Develop and evaluate multivariate functional mark distance correlation

Develop a worked example of genuinely multivariate functional marks, combining several function-valued marks within the proposed mark distance correlation framework, and systematically study the statistical power of the resulting characteristic.

Background

The paper extends distance correlation to marked spatial point processes and demonstrates applications to scalar, multivariate, mixed-type, and single functional marks. For several function-valued marks attached to each point, the authors note that the construction is conceptually straightforward, for example through a suitably weighted fusion of the component-wise L2 metrics, but no empirical demonstration or systematic power analysis is provided.

The unresolved work is therefore to implement and illustrate the genuinely multivariate functional-mark extension and to assess how well the resulting mark distance characteristic detects dependence under relevant spatial and functional dependence structures. This is an explicitly deferred research problem rather than a result established in the paper.