,
then almost every linear transformation of sufficient rank again leads to an
embedding. A specific choice of linear transformation is known as
principal component analysis, singular value decomposition, empirical
orthogonal functions, Karhunen-Loéve decomposition, and probably a few other
names. The technique is fairly widely used, for example to reduce multivariate
data to a few major modes. There is a large literature, including textbooks
like that by Jolliffe [31]. In the context of nonlinear signal processing,
the technique has been advocated among others by Broomhead and King [32].
The idea is to introduce a new set of orthonormal basis vectors in embedding
space such that projections onto a given number of these directions preserve
the maximal fraction of the variance of the original vectors. In other words,
the error in making the projection is minimized for a given number of
directions. Solving this minimization problem [31] leads to an eigenvalue
problem. The desired principal directions can be obtained as the
eigenvectors of the symmetric autocovariance matrix that correspond to the
largest eigenvalues. The alternative and formally equivalent approach via the
trajectory matrix is used in Ref. [32]. The latter is numerically more
stable but involves the singular value decomposition of an 
matrix
for N data points embedded in m dimensions, which can easily exceed
computational resources for time series of even moderate length [33].
In almost all the algorithms described below, simple time delay embeddings can
be substituted by principal components. In the TISEAN project (routines
pca, pc), principal components are only provided as a stand-alone
visualization tool and for linear filtering [34], see
Sec. 
below. In any case, one first has to choose an
initial time delay embedding and then a number of principal components to be
kept. For the purpose of visualization, the latter is immediately restricted
to two or at most three. In order to take advantage of the noise averaging
effect of the principal component scheme, it is advisable to choose a much
shorter delay than one would for an ordinary time delay embedding, while at
the same time increasing the embedding dimension. Experimentation is
recommended. Figure shows the contributions of the first two
principal components to the magneto-cardiogram shown in Fig. .
Figure: Phase space representation of a human magneto-cardiogram using the two largest principal components. An initial embedding was chosen in m=20 dimensions with a delay ofms. The two components cover 70% of the variance of the initial embedding vectors.
