Bayesian Estimation of Time Series Lags and Structure
Jeffrey D. Scargle
TL;DR
The paper addresses estimating time-series lags and extracting structure from data weighted over a range of the independent variable. It develops a Bayesian framework in which the lag posterior for time-tagged data is proportional to $e^{c_1(a)\\gamma_{X,Y}(\\tau)}$ and, for evenly spaced data, to $e^{K_1(a)\\gamma_{X,Y}(\\tau)}$, with the cross-correlation $\\gamma_{X,Y}(\\tau)$ as the sufficient statistic. It then extends Bayesian Blocks to data with normal errors and weighted measurements, yielding a Gaussian posterior with a quadratic form $H$ in terms of block heights $B_j$ and inner products $G_j(n)$, enabling joint lag and structure estimation. The work provides practical, algorithmic tools for lag detection and time-series structure reconstruction in astronomy, with plans to release software implementations and extend the framework to scale-sensitive analyses like the scalegram.
Abstract
This paper derives practical algorithms, based on Bayesian inference methods, for several data analysis problems common in time series analysis of astronomical and other data. One problem is the determination of the lag between two time series, for which the cross-correlation function is a sufficient statistic. The second problem is the estimation of structure in a time series of measurements which are a weighted integral over a finite range of the independent variable.