# KMAc: KMAc (the unconditional version of graph-based KPC) with... In KPC: Kernel Partial Correlation Coefficient

## Description

Calculate \hat{η}_n (the unconditional version of graph-based KPC) using directed K-NN graph or minimum spanning tree (MST).

## Usage

 1 2 3 4 5 6 KMAc( 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. Gaussian kernel: rbfdot(sigma = 1), linear kernel: 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, ties will be broken at random. MST is found using package emstreeR. In particular,

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

where G_n denotes a MST or K-NN graph on X_1,… , X_n, \mathcal{E}(G_n) denotes the set of edges of G_n and (i,j)\in\mathcal{E}(G_n) implies that there is an edge from X_i to X_j in G_n. Euclidean distance is used for computing the K-NN graph and the MST.

## Value

The algorithm returns a real number ‘KMAc’, the empirical kernel measure of association

## References

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

KPCgraph, Klin
 1 2 library(kernlab) KMAc(Y = rnorm(100), X = rnorm(100), k = rbfdot(1), Knn = 1)