## 3 Methodology and Details of Data Analysis  
### Least Squares Spectral Analysis (LSSA) Method  
Due to the nature of the measurements, the observed time series are assumed to be composed of two main constituents: signal and noise. The noise can be random and systematic. Uncorrelated random noise with constant spectral density, which is known as white noise, is assumed to be ideal. However, in practice, the observables mostly include non-white random noise, which is a band-limited random function of time. Systematic noise may also have systematic and non-systematic parts and can be modeled with certain mathematical forms. In contrast, non-systematic noise can include datum shifts and trends, which renders the series non-stationary. When analyzing the time series, a possible trend is generally identified and removed to avoid non-systematic noise [34,36], and de-trended data consists of superimposed signals. Depending on the structure of the analyzed data, in some cases a particular signal of a time series can be dominant and can be modeled using simple techniques such as a parametric model or polynomial fitting. However, the measurements mostly have irregular distributions with unequal intervals, and in such cases spectral techniques are used to analyze the data as a sum of periodic signalsand to estimate their periods and amplitudes. This can be done either using non-parametric methods, such as Fourier transform based algorithms, which are commonly applied to longer time series with equidistant data, or using parametric methods, such as Least Squares Spectral Analysis, a method to decompose a signal into a sum of trigonometric terms using least squares techniques, even for short time series with non-equidistant data. Spectral Analysis by Least Squares was first developed and applied by [PERSON] in 1969 [PERSON] (1969, 1970) to overcome the inherent limitations of classical Fourier methods. Notable advantages provided by LSSA are: (_i_) the systematic noise (colored or otherwise) can be rigorously suppressed without causing any shift in the existing spectral peaks [PERSON] (1969), (_ii_) time series with unequally-spaced values can also be analyzed without pre-processing [PERSON] (1969), (_iii_) time series with an associated covariance matrix can be analyzed and (_iv_) statistical tests on the significance of spectral peaks can be performed, which makes the method promising and powerful for evaluating the time series. In LSSA, an observed time series is considered to be a function of time \(t_{i}\) and is represented by \(f=f(t)=\{f_{i}\}\), \(i=1,2,\ldots,n\). As the main objective of LSSA is to detect the periodic signals in \(f\), especially when both random and systematic noise is present, \(f\) can be modeled with function \(g\) as follows:  
\[g=\Phi\,x \tag{1}\]  
where \(\Phi\) is a matrix of known base functions and \(x\) is the vector of unknown parameters. Here it is not necessary for the time series to be equally-spaced. However, the observations \(f_{i}\) are assumed to possess a fully populated covariance matrix \(\mathrm{C}_{f}\). To estimate the model parameters \(x\), the standard least squares method (e.g., [PERSON] (1969)) is used, in which the difference between \(g\) and \(f\) is minimized in the least squares sense. The estimate of model parameters can be obtained as follows:  
\[\hat{x}=(\Phi^{T}\mathrm{C}_{f}^{-1}\Phi)^{-1}\,\Phi^{T}\mathrm{C}_{f}^{-1}f \tag{2}\]  
\[\widehat{g}=\[PERSON]\,\hat{x}=\[PERSON](\Phi^{T}\mathrm{C}_{f}^{-1}\[PERSON])^{-1}\,\[PERSON]^{T }\mathrm{C}_{f}^{-1}f \tag{3}\]  
In the least squares method, the model parameters are determined to minimize the difference between \(\widehat{g}\) and \(f\).