## 3 Example and Discussion  
We consider, as an example of a dataset for which the distribution is far from normal, a sequence of 85,284 radon decay measurements acquired at the Geological Survey of Israel (GSI) [5,6]. The measurements, registered by universal time, were acquired at 1-hour intervals between day 86 of 2007 and day 312 of 2016, for local hour of day in the range 10 pm to 2 am [6]. The distribution of measurements, shown as a histogram in Figure 1a, is obviously far from normal.  
A power spectrum was computed by the following likelihood procedure [4]:  
\[S=\frac{1}{2\sigma^{2}}\sum_{n=1}^{N}x_{n}^{2}-\frac{1}{2\sigma^{2}}\sum_{n=1}^{N }(x_{n}-X_{n})^{2}, \tag{7}\]  
where \(\sigma\) is the standard deviation of the measurements,  
\[X_{n}=A\ e^{2\pi vt_{n}}+A^{*}e^{-2\pi vt_{n}} \tag{8}\]  
and, for each frequency, the complex amplitude \(A\) is adjusted to maximize the power \(S\). This choice can be made by noting that the complex amplitude that maximizes \(S\) is the amplitude that is unchanged by arbitrary small perturbations in the complex power, so that we can determine the appropriate complex amplitude by the operation  
\[\frac{\partial S}{\partial A^{*}}=0. \tag{9}\]  
This likelihood procedure yields exactly the same power as would be derived from the Lomb-Scargle procedure but it also yields--from the complex amplitude \(A\)--the amplitude and phase for each frequency. The power spectrum computed in this way is shown in Figure 1c.  
The distribution of measurements, normalized according to the RONO procedure, is shown in Figure 1b. It is indeed precisely normal.  
Figure 1: (**a**,**b**) Histograms of midnight Geological Survey of Israel (GSI) gamma measurements without (**a**) and with (**b**) rank-order normalization (RONO) normalization. (**c**,**d**) Power spectrum formed from the 4-h band of measurements centered on midnight for the frequency band 0–6 year\({}^{-1}\) based on raw data (**c**) and on RONO-normalized data (**d**).  
The power spectrum formed from the normalized measurements, shown in Figure 1d, is visually indistinguishable from Figure 1c. Table 1 lists the top 20 peaks in the power spectrum formed from the original dataset and the top 20 peaks in the power spectrum formed from the RONO-normalized dataset. We see that the frequencies are exactly the same and the powers differ by only a few percent.  
This result suggests that a typical power-spectrum analysis may be less sensitive to departures from normality of the dataset than one might expect.  
Further development of the RONO procedure could be to implement this approach as a data-preprocessing step for time-frequency analysis methods (e.g., wavelet analysis, Stockwell-transform) or for correlation analysis methods (e.g., wavelet coherence, linear and non-linear correlation analysis).  
**Author Contributions:** Conceptualization, software, visualization, writing--original draft preparation, writing--review and editing; P.S.; visualization, writing--review and editing, F.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Acknowledgments:** PS is indebted to [PERSON] and [PERSON] for many helpful conversations concerning time series analysis.  
**Conflicts of Interest:** The authors declare no conflict of interest.