  Motivated by the pressing needs for capturing complex but interpretable
variable relationships in scientific research, here we generalize the squared
Pearson correlation to capture a mixture of linear dependences between two
real-valued random variables, with or without an index variable that specifies
the line memberships. We construct generalized Pearson correlation squares by
focusing on three aspects: the exchangeability of the two variables, the
independence of parametric model assumptions, and the availability of
population-level parameters. For the computation of the generalized Pearson
correlation square from a sample without line-membership specification, we
develop a K-lines clustering algorithm, where K, the number of lines, can be
chosen in a data-adaptive way. With our defined population-level generalized
Pearson correlation squares, we derive the asymptotic distributions of the
sample-level statistics to enable efficient statistical inference. Simulation
studies verify the theoretical results and compare the generalized Pearson
correlation squares with other widely-used association measures in terms of
power. Gene expression data analysis demonstrates the effectiveness of the
generalized Pearson correlation squares in capturing interpretable gene-gene
relationships missed by other measures. We implement the estimation and
inference procedures in an R package gR2.
