# A Survey on Change Detection and Time Series Analysis with Applications
## 2 Decomposition Methods into Frequency Domain
### Recent Methods of Mitigating Spectral Leakages in Spectrum  
Since the sinusoidal basis functions in the Fourier transform are not orthogonal over irregularly spaced series, spectral leakages appear in the Fourier spectrum, that is, power or energy leaks from one spectral peak into another [19; 20]. In LSSA, since the selected frequencies are examined one by one (out-of-context), spectral leakages still appear, which can result in a poor signal estimation, although the non-orthogonality between the sine and cosine basis functions is taken into account for each frequency.  
To visualize the spectral leakages, a synthetic unequally spaced time series is generated, as described below, that may represent a geophysical time series for one year. Suppose that \(f(t_{\ell})=1.5\sin(2\pi\cdot 6t_{\ell})+\sin(2\pi\cdot 15t_{\ell})\) and \(t_{\ell}\) (\(\ell=1,...,70\)) are random numbers in \([0,1]\) generated by the MATLAB command \"rand\" and sorted in ascending order (Figure 0(a)). For example, \(f(t)\) may be an ambient temperature in degree Celsius (\({}^{\circ}\)C) at time \(t\), and so the amplitude will be in \({}^{\circ}\)C in Figure 1. The time series contains two sine waves of frequencies 6 and 15 cycles per year with amplitudes 1.5 and 1, respectively. The ideal spectrum in terms of amplitudes is shown in Figure 0(b), which clearly shows the spectral peaks corresponding to the two sine waves. LSS of this time series calculated by Equation (7) (suppose \(\mathbf{P}=\mathbf{I}\), the identity matrix) is illustrated in Figure 0(c). It can be seen that the spectral peaks corresponding to the two sine waves are significant.  
In LSSA, the estimated amplitudes of this time series are obtained by Equation (5), then the square root of \(\mathbf{\xi}^{\mathrm{T}}\mathbf{\xi}\) is calculated, and its corresponding spectrum is illustrated in Figure 0(d). Alternatively, the absolute values of the Fourier coefficients obtained from Equation (4) (Figure 0(e)) show severe spectral leakages (see arrows). The absolute value of Fourier coefficients is approximately half of the amplitudes shown in Figure 0(d) but with more severe spectral leakages. Note that DFT of a real time series has an identical response to both positive and negative frequencies. In solving physical problems, negative frequencies in DFT have no meaning; however, it is mathematically a useful concept that can simplify the frequency domain representation of signals.  
The LSS peaks are cleaner and sharper in the percentage variance mode (Figure 0(c)) than amplitude mode (Figure 0(d)), and also, the statistics of the significant peaks are on the percentage variance and not on the amplitude. This is a fundamental and crucial approach in the development of the antileakage LSSA algorithm [19]. Moreover, the identification of signal frequencies in DFT is more difficult and challenging compared to LSSA because DFT does not consider the correlations among the sine and cosine basis functions [3; 14].  
One of the purposes of mitigating spectral leakages in the spectrum is regularization. Regularization is a process of introducing additional information in order to solve an ill-posed problem or to prevent over-fitting. This process usually uses the spectrum of an equally or unequally spaced time series in an inverse mode to produce an equally spaced time series that will have the same spectrum as the original time series but with attenuated random noise. In other words, regularization places a time series onto any desired equally spaced series. The same concept is true for data series, i.e., a sequence of data points that are distance indexed, so instead of time and frequency, distance and wavenumber are used, respectively.