gremlin97/RemoteSensingCorpus · Datasets at Hugging Face

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assume that trend and seasonal components are smooth and slowly
changing, and so these are not directly applicable to the problem of
identifying change. For example, the Seasonal-Trend decomposition
procedure (STL) is capable of ﬂexibly decomposing a series into trend,
seasonal and remainder components based on a LOcally wEighted
regreSsion Smoother (LOESS) ( Cleveland et al., 1990 ). This smoothing
prevents the detection of changes within time series.
2.1. Decomposition model
We propose an additive decomposition model that iteratively ﬁts a
piecewise linear trend and seasonal model. The general model is of the
form Yt=Tt+St+et,t=1,…,n, where Ytis the observed data at time
t,Ttis the trend component, Stis the seasonal component, and etis the
remainder component. The remainder component is the remaining
variation in the data beyond that in the seasonal and trend
components ( Cleveland et al., 1990 ). It is assumed that Ttis piecewise
linear, with break points t1⁎,…,tm⁎and de ﬁnet0⁎=0, so that:
Tt=αj+βjt ð1Þ
fortj−1⁎<t≤tj⁎and where j=1,…,m. The intercept and slope of
consecutive linear models, αjandβj, can be used to derive the
magnitude and direction of the abrupt change (hereafter referred to
as magnitude) and slope of the gradual change between detected
break points. The magnitude of an abrupt change at a breakpoint is
derived by the difference between Ttattj−1⁎andtj⁎, so that:
Magnitude = αj−1−αj/C16/C17
+βj−1−βj/C16/C17
t ð2Þ
and the slopes of the gradual change before and after a break point are
βj−1andβj. This technique represents a simple yet robust way to
characterize changes in time series. Piecewise linear models, as a
special case of non-linear regression ( Venables and Ripley, 2002 ), are
often used as approximations to complex phenomena to extract basic
features of the data ( Zeileis et al., 2003 ).
Similarly, the seasonal component is ﬁxed between break points, but
can vary across break points. Furthermore, the seasonal break points
may occur at different times from the break points detected in the trend
component. Let the seasonal break points be given by t1#,…,tp#,a n d
deﬁnet0#=0. Then for tj−1#<t≤tj#, we assume that:
St=γi;j if time tis in season i;i=1 ;…;s−1;
−Ps−1
i=1γi;jif time tis in season 0 ;(
ð3Þ
where sis the period of seasonality (e.g. number of observations per
year) and γi,jdenotes the effect of season i. Thus, the sum of the seasonal
component, Stacross ssuccessive times is exactly zero for tj−1#<t≤tj#.
This prevents apparent changes in trend being induced by seasonal
breaks happening in the middle of a seasonal cycle. The seasonal term
can be re-expressed as:
St=Xs−1
i=1γi;jdt;i−dt;0/C16/C17
ð4Þ107 J. Verbesselt et al. / Remote Sensing of Environment 114 (2010) 106 –115
where dt,i=1 when tis in season iand 0 otherwise. Therefore, if tis in
season 0, then dt,i−dt,0=−1. For all other seasons, dt,i−dt,0=1when t
is in season i≠0.dt,iis often referred to as a seasonal dummy variable
(Makridakis et al., 1998 , pp. 269 –274); it has two allowable values (0
and 1) to account for the seasons in a regression model. The regression
model expressed by Eq. (4)can also be interpreted as a model without
intercept that contains s−1 seasonal dummy variables.
2.2. Iterative algorithm to detect break points
Our method is similar to that proposed by Haywood and Randal
(2008) for use with monthly tourism data. Following Haywood and
Randal (2008) , we estimate the trend and seasonal components
iteratively. However, we differ from their method by: (1) using STL to
estimate the initial seasonal component Sˆt; (2) using a robust procedure
when estimating the coef ﬁcients αj,βjandγi,j; (3) using a preliminary
structural change test; and 4) forcing the seasonal coef ﬁcients to always
sum to 0 (rather than adjusting them afterward). An alternative
approach proposed by Shao and Campbell (2002) combines the
seasonal and trend term in a piecewise linear regression model without
iterative decomposition. This approach does not allow for an individual
estimation of breakpoints in the seasonal and trend component.
Sequential test methods for detecting break points (i.e. abrupt
changes) in a time series have been developed, particularly within
econometrics ( Bai and Perron, 2003; Zeileis et al., 2003 ). These
methods also allow linear models to be ﬁtted to sections of a time
series, with break points at the times where the changes occur. The
optimal position of these breaks can be determined by minimizing the
residual sum of squares, and the optimal number of breaks can be
determined by minimizing an information criterion. Bai and Perron
(2003) argue that the Akaike Information Criterion usually over-
estimates the number of breaks, but that the Bayesian Information
Criterion (BIC) is a suitable selection procedure in many situations
(Zeileis et al., 2002; Zeileis et al., 2003; Zeileis and Kleiber, 2005 ).
Before ﬁtting the piecewise linear models and estimating the break-
points it is recommended to test whether breakpoints are occurring in
the time series ( Bai and Perron, 2003 ). The ordinary least squares
(OLS) residuals-based MOving SUM (MOSUM) test, is selected to test
for whether one or more breakpoints are occurring ( Zeileis, 2005 ). If
the test indicates signi ﬁcant change ( P<0.05), the break points are
estimated using the method of Bai and Perron (2003) , as implemented
byZeileis et al. (2002) , where the number of breaks is determined by
the BIC, and the date and con ﬁdence interval of the date for each break
are estimated.
The iterative procedure begins with an estimate of S ̂tby using the
STL method, where S ̂tis estimated by taking the mean of all seasonal
sub-series (e.g. for a monthly time series the ﬁrst sub-series contains
the January values). Then it follows these steps.
Step 1 If the OLS-MOSUM test indicates that breakpoints are
occurring in the trend component, the number and position
of the trend break points ( t1⁎,…,tm⁎) are estimated from the
seasonally adjusted data, Yt−S ̂t.

assume that trend and seasonal components are smooth and slowly

changing, and so these are not directly applicable to the problem of

identifying change. For example, the Seasonal-Trend decomposition

procedure (STL) is capable of ﬂexibly decomposing a series into trend,

seasonal and remainder components based on a LOcally wEighted

regreSsion Smoother (LOESS) ( Cleveland et al., 1990 ). This smoothing

prevents the detection of changes within time series.

2.1. Decomposition model

We propose an additive decomposition model that iteratively ﬁts a

piecewise linear trend and seasonal model. The general model is of the

form Yt=Tt+St+et,t=1,…,n, where Ytis the observed data at time

t,Ttis the trend component, Stis the seasonal component, and etis the

remainder component. The remainder component is the remaining

variation in the data beyond that in the seasonal and trend

components ( Cleveland et al., 1990 ). It is assumed that Ttis piecewise

linear, with break points t1⁎,…,tm⁎and de ﬁnet0⁎=0, so that:

Tt=αj+βjt ð1Þ

fortj−1⁎<t≤tj⁎and where j=1,…,m. The intercept and slope of

consecutive linear models, αjandβj, can be used to derive the

magnitude and direction of the abrupt change (hereafter referred to

as magnitude) and slope of the gradual change between detected

break points. The magnitude of an abrupt change at a breakpoint is

derived by the difference between Ttattj−1⁎andtj⁎, so that:

Magnitude = αj−1−αj/C16/C17

+βj−1−βj/C16/C17

t ð2Þ

and the slopes of the gradual change before and after a break point are

βj−1andβj. This technique represents a simple yet robust way to

characterize changes in time series. Piecewise linear models, as a

special case of non-linear regression ( Venables and Ripley, 2002 ), are

often used as approximations to complex phenomena to extract basic

features of the data ( Zeileis et al., 2003 ).

Similarly, the seasonal component is ﬁxed between break points, but

can vary across break points. Furthermore, the seasonal break points

may occur at different times from the break points detected in the trend

component. Let the seasonal break points be given by t1#,…,tp#,a n d

deﬁnet0#=0. Then for tj−1#<t≤tj#, we assume that:

St=γi;j if time tis in season i;i=1 ;…;s−1;

−Ps−1

i=1γi;jif time tis in season 0 ;(

ð3Þ

where sis the period of seasonality (e.g. number of observations per

year) and γi,jdenotes the effect of season i. Thus, the sum of the seasonal

component, Stacross ssuccessive times is exactly zero for tj−1#<t≤tj#.

This prevents apparent changes in trend being induced by seasonal

breaks happening in the middle of a seasonal cycle. The seasonal term

can be re-expressed as:

St=Xs−1

i=1γi;jdt;i−dt;0/C16/C17

ð4Þ107 J. Verbesselt et al. / Remote Sensing of Environment 114 (2010) 106 –115

where dt,i=1 when tis in season iand 0 otherwise. Therefore, if tis in

season 0, then dt,i−dt,0=−1. For all other seasons, dt,i−dt,0=1when t

is in season i≠0.dt,iis often referred to as a seasonal dummy variable

(Makridakis et al., 1998 , pp. 269 –274); it has two allowable values (0

and 1) to account for the seasons in a regression model. The regression

model expressed by Eq. (4)can also be interpreted as a model without

intercept that contains s−1 seasonal dummy variables.

2.2. Iterative algorithm to detect break points

Our method is similar to that proposed by Haywood and Randal

(2008) for use with monthly tourism data. Following Haywood and

Randal (2008) , we estimate the trend and seasonal components

iteratively. However, we differ from their method by: (1) using STL to

estimate the initial seasonal component Sˆt; (2) using a robust procedure

when estimating the coef ﬁcients αj,βjandγi,j; (3) using a preliminary

structural change test; and 4) forcing the seasonal coef ﬁcients to always

sum to 0 (rather than adjusting them afterward). An alternative

approach proposed by Shao and Campbell (2002) combines the

seasonal and trend term in a piecewise linear regression model without

iterative decomposition. This approach does not allow for an individual

estimation of breakpoints in the seasonal and trend component.

Sequential test methods for detecting break points (i.e. abrupt

changes) in a time series have been developed, particularly within

econometrics ( Bai and Perron, 2003; Zeileis et al., 2003 ). These

methods also allow linear models to be ﬁtted to sections of a time

series, with break points at the times where the changes occur. The

optimal position of these breaks can be determined by minimizing the

residual sum of squares, and the optimal number of breaks can be

determined by minimizing an information criterion. Bai and Perron

(2003) argue that the Akaike Information Criterion usually over-

estimates the number of breaks, but that the Bayesian Information

Criterion (BIC) is a suitable selection procedure in many situations

(Zeileis et al., 2002; Zeileis et al., 2003; Zeileis and Kleiber, 2005 ).

Before ﬁtting the piecewise linear models and estimating the break-

points it is recommended to test whether breakpoints are occurring in

the time series ( Bai and Perron, 2003 ). The ordinary least squares

(OLS) residuals-based MOving SUM (MOSUM) test, is selected to test

for whether one or more breakpoints are occurring ( Zeileis, 2005 ). If

the test indicates signi ﬁcant change ( P<0.05), the break points are

estimated using the method of Bai and Perron (2003) , as implemented

byZeileis et al. (2002) , where the number of breaks is determined by

the BIC, and the date and con ﬁdence interval of the date for each break

are estimated.

The iterative procedure begins with an estimate of S ̂tby using the

STL method, where S ̂tis estimated by taking the mean of all seasonal

sub-series (e.g. for a monthly time series the ﬁrst sub-series contains

the January values). Then it follows these steps.

Step 1 If the OLS-MOSUM test indicates that breakpoints are

occurring in the trend component, the number and position

of the trend break points ( t1⁎,…,tm⁎) are estimated from the

seasonally adjusted data, Yt−S ̂t.