Fig 1. The liquid association coefficient (LAC).

(a) Illustration of LAC using examples. Left column: dynamic correlation with an unknown conditioning factor. When the factor is low, x and y are negatively correlated; when the factor is high, x and y are positively correlated. Second left column: independent case. Right two columns: correlated case. In all the cases, the marginal distribution of X and Y are standard normal. (b) Empirical distributions of LAC score under conditions of dynamic correlation, simple correlation, or independence. The densities are based on 1000 simulations. In the dynamic correlation cases, one-third of the data points follow a bivariate normal distribution with mean $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix $\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)$, one-third follow a bivariate normal distribution with mean $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix $\left(\begin{array}{cc}1& -\rho \\ -\rho & 1\end{array}\right)$, and another one-third follow independent standard normal distributions. In the correlated case, all data points follow a bivariate normal distribution with mean $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix $\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)$.