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Broomhead and King [Broomhead and King, 1986] and Albano et. al [Albano et al., 1988]
claim that they can improve the dimension
estimation by rotating the phase space according to their singular
vectors. The dimension and entropy are invariants under such a transformation.
The singular values () measure the importance of the observations
in the direction of the principal components.
By visual inspection of the singular values, one may
identify a ``noise floor'', and singular vectors, belonging to such
small singular values should be discarded to reduce the effects of noise.
A potential disadvantage is that the
maximum embedding dimension (after the rotation)
is limited to the number of ``good'' singular
The ``normalized'' singular values are given by
where is the dimension for the reconstruction of phase space
Though we will compute normalized singular values
in section 7.4, we will not go in further detail.