## 3 Methodology and Details of Data Analysis
### Least Squares Spectral Analysis (LSSA) Method
Using the standard least squares Vanicek (1969), we obtain:  
\[\hat{\
u}=f-\widehat{g}=f-\[PERSON](\Phi^{T}\mathrm{C}_{f}^{-1}\[PERSON])^{-1}\,\[PERSON]^{T }\mathrm{C}_{f}^{-1}f \tag{4}\]  
From the projection theorem, it is known that \(\hat{\
u}\perp\widehat{g}\), which means that \(f\) has been decomposed into a signal \(\widehat{g}\) and noise \(\hat{\
u}\) (residual series). Thus, to describe how \(\widehat{g}\) represents \(f\), a fractional measure \(s\) that is the ratio of the length of this orthogonal projection to the length of \(f\) is used:  
\[s=\frac{f^{T}\mathrm{C}_{f}^{-1}\hat{g}}{f^{T}\mathrm{C}_{f}^{-1}f} \tag{5}\]  
In spectral analysis, the hidden periodic signals that are expressed in terms of sine and cosine base functions are sought. Therefore, if a set of spectral frequencies (\(\omega_{i}\), \(i=1,\,2,\ldots,m\)) are specified, then we can express the signals as:  
\[\widehat{g}(\omega_{i})=\ \hat{x}_{1i}\ \ \cos\omega_{i}t+\hat{x}_{2i}\ \ \sin\omega_{i}t \tag{6}\]Let \(\hat{x}=[\hat{x}_{1i}\,\hat{x}_{2i}\ ]^{T}\) and \(\Phi=[\ \cos\omega_{i}t\,\sin\omega_{i}t]\). Then \(\hat{x}\) can be determined from Equation (1). For different frequencies \(\omega_{i}\), \(i=1,\,2,\ldots,m\), different spectral values can be obtained. Then, the least squares spectrum is defined by:  
\[s\left(\omega_{i}\right)=\frac{f^{T}\mathrm{C}_{f}^{-1}\hat{s}\left(\omega_{i} \right)}{f^{T}\mathrm{C}_{f}^{-1}f}=\left[1+\frac{Q_{n}}{Q_{s}}\right]^{-1}, \quad i=1,\,2,\ldots,m \tag{7}\]  
where \(Q_{n}\) and \(Q_{s}\) are the quadratic norms of the noise and signal, respectively [34]. Equation (7) describes the least squares spectrum. Obviously, the least squares spectrum of \(f\) is the collection of the spectral values for all desired frequencies \(\omega_{i}\), \(i=1,\,2,\ldots,m\). The greater the spectral value at a frequency \(\omega_{i}\), the more powerful \(f\) is at this frequency [34]. Given Equation (7), statistically significant spectral peaks satisfy the following inequality:  
\[s\left(\omega_{i}\right)\geq\left[1+\frac{v}{2}F_{v,2,\alpha}\right]^{-1} \tag{8}\]  
It is obvious from Equation (8) that the least squares spectrum follows the Fisher distribution with \(v\) degree of freedom and \(\alpha\) level of significance [34]. Also, Equation (7) can be further developed into other forms to provide the researcher with more familiar spectral representations, such as power spectral density (PSD) in decibels (dB) or in unit\({}^{2}\)/frequency, where 'unit' is the unit of the time series values [37]. In [34], the least squares PSD in decibels is given by:  
\[\mathrm{PSD}_{\mathrm{LS}}=10\log\left[\frac{s}{1-s}\right]\left(\mathrm{dB}\right) \tag{9}\]  
Solving Equation (7) with respect to \(Q_{s}\) and dividing by the frequency \(f\), the classical least squares spectrum can be transformed into the least squares PSD in unit\({}^{2}\)/\(f\). \[\mathrm{PSD}_{\mathrm{LS}}=\frac{Q_{n}}{f}\left[\frac{s}{1-s}\right]\left( \mathrm{unit}^{2}\right/f\right) \tag{10}\]  
The least squares PSD in Equations (9) and (10) are equivalent to those determined from the Fast Fourier Transform (FFT) when the series is equally spaced and equally weighted. In addition, these equations can also be used to calculate the power spectra of any series, without the restrictive conditions of the FFT, which is the advantage of the LSSA method [37]. Generally speaking, the observed time series may include trigonometric base functions (Equation (6)) to describe the periodic components of the series, along with random walk and auto-regressive components. When the calculation of the least squares spectrum is carried out, there will be a simultaneous least squares solution for the parameters of the process. This approach is represented as a rigorous approach to the problem of hidden periodicities, where the parameters of the assumed linear system driven by noise are determined simultaneously with the amplitudes and phase of the periodic components, as well as with other parameters that describe systematic noise [34,37,58].