KMAc {KPC}

KMAc (the unconditional version of graph-based KPC) with geometric graphs.

Description

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

Usage

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". A small Knn (e.g., Knn=1) is recommended for an accurate estimate of the population KMAc.

Details

\hat{\eta}_n is an estimate of the population kernel measure of association, based on data \{(X_i,Y_i)\}_{i=1}^n from \mu. For K-NN graph, ties will be broken at random. MST is found using package emstreeR. In particular,

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

where G_n denotes a MST or K-NN graph on X_1,\ldots , 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>.

See Also

KPCgraph, Klin

Examples

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