Bayesian Nonparametric Spectral Estimation
Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this context, we propose a joint probabilistic model for signals, observations and spectra, where SE is addressed as an exact inference problem. Assuming a Gaussian process prior over the signal, we apply Bayes' rule to find the analytic posterior distribution of the spectrum given a set of observations. Besides its expressiveness and natural account of spectral uncertainty, the proposed model also provides a functional-form representation of the power spectral density, which can be optimised efficiently.
Comparison with previous approaches, in particular against Lomb-Scargle, is addressed theoretically and also experimentally in three different scenarios. Code and demo available at github.com/GAMES-UChile.
Spectral estimation is the task of finding the power associated with a signal across the frequency range of the signal. In classical signal processing applications, this is a solved problem when the time series is fully observed and equally spaced in time. However, in many cases, there are either missing observations, or the data is unevenly spaced in time. Here, probabilistic methods can be extremely valuable, as we can use nonparametric methods to provide better interpolation of the frequency domain, whilst handling the uncertainty in a principled way. This paper proposes a method to tackle this problem, with exact inference.  A crucial component of this paper is windowing of the signal: without it, the Fourier transform of the GP would not exist. The authors here chose a version of the exponential window, which I presume is chosen for mathematical convenience rather than for any particular property (such as spectral leakage properties). From here, paper follows through the mechanics of deriving the local-spectrum GP using this windowing technique, the full derivation of which appears in Appendix 1.1.   As I understand it, inference proceeds by sampling the time-domain GP, and then sampling in the spectral domain conditioned on the observations and given the time-domain functions.   The final part is to compute the spectral energy, which is the (random) squared norm of the real and imaginary parts, which since they are independent GPs, results in a Chi-squared process.  The rest of the paper is concerned with model consistency, and showing that the proposed model forms the limit of the popular Lomb-Scargle method (used for non-uniformly sampled data) as the number of sinusoids tends to infinity. Empirical analysis demonstrates that the proposed method is competitive with the Lomb-Scargle method, whilst being able to proved uncertainty estimates.  Overall this is a well written paper, and provides significant contributions to the field of Bayesian spectral analysis.  Specific comments: I find the use of the same K to represent the kernel function (of two variables) and the stationary kernel function (taking the difference between variables as input) confusing - perhaps $K_S$ could be used? When is K a function of 1 and two variables? (e.g. in 64 it is again used as $K(T)$). 71 “the signal and its spectrum are jointly Gaussian”. Is the spectrum Complex not Gaussian? What does it mean to say that they are jointly Gaussian? Eq 21, t on LHS is a vector 212 Fig 5.1 -> Fig 4? 233 what does “training” mean in the context of the Lomb-Scargle method? 252 available -> able  Note: Thanks to the authors for the clarifications in the feedback.