# A Survey on Change Detection and Time Series Analysis with Applications

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## Abstract

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## 1. Introduction

- In Section 2, several popular frequency decomposition methods are briefly reviewed, such as the Fourier transform and least-squares spectral analysis along with their modifications.
- In Section 3, several popular time-frequency decomposition methods are discussed, including the short-time Fourier and wavelet transforms, Hilbert–Huang transform, constrained least-squares spectral analysis, and least-squares wavelet analysis, and then two methods of analyzing multiple time series together are reviewed.
- In Section 4, several change or breakpoint detection methods within non-stationary time series are studied.
- In Section 5, many applications of the methods in applied sciences along with other techniques of time series analysis are briefly mentioned.
- Finally, the conclusion, findings, and limitations of this investigation are briefly summarized in Section 6.

## 2. Decomposition Methods into Frequency Domain

#### 2.1. Fourier Transform

#### 2.2. Least-Squares Spectral Analysis (LSSA)

#### 2.3. Recent Methods of Mitigating Spectral Leakages in Spectrum

#### 2.3.1. Antileakage and Arbitrary Sampled Fourier Transforms

#### 2.3.2. Interpolation by Matching Pursuit (IMAP)

#### 2.3.3. AntiLeakage Least-Squares Spectral Analysis (ALLSSA)

#### 2.4. Recent Methods of Mitigating Spectral Leakages in Spectrum Beyond Aliasing

#### 2.4.1. Interpolation by Matching Pursuit (MIMAP)

#### 2.4.2. Multichannel AntiLeakage Least-Squares Spectral Analysis (MALLSSA)

## 3. Decomposition Methods into Time-Frequency Domain

#### 3.1. Short-Time Fourier Transform (STFT)

#### 3.2. Continuous Wavelet Transform (CWT)

#### 3.3. Hilbert–Huang Transform (HHT)

#### 3.4. Constrained Least-Squares Spectral Analysis (CLSSA)

#### 3.5. Weighted Wavelet Z-Transform (WWZ)

#### 3.6. Least-Squares Wavelet Analysis (LSWA)

#### 3.7. Methods of Analyzing Multiple Time Series Together

#### 3.7.1. Cross-Wavelet Transform (XWT)

#### 3.7.2. Least-Squares Cross Wavelet Analysis (LSCWA)

## 4. Change or Breakpoint Detection within Non-Stationary Time Series

#### 4.1. Breaks for Additive Seasonal and Trend (BFAST)

#### 4.2. BFAST Monitor

#### 4.3. Continuous Change Detection and Classification (CCDC)

#### 4.4. Jumps Upon Spectrum and Trend (JUST)

#### 4.5. JUST Monitor

## 5. Other Methods and Applications

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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 | Discrete Wavelet Transform |

EMD | Empirical Mode Decomposition |

EWMACD | Exponentially Weighted Moving Average Change Detection |

EWT | Empirical Wavelet 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 |

## Appendix A. Flowcharts of ALFT, ASFT, IMAP, ALLSSA, and LSWA

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**Figure 1.**Spectral leakages in DFT and LSSA for unequally spaced time series. Arrows show many of the leakages.

**Figure 2.**(

**a**) Two aliased sinusoids with identical values at sampling frequency 32 Hz, and (

**b**) their gradients with many different values at sample positions.

**Figure 4.**Morlet wavelet of frequency 10 Hz and its analyses (1000 samples per second). (

**a**) Sinusoids of frequency 10 Hz, (

**b**) DFT spectrum of panel a, (

**c**) a Gaussian function, (

**d**) DFT spectrum of panel c, (

**e**) a Morlet wavelet, and (

**f**) DFT spectrum of panel e.

**Figure 5.**Simulation of a 16-day NDVI time series: (

**a**) seasonal component, (

**b**) trend component with an abrupt change in 2019 with a negative magnitude, (

**c**) noise component, (

**d**) the sum of seasonal, trend, and noise components, (

**e**) $30\%$ of the observations are randomly removed, and (

**f**) illustration of near-real-time monitoring. The breakpoint or jump is shown by a red star. The red vertical bars are the error bars. The black curve is the result of an over-fitting issue that arises when the sinusoids of frequencies 1,2,3 cycles per year are forced to be fitted to the stable time series segment, so when the result is forecast into the monitoring period, shown in the gray background, the breakpoint may not be correctly detected.

Field | Data/Time Series | Description | References |
---|---|---|---|

Geodesy | Very Long Baseline Interferometry (VLBI) and Temperature Data—GRACE and GOCE Data | LSSA, LSWA, and LSCWA are applied to VLBI and temperature time series; the methods are also applied to study the behavior of GRACE stripes in space-frequency domains | [43,60,94,95] |

Geophysics | Marine Seismic and Multicomponent Streamer Data | ALFT, ASFT, IMAP, MIMAP, ALLSSA and MALLSSA are applied to regularize coarsely sampled seismic data | [19,20,21,22,23,24,25,26,27] |

Remote Sensing | MODIS and Landsat Imagery | BFAST, BFAST Monitor, CCDC, DBEST, EWMACD, JUST, and JUST Monitor as well as LSWA and LSCWA and other similar methods are applied for vegetation and drought monitoring, forecasting forest fire danger conditions, and crop yield forecasting | [65,66,71,73,76,77,89,90,91,92,96,97,98,99,100] |

Astronomy | Telescope Observations—AAVSO | WWA, WWZ, HHT, LSSA, ALLSSA, and LSWA are applied to study the orbital, pulsation, and spin periods of variable stars and others | [42,51,101] |

Hydrology | Streamflow and Climate Data | CWT, XWT, HHT, LSWA, LSCWA, and other similar methods are applied to study inter- and intra-annual fluctuations in streamflow and climate time series for flow forecasting and other purposes | [102,103,104,105,106,107] |

Finance | Stock exchange and price series; daily Eurozone stock market | CWT, DWT, maximal overlap DWT, and wavelet coherency are applied to analyze financial time series | [108,109,110,111] |

Medicine | Electroencephalogram (EEG) and Electrocardiogram (ECG) Signals | CWT, DWT, EMD, and EWT are applied to the EEG and ECG signals to study brain and heart activities | [112,113,114] |

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**MDPI and ACS Style**

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

**AMA Style**

Ghaderpour E, Pagiatakis SD, Hassan QK.
A Survey on Change Detection and Time Series Analysis with Applications. *Applied Sciences*. 2021; 11(13):6141.
https://doi.org/10.3390/app11136141

**Chicago/Turabian Style**

Ghaderpour, Ebrahim, Spiros D. Pagiatakis, and Quazi K. Hassan.
2021. "A Survey on Change Detection and Time Series Analysis with Applications" *Applied Sciences* 11, no. 13: 6141.
https://doi.org/10.3390/app11136141