Curve Correlation

Curve correlation was recently proposed as a way to measure association between time courses that are noisy and may be observed irregularly and at disparate time points. The crux of the method is basis-function smoothing of the time series, which can effectively mitigate the well-known attenuation problem for correlation estimation with noisy data. This technique is reminiscent of functional data analysis, but treats the observations, rather than the variables, as lying along a continuum. This paper provides an in-depth examination of curve correlation. Whereas in the classical setting the population correlation is the estimand and the sample correlation is its estimate, we show how the curve correlation plays both roles: it may function either as an estimand, since it is not observed directly, and as an estimate of an underlying stochastic process correlation. We contrast curve correlation with the related idea of dynamic correlation, investigate boot-strap and posterior simulation approaches to interval estimation, and derive a formula for the variance of the curve correlations arising from a bivariate Gaussian process. Illustrative examples are provided from accelerometry, meteorology, international development and election outcomes.