  As the most commonly used correlation coefficient, Pearson's r is generally
considered as a measure only for linear dependence. The use of Pearson's r has
been strongly discouraged for nonlinear relationships. Recent research shows
that this opinion is myth and Pearson's r should not be ruled out a priori for
measuring nonlinear monotone relationships. We find that Pearson's r does have
potential to measure nonlinear dependence as long as the relationships are
monotone, it just underestimates them. We derive a new inequality which is
tighter than the famous Cauchy-Schwarz Inequality, leverage it to adjust the
underestimation, and propose a new correlation coefficient, i.e., rearrangement
correlation. The rearrangement correlation is able to capture arbitrary
monotone relationships, both linear and nonlinear ones. It reverts to Pearson's
r in linear scenarios. Simulation experiments and real-life investigations show
that the rearrangement correlation is more accurate in measuring nonlinear
monotone dependence than the three classical correlation coefficients, and
other recently proposed dependence measures.
