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 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 is addressed theoretically, showing that the proposed method is an infinite-dimensional variant of the Lomb-Scargle approach, and also empirically through three experiments.

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.

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 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.