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In this paper, the mathematical derivation of the underlying probability distribution function for the normalized least-squares wavelet spectrogram is presented. The impact of empirical and statistical weights on the estimation of the spectral peaks and their significance are demonstrated from the statistical point of view both theoretically and practically. The simulation results show an improvement of approximately 0. 02 mm (RMSE) for annual signal estimation when statistical weights are considered in the least-squares wavelet analysis (LSWA). The weighted LSWA estimates the signals more accurately than the ordinary LSWA for different percentage amount of missing data. As a real-world application, Global Navigation Satellite Systems (GNSS) time series for a station in Rome, Italy are analyzed. The analyses of the GNSS time series provided by different agencies for the same station reveal statistically significant annual peaks, more significant in 2010 but less significant between 2018 and 2020, while the higher frequency components show different spectral patterns over time. A declining trend of approximately − 0. 42 mm/year since 2004 is estimated for the GNSS height time series, likely due to gradual land subsidence. The results not only highlight the advantages of LSWA but can also help to better understand the uncertainties involved in signal estimation. • The Least-Squares Wavelet Analysis (LSWA) of GNSS time series in Rome is presented. • The mathematical derivation of stochastic surfaces for spectrograms in LSWA is shown. • Considering statistical weights improved the accuracy of signal estimation. • Jumps and missing data affect signal estimation but can be greatly treated by LSWA. • Annual signals were more significant in 2010 but less between 2018 and 2020. [ABSTRACT FROM AUTHOR]

With the advent of the digital computer, time series analysis has gained wide attention and is being applied to many fields of science. This paper reviews many traditional and recent techniques for time series analysis and change detection, including spectral and wavelet analyses with their advantages and weaknesses. First, Fourier and least-squares-based spectral analysis methods and spectral leakage attenuation methods are reviewed. Second, several time-frequency decomposition methods are described in detail. Third, several change or breakpoints detection methods are briefly reviewed. Finally, some of the applications of the methods in various fields, such as geodesy, geophysics, remote sensing, astronomy, hydrology, finance, and medicine, are listed in a table. The main focus of this paper is reviewing the most recent methods for analyzing non-stationary time series that may not be sampled at equally spaced time intervals without the need for any interpolation prior to the analysis. Understanding the methods presented herein is worthwhile to further develop and apply them for unraveling our universe. [ABSTRACT FROM AUTHOR]

Change detection within non-stationary and unequally spaced remote sensing time series has become a key methodology for a broad range of environmental applications. A new method of analysing vegetation variation over lands is proposed. Four regions in northern Tunisia with various characteristics are selected, and a non-stationary and unequally spaced Normalized Difference Vegetation Index (NDVI) time series is obtained for each region since 2000. The Landsat 7 remote sensing satellite imagery with insignificant cloud-shadow coverage is used to calculate the NDVI after atmospheric correction. The Least-Squares Wavelet (LSWAVE) software is implemented to rigorously analyse each NDVI time series and study the relationship between the vegetation of olive trees and temperature/precipitation in one of the regions. To investigate possible effects of temperature on the green cover caused by increasing water salinity, the coherency between the NDVI and sea surface temperature time series is also shown in the region of Lake Ichkeul in Tunisia. [ABSTRACT FROM AUTHOR]

The least-squares wavelet analysis, an alternative to the classical wavelet analysis, was introduced in order to analyze unequally spaced and non-stationary time series exhibiting components with variable amplitude and frequency over time. There are a few methods such as cross-wavelet transform and wavelet coherence that can analyze two time series together. However, these methods cannot generally be used to analyze unequally spaced and non-stationary time series with associated covariance matrices that may have trends and/or datum shifts. A new method of analyzing two time series together, namely the least-squares cross-wavelet analysis, is developed and applied to study the disturbances in the gravitational gradients observed by GOCE satellite that arise from plasma flow in the ionosphere represented by Poynting flux. The proposed method also shows its outstanding performance on the Westford-Wettzell very long baseline interferometry baseline length and temperature series. [ABSTRACT FROM AUTHOR]