Noise robust estimates of correlation dimension and $K_2$ entropy

Using Gaussian kernels to define the correlation sum we derive simple formulas that correct the noise bias in estimates of the correlation dimension and $K_2$ entropy of chaotic time series. The corrections are only based on the difference of correlation dimensions for adjacent embedding dimensions and hence preserve the full functional dependencies on both the scale parameter and embedding dimension. It is shown theoretically that the estimates, which are derived for additive white Gaussian noise, are also robust for moderately colored noise. Simulations underline the usefulness of the proposed correction schemes. It is demonstrated that the method gives satisfactory results also for non-Gaussian and dynamical noise.