Singular System Analysis

The two techniques presented above solved only partly the problem of determining the delay time $\tau$ and none of it addressed the virulent problem of noise. Therefore we now turn to the singular system analysis [7] which shows a way how to determine the embedding dimension and the delay time reliably even in the presence of noise. We proceed in three steps: In the first step (in section 3.4.1), we discuss a method to derive a suitable embedding dimension from a finite data set which is measured with infinite precision. Then, in section 3.4.2, we generalize the result of the first step to noisy data. The third step (in the second part of section 3.4.2) gives a slightly more sophisticated method of construction of the vectors $x_i$. In section 3.4.3, we again consider the problem of choosing $\tau$ and the embedding dimension.