(36) | |||

(37) |

The elements of are called the

Recall that is by definition an orthonormal basis of . So the -matrix can be interpreted as the projection of the trajectory matrix onto this new basis. Then is the trajectory matrix expressed in the basis , and in the above equation is the covariance matrix in the same new basis. If one considers the general form of a covariance matrix, , then it is clear that measures the correlation of all the vectors , averaged over the entire trajectory. Thus the fact that the product gives a diagonal matrix (eq. (40)) shows that

(39) |

(40) |

( is the -matrix formed of the eigenvectors of .) We will see below that this decomposition is useful, because appears as one of the factors constituting . We want to find those entries of which are obviously non-zero only due to the effects of noise. One possibility to do this is to measure, in addition to the true time series, a time series which consists only of the noise and then to compute the corresponding mean square projections onto the -directions (which we found for the true time series). These quantities are to be compared with the respective (of the true time series) and if both are found to be of the same order of magnitude then we know that this particular is only non-zero due to the noise; in other words, the corresponding direction is noise-dominated. A straightforward strategy (which uses the special form of eq. (43)) to get rid of this most significant influence of noise is to set those selected entries equal to zero

(Notice that the are

This equation is nice to work with since the and are easy to compute by diagonalization of the covariance matrix. The sum in eq. (47) will run over summands, and of course we expect . We relabel the such that the first of them correspond to those eigenvalues which are not noise-dominated. Then we know, after having eliminated the effect of noise as far as possible, that the trajectory is confined to a -dimensional subspace of which is spanned by . So we can take as the embedding space instead of . Finally we get the following vectors on the trajectory in :

(46) |

- ... uncorrelated
^{13} - This result justifies the choice in section 3.1: since in a well-chosen basis () the vectors which form the trajectory are uncorrelated, the lag-time does not influence our results and we can choose it at our convenience.
- ...
zero
^{14} -
One should be aware of the fact that this strategy does not remove
*all*effects of the noise on the trajectory: The noise contribution to those which are non-zero according to eq. (46) remains unchanged. It is not totally clear to me why Broomhead and King in [7] do not propose to subtract the noise floor from*all*eigenvalues by replacing in eq. (43) with (or with in the case of white noise); this would (on the average!) remove the noise floor from all eigenvalues. Maybe the reason for not doing this are:

we would not get eq. (47) in this nice form which is easy to handle numerically;

the effect of noise on the first components is assumed to be not very large;

for the calculation of the numerical values of the do not matter anyhow, apart from being above or in the noise floor.