KMAc | R Documentation |
Calculate \hat{η}_n (the unconditional version of graph-based KPC) using directed K-NN graph or minimum spanning tree (MST).
KMAc( Y, X, k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)), Knn = 1 )
Y |
a matrix of response (n by dy) |
X |
a matrix of predictors (n by dx) |
k |
a function k(y, y') of class |
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. |
\hat{η}_n is an estimate of the population kernel measure of association, based on data \{(X_i,Y_i)\}_{i=1}^n from 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.
The algorithm returns a real number ‘KMAc’, the empirical kernel measure of association
Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using kernels and geometric graphs” <arXiv:2010.01768>.
KPCgraph
, Klin
library(kernlab) KMAc(Y = rnorm(100), X = rnorm(100), k = rbfdot(1), Knn = 1)
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