Description Usage Arguments Details Value References See Also Examples

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))

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`Y` |
a matrix of response (n by dy) |

`X` |
a matrix of predictors (n by dx) |

`k` |
a function |

`Knn` |
the number of K-nearest neighbor to use; or "MST". |

*\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.

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.

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

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