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 (2022).
“Kernel partial correlation coefficient – a measure of
conditional dependence” (URL listed below). 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.
