  Building upon the Chatterjee correlation (2021: J. Am. Stat. Assoc. 116,
p2009) for two real-valued variables, this study introduces a generalized
measure of directed association between two vector variables, real or
complex-valued, and of possibly different dimensions. The new measure is
denoted as the "distance-based Chatterjee correlation", owing to the use here
of the "distance transformed data" defined in Szekely et al (2007: Ann.
Statist. 35, p2769) for the distance correlation. A main property of the new
measure, inherited from the original Chatterjee correlation, is its predictive
and asymmetric nature: it measures how well one variable can be predicted by
the other, asymmetrically. This allows for inferring the causal direction of
the association, by using the method of Blobaum et al (2019: PeerJ Comput. Sci.
1, e169). Since the original Chatterjee correlation is based on ranks, it is
not available for complex variables, nor for general multivariate data. The
novelty of our work is the extension to multivariate real and complex-valued
pairs of vectors, offering a robust measure of directed association in a
completely non-parametric setting. Informally, the intuitive assumption used
here is that distance correlation is mathematically equivalent to Pearson's
correlation when applied to "distance transformed" data. The next logical step
is to compute Chatterjee's correlation on the same "distance transformed" data,
thereby extending the analysis to multivariate vectors of real and complex
valued data. As a bonus, the new measure here is robust to outliers, which is
not true for the distance correlation of Szekely et al. Additionally, this
approach allows for inference regarding the causal direction of the association
between the variables.
