Klin: A near linear time analogue of KMAc In KPC: Kernel Partial Correlation Coefficient

Description

Calculate \hat{η}_n^{\mbox{lin}} (the unconditional version of graph-based KPC) using directed K-NN graph or minimum spanning tree (MST). The computational complexity is O(nlog(n))

Usage

 1 2 3 4 5 6 Klin( Y, X, k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)), Knn = 1 )

Arguments

 Y a matrix of response (n by dy) X a matrix of predictors (n by dx) k a function k(y, y') of class kernel. It can be the kernel implemented in kernlab e.g. rbfdot(sigma = 1), vanilladot() Knn the number of K-nearest neighbor to use; or "MST".

Details

\hat{η}_n is an estimate of the population kernel measure of association, based on data (X_1,Y_1),… ,(X_n,Y_n)\sim μ. For K-NN graph, \hat{η}_n can be computed in near linear time (in n). In particular,

\hat{η}_n^{\mbox{lin}}:=\frac{n^{-1}∑_{i=1}^n d_i^{-1}∑_{j:(i,j)\in\mathcal{E}(G_n)} k(Y_i,Y_j)-(n-1)^{-1}∑_{i=1}^{n-1} k(Y_i,Y_{i+1})}{n^{-1}∑_{i=1}^n k(Y_i,Y_i)-(n-1)^{-1}∑_{i=1}^{n-1} k(Y_i,Y_{i+1})}

, where all symbols have their usual meanings as in the definition of \hat{η}_n. Euclidean distance is used for computing the K-NN graph and the MST.

Value

The algorithm returns a real number ‘Klin’: an empirical kernel measure of association which can be computed in near linear time when K-NN graphs are used.

References

Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using kernels and geometric graphs” <arXiv:2010.01768>.