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Fig. 9.
A landmark approximation.
great importance to be identified as a landmark. The specific form used
in the paper defines an
th order landmark of a one-dimensional function
to be a point where the function's
n
th derivative is zero. Thus, first-order
landmarks are extrema, second-order landmarks are inflection points, and so
forth. A smothing technique is also introduced, which lets certain landmarks
be overshadowed by others. For instance, local extrema representing small
fluctuations may not be as important as a global maximum or minimum.
Figure 9 shows an approximated time sequence, reconstructed from a
twelve-dimensional landmark signature.
One of the main contributions of Perng et al . (2000) is to show that for
suitable selections of landmark features, the model is invariant with respect
to the following transformations:
n
Shifting
Uniform amplitude scaling
Uniform time scaling
Non-uniform time scaling (time warping)
Non-uniform amplitude scaling
It is also possible to allow for several of these transformations at once,
by using the intersection of the features allowed for each of them. This
makes the method quite flexible and robust, although as the number of
transformations allowed increases, the number of features will decrease;
consequently, the index will be less precise.
A particularly simple landmark based method (which can be seen as a
special case of the general landmark method) is introduced by Kim et al .
(2001). They show that by extracting the minimum, maximum, and the
first and last elements of a sequence, one gets a (rather crude) signature
that is invariant to time warping. However, since time warping distance
does not obey the triangle inequality [Yi et al . (1998)], it cannot be used
directly. This problem is solved by developing a new distance measure that
underestimates the time warping distance while simultaneously satisfying
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