different behaviors from the predictors. Also the simplifying assumption is made that the
vector observations z(n) exactly describe the underlying system dynamics, when they
are actually computed with observed values x(n) that could be corrupted by noise, so any
disturbance is actually considered part of the system's dynamics to be modeled.
Choice of embedding. Arguably "the question of what is a good embedding cannot
be answered separately from the question of what kind of model one is trying to build"
[20], but for coherence purposes, the embedding is chosen to be the same for all models.
It is also set to be the order of the linear filter, since the output of a linear filter is nothing
else than a linear projection of the state space input trajectory on the hyperplane defined
by the filter weights. Two parameters have then to be determined: the embedding
dimension K and the lag r.
The return maps of the data (x(n) plotted versus x(n- )), for z = 1 (Figure3-5) and
z = 2 (Figure3-6), show how the lag impacts the reconstruction space: the higher the lag,
the further the trajectories are from each other, but they still intersect at the same points.
The choice of the lag parameter is only limited by the fact that r has to be such that x(n)
and x(n- r) are sufficiently independent from each other to define distinct dimensions in
state space, but still somehow correlated so as to carry information about the systems
dynamics.
In this thesis the focus is set on segmenting the data, with models accurate and
specific enough to discriminate between them, more than achieving a perfect embedding
and prediction, so we chose the commonly used lag r = 1 and we will not further
consider methods for finding the perfect embedding, but the possibility of fine tuning this
parameter should be kept in mind.