Fluid Correlation: A Novel Nonparametric Metric to Assess the Dynamic Association

The dynamic association between stochastic processes provides crucial information to science and industrial enterprises. Classic methods only provide a fixed and static metric that represents the global association of the stochastic processes. The metric that characterizes the dynamic association is still lacking. Developing such a metric is challenging since the temporal dependence of the stochastic processes could introduce an additional layer of complexity. To surmount the challenge, we develop a method for characterizing the dynamic association between two stochastic processes by delineating the varying factors that may affect the association of the stochastic processes. This is made possible via a novel concept of <i>fluid correlation</i>, which is derived under the concurrent regression models. We propose a penalized total least squares method for estimating the fluid correlation in a reproducing kernel Hilbert space. Asymptotic Bayesian confidence bands are obtained for the uncertainty quantification. The capability of the fluid correlation is demonstrated in constructing a microbial dynamic association, which is particularly informative for the understanding of the microbial ecosystem that dynamically changes with environmental factors.