2.3.1. General Idea of the Method
Target of change detection. In general the aim of change detection is to obtain details of “change/no change” or “from–to” information between the detection dates [
10]. “Change/no change” information can be used to determine the location of land cover change. We can just classify land cover types in the changed area to acquire “from–to” information over the study period. Thus, change detection efficiency can be improved greatly. In this study the target of change detection is to obtain “change/no change” information as both BFAST and DBEST did.
Categories of change detection. As discussed in
Section 1, land cover changes were classified into SC and DC, and the DC category was classified further into AC and TC. SCs are temporal and caused mainly by events such as floods or severe drought. Their NDVIs will be remarkably different from the normal values and will reverse within a few years. DCs are long-lasting and their NDVIs will not reverse any more over the study period.
ACs are usually related with land cover type changes and mainly caused by human activities such as reforestation, deforestation, urbanization, and land reclamation. Their NDVIs will change abruptly. For TCs, land cover types generally remain the same and are often related with changes in the status of the growing plants or in the species composite due to global warming, soil erosion [
12,
13,
14,
15], desertification, or water erosion. Their NDVIs will change slowly but continuously. It should be pointed that TCs will not be analyzed further if ACs happened in a trajectory because we believe that the TCs were affected greatly by ACs in these trajectories.
Framework of Change detection. Different land cover types present different AA-NDVI trajectories (
Figure 2). Changes in land cover type or in the status of a land cover type can be illustrated by the change of the AA-NDVI trajectory. The general idea behind the method is to distinguish different types of land cover change based on changes of the AA-NDVI trajectories using different statistical methods.
The NDVIs of SCs can be considered outliers since their NDVIs were remarkably different from the normal value. Grubbs’ test [
65] was used to determine these outliers in each of the AA-NDVI trajectories.
AA-NDVI trajectories of ACs cannot be detected just based on one method. The AA-NDVI trajectories before or after an AC point can be grouped into two classes: “no trend” and “with trend”. The “no trend” AA-NDVI trajectories generally belong to land cover types that have a stable status, e.g., natural vegetation such as grassland or forest in the climax stage of succession, or artificial land cover such as cropland (their pixel AA-NDVIs fluctuate with their averages). The “with trend” AA-NDVI trajectories generally belong to land cover types that reflect continuous positive or negative trends, e.g., forest or bush that is continually growing or being degraded. For an AA-NDVI trajectory whose subtrajectories before or after a specific point both indicated “no trend” (type NT abrupt land cover change), the BF test method, which can distinguish the mean jump in a time series [
36,
37], was used to determine whether an AC points existed.
For an AA-NDVI trajectory in which one or both subtrajectories before or after a specific point indicated “with trend” (type WT abrupt land cover change). Both the Tomé’s trend detection method [
38,
39] and the Chow test based on the difference of slopes of two segments [
40] were used to determine whether an AC point existed. This method just tried to decrease the omission of land cover change which cannot be detection by BF test due to the similar mean of AA-NDVI between two segments of NDVIs in this kind of time series.
All of the trajectories that did not indicate ACs, whether they indicate a positive trend (vegetation greenness increase) or a negative trend (vegetation greenness decrease) will be determined. In this study, the SMK method was used to determine whether an AA-NDVI trajectory had a significant trend.
This study focused on directional land cover changes, and thus other statuses of land cover change were flagged as “no change” (NC). First, we eliminated the impact of SCs. Then, ACs were detected using the BF test or the combination of the Tomé’s method and Chow test. Finally, for all pixels indicating no AC in land cover, it was determined whether their trajectories indicated TCs (
Figure 3).
2.3.2. Eliminating the Impact of Short-Lived Land Cover Changes
In this study, the Grubbs’ test [
65] was used to determine SCs by considering them outliers in each AA-NDVI trajectory. The Grubbs’ test can be defined as follows:
where
is the
i-th NDVI value in an AA-NDVI trajectory, and
and
are the mean and standard deviation, respectively, of the AA-NDVIs. The hypothesis of no outliers is rejected if
where
t is the percent point function of the
t distribution and
N is the total number of time series. In this study, the significance level was 95% (
p = 0.05).
The Grubbs’ test detects one outlier at a time. For multiple outliers, a detected outlier was deleted and the Grubbs’ test run again, following which the process was repeated until no further outliers were detected. The detected NDVIs as outliers were considered SCs and their NDVIs were replaced by the maximum (if the outlier was the maximum in the time series) or by the minimum NDVIs (if the outlier was the minimum in the time series), except for those NDVIs determined as outliers in the AA-NDVI trajectories.
2.3.3. Abrupt Land Cover Change Detection
- (1)
Type NT Abrupt Land Cover Change Detection
The BF test provides a method with which to distinguish the mean jump points in a time series dataset with a small number of samples [
36,
37]. Compared with other methods, it does not require strict normality or heteroscedasticity in the original dataset [
66]. The entire AA-NDVI trajectory can be treated as a time series:
x = {
,
,
}. The AA-NDVI trajectory can be divided into
m parts by (
m – 1) abrupt points, following which the
F statistics were calculated, as in Equations (3)–(7):
where
The
F statistics in the BF test are subject to the
F-distribution with
degrees of freedom (Equations (8) and (9)), where
m is the number of parts divided by the abrupt points (
m is 2 when there is only one jump point
I),
is the number of samples in the
i-th part,
N is the total number of samples (
N was 14 in this study),
is the mean of the samples in the
i-th part,
is the mean of the total samples, and
is the variance of the samples in the
i-th part.
where
F statistics was computed for all possible locations of abrupt points and the locations of abrupt points with the maximum F statistics () were selected as possible abrupt points.
In this study, the user needs to decide the minimum interval of two potential abrupt points (
i.e., two years referenced as [
32,
34]) and thus the maximum number of abrupt points will be determined. Users will conduct the process based on Equations (1)–(9) starting with the case of one abrupt point and continuing up to the maximum number of abrupt points. The best segmentation was selected based on the abrupt points with the maximum
(
).
The hypothesis of no mean jump is rejected if
where
is the maximum
F statistics among those at all locations of the abrupt point
I (α
= 0.05 in this study).
Because the variation of the AA-NDVIs was high and the difference of the means of some land cover types relatively low, the risk of type-II errors in the hypothesis test was high. A threshold of the difference of the NDVI means of two adjacent subparts was used to reduce this type of risk:
where
is the NDVIs in each subpart,
I is the location of the abrupt point, and
and
are the numbers of samples in two adjacent subparts. The threshold was developed referring to the principle of Zhu
et al. [
18], which was proposed for monitoring abrupt land cover changes:
where
and
represent the standard deviations of two adjacent subparts.
- (2)
Type WT Abrupt Land Cover Change Detection
Tomé’s method is a piecewise fitting method that can determine the detailed trend change within a time series [
38,
39]. Because the number of abrupt points in a time series is set arbitrarily using Tomé’s method, we used the Chow test [
40] to determine whether the potential abrupt points were also abrupt points based on Tomé’s method, such that abrupt land cover changes could be detected automatically.
Suppose there is one breakpoint in a time series
y = {
,
,
} and the location of the abrupt point may be
(
). The entire time series can be divided into two segments that could be expressed by two linear equations (Equations (13) and (14)):
where
N is the total number within the entire time series, and
,
,
, and
are parameters. Because the two segments are continuous, we can obtain Equation (15):
Therefore, there are three unknown solutions for this series of linear equations. Suppose
is the vector solution,
s = {
,
,
}, and A is a rectangular matrix (
) with its first line equal to {
} and its second line equal to {
}. Then, the solutions can be obtained by solving Equation (16):
Thus, we can determine the possible locations of the abrupt points. The same strategy can also be used when there is more than one abrupt point. When the user decides the minimum interval of two potential abrupt points and thus the maximum number of abrupt points will be also determined. The most possible number of abrupt points can be determined based on minimum
with different numbers of abrupt points. Tomé’s method suggests that users start with the case of one abrupt point and continue up to the presupposed maximum number of abrupt points, such that all of the abrupt points could be determined automatically. In this study, the Chow test, which was used to determine whether the two segments separated by abrupt points were significantly different, was calculated as shown in Equation (17):
where
is the sum of the squared residuals over the entire time series,
and
are the sums of the squared residuals of two segments before and after the potential abrupt point, respectively,
N is the total number of dates in the two segments,
k is the cost in degrees of freedom, and (
N −
2k) is the remaining degrees of freedom.
subjects to the
F-distribution with (
k, (
N −
2k)) degrees of freedom. The hypothesis of “no trend” difference before and after the abrupt point is rejected if
We concluded there was significant difference in the AA-NDVI trends before and after the abrupt point and considered it a real abrupt point (α = 0.05 in this study).
2.3.4. Trend Land Cover Change Detection
In this study, the SMK method was used to determine whether an AA-NDVI trajectory indicated significant trend. SMK is a non-parametric method that is robust against non-normality of a dataset [
67]. The trend slope can be defined as in Equation (19):
where
and
are NDVIs in a time series of AA-NDVIs and
is the trend slope of the time series. The condition
> 0 means a positive trend of vegetation growth during the study period, while
< 0 means a negative trend. The significance of the slope can be determined using the Mann-Kendall test as in Equations (20)–(24):
where
where
and
are NDVIs in an AA-NDVI series and
is the number of dates. The hypothesis of no AA-NDVI trend is rejected if
Finally, the NDVI change rate over the study period, which directly expresses the magnitude of trend change, can be determined using Equations (25) and (26):
where
is the NDVI change rate (percentage of AA-NDVI increase over the study period),
is the mean of the time series of NDVIs,
is the mean value of the dates, and
and
are the start and end years of the time series, respectively. The area of TC can be acquired using Equation (27):
where
is the change rate threshold, which was set as 10 in this study.