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Ghaderpour, E. Time Series Analysis. Encyclopedia. Available online: https://encyclopedia.pub/entry/11685 (accessed on 03 December 2023).

Ghaderpour E. Time Series Analysis. Encyclopedia. Available at: https://encyclopedia.pub/entry/11685. Accessed December 03, 2023.

Ghaderpour, Ebrahim. "Time Series Analysis" *Encyclopedia*, https://encyclopedia.pub/entry/11685 (accessed December 03, 2023).

Ghaderpour, E.(2021, July 05). Time Series Analysis. In *Encyclopedia*. https://encyclopedia.pub/entry/11685

Ghaderpour, Ebrahim. "Time Series Analysis." *Encyclopedia*. Web. 05 July, 2021.

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With the advent of the digital computer, time series analysis has gained wide attention and is being applied to many fields of science. In this survey entry, many traditional and recent techniques for time series analysis and change detection are reviewed, including spectral and wavelet analyses with their advantages and weaknesses. The main focus of the survey is the analysis of unevenly sampled or unequally spaced time series.

applied sciences
change detection
Fourier transform
least-squares
non-stationary
spectral analysis
time series
trend analysis
unequally spaced
wavelet analysis

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.

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: 10.3390/app11136141

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

- extract useful information from a time series theoretically and empirically,
- attenuate noise and regularize time series,
- detect and classify changes in the time series components, and
- 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

There are many more robust time series analysis techniques proposed by researchers that were not mentioned in this survey. In many practical applications, time series contain seasonal and trend components. Simultaneous estimation of the statistically significant components can provide more accurate and reliable estimates for the time series components

and thus are more appropriate for change detection and monitoring. The time series components can also be estimated more accurately when considering the observational uncertainties. Therefore, the observations with higher uncertainties hold less weight during the analysis and vice versa. Computational complexity optimization is another major challenge in time series analysis when dealing with big data sets. An inappropriate algorithm modification for reducing the computational cost can produce unreliable and inaccurate results. Each method presented herein has advantages and weaknesses, and there is plenty of room for researchers and scientists to expand and improve the existing methods.

and thus are more appropriate for change detection and monitoring. The time series components can also be estimated more accurately when considering the observational uncertainties. Therefore, the observations with higher uncertainties hold less weight during the analysis and vice versa. Computational complexity optimization is another major challenge in time series analysis when dealing with big data sets. An inappropriate algorithm modification for reducing the computational cost can produce unreliable and inaccurate results. Each method presented herein has advantages and weaknesses, and there is plenty of room for researchers and scientists to expand and improve the existing methods.

Finally, we conclude this by the following quote from Nikola Tesla (1856–1943):

*If you want to find the secrets of the universe, think in terms of energy, frequency, and vibration. We are just waves in time and space, changing continuously, and the illusion of individuality is produced through the concatenation of the rapidly succeeding phases of existence. What we define as likeness is merely the result of the symmetrical arrangement of molecules that compose our bodies.*

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