# A Survey on Change Detection and Time Series Analysis with Applications
## 3 Decomposition Methods into Time-Frequency Domain
### Continuous Wavelet Transform (CWT)
For a time series, a scalogram can be obtained by calculating wavelet power \(|\hat{f}_{\psi}(t_{j},s)|^{2}\) for all pairs \((t_{j},s)\). The time and frequency resolutions of CWT depend on the scale \(s\). It can be seen from Figure 3c that CWT has high-frequency resolution at low frequencies and high-time resolution at high frequencies [1]. The Discrete Wavelet Transform (DWT) of a time series is calculated by passing it through a series of filters, calculated differently from CWT for different purposes [1; 34; 39]. CWT is more useful for feature extraction and time series analysis due to its fine resolution; whereas, DWT is a fast transform that represents a time series in a more redundant form and often used for data compression and noise attenuation of two-dimensional signals or images [1; 34]. Note that unequally spaced wavelets are not defined, so CWT and DWT are not defined for an unequally spaced time series unless one interpolates to fill in the missing data or shrinks the wavelet [40; 41]. However, interpolation may produce undesirable biases in spectrograms, especially for non-stationary time series [42; 43]. Moreover, the covariance matrix associated with the time series is not considered in STFT and CWT [43]. There are a bewildering variety of wavelets to use in CWT, such as Biorthogonal, Coiflets, Daubechies, Harr, Meyer, Morlet, and Symlet, and if one uses an inappropriate wavelet to correlate with a time series, then the CWT spectrogram may give misleading and nonphysical results [36; 39; 44; 45]. In order to study the periodicity of the constituents and to detect the periodic fluctuations in a time series, one may choose a (mother) wavelet, which fluctuates based on the periodic sinusoidal basis functions [31; 42]. A simple and commonly used wavelet possessing this property is the Morlet wavelet defined as  
\[\psi(\eta)=\pi^{-1/4}e^{-\eta^{2}/2}e^{2\pi i\omega_{0}\eta}, \tag{12}\]  
where \(\eta\) may represent time, and the cyclic frequency \(\omega_{0}\) is chosen to be \(6/(2\pi)\) cycles per unit time to satisfy the wavelet admissibility condition of the mother wavelet and provides a good balance between time and frequency resolution [35; 37; 38]. The condition \(\int_{0}^{\infty}(|\Psi(\omega)|^{2}/\omega)\ \mathrm{d}\omega<\infty\) is called the wavelet admissibility condition that is a necessary condition for the inverse of CWT, where \(\Psi\) is the Fourier transform of \(\psi\)[1]. For this choice of \(\omega_{0}\), the Fourier period of the Morlet wavelet (\(T=1/\omega_{0}\)) is almost equal to 1.03 s.  
To match the inverse of scale factor \(s\) exactly to frequency, [PERSON] [42] applied a simple transformation to define the Morlet wavelet as  
\[\psi(z)=e^{-cz^{2}}e^{iz}=e^{-c(2\pi\omega(t-\tau))^{2}}e^{2\pi i\omega(t- \tau)}, \tag{13}\]  
where \(\omega\) is the cyclic frequency and \(z=2\pi\omega_{0}\eta\), so \(c=1/(8\pi^{2}\omega_{0}^{2})\), disregarding coefficient \(\pi^{-1/4}\) in Equation (12) because it does not affect the results of the least-squares methods. For MATLAB users, the command \"scal2 fra\" may be used to convert the scales to frequencies in CWT [36]. Note that when \(\omega_{0}=6/(2\pi)\), \(c=1/72\), that is, approximately close to \(c=0.0125\) used in [42]. Figure 4 shows the construction and analyses of a Morlet wavelet given by Equation (13) for cyclic frequency \(\omega=10\) cycles per second (Hertz or Hz). Figure 4a,c are generated using \(e^{2\pi i\omega(t-\tau)}\) and \(e^{-c(2\pi i\omega(t-\tau))^{2}}\), respectively, which are the two components of the Morlet wavelet shown in Figure 4e. Comparing Figure 4b-f, one can see that the bandwidth of the Morlet wavelet is larger (poor frequency resolution) than the pure sinusoids. There are several definitions for bandwidth in the literature. The bandwidth is typically understood as the frequency interval in the spectrum where the main part (95% or 99%) of its power is located; others use 50% to be compatible with the definition of the cutoff frequency of a filter: \(-3\) decibel (dB) half-power point [1,46].