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two consecutive space-time points, given the speed bound, are located within
them.
A rigorous analysis of a moving object's trajectory requires that both the data
model and the query language account for uncertainty. That means the language
constructs must be aware that the data being queried are uncertain. Typical
queries onmoving object data ask for the objects inside a region sometime during
a time interval, or for the ones always inside a region during a time interval.
For example, taking into account the uncertainty of the objects' position, one
may query the objects that were possibly inside the region or the ones that were
definitely there. For example, we may be interested in a query like: “Give me
the current location of a bus that will possibly be at the corner of Avenue A and
Avenue B at some time between 4:00 P.M. and 4:30 P.M.”
In the remainder of the section we study two models for considering uncer-
tainty in trajectories. We also discuss uncertainty in road networks, and conclude
the section studying how uncertainty is accounted for in trajectory clustering.
5.3.1 A Simple Model for Trajectory Uncertainty
Let
2
the 2D real plane. We consider
objects moving in a subset of the 2D (
x,y
) space
denote the set of the real numbers, and
R
R
2
and describe this movement
R
2
, where
t
represents time. Moving objects (which
hereon we assume to be points) produce, as we have already seen in this topic,
the kind of curves that we denote as
trajectories
. In practice, trajectories are
only known at discrete moments in time, and given as sequences of the form
S
={
(
x
0
,y
0
,t
0
)
,
(
x
1
,y
1
,t
1
)
,...,
(
x
N
,y
N
,t
N
)
}
. Given a trajectory
T
between
times
t
1
and
t
N
, the expected location of the object at a point in time t
between
t
i
and
t
i
+
1
(1
≤
i<N
) could be obtained through linear interpolation
between (
x
i
,y
i
)and(
x
i
+
1
,y
i
+
1
)
.
Note that in its general form, a trajectory can represent both the past and
future motion of objects. For future movement one can think of the trajec-
tory as a set of points describing the motion plan of the object. The most
common assumption is that we have a set of points that the object is going
to visit, and that between the points the object is moving along the shortest
path.
This simplemodel allows defining the notion of
uncertain trajectory
, obtained
by associating an uncertainty threshold
r
with each line segment of the trajectory.
For a given motion plan, the moving object associated with, for instance, a GPS
device will update a server if and only if it deviates from its expected location
(according to the trajectory) by
r
or more. In practice, a GPS update is sent
at certain predefined intervals; therefore, the location of the object is known,
and by linear interpolation, the object's expected location can be computed at
in the (
t,x,y
) space
R
×
R
