Fig. 5. The VZA stratified DNB time series and high-resolution
images of a riverside esplanade with new pedestrians built in 2016 near
Charles River in Massachusetts, U.S. (a) DNB time series at the
riverside esplana. The red, green, and blue dots indicate the DNB
observations within different VZA intervals. (b) The high-resolution
Google Earth image in June 2015 of the selected pixel in Fig. 5a. The
red rectangle represents the location and size of the selected pixel.
(c) The high-resolution Google Earth image in June 2017 of the selected
pixel in Fig. 5a. The red rectangle represents the location and size of
the selected pixel.
3.3. Continuous monitoring of NTL change
The VZA-COLD algorithms were applied to estimate time series models from
DNB observations collected within different VZA intervals while
collectively identifying NTL changes across all VZA strata. For DNB
observations stratified within each VZA interval, an individual harmonic
model was estimated to capture the seasonality and trend of the DNB time
series, which could greatly reduce the impact from the intra-annual
(e.g., vegetation phenology and snow) and inter-annual (e.g., gradual
economic growth and vegetation long-term growth) changes. We tested the
models with unimodal, bimodal, and trimodal seasonality (4, 6, and 8
coefficients) based on the Ordinary Least Square (OLS) regression, Least
Absolute Shrinkage and Selection Operator (LASSO) regression
(Tibshiranit, 1996), and robust regression (Hampel et al., 2011; Zhu et
al., 2012) to explore the optimal combinations for modeling the daily
DNB time series. We observed that models with a single-term harmonics
model (Eq. 1) and based on robust regression had the best results for
our calibration samples and were more robust to outliers and less likely
to be overfitted. Therefore, the single-term harmonic model (Eq. 1)
estimated based on robust regression was selected to predict the overall
DNB magnitude, intra-annual seasonality, and inter-annual trends, which
would be used in continuous monitoring of NTL changes.
\({\hat{\rho}}_{i,x}=\ a_{0}+\ a_{1}\cos\ \left(\frac{2\pi}{T}x\right)\ +\ b_{1}\sin\ \left(\frac{2\pi}{T}x\right)\ +\ c_{1}x\)(1)
where,
\({\hat{\rho}}_{i,x}\): Predicted DNB value for the i th VZA
interval at Julian date x.
x : Julian date.
\(T\): Number of days per year (\(T\) = 365.25).
\(a_{0}\): Coefficient for overall value for the DNB.
\(a_{1},\ b_{1}\): Coefficient for intra-annual change for the DNB.
\(c_{1}\): Coefficient for inter-annual change (slope) for the DNB.
Continuous change detection was conducted based on the models estimated
from each VZA stratification following the COLD algorithms (Zhu et al.,
2020), by comparing the actual observations with the model predictions.
Breakpoints were identified based on the number of consecutive anomaly
observations beyond the applied change probability thresholds. The
VZA-COLD made three major changes compared with the original COLD
algorithm. First, due to the high temporal frequency and the large
fluctuations observed with the daily DNB time series, VZA-COLD would
tolerate one of the observations (except for the first one) not showing
up as an anomaly in the consecutive anomaly test. Second, VZA-COLD
detected changes based on a set of four time series models estimated
from observations of different VZA intervals (instead of observations
from different spectral bands), and when a breakpoint was identified by
any of the VZA interval models, it would be applied to all the four
VZA-stratified models, and thus, dividing their time segments with the
same break time. Third, the optimal parameters for the change detection
process were also different from the default COLD algorithm. We tested
the number of consecutive anomaly observations to confirm a change from
12 to 16, along with the change probabilities of 70%, 75%, 80%, 85%,
and 90%, and analyzed the performance metrics of omission rate,
commission rate, and F1 score based on the calibration samples (Fig. 6).
According to the results, the consecutive anomaly observation of 14 and
75% change probability were selected to detect NTL changes.