*3.5.3. Influence of sampling interval on the principal component spectrum of chaotic time series[49]*

The Lorenz chaotic system is considered to give the state variable *x* in order to study the influence of sampling interval on the principal component spectrum. The principal component spectrum slant and have no floor for the chaotic time series *x* with τ=0.005 (see Fig. 16a). When τ=0.1 (see Fig. 16b), the principal component spectrum are basically similar

**Figure 16.** The principal component analysis of gaussian noise and Lorenz chaos time series by different sampling intervals based on SVD, *d*=3 : 2 : 23, abscissa is *d*, ordinate is log( / ( )) *<sup>i</sup> <sup>i</sup>* σ *tr* σ

to those in the Figure 16a. When τ=1, each line is separated from each other and tends to horizontal line in the case of different embedding dimensions(see Fig. 16c). It shows that the distribution of the total energy has little difference in each principal axis, like the Gaussian noise (see Fig. 16d). For the Gaussian noise, its principal component spectrum curves are horizontal lines, where *N*=10000. It shows that every principal component is equal to each other. The energy distributes into every principal axis averagely. Therefore, it can be seen that sampling interval affects the determination of embedding dimension. When the sampling interval is not undersampling, the determination of embedding dimension depends the amount of signal-noise-ratio. In the case of undersampling, the chaotic time series is similar to noise so that the embedding dimension seems to be estimated as 1.
