# A Survey on Change Detection and Time Series Analysis with Applications
## 3 Decomposition Methods into Time-Frequency Domain
### Methods of Analyzing Multiple Time Series Together  
#### 3.7.1 Cross-Wavelet Transform (XWT)  
To examine two equally spaced time series together having the same sampling rates and study the coherency and phase differences between their corresponding constituents, [PERSON] and [PERSON] [38; 47] proposed a method called the Cross-Wavelet Transform (XWT). Briefly, suppose that \(\mathbf{f_{1}}\) and \(\mathbf{f_{2}}\) are two equally spaced time series of size \(n\). For each pair \((t_{j},s)\), let \(\hat{f}_{1\psi}(t_{j},s)\) and \(\hat{f}_{2\psi}(t_{j},s)\) be their corresponding CWT coefficients calculated by Equation (11). XWT is defined as  
\[X_{\psi}(t_{j},s)=\left(\hat{f}_{1\psi}(t_{j},s)\right)\left(\hat{f}_{2\psi}^{ *}(t_{j},s)\right), \tag{21}\]  
where \(\hat{f}_{2\psi}^{*}(t_{j},s)\) is the complex conjugate of \(\hat{f}_{2\psi}(t_{j},s)\), and the XWT power is defined as \(|X_{\psi}(t_{j},s)|\)[37; 38]. The argument \(\arg[X_{\psi}(t_{j},s)]\) defines the phase differences (local relative phase or coherence phase) between the constituents of the two series [37; 38]. Note that the argument of a complex number \(z=a+ib\) is \(\arg[z]=\tan^{-1}(b/a)\).  
[PERSON] and [PERSON] [38] showed that if a time series is derived from a population of random variables following the multidimensional normal distribution, then the real and imaginary parts of the Fourier coefficients and thus the CWT coefficients are also normally distributed. Therefore, the continuous wavelet power, that is, the sum of squares of two normally distributed random variables, follows \(\chi_{2}^{2}\) (chi-squared distribution with two degrees of freedom). Now, if both time series are independently derived from populations of random variables following the multidimensional normal distribution, then the underlying probability distribution function of the squared absolute value of XWT, i.e., \(|X_{\psi}(t_{j},s)|^{2}=|\hat{f}_{1\psi}(t_{j},s)|^{2}|\hat{f}_{2\psi}(t_{j}, s)|^{2}\), is in fact the product of two chi-squared distributions each with two degrees of freedom. The underlying probability distribution function of the XWT power (\(|X_{\psi}(t_{j},s)|\)) is \(f(z)=zK_{0}(z)\), where \(K_{0}(z)\) is the modified Bessel function of order zero defined as \(K_{0}(z)=\int_{0}^{\infty}\cos\,(z\sinh t)\,\,\mathrm{d}t=\int_{0}^{\infty}\cos \,(xt)/\sqrt{t^{2}+1}\,\,\mathrm{d}t\)[58; 59].  
Because in the locations toward the edges of the CWT spectrogram, the wavelet is not completely localized, edge artifacts occur in those locations. The cone of influence (COI) in CWT or XWT is defined as the area outside which the edge effects can be neglected [37; 38].  
Although CWT and XWT are well-established methods of time-frequency analysis, they cannot be used directly for analyzing unequally spaced series with associated covariance matrices. Furthermore, these methods cannot handle the constituents of known forms, such as trends and/or datum shifts, and thus are not appropriate for analyzing time series in many geodetic and geophysical applications.