Conditional local distance correlation for manifold-valued data.

Area of Science:

• Statistics
• Data Science
• Medical Imaging

Background:

• Manifold-valued data is prevalent in diverse scientific fields, including medical imaging, computational biology, and computer vision.
• Characterizing nonlinear associations in such complex data is challenging, especially when considering conditional dependencies.

Purpose of the Study:

• To introduce a novel conditional local distance correlation measure for assessing nonlinear associations between manifold-valued data (X) and other variables (Y), conditional on a third set of variables (Z).
• To develop a computationally efficient estimation procedure and a bootstrap method for hypothesis testing of conditional independence.

Main Methods:

• The proposed measure is based on the intrinsic distances within the data spaces, avoiding parametric assumptions or tangent space projections.
• A fast estimation algorithm is developed for the nonlinear association measure.
• A bootstrap methodology is employed to determine the asymptotic distribution and p-value for conditional independence testing.

Main Results:

• The developed method effectively quantifies nonlinear associations in manifold-valued data.
• The estimation procedure is computationally efficient.
• The bootstrap method provides reliable p-values for hypothesis testing, as validated by simulation studies and real-world data analysis.

Conclusions:

• The introduced conditional local distance correlation measure offers a powerful tool for analyzing complex relationships in manifold-valued data.
• The method is robust, computationally efficient, and suitable for applications in medical imaging and other related fields.
• This work facilitates a deeper understanding of conditional independence in high-dimensional, manifold-structured datasets.