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Estimating Red Noise Spectrum of Time Series Using Bayesian Inference

Spectral analysis of a time series is to reveal the frequency components in the series and estimate the noise level for characterizing the spectral densities. The frequency components and spectral densities contain abundant and complex physical information. However, the background spectrum of the time series in many research fields, such as climatology, geography, and astronomy, is usually dominated by red noises, rather than random white noise. A common practice estimating red noise is to calculate the periodogram and then to fit the red noise spectrum. Markov Chain Monte Carlo (MCMC) sampling is usually employed to infer the significance of the peaks on top of an underlying red noise continuum in a power spectral density (PSD). In the study, we proposed a novel method based on Hamilton Monte Carlo (HMC) to estimate and assess the significance for discriminating them whether physical processes or noise. We first used synthetic signals dominated by red noise to illustrate the process estimating the red noise spectrum. The root-mean-square error of parameter estimation is only 1.08, indicating that the HMC method has a more high fitting accuracy to parameter estimation. We further checked the reliability of the method using climatology and astronomy data. The results prove that the proposed method estimating red noise with MCMC is effective and credible.