9.3 Filtered delay embeddings Higher order differential equation can be converted to a set of first order differential equation using additional variables. Derivative coordinate One should form the adequate differences between sucessive observations, Let be the clean variable with autocorrelation function and variance . The observed data has relative noise level . Thus, the relative noise level of the first derivative

9.3.2 Principle component analysis The data with a high sampling rate may contain high redundancy. This is reduced by increasing the sampling rate, or low pass filter. Principal component analysis(PCA) Characterize the time series by its most relevant components in a delay embedding space The set of all delay vectors forms an irregular cloud in Allows for the computation of a series of one dimensional subspaces ordered according to their relevance to the data. The eigenvalues are the squared lengths of the semi-axes of the hyper-ellipsoid which best fits the cloud of data points, and the corresponding eigenvectors give the directions of the axes. The most relevant directions in space thus given by the vectors corresponding to the largest eigenvalues. If there are very small eigenvalues, the corresponding directions may be neglected.

9.3.2 Principle component analysis The data are represented by directional vectors. The components of the new vectors are the projections of the old ones onto the eigenvectors. If one decides that the most relevant eigenvectors are enough to describe the signal, just truncates the new vectors after the component. , this transformation turns into a Fourier transform. For small , the PCA selects the relevant structures in space. PCA is a linear method. The only nonlinear step is the determination and ordering of the eigenvectors and eigenvalues. PCA was also used for dimension estimates; Noise floor: The number of significant eigenvalues of the covariance matrix mirrors the dimension of the subspace which contains attractor. All eigenvalues smaller than this floor are considered to reflect mere noise directions.