Statistical moments of Gaussian kernel correlation sum and weighted least square estimator of correlation dimension and noise level

In this paper, using non-central chi-squared distribution and polynomial chaos decomposition of a log-normal random variable, we derive the exact expressions for the covariance and variance of the Gaussian kernel correlation sum (Gkcs). The obtained results are combined with U-statistics theory and non-linear models theory to construct the exact confidence intervals for the correlation dimension and for the noise level in the case where deterministic time series are corrupted by additive Gaussian noise. The theoretical results are tested on two continuous chaotic dynamics corrupted by Gaussian noise for different values of signal to noise ratio (SNR). An application to a real data time series has been also conducted.