13 - Bayesian revolution in spectral analysis

Summary

Overview

Science is all about identifying and understanding organized structures or patterns in nature. In this regard, periodic patterns have proven especially important. Nowhere is this more evident than in the field of astronomy. Periodic phenomena allow us to determine fundamental properties like mass and distance, enable us to probe the interior of stars through the new techniques of stellar seismology, detect new planets, and discover exotic states of matter like neutron stars and black holes. Clearly, any fundamental advance in our ability to detect periodic phenomena will have profound consequences in our ability to unlock nature's secrets. The purpose of this chapter is to describe advances that have come about through the application of Bayesian probability theory, and provide illustrations of its power through several examples in physics and astronomy. We also examine how non-uniform sampling can greatly reduce some signal aliasing problems.

New insights on the periodogram

Arthur Schuster introduced the periodogram in 1905, as a means for detecting a periodicity and estimating its frequency. If the data are evenly spaced, the periodogram is determined by the Discrete Fourier Transform (DFT), thus justifying the use of the DFT for such detection and measurement problems. In 1965, Cooley and Tukey introduced the Fast Discrete Fourier Transform (FFT), a very efficient method of implementing the DFT that removes certain redundancies in the computation and greatly speeds up the calculation of the DFT.