Getting an Error Estimate for the Spectrum

Students in Physics 3730 may skip this step.

The Fourier spectrum you get shows peaks. Are they meaningful, or are they simply random noise fluctuations? To find out, it is necessary to do an error analysis to determine how noise in the measurement of the light curve translates to variations in the spectrum. To do this we use an assumed probability distribution of the intensities in each bin to generate a set of random artificial data sets and run the Fourier analysis on each. The variation in the resulting spectrum tells us how much can be attributed to noise. This type of analysis is sometimes called a “bootstrap” analysis.

To be more specific, consider the signal in just one bin. If the central value of the signal is $I_0$ and the standard deviation is $\sigma$, we model the probability of getting a signal $I$ as a Gaussian:

\begin{displaymath}
P(I) = \frac{1}{\sqrt{2 \pi}\sigma} \exp[-(I-I_0)^2/(2\sigma^2)].
\end{displaymath} (15)

So for the artificial intensity value in this bin, we choose a random value of $I$ based on this probability distribution. We do the same for each bin to generate a complete artificial data set. We then run the artificial data through the Fourier analysis and get a new spectrum. We repeat several times. We can then average the spectra and determine its error at each frequency. That is, we get the mean value and standard deviation for the spectrum at each value of the frequency. This should help us decide what structure is noise and what is significant.

Details are given in the Appendix ( 8).