KPC: Kernel Partial Correlation Coefficient

Implementations of two empirical versions the kernel partial correlation (KPC) coefficient and the associated variable selection algorithms. KPC is a measure of the strength of conditional association between Y and Z given X, with X, Y, Z being random variables taking values in general topological spaces. As the name suggests, KPC is defined in terms of kernels on reproducing kernel Hilbert spaces (RKHSs). The population KPC is a deterministic number between 0 and 1; it is 0 if and only if Y is conditionally independent of Z given X, and it is 1 if and only if Y is a measurable function of Z and X. One empirical KPC estimator is based on geometric graphs, such as K-nearest neighbor graphs and minimum spanning trees, and is consistent under very weak conditions. The other empirical estimator, defined using conditional mean embeddings (CMEs) as used in the RKHS literature, is also consistent under suitable conditions. Using KPC, a stepwise forward variable selection algorithm KFOCI (using the graph based estimator of KPC) is provided, as well as a similar stepwise forward selection algorithm based on the RKHS based estimator. For more details on KPC, its empirical estimators and its application on variable selection, see Huang, Z., N. Deb, and B. Sen (2020). “Kernel partial correlation coefficient – a measure of conditional dependence” <arXiv:2012.14804>. When X is empty, KPC measures the unconditional dependence between Y and Z, which has been described in Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using kernels and geometric graphs” <arXiv:2010.01768>, and it is implemented in the functions KMAc() and Klin() in this package. The latter can be computed in near linear time.

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