# A Survey on Change Detection and Time Series Analysis with Applications
## 3 Decomposition Methods into Time-Frequency Domain
### Constrained Least-Squares Spectral Analysis (CLSSA)  
[PERSON] et al. [55] introduced CLSSA as a seismic spectral decomposition method and showed that it has spectral resolution advantages over the conventional methods, such as CWT. Suppose that \(\mathbf{\Phi}_{w}\) is a design matrix consisting of the sinusoidal functions truncated by the end points of a window in the time domain. More precisely, let  
\[\mathbf{\Phi}_{w}=\begin{bmatrix}e^{2\pi i\omega_{1}(t_{1}-\tau)}&e^{2\pi i\omega_ {2}(t_{1}-\tau)}&\cdots&e^{2\pi i\omega_{n}(t_{1}-\tau)}\\ e^{2\pi i\omega_{1}(t_{2}-\tau)}&e^{2\pi i\omega_{2}(t_{2}-\tau)}&\cdots&e^{2 \pi i\omega_{n}(t_{2}-\tau)}\\ \vdots&\vdots&\cdots&\vdots\\ e^{2\pi i\omega_{1}(t_{m}-\tau)}&e^{2\pi i\omega_{2}(t_{m}-\tau)}&\cdots&e^{2 \pi i\omega_{n}(t_{m}-\tau)}\end{bmatrix}, \tag{14}\]  
be a \(m\times n\) matrix, where \(m\) is the number of data points within the time window, \(n\) is the number of frequencies (equal to the size of the entire data series according to DFT), and \(\tau\) is the center of the window (the translating parameter). Now consider system (model) \(\mathbf{\Phi}_{w}\mathbf{u}=\mathbf{y}\), where \(\mathbf{y}\) is the windowed seismic data series (complex segment) of size \(m\) and \(\mathbf{u}\) is the column vector of unknown Fourier coefficients. Since \(m\) is less than \(n\), this system is underdetermined, and specific constraints are needed to estimate the unknowns and achieve a unique solution. Note that an underdetermined system is a system that has more unknowns than equations.  
The CLSSA uses two diagonal matrices \(\mathbf{W_{u}}\) and \(\mathbf{W_{y}}\), which are the model and data weights, respectively, to constrain the underdetermined system. Matrix \(\mathbf{W_{u}}\) changes in iteration (starts with \(\mathbf{I}\)), and \(\mathbf{W_{y}}\) is constant throughout the iteration. Applying these two matrices, the underdetermined system becomes \(\mathbf{W_{y}}\mathbf{\Phi}_{w}\mathbf{W_{u}}\mathbf{W_{u}}^{-1}\mathbf{u}= \mathbf{W_{y}}\mathbf{y}\). Let \(\mathbf{\[PERSON]}_{\mathbf{w}}=\mathbf{W_{y}}\mathbf{\[PERSON]}_{w}\mathbf{W_{u}}\) and \(\mathbf{u_{w}}=\mathbf{W_{u}}^{-1}\mathbf{u}\). The CLSSA minimizes the following cost function  
\[\mathrm{Y}=\|\mathbf{\[PERSON]}_{\mathbf{w}}\mathbf{u_{w}}-\mathbf{W_{y}}\mathbf{y}\|^ {2}+\alpha\|\mathbf{u_{w}}\|^{2}, \tag{15}\]  
where \(\|.\|\) is the L2 norm and parameter \(\alpha\) is selected empirically which varies at each iteration step. Note that for a \(n\)-dimensional complex vector \(\mathbf{z}=[z_{1},\ldots,z_{n}]\), the L2 norm of \(\mathbf{z}\) is \(\|\mathbf{z}\|=\sqrt{|z_{1}|^{2}+\cdots+|z_{n}|^{2}}=\sqrt{z_{1}z_{1}+\cdots+ z_{n}z_{n}}\). The solution of Equation (15) is  
\[\mathbf{\hat{u}_{w}}=\mathbf{\Phi}_{\mathbf{w}}^{*}(\mathbf{\Phi}_{\mathbf{w}}\mathbf{ \Phi}_{\mathbf{w}}^{*}+\alpha\mathbf{I})^{-1}\mathbf{W_{y}}\mathbf{y}, \tag{16}\]  
where \(*\) is the conjugate transpose. The estimated Fourier coefficients are \(\mathbf{\hat{u}}=\mathbf{W_{u}}\mathbf{\hat{u}_{w}}\).  
The approach of constrained least-squares solutions incorporated to bypass the problem of a singular normal equation matrix (rank-deficient matrix) has a much longer history. In the 1970s, there were many attempts to solve underdetermined problems in geodesy, resulting in minimal constraints and inner constraints, etc., which in the end satisfied the least-squares g-inverse and Moore-Penrose inverse, etc. [5; 56].  
Although CLSSA results in a time-frequency analysis with improved frequency resolution compared to STFT and CWT, no detailed discussion and analyses are provided for analyzing unequally spaced time series by [PERSON] [57]. CLSSA may not be able to handle the constituents of known forms, such as trends and/or datum shifts, and so it may not be appropriate for the detection of hidden signatures in time series. Furthermore,no discussion is made for incorporating the covariance matrix in the model as well as the rigorous statistical analyses for a possible test on the significance of spectral peaks.