  A time series consists of a series of values or events obtained over repeated
measurements in time. Analysis of time series represents and important tool in
many application areas, such as stock market analysis, process and quality
control, observation of natural phenomena, medical treatments, etc. A vital
component in many types of time-series analysis is the choice of an appropriate
distance/similarity measure. Numerous measures have been proposed to date, with
the most successful ones based on dynamic programming. Being of quadratic time
complexity, however, global constraints are often employed to limit the search
space in the matrix during the dynamic programming procedure, in order to speed
up computation. Furthermore, it has been reported that such constrained
measures can also achieve better accuracy. In this paper, we investigate two
representative time-series distance/similarity measures based on dynamic
programming, Dynamic Time Warping (DTW) and Longest Common Subsequence (LCS),
and the effects of global constraints on them. Through extensive experiments on
a large number of time-series data sets, we demonstrate how global constrains
can significantly reduce the computation time of DTW and LCS. We also show
that, if the constraint parameter is tight enough (less than 10-15% of
time-series length), the constrained measure becomes significantly different
from its unconstrained counterpart, in the sense of producing qualitatively
different 1-nearest neighbor graphs. This observation explains the potential
for accuracy gains when using constrained measures, highlighting the need for
careful tuning of constraint parameters in order to achieve a good trade-off
between speed and accuracy.
