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means of a representative object. The difference is that now the representative
is a probability distribution of trajectories that fits well with the trajectories in
its cluster. Another well-known statistical tool often adopted when dealing with
trajectories is hidden Markov models (HMMs). The basic approach, here, con-
sists of modeling a trajectory as a sequence of transitions between spatial areas.
Then, a cluster of trajectories is modeled by means of a Markov model (i.e., the
set of transition probabilities between all possible pairs of regions) that better
fits the trajectories.
Other examples of trajectory-oriented clustering methods can arise by adding
novel dimensions to the clustering problem. For instance, in the literature the
problem was investigated of finding clusters by means of a distance-based clus-
teringmethod (a density-based one, more exactly, though a similar process might
be easily replicated for other approaches) when it is not known in advance the
time interval to consider for clustering. For instance, we might expect that rush
hours in urban traffic data exhibit cluster structures that are better defined than
what happens in random periods of the day. The problem, then, becomes to find
both the optimal time interval (rush hours were just a guess to be confirmed)
and the corresponding optimal cluster structure. The solution proposed, named
time-focused trajectory clustering , adopts a trajectory distance computed as the
average spatial distance between the trajectories within a given time interval,
which is a parameter of the distance. Then, for each time interval T , the algo-
rithm can be run focusing on the trajectory segments laying within T .The
quality of the resulting clusters is evaluated in terms of their density, and a
simple procedure is provided to explore only a reasonable subset of the possi-
ble values of T . A sample result of the process is given in Figure 6.7 ,which
depicts a set of trajectories forming three clusters (plus some noise) and shows
the optimal time interval (that where the clusters are clearest) as dark trajectory
segments.
6.3.2 Trajectory Classification
Clustering is also known as unsupervised classification, since the objective is
to find a way to put objects into groups without any prior knowledge of which
groups might exist, and what their objects look like. In several contexts such
knowledge is available, more exactly in the form of a set of predefined classes
and a set of objects that are already labeled with the class they belong to -
the so-called training set . The problem, here, becomes finding rules to classify
new objects in a way that is coherent with the prior knowledge, that is, they
fit well with the training set. For instance, we might have access to a set of
vehicle trajectories that were manually labeled with the vehicle type (car, truck,
motorbike), and we would like to find a way to automatically label another,
much larger set of new trajectories.
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