Extracting colored noise statistics in time series via negentropy

In the analysis of some specific time series (e.g., Global Positioning System coordinate time series, chaotic time series, human brain imaging), the noise is generally modeled as a sum of a power-law noise and white noise. Some existing softwares estimate the amplitude of the noise components using convex optimization (e.g., Levenberg-Marquadt) applied to a log-likelihood cost function. This work studies a novel cost function based on an approximation of the negentropy. Restricting the study to simulated time series with flicker noise plus white noise, we demonstrate that this cost function is convex. Then, we show thanks to numerical approximations that it is possible to obtain an accurate estimate of the amplitude of the colored noise for various lengths of the time series as long as the ratio between the colored noise amplitude and the white noise is smaller than 0.6. The results demonstrate that with our proposed cost function we can improve the accuracy by around 5% when compared with the log-likelihood ones with simulated time series shorter than 1400 samples.