## 3 Methods
### The Correlogram-Based Periodogram  
The power spectrum of a time series represents the frequency distribution of series' power at different frequency bands. It is commonly employed to discern cyclic patterns within time series data, as manifested by peaks in the power spectrum corresponding to the periods of these cycles. In the case of purely random processes, such as white noise, the power spectrum remains approximately constant across all frequencies; this signifies that the series is entirely uncorrelated, with each value being completely independent of the others.  
In this study, since the time series we will analyze are short, we will employ the correlogram-based periodogram, which has proven to be an efficient method for estimating the power spectrum of short time series with only a few dozen samples [43].  
The simple model of periodic series is:  
\[y_{n}=\beta\cos(\omega t+\phi)+\epsilon_{n} \tag{7}\]  
with \(\beta>0\), \(0<\omega<\pi\), \(\phi\) distributed as a uniform random variable in \((-\pi,\pi]\), and \(\{\epsilon_{n}\}\) denoting a sequence of uncorrelated random variables with zero mean and standard deviation \(\sigma\), independent of \(\phi\).  
The classical periodogram is given by  
\[I(\omega)=\frac{1}{N}\bigg{|}\sum_{n=1}^{N}y_{n}e^{-i\omega n}\bigg{|}^{2},0 \leq\omega\leq\pi \tag{8}\]  
with \(N\) being the length of the series, and it is calculated for  
\[\omega_{l}=\frac{2\pi l}{N},\,l=0,1,\ldots,a \tag{9}\]  
where \(a=\left[\frac{N-1}{2}\right]\) and \([x]\) indicates the integer part of \(x\). If a strong cycle with frequency \(\omega_{0}\) modulates the series, the periodogram peaks at \(\omega_{0}\), while, if the series is uncorrelated (\(\beta=0\)), according to [PERSON] [44], the classical periodogram is uniformly distributed across the frequencies. To improve the detection of periodic components in a time series, [PERSON] et al. [43] introduced the correlogram-based periodogram, which has been shown to be a more effective tool for identifying cyclic patterns compared to the traditional approach. The periodogram \(I(\omega)\) is equivalent to the correlogram:  
\[S(\omega)=\sum_{k=-N+1}^{N-1}\hat{r}(k)e^{-i\omega k} \tag{10}\]  
where  
\[\hat{r}(m)=\frac{1}{N}\sum_{k=1}^{N-m}y_{k}y_{k+m} \tag{11}\]  
estimates the autocorrelation function. By using the sample autocorrelation function  
\[\rho(m)=\frac{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sigma _{x}\sigma_{y}} \tag{12}\]  
where \(\bar{x}\) indicates the mean, the correlogram-based periodogram proposed by [PERSON] et al. [43] for estimating the power spectral density of a series is the following:  
\[\hat{S}(\omega)=2 Re\left(\sum_{k=0}^{L}\tilde{\rho}(k)e^{-i\omega k}\right)- \tilde{\rho}(0). \tag{13}\]where the real part of \(x\) is indicated by \(Re(x)\), the series' length by \(N\), and the biased Spearman rank correlation coefficient by \(\bar{p}(m)\). If the classical periodogram is always positive, the correlogram-based periodogram could be negative; therefore, the absolute value is taken.  
To test the significance of a frequency \(\omega_{l}\), the \(g\)-statistics could be used:  
\[g=\frac{\left|\bar{S}(\omega_{l})\right|}{\sum_{i=1}^{d}\left|\bar{S}(\omega_{i })\right|}. \tag{14}\]  
A large value of \(g\) at a specific frequency indicates that this frequency carries substantial power in the signal, suggesting it is statistically significant. The statistical significance of a tested frequency can be evaluated using a simulation-based approach [43]. Specifically, by employing Equation (7), we generate a set of random time series, and for each simulated series, the corresponding \(g\)-value is computed using Equation (14). The collection of these \(g\)-values is then used to approximate the distribution of the \(g\)-statistic, from which the associated \(p\)-value is estimated. A low \(p\)-value indicates a strong presence of the tested frequency in the original series.