KPCgraph {KPC}

Kernel partial correlation with geometric graphs

Description

Calculate the kernel partial correlation (KPC) coefficient with directed K-nearest neighbor (K-NN) graph or minimum spanning tree (MST).

Usage

KPCgraph(
  Y,
  X,
  Z,
  k = kernlab::rbfdot(1/(2 * stats::median(stats::dist(Y))^2)),
  Knn = 1,
  trans_inv = FALSE
)

Arguments

Y

a matrix (n by dy)

X

a matrix (n by dx) or NULL if X is empty

Z

a matrix (n by dz)

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

a positive integer indicating the number of nearest neighbor to use; or "MST". A small Knn (e.g., Knn=1) is recommended for an accurate estimate of the population KPC.

trans_inv

TRUE or FALSE. Is k(y, y) free of y?

Details

The kernel partial correlation squared (KPC) measures the conditional dependence between Y and Z given X, based on an i.i.d. sample of (Y, Z, X). It converges to the population quantity (depending on the kernel) which is between 0 and 1. A small value indicates low conditional dependence between Y and Z given X, and a large value indicates stronger conditional dependence. If X == NULL, it returns the KMAc(Y,Z,k,Knn), which measures the unconditional dependence between Y and Z. Euclidean distance is used for computing the K-NN graph and the MST. MST in practice often achieves similar performance as the 2-NN graph. A small K is recommended for the K-NN graph for an accurate estimate of the population KPC, while if KPC is used as a test statistic for conditional independence, a larger K can be beneficial.

Value

The algorithm returns a real number which is the estimated KPC.

See Also

KPCRKHS, KMAc, Klin

Examples

library(kernlab)
n = 2000
x = rnorm(n)
z = rnorm(n)
y = x + z + rnorm(n,1,1)
KPCgraph(y,x,z,vanilladot(),Knn=1,trans_inv=FALSE)

n = 1000
x = runif(n)
z = runif(n)
y = (x + z) %% 1
KPCgraph(y,x,z,rbfdot(5),Knn="MST",trans_inv=TRUE)

discrete_ker = function(y1,y2) {
    if (y1 == y2) return(1)
    return(0)
}
class(discrete_ker) <- "kernel"
set.seed(1)
n = 2000
x = rnorm(n)
z = rnorm(n)
y = rep(0,n)
for (i in 1:n) y[i] = sample(c(1,0),1,prob = c(exp(-z[i]^2/2),1-exp(-z[i]^2/2)))
KPCgraph(y,x,z,discrete_ker,1)
##0.330413