Bayesian interpretation of periodograms

TL;DR

This paper revisits spectral analysis using a Bayesian framework, interpreting periodograms as regularized least squares solutions and proposing a practical maximum likelihood method for hyperparameter and window selection.

Contribution

It introduces a Bayesian interpretation of periodograms within a regularization framework and develops a maximum likelihood approach for hyperparameter and window selection.

Findings

Periodograms can be derived as regularized least squares minimizers.

Bayesian interpretation provides new insights into spectral analysis.

Maximum likelihood method effectively selects hyperparameters and windows.

Abstract

The usual nonparametric approach to spectral analysis is revisited within the regularization framework. Both usual and windowed periodograms are obtained as the squared modulus of the minimizer of regularized least squares criteria. Then, particular attention is paid to their interpretation within the Bayesian statistical framework. Finally, the question of unsupervised hyperparameter and window selection is addressed. It is shown that maximum likelihood solution is both formally achievable and practically useful.