# Trend Change Detection in NDVI Time Series: Effects of Inter-Annual Variability and Methodology

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. GIMMS NDVI3g Dataset

#### 2.2. Breakpoint Detection Algorithm

#### 2.3. Methods for Trend Estimation

#### 2.3.1. Trend Estimation on Annual Aggregated Time Series (Method AAT)

#### 2.3.2. Trend Estimation Based on a Season-Trend Model (Method STM)

_{1}is the intercept and α

_{2}the slope of the trend, γ are the amplitudes and δ the phases of k harmonic terms and ε is the residual error [44]. The frequency f is the number of observation per year (i.e., 12 for monthly observations). Parameters α

_{1}, α

_{2}are estimated using ordinary least squares (OLS) regression whereby the derived time series segments are considered as categorical interaction term with the trend slope α

_{2}. The significance of the trend in each segment is estimated from a t-test on the interaction parameter of the regression between time series segment and α

_{2}.

#### 2.3.3. Trend Estimation on De-Seasonalized Time Series

_{2}is estimated using OLS from the seasonal-adjusted time series:

#### 2.4. Simulation of Surrogate Time Series

#### 2.4.1. Estimation of Inter-Annual Variability, Seasonality and Short-Term Variability from Observed Time Series

- (1)
- The mean of each NDVI time series was calculated.
- (2)
- In the second step, monthly values were averaged to annual values and the trend was calculated according to method AAT but without computing breakpoints. Hence, the slope of the annual NDVI trend over the full length of the time series was estimated.
- (3)
- To estimate the inter-annual variability, the standard deviation and range of the annual anomalies were calculated. The mean of the time series and the derived trend component from step (2), were subtracted from the annual values to derive the trend-removed and mean-centred annual values (annual anomalies). If the trend slope was not significant (p > 0.05), only the mean was subtracted. The standard deviation and the range of the annual anomalies were computed as measures for the inter-annual variability of the time series.
- (4)
- In the next step, the range of the seasonal cycle was estimated. The mean, the trend component and the annual anomalies were subtracted from the original time series to calculate a detrended, centered and for annual anomalies adjusted time series. Based on this time series the seasonal cycle was estimated as the mean seasonal cycle and the range was computed.
- (5)
- In the last step, the standard deviation and the range of the short-term anomalies were computed. Short-term anomalies were computed by subtracting the mean, the trend component, the annual anomalies and the mean seasonal cycle from the original time series. The result is the remainder time series component. The standard deviation of the remainder time series component is a measure of short-term variability.

#### 2.4.2. Surrogate Time Series and Factorial Experiment

- (1)
- Trend: Time series with strong and weak positive, strong and weak negative and without a trend were created. Different magnitudes of trend slopes were derived from the 1% percentile of the observed distribution of trend slopes (strong decrease), 25% percentile (weak decrease), median (no trend), 75% percentile (weak increase) and 99% percentile (strong increase), respectively.
- (2)
- Inter-annual variability: Time series with low, medium and high inter-annual variability were created based on normal-distributed random values with zero mean and a standard deviation according to the 1%, 50% and 99% percentiles of the observed distribution of the standard deviation of annual anomalies. Values outside the observed ranges of inter-annual variability were set to the minimum or maximum of the observed distribution, respectively.
- (3)
- Seasonality: Seasonal cycles based on a harmonic model with low, medium, and high amplitudes were created according to the observed 1%, 50% and 99% percentiles of the distribution of seasonal ranges.
- (4)
- Short-term variability: Different levels of short-term variability were created based on normal-distributed random values with zero mean and a standard deviation according to the 1%, 50% and 99% percentiles of the observed distribution of the standard deviation of remainder time series values.

- (1)
- Type of trend and number of breakpoints/segments (maximum 2 breakpoints = maximum 3 segments per time series with positive, negative or no trend = 27 possibilities),
- (2)
- Trend magnitude (weak, strong),
- (3)
- Inter-annual variability (low, medium, high),
- (4)
- Short-term variability (low, medium, high),
- (5)
- Type of trend change (gradual, abrupt) and
- (6)
- Range of seasonal cycle (low, medium, high).

#### 2.5. Evaluation of Method Performances

#### 2.5.1. Evaluation of Breakpoints

#### 2.5.2. Evaluation of Trend Slopes and Significances

- N3: significant negative trend (slope < 0 and p ≤ 0.05)
- N2: non-significant negative trend (slope < 0 and 0.05 < p ≤ 0.1)
- N1: no trend with negative tendency (slope < 0 and p > 0.1)
- P1: no trend with positive tendency (slope > 0 and p > 0.1)
- P2: non-significant positive trend (slope > 0 and 0.05 < p ≤ 0.1)
- P3: significant positive trend (slope > 0 and p ≤ 0.05).

#### 2.5.3. Evaluation of the Overall Performance for Trend and Breakpoint Estimation

#### 2.6. Application to Real Time Series of Alaska: Ensemble of NDVI Trends

_{m}of a method m, weighted by the length l of the corresponding time series segment seg and the p-value p of the trend in the segment expressed as significance:

_{m}.

## 3. Results

#### 3.1. Observed and Simulated Properties of NDVI Time Series

#### 3.2. Evaluation of Estimated Breakpoints

#### 3.3. Evaluation of Estimated Trends

#### 3.4. Effects on the Overall Performance of the Methods

#### 3.5. Multi-Method Ensemble of Breakpoint and Trend Estimates in Alaska

## 4. Discussion

#### 4.1. Effect of Temporal Resolution on Method Performance

#### 4.2. Effect of Trend Changes on Method Performance

#### 4.3. Effect of Inter-Annual Variability on Method Performance

#### 4.4. Plausibility of Trend and Breakpoint Estimates in Alaska

#### 4.5. Practical Recommendations

## 5. Conclusions

## Acknowledgments

**Conflict of Interest**The authors declare no conflict of interest.

## References and Notes

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**Figure 1.**Estimated time series components for a random-selected example grid cell in central Alaska (3*3 grid cells averaged around central pixel 146.424°W, 64.762°N). The upper panel shows the original Normalized Difference Vegetation Index (NDVI) time series with its mean value (red line). The next panels show the estimated trend, inter-annual variability (IAV) (i.e., annual anomalies), seasonality (i.e., mean seasonal cycle) and short-term variability (remainder component), respectively. The sum of mean, trend, IAV, seasonal and remainder component equals the original time series.

**Figure 2.**Spatial and statistical distributions of NDVI time series properties in Alaska and time series components of the simulated NDVI time series. The left panel shows from top to bottom maps of the following time series properties: mean annual NDVI, slope of the annual trend (ΔNDVI/year), standard deviation of the inter-annual variability (iav), range of the seasonal cycle (seas), and the standard deviation of the remainder component (rem). The middle panel shows the statistical distribution of these properties, respectively. The right panel shows examples of the respective surrogate time series components.

**Figure 3.**Examples of simulated time series with different components of trend, IAV, seasonal and remainder referring to the simulated trend, inter-annual variability, seasonal and remainder time series components, respectively. The sum of these time series components gives the total simulated surrogate NDVI time series (upper panel). Left: time series with one breakpoint and gradual change (e.g., caused by gradual changes in environmental conditions), no trend in the first segment and decreasing trend in the second segment, medium inter-annual variability, medium seasonality and medium short-term variability. Right: Time series with one breakpoint and abrupt change (e.g., caused by a few years with exceptional favourable growing conditions), increasing trend in first segment and decreasing trend in second segment, high inter-annual variability, medium seasonality and low short-term variability.

**Figure 4.**Frequencies of differences between real and estimated number of breakpoints for the different methods from all experiments (blue indicates underestimation, red overestimation of the number of real breakpoints). (

**a**) Performance of the methods in all experiments. (

**b**) Grouped by trend magnitude. (

**c**) Grouped by inter-annual variability. (

**d**) Grouped by short-term variability. (

**e**) Grouped by the real number of breakpoints. (

**f**) Grouped by the type of trend change. (

**g**) Grouped by the range of the seasonal cycle.

**Figure 5.**Distribution of the temporal absolute difference between real and estimated breakpoints. (

**a**) Performance of the methods in all experiments. (

**b**) Grouped by trend magnitude. (

**c**) Grouped by inter-annual variability. (

**d**) Grouped by short-term variability. (

**e**) Grouped by the real number of breakpoints. (

**f**) Grouped by the type of trend change. (

**g**) Grouped by the range of the seasonal cycle. + denotes the mean of the distribution. The difference is only based on detected breakpoints. As method AAT detected fewer breakpoints, it has a much smaller sample size (n = 42) than the other methods (STM n = 732, MAC n = 1,368, SSA n = 1,380).

**Figure 6.**Comparison of real and estimated slopes from different methods, all time series segments and all experiments. Slopes are coloured blue if both real and estimated slopes are not significant, green if only the real or estimated slope was significant and red if both slopes were significant (0.95 significance level).

**Figure 7.**Distribution of the root mean square error (RMSE) between real and estimated trend component for the different methods from all experiments. (

**a**) Performance of the methods in all experiments. (

**b**) Grouped by trend magnitude. (

**c**) Grouped by inter-annual variability. (

**d**) Grouped by short-term variability. (

**e**) Grouped by the real number of breakpoints. (

**f**) Grouped by the type of trend change. (

**g**) Grouped by the range of the seasonal cycle.

**Figure 8.**Ensemble of breakpoint and trend estimates from all methods in Alaska. AAT, STM, MAC and SSA are the four applied trend methods. ‘all’ indicates that all NDVI values were used (i.e., including interpolated and snow-affected observations). ‘ex’ snow-affected values were excluded from trend analysis. ‘peak’ trend was computed only on annual peak NDVI. (

**a**) Mean number of detected breakpoints from all methods. (

**b**) Uncertainty of the number of detected breakpoints (standard deviation of number of breakpoints from all methods). (

**c**) Number of detected breakpoints grouped by method. (

**d**) mean duration of greening trends (years) with associated uncertainties (

**e**) and distribution of greening duration per each method (

**f**). (

**g**) mean duration of browning trends (years) with associated uncertainties (

**h**) and distribution of browning duration per each method (

**i**). (

**j**) Multi-method mean trend slope (ΔNDVI/year) with associated uncertainties (

**k**) and distributions per each method (

**l**).

**Figure 9.**Comparison of detected breakpoints with temporal fire and climate patterns in Alaska. (

**a**) Time series of the total number of detected breakpoints per year for each method. (

**b**) Total annual burnt area and annual anomaly of flagged GIMMS NDVI3g pixels with reduced quality. (

**c**) Annual temperature and precipitation anomalies (baseline 1982–2009) averaged for Alaska.

**Figure 10.**Comparison of detected breakpoints of the year 2004 from four different methods with 2004 burnt areas and in situ photos (taken in 2008). NBP is the total number of breakpoints that was detected in 2004 in this region. BPinBA denotes the percentage of breakpoints that was found inside a burnt area polygon. For methods AAT-ex and AAT-peak breakpoints for both 2003 and 2004 are shown to compensate for the lower breakpoint timing precision of these methods. All breakpoints that were found by AAT-peak were found at least also by one other method.

**Table 1.**Normalized confusion matrices of estimated and real trend classes for each method. N3: significant negative trend, N2: non-significant negative trend, N1: no trend with negative tendency, P1: no trend with positive tendency, P2: non-significant positive trend, P3: significant positive trend. ToAcc: total normalized accuracy, Kappa: Kappa coefficient.

Method AAT | Real.N3 | Real.N2 | Real.N1 | Real.P1 | Real.P2 | Real.P3 | Sum |

Est.N3 | 55.24 | 11.18 | 15.57 | 8.44 | 5.95 | 3.62 | 100.00 |

Est.N2 | 12.48 | 43.27 | 26.76 | 11.11 | 0.00 | 6.38 | 100.00 |

Est.N1 | 13.27 | 14.55 | 24.55 | 17.46 | 17.74 | 12.42 | 100.00 |

Est.P1 | 10.37 | 10.57 | 15.29 | 24.43 | 24.85 | 14.49 | 100.00 |

Est.P2 | 5.54 | 13.72 | 11.98 | 22.01 | 31.01 | 15.74 | 100.00 |

Est.P3 | 3.09 | 6.70 | 5.85 | 16.56 | 20.45 | 47.34 | 100.00 |

Sum | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 600.00 |

ToAcc = 37.64, Kappa = 0.25 | |||||||

Method STM | Real.N3 | Real.N2 | Real.N1 | Real.P1 | Real.P2 | Real.P3 | Sum |

Est.N3 | 47.90 | 20.68 | 13.18 | 7.58 | 6.07 | 4.59 | 100.00 |

Est.N2 | 20.58 | 32.21 | 14.54 | 11.05 | 15.14 | 6.48 | 100.00 |

Est.N1 | 14.60 | 18.92 | 22.68 | 14.79 | 18.47 | 10.54 | 100.00 |

Est.P1 | 10.37 | 11.09 | 20.22 | 25.48 | 17.20 | 15.65 | 100.00 |

Est.P2 | 1.15 | 8.16 | 19.91 | 25.21 | 30.22 | 15.35 | 100.00 |

Est.P3 | 5.41 | 8.94 | 9.47 | 15.89 | 12.89 | 47.39 | 100.00 |

Sum | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 600.00 |

ToAcc = 34.31, Kappa = 0.21 | |||||||

Method MAC | Real.N3 | Real.N2 | Real.N1 | Real.P1 | Real.P2 | Real.P3 | Sum |

Est.N3 | 48.08 | 22.05 | 11.81 | 7.56 | 4.37 | 6.13 | 100.00 |

Est.N2 | 13.24 | 29.18 | 14.06 | 10.84 | 26.98 | 5.69 | 100.00 |

Est.N1 | 15.15 | 19.12 | 27.35 | 14.75 | 10.87 | 12.76 | 100.00 |

Est.P1 | 10.91 | 16.97 | 15.94 | 25.79 | 18.33 | 12.06 | 100.00 |

Est.P2 | 7.14 | 4.45 | 22.71 | 26.33 | 21.16 | 18.22 | 100.00 |

Est.P3 | 5.48 | 8.23 | 8.13 | 14.73 | 18.29 | 45.15 | 100.00 |

Sum | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 600.00 |

ToAcc = 32.79, Kappa = 0.19 | |||||||

Method SSA | Real.N3 | Real.N2 | Real.N1 | Real.P1 | Real.P2 | Real.P3 | Sum |

Est.N3 | 48.08 | 17.79 | 14.94 | 6.76 | 6.21 | 6.22 | 100.00 |

Est.N2 | 9.07 | 37.14 | 19.66 | 11.88 | 18.57 | 3.69 | 100.00 |

Est.N1 | 15.20 | 20.16 | 24.72 | 14.54 | 14.08 | 11.31 | 100.00 |

Est.P1 | 13.80 | 9.59 | 13.99 | 24.52 | 24.76 | 13.34 | 100.00 |

Est.P2 | 7.88 | 6.19 | 18.57 | 25.12 | 22.70 | 19.55 | 100.00 |

Est.P3 | 5.98 | 9.13 | 8.12 | 17.17 | 13.69 | 45.90 | 100.00 |

Sum | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 600.00 |

ToAcc = 33.84, Kappa = 0.21 |

**Table 2.**Analysis of variance table for the RMSE between real trend and estimated trend. IAV and STV denote inter-annual and short-term variability, respectively.

Factor | Df | Sum Sq | Mean Sq | F value | P (>F) | Sum Sq/Total Sq (%) |
---|---|---|---|---|---|---|

IAV | 2 | 0.2136 | 0.1068 | 4096.7 | <2.2e-16 | 30.73 |

Type of change | 1 | 0.0511 | 0.0511 | 1959.3 | <2.2e-16 | 7.35 |

IAV * Method | 6 | 0.0469 | 0.0078 | 299.8 | <2.2e-16 | 6.75 |

Trend magnitude | 1 | 0.0252 | 0.0252 | 966.5 | <2.2e-16 | 3.62 |

IAV * STV | 4 | 0.0153 | 0.0038 | 146.8 | <2.2e-16 | 2.20 |

Type of change * Method | 3 | 0.0121 | 0.0040 | 154.6 | <2.2e-16 | 1.74 |

Method | 3 | 0.0108 | 0.0036 | 137.8 | <2.2e-16 | 1.55 |

Trend magnitude * Method | 3 | 0.0073 | 0.0024 | 93.9 | <2.2e-16 | 1.06 |

Number of breakpoints | 2 | 0.0072 | 0.0036 | 137.6 | <2.2e-16 | 1.03 |

STV * Type of change | 2 | 0.0061 | 0.0030 | 116.5 | <2.2e-16 | 0.87 |

STV | 2 | 0.0024 | 0.0012 | 46.1 | <2.2e-16 | 0.35 |

Trend magnitude * Type of change | 1 | 0.0022 | 0.0022 | 86.3 | <2.2e-16 | 0.32 |

Trend magnitude * Number of breakpoints | 2 | 0.0022 | 0.0011 | 41.7 | <2.2e-16 | 0.31 |

Type of change * Number of breakpoints | 1 | 0.0022 | 0.0022 | 83.4 | <2.2e-16 | 0.31 |

Trend magnitude * STV | 2 | 0.0022 | 0.0011 | 41.3 | <2.2e-16 | 0.31 |

IAV * Type of change | 2 | 0.0005 | 0.0003 | 10.4 | 2.962E-05 | 0.08 |

Seasonality * Number of breakpoints | 4 | 0.0005 | 0.0001 | 5.0 | 4.945E-04 | 0.08 |

STV * Number of breakpoints | 4 | 0.0005 | 0.0001 | 4.5 | 1.238E-03 | 0.07 |

Trend magnitude * IAV | 2 | 0.0004 | 0.0002 | 7.4 | 0.001 | 0.06 |

STV * Method | 6 | 0.0003 | 0.0001 | 2.1 | 4.948E-02 | 0.05 |

Trend magnitude * Seasonality | 2 | 0.0002 | 0.0001 | 4.6 | 0.010 | 0.03 |

Seasonality * Type of change | 2 | 0.0002 | 0.0001 | 4.2 | 1.526E-02 | 0.03 |

IAV * Number of breakpoints | 4 | 0.0002 | 0.0001 | 2.0 | 8.910E-02 | 0.03 |

Seasonality | 2 | 0.0000 | 0.0000 | 0.2 | 0.799 | 0.00 |

Residuals | 10,952 | 0.2855 | 0.0000 | 41.07 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Forkel, M.; Carvalhais, N.; Verbesselt, J.; Mahecha, M.D.; Neigh, C.S.R.; Reichstein, M. Trend Change Detection in NDVI Time Series: Effects of Inter-Annual Variability and Methodology. *Remote Sens.* **2013**, *5*, 2113-2144.
https://doi.org/10.3390/rs5052113

**AMA Style**

Forkel M, Carvalhais N, Verbesselt J, Mahecha MD, Neigh CSR, Reichstein M. Trend Change Detection in NDVI Time Series: Effects of Inter-Annual Variability and Methodology. *Remote Sensing*. 2013; 5(5):2113-2144.
https://doi.org/10.3390/rs5052113

**Chicago/Turabian Style**

Forkel, Matthias, Nuno Carvalhais, Jan Verbesselt, Miguel D. Mahecha, Christopher S.R. Neigh, and Markus Reichstein. 2013. "Trend Change Detection in NDVI Time Series: Effects of Inter-Annual Variability and Methodology" *Remote Sensing* 5, no. 5: 2113-2144.
https://doi.org/10.3390/rs5052113