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Singular value decomposition

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 ($\sigma_i$) 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 values. The ``normalized'' singular values are given by
\log(\sigma_i) / \sum_{j=1}^{d} \sigma_j \quad i=1,\;2,\;\ldots,\; d
\end{displaymath} (6.3)

where $d$ is the dimension for the reconstruction of phase space before rotation. Though we will compute normalized singular values in section 7.4, we will not go in further detail.