1. Introduction

A time series is an ordered sequence of data points measured at discrete time intervals that may not be equally spaced or may contain data gaps, and so the time series is unequally spaced or unevenly sampled. In certain experiments, measurements may also have uncertainties introduced by random noise and thus may also be unequally weighted.

In many areas of research, such as geodesy, geophysics, geodynamics, astronomy, and speech communications, researchers deal with series of data points measured over time or distance to study the periodicity and/or power of certain constituents. For example, in a speech record, one may be interested in certain signals in the record, such as the voice of a person obscured by other simultaneously recorded constituents, noise from musical instruments, and the environment. Furthermore, the physical series of data are usually a combination of various sinusoidal waves, such as light emitted from a star, sound waves, ocean waves, and thus certain phenomena may be studied more accurately by decomposing the data into a new domain, e.g., frequency or time-frequency domains. The decomposition of a series of data points into sinusoidal waves of various frequencies may be imagined as dispersing sunlight to waves of different frequencies representing different colors when it passes through a triangular prism, e.g., a rainbow.

If all statistical properties of a time series, i.e., the mean, variance, all higher-order statistical moments, and auto-correlation function, do not change in time, then the time series is called stationary. A non-stationary time series may contain systematic noise, such as trends, datum shifts, and jumps, indicating that its mean value is not constant in time. In certain fields, such as geodesy, geophysics, and astronomy, a time series is usually associated with a covariance matrix, which means that the time series is stationary or non-stationary in its second statistical moments. In geodynamics applications, seismic noise may contaminate the time series of interest or certain components of the time series may exhibit variable frequency, such as linear, quadratic, exponential, or hyperbolic chirps.

To solve problems regarding heat transfer and vibrations, Joseph Fourier introduced the Fourier series and showed that certain functions can be written as an infinite sum of harmonics or sinusoids. The Fourier transform basically decomposes a time series into the frequency domain using sinusoidal basis functions. The amplitudes of the sinusoids for a set of frequencies generate a spectrum like the light spectrum. Since then, numerous Fourier-based methods have been proposed by researchers for various purposes, such as time series analysis.

2. Related Material

The following open-access article reviews many current and traditional time series analysis methods listed below and briefly mentions their applications in various fields of science and engineering, such as geodesy, geophysics, remote sensing, astronomy, hydrology, finance, and medicine.

Ghaderpour, E.; Pagiatakis, S.D.; Hassan, Q.K. A Survey on Change Detection and Time Series Analysis with Applications. Appl. Sci. 2021, 11, 6141. DOI: DOI: 10.3390/app1113614110.3390/app11136141

Questions discussed in the review paper above include how one may:

1. extract useful information from a time series theoretically and empirically,
2. attenuate noise and regularize time series,
3. detect and classify changes in the time series components, and
4. analyze unequally spaced time series without any interpolations while considering the observational uncertainties.

The reviewed methods with their abbreviations are listed below:

ALFT Anti-Leakage Fourier Transform

ALLSSA Anti-Leakage Least-Squares Spectral Analysis

ASFT Arbitrary Sampled Fourier Transform

BFAST Breaks For Additive Seasonal and Trend

CCDC Continuous Change Detection and Classification

CLSSA Constrained Least-Squares Spectral Analysis

CWT Continuous Wavelet Transform

XWT Cross Wavelet Transform

DBEST Detecting Breakpoints and Estimating Segments in Trend

DFT Discrete Fourier Transform

DTFT Discrete-Time Fourier Transform

DWT DiscreteWavelet Transform

EMD Empirical Mode Decomposition

EWMACD Exponentially Weighted Moving Average Change Detection

EWT EmpiricalWavelet Transform

HHT Hilbert–Huang Transform

IMAP Interpolation by MAtching Pursuit

JUST Jumps Upon Spectrum and Trend

LSCWA Least-Squares Cross Wavelet Analysis

LSSA Least-Squares Spectral Analysis

LSWA Least-Squares Wavelet Analysis

MALLSSA Multichannel Anti-Leakage Least-Squares Spectral Analysis

MIMAP Multichannel Interpolation by MAtching Pursuit

OLS Ordinary Least-Squares

OLS-MOSUM Ordinary Least-Squares Residuals-Based Moving Sum

STFT Short-Time Fourier Transform

WWA Weighted Wavelet Amplitude

WWZ Weighted Wavelet Z-Transform