For each frequency ω, i.e., for each mass mϕ, we performed the linear least-squares analysis of the normalized signal Embedded Image composed of the Embedded Image contributions from all the four laboratories. We fit to the function R(t) = B + A(ω)cos(ωt + δ), where B is a constant offset (the contributions from different laboratories are weighted with inverse variance). In the left panel in Fig. 4, the square of the fitted amplitude is shown as a function of ω. To interpret the obtained result, one has to verify whether the fitted amplitude considerably exceeds the noise level. First, let us consider the simplest case when the noise of Embedded Image is white. The square amplitude of the harmonic component, A2(ω), can be treated as a new random variable. For the case of white noise [N samples of normalized signal Embedded Image with SD σ], its expected value does not depend on ω, 〈A2(ω)〉 = 4σ2/N, and the cumulative probability distribution function (CDF) can be expressed by a simple analytical formula (37)Embedded Image(9)Then, the A2 corresponding to X CL can be expressed asEmbedded Image(10)

The statistical interpretation of Eq. 10 is that the probability that Embedded Image is X. For instance, for X = 95%, the corresponding A2 level is Embedded Image. Equation 10 concerns a single frequency (one of the N/2 frequency channels available in the spectrum). That is, 1 per 20 frequency channels on average should exceed the Embedded Image level. Beyond the 95% CL criterion, we also defined a detection threshold for Nf frequency channels, Embedded Image, in a similar manner as performed in (13) and (37): The random variable, A2, exceeds the Embedded Image level for any frequency channel, on average, only 0.05 times per measurement. That is, if the measurement were repeated 100 times, then the noise would be interpreted as a positive detection, on average, only five times. To make this definition unique, we set an additional condition that the probability of exceeding the detection threshold is the same for all the frequencies. For the case of white noise, the detection threshold defined here can be expressed asEmbedded Image(11)

As an Nf, we took the total considered frequency range (see Fig. 4) divided by Δω = 2π/Ttot, where Ttot is a total time of our measurements (i.e., the difference between the time of the last and first samples in the data combined from all the laboratories). Equations 10 and 11 are strictly valid for white noise. We tested with the Monte Carlo simulations, however, that Eqs. 10 and 11 well approximate the CL and detection threshold for other types of noise; for this, one should replace the white noise parameter 〈A2〉 with the noise model function 〈A2〉(ω). For instance, in the case of pink noise, 〈A2〉(ω) ∝ 1/ω, the A2 corresponding to 5 and 99% CLs calculated as Embedded Image is indistinguishable from accurate Monte Carlo simulations at the scale of Fig. 4. A slight difference occurs for the detection threshold, but it can be easily eliminated with a numerically determined correction factorEmbedded Image(12)

We used the above expressions to determine the CLs and detection threshold. The periodogram in the left panel in Fig. 4 shows that in our measurements, the power distribution in the collected data is consistent with pink noise (37).

Note: The content above has been extracted from a research article, so it may not display correctly.



Q&A
Please log in to submit your questions online.
Your question will be posted on the Bio-101 website. We will send your questions to the authors of this protocol and Bio-protocol community members who are experienced with this method. you will be informed using the email address associated with your Bio-protocol account.



We use cookies on this site to enhance your user experience. By using our website, you are agreeing to allow the storage of cookies on your computer.