2.5 Phase space methods

Phase space methods rely on the embedding of the data in a so-called phase space, by the use of delayed vector methods, that is, <sup>y</sup><sup>∈</sup> <sup>R</sup><sup>N</sup>�<sup>1</sup> ! <sup>X</sup> <sup>∈</sup> <sup>R</sup>ð Þ� <sup>N</sup>�mþ<sup>1</sup> <sup>m</sup> <sup>¼</sup> ½ � xð Þt ; xð Þ t � k …xð Þ t � k mð Þ � 1 . Embedding can also be done by the use of principal components or singular value decomposition of <sup>X</sup> <sup>∈</sup> <sup>R</sup>ð Þ� <sup>N</sup>�mþ<sup>1</sup> <sup>m</sup>, where <sup>k</sup> <sup>¼</sup> 1 and m is comparatively large. In the latter case, the scores of the eigenvectors would represent an orbit or attractor with some geometrical structure, depending on the frequencies with which different regions of the phase space are visited. The topology


Table 1.

Data preprocessing methodologies for multiscale process monitoring.
