Least-Squares Wavelet Analysis of Unequally Spaced and Non-stationary Time Series and Its Applications

Abstract

Least-squares spectral analysis, an alternative to the classical Fourier transform, is a method of analyzing unequally spaced and non-stationary time series in their first and second statistical moments. However, when a time series has components with low or high amplitude and frequency variability over time, it is not appropriate to use either the least-squares spectral analysis or Fourier transform. On the other hand, the classical short-time Fourier transform and the continuous wavelet transform do not consider the covariance matrix associated with a time series nor do they consider trends or datum shifts. Moreover, they are not defined for unequally spaced time series. A new method of analyzing time series, namely, the least-squares wavelet analysis is introduced, which is a natural extension of the least-squares spectral analysis. This method decomposes a time series to the time–frequency domain and obtains its spectrogram. In addition, the probability distribution function of the spectrogram is derived that identifies statistically significant peaks. The least-squares wavelet analysis can analyze any non-stationary and unequally spaced time series with components of low or high amplitude and frequency variability, including datum shifts, trends, and constituents of known forms, by taking into account the covariance matrix associated with the time series. The outstanding performance of the proposed method on synthetic time series and a very long baseline interferometry series is demonstrated, and the results are compared with the weighted wavelet Z-transform.

Appendix A: Derivation of Eq. (10)

A similar methodology is used as in Wells et al. (1985) and Craymer (1998) to obtain Eq. (10). From the model $\mathbf {y}=\overline{\mathbf {\Phi }}\ \mathbf {\overline{c}}=\underline{\mathbf {\Phi }}\ \mathbf {\underline{c}}+ \mathbf {\Phi }\ \mathbf {c}$, estimate $\mathbf {\overline{c}}$ using the least-squares method as $\hat{\mathbf {\overline{c}}}=\mathbf {\overline{N}}^{-1}\overline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}} \mathbf {y}$, where $\mathbf {\overline{N}}=\overline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}}\overline{\mathbf {\Phi }}$. Now write $\mathbf {\overline{N}}^{-1}$ as

$$\begin{aligned} \mathbf {\overline{N}}^{-1}= \left[ \begin{array}{rr} \underline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}}\underline{\mathbf {\Phi }} &{} \ \ \ \underline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}}\mathbf {\Phi }\\ \mathbf {\Phi }^\mathrm{T} \mathbf {P}_{\mathbf {y}} \underline{\mathbf {\Phi }} &{} \ \ \ \mathbf {\Phi }^\mathrm{T} \mathbf {P}_{\mathbf {y}} \mathbf {\Phi } \end{array} \right] ^{-1}\!=\left[ \begin{array}{ll} \mathbf {M_1} &{} \ \mathbf {M_2} \\ \mathbf {M_3} &{} \ \mathbf {M_4} \end{array} \right] \!, \end{aligned}$$

where

$$\begin{aligned} \mathbf {M_1}&=\Big (\mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}-\mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }\ (\mathbf {\Phi }^\mathrm{T} \mathbf {P}_{\mathbf {y}} \mathbf {\Phi })^{-1}\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}\Big )^{-1}, \\ \mathbf {M_2}&=-\mathbf {M_1}\ \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }\ (\mathbf {\Phi }^\mathrm{T} \mathbf {P}_{\mathbf {y}} \mathbf {\Phi })^{-1},\\ \mathbf {M_3}&=-\mathbf {M_4}\ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1}, \\ \mathbf {M_4}&=\Big (\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }-\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1}\mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }\Big )^{-1}, \end{aligned}$$

and $\mathbf {\underline{N}}=\underline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}}\underline{\mathbf {\Phi }}$. Thus, from

$$\begin{aligned} \left[ \begin{array}{cc} \mathbf {x} &{} \\ \hat{\mathbf {c}} \end{array} \right] \!=\hat{\mathbf {\overline{c}}}=\mathbf {\overline{N}}^{-1}\overline{\mathbf {\Phi }}^\mathrm{T} \mathbf {P}_{\mathbf {y}} \mathbf {y}=\left[ \begin{array}{ll} \mathbf {M_1} &{} \ \mathbf {M_2} \\ \mathbf {M_3} &{} \ \mathbf {M_4} \end{array} \right] \left[ \begin{array}{cc} \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y} &{} \\ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y} \end{array} \right] \!, \end{aligned}$$

one obtains

$$\begin{aligned} \hat{\mathbf {c}}&=\mathbf {M_3} \ \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y}+ \mathbf {M_4} \ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y} \\&=-\mathbf {M_4}\ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1} \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y}+\mathbf {M_4} \ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y}\\&= \mathbf {M_4} \ \mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\Big (\mathbf {y}-\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1} \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y}\Big )\\&= \mathbf {N}^{-1}\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\hat{\mathbf {g}}. \end{aligned}$$

where $\mathbf {N}=\mathbf {M_4}^{-1}=\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }-\mathbf {\Phi }^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1}\mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}}\mathbf {\Phi }$, and $\hat{\mathbf {g}}=\mathbf {y}-\mathbf {\underline{\Phi }}\ \mathbf {\underline{N}}^{-1} \mathbf {\underline{\Phi }}^\mathrm{T}\mathbf {P}_{\mathbf {y}} \mathbf {y}$ is the estimated residual segment. Note that for each frequency, $\mathbf {N}$ is a square matrix of order two, and so its inverse is computationally efficient.