Kernel Integrated $R^2$: A Measure of Dependence
This work addresses the problem of measuring statistical dependence for complex data types (multivariate, functional, structured) for researchers and practitioners needing robust dependence analysis beyond scalar responses.
This paper introduces kernel integrated R^2, a new statistical dependence measure that extends integrated R^2 to handle multivariate, functional, and structured data by incorporating reproducing kernel Hilbert spaces. It is shown to be zero if and only if independence holds, and one if the response is almost surely a measurable function of covariates. Numerical experiments demonstrate competitive power against state-of-the-art dependence measures.
We introduce kernel integrated $R^2$, a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated $R^2$ with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated $R^2$ from scalar responses to responses taking values on general spaces equipped with a characteristic kernel, allowing to measure dependence of multivariate, functional, and structured data, while remaining sensitive to tail behaviour and oscillatory dependence structures. We establish that (i) this new measure takes values in $[0,1]$, (ii) equals zero if and only if independence holds, and (iii) equals one if and only if the response is almost surely a measurable function of the covariates. Two estimators are proposed: a graph-based method using $K$-nearest neighbours and an RKHS-based method built on conditional mean embeddings. We prove consistency and derive convergence rates for the graph-based estimator, showing its adaptation to intrinsic dimensionality. Numerical experiments on simulated data and a real data experiment in the context of dependency testing for media annotations demonstrate competitive power against state-of-the-art dependence measures, particularly in settings involving non-linear and structured relationships.