# A Survey on Change Detection and Time Series Analysis with Applications
## 4 Change or Breakpoint Detection within Non-Stationary Time Series
### Breaks for Additive Seasonal and Trend (BFAST)  
The BFAST is based on an iterative process that decomposes a time series into trend, seasonal, and remainder components [65; 66]. The trend component between any two consecutive jump locations is assumed to be linear, and the seasonal component is estimated by summing the cosine and sine functions of fixed frequencies. For example, in remote sensing applications, the frequencies are 1,2,3 cycles per year. The Ordinary Least-Squares (OLS) residual segment is obtained by subtracting the estimated trend and/or sinusoidal functions of fixed frequencies via the least-squares fit from the segment.  
The iterative process begins by estimating the seasonal component via a standard season-trend decomposition. The Ordinary Least-Squares Residuals-Based Moving Sum (OLS-MOSUM) is applied to investigate whether there exist any breakpoints in the trend component. If the OLS-MOSUM test shows that breakpoints are happening in the trend component, then their numbers and locations are estimated by the Least-Squares from the seasonally adjusted data, i.e., from the time series minus the estimated seasonal component. Given the breakpoints, the intercepts and slopes of the linear trends are estimated via a robust regression based on M-estimation. If the OLS-MOSUM test indicates breakpoints are occurring in the seasonal component, then their numbers and locations are estimated by the Least-Squares from the de-trended data, i.e., from the time series minus the estimated trend component. The amplitudes of the sinusoidal functions are estimated using a robust regression based on M-estimation to estimate the seasonal component. This process is iterated until the numbers and positions of the breakpoints no longer change.  
The BFAST R-code is available online [67]. The BFAST is a popular change detection method applied in many applications. For example, [PERSON] et al. in [68] showed that BFAST rarely failed to detect breakpoints. Since BFAST analyzes the entire time series by implementing a harmonic seasonal model, it is robust to noise and minimally influenced by outliers [66]. Though BFAST is a robust and popular change detection method, it also has weaknesses. For example, [PERSON] et al. in [69] demonstrated that BFAST performed very poorly at detecting seasonal changes, such as a change in amplitude or change in the number of seasons. They showed that BFAST is affected by missing data due to the use of linear interpolation to fill in the missing values. [PERSON] and [PERSON] in [70] showed that  
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.  
BFAST did not clearly detect changes caused by fires in semi-arid regions. Moreover, BFAST does not consider the observational uncertainties, e.g., the error bars shown in Figure 5f.