# A Survey on Change Detection and Time Series Analysis with Applications
## 3 Decomposition Methods into Time-Frequency Domain
### Weighted Wavelet Z-Transform (WWZ)  
[PERSON] [42] proposed a robust method of analyzing astronomical time series that are non-stationary and unequally spaced, namely, the Weighted Wavelet Z-transform (WWZ). In this section, the same notation is used as in [42] to define the WWZ. Suppose that \(\mathbf{f}=\big{[}f(t_{j})\big{]}\) is a time series of \(n\) data points. For each \(\tau\) (the window location; can be the \(t_{j}\) or equally spaced times) and each \(\omega_{k}\in\Omega\), let \(\mathbf{\Phi}_{1}=[\mathbf{1}]\)  
\[\mathbf{\Phi}_{2} =\big{[}\cos\big{(}2\pi\omega_{k}(t_{1}-\tau)\big{)},\ldots,\cos \big{(}2\pi\omega_{k}(t_{n}-\tau)\big{)}\big{]}, \tag{17}\] \[\mathbf{\Phi}_{3} =\big{[}\sin\big{(}2\pi\omega_{k}(t_{1}-\tau)\big{)},\ldots,\sin \big{(}2\pi\omega_{k}(t_{n}-\tau)\big{)}\big{]}, \tag{18}\]  
where \([\mathbf{1}]\) is the all-ones vector of dimension \(n\). The inner product of two vectors \(\mathbf{u}=[u(t_{1}),\ldots,u(t_{n})]\) and \(\mathbf{v}=[v(t_{1}),\ldots,v(t_{n})]\) is defined as  
\[\langle\mathbf{u}|\mathbf{v}\rangle=\frac{\sum_{i=1}^{n}w_{i}(t_{i})v(t_{i})} {\sum_{i=1}^{n}w_{i}}, \tag{19}\]  
where \(w_{i}\) is the statistical weight chosen as \(w_{i}=e^{-c(2\pi\omega_{k}(t_{i}-\tau))^{2}}\), and \(c\) is a window constant, which may be selected to be \(c=0.0125\), as described in Section 3.2, see Equation (13). The constant \(c\) determines how rapidly the analyzing wavelet decays [42]. Let \(\mathbf{S}\) be the square matrix of order three whose entries are \(\langle\mathbf{\Phi}_{a}|\mathbf{\Phi}_{b}\rangle\) (\(1\leq a,b\leq 3\)). The Weighted Wavelet Z-transform (WWZ) (power) is defined as  
\[Z=\frac{(N_{\text{eff}}-3)V_{y}}{2(V_{x}-V_{y})}, \tag{20}\]  
where \(N_{\text{eff}}=(\sum_{i=1}^{n}w_{i})^{2}/(\sum_{i=1}^{n}w_{i}^{2})\) is the effective number of data points, \(V_{x}=\langle\mathbf{f}|\mathbf{f}\rangle-\langle\mathbf{1}|\mathbf{f}\rangle^ {2}\) is the weighted variation of the data, and \(V_{y}=\sum_{a,b}\mathbf{S}_{ab}^{-1}\langle\mathbf{\Phi}_{a}|\mathbf{f} \rangle\langle\mathbf{\Phi}_{b}|\mathbf{f}\rangle-\langle\mathbf{1}|\mathbf{f} \rangle^{2}\) is the weighted variation of the model function obtained by the least-squares method [42]. Assuming the normality condition, [PERSON] also showed that Equation (20) follows the F-distribution with \(N_{\text{eff}}-3\) and two degrees of freedom and expected value of one [42].  
The WWZ is based on the least-squares optimization that is a robust method of analyzing non-stationary and unequally spaced time series; however, WWZ is a poor estimator of amplitude [42], and the spectral peaks in WWZ lose their power toward higher frequencies similar to CWT. Therefore, it may not detect possible hidden signatures in time series. The latter shortcoming is caused mainly by the effective number of data points that decreases when the frequency increases, making the numerator of Equation (20) smaller. Note that the term \(V_{y}/(V_{x}-V_{y})\) in Equation (20) is the estimated signal-to-noise ratio [42]. Subtracting \(V_{y}\) from \(V_{x}\) in the denominator of Equation (20) significantly alters the true power of the signals that may not allow one to see the behavior of the time series, especially when searching for possible hidden signals with low power in a time series. Furthermore, in WWZ, constituents of known forms and the covariance matrix associated with the time series are not considered, which may potentially alter the results. After the signal period is estimated via WWZ, the signal amplitude may be obtained via the Weighted Wavelet Amplitude (WWA) that is the square root of the sum of squares of the estimated cosine and sine coefficients [42].