Efficient implementation of the Gaussian kernel algorithm in estimating invariants and noise level from noisy time series data

We describe an efficient algorithm which computes the Gaussian kernel correlation integral from noisy time series; this is subsequently used to estimate the underlying correlation dimension and noise level in the noisy data. The algorithm first decomposes the integral core into two separate calculations, reducing computing time from ${O(N}^2\times N_b)$ to ${O(N}^2{+N}_b^2).\mathrm{}$ With other further improvements, this algorithm can speed up the calculation of the Gaussian kernel correlation integral by a factor of $\gamma \sim (2\mathrm{}–{10)N}_b.$ We use typical examples to demonstrate the use of the improved Gaussian kernel algorithm.