is very helpful. My apologies for taking up your time when you have so much to do.
Roy
Hi Roy
Oh, no problem at all! I’m happy to help and encourage questions! I was just concerned about the length of time it took for me to reply.
David
Hi David,
No need for any concern and thanks again.
Roy
