dcov.gamma: Test to check the independence between two variables x and y...

In kpcalg: Kernel PC Algorithm for Causal Structure Detection

Description

Test to check the independence between two variables x and y using the Distance Covariance. The dcov.gamma() function, uses Distance Covariance independence criterion with gamma approximation to test for independence between two random variables.

Usage

dcov.gamma(x, y, index = 1, numCol = 100)

Arguments

x	data of first sample
y	data of second sample
index	exponent on Euclidean distance, in (0,2]
numCol	Number of columns used in incomplete Singular Value Decomposition

Details

Let x and y be two samples of length n. Gram matrices K and L are defined as: K_{i,j} =|x_i-x_j|^s and L_{i,j} =|y_i-y_j|^s, where 0<s<2. H_{i,j} = delta_{i,j} - 1/n. Let A=HKH and B=HLH, then nV^2 = ∑ A_{i,j} B_{i,j} \n^2. For more detail: dcov.test in package energy. Gamma test compares nV^2_n(x,y) with the alpha quantile of the gamma distribution with mean and variance same as nV^2_n under independence hypothesis.

Value

dcov.gamma() returns a list with class htest containing

method	description of test
statistic	observed value of the test statistic
estimate	nV^2(x,y)
estimates	a vector: [nV^2(x,y), mean of nV^2(x,y), variance of nV^2(x,y)]
replicates	replicates of the test statistic
p.value	approximate p-value of the test
data.name	desciption of data

Author(s)

Petras Verbyla ([e-mail]) and Nina Ines Bertille Desgranges

References

A. Gretton et al. (2005). Kernel Methods for Measuring Independence. JMLR 6 (2005) 2075-2129.

G. Szekely, M. Rizzo and N. Bakirov (2007). Measuring and Testing Dependence by Correlation of Distances. The Annals of Statistics 2007, Vol. 35, No. 6, 2769-2794.

See Also

hsic.perm, hsic.clust, hsic.gamma, dcov.test, kernelCItest

Examples

library(energy)
set.seed(10)
#independence
x <- runif(300)
y <- runif(300)

hsic.gamma(x,y)
hsic.perm(x,y)
dcov.gamma(x,y)
dcov.test(x,y)

#uncorelated but not dependent
z <- 10*(runif(300)-0.5)
w <- z^2 + 10*runif(300)

cor(z,w)
hsic.gamma(z,w)
hsic.perm(z,w)
dcov.gamma(z,w)
dcov.test(z,w)

Example output

	HSIC test of independence

data:  Gamma approximation
HSIC = 3.9419e-05, p-value = 0.4951
sample estimates:
        HSIC 
3.941891e-05 


	HSIC test of independence

data:  Permutation approximation
HSIC = 3.9419e-05, p-value = 0.5149
sample estimates:
        HSIC 
3.941891e-05 


	dCov test of independence

data:  index 1, Gamma approximation
nV^2 = 0.070484, p-value = 0.7095
sample estimates:
      dCov 
0.01532792 


	Specify the number of replicates R (R > 0) to perform the test of
	independence

data:  index 1, replicates 0
nV^2 = 0.071288, p-value = NA
sample estimates:
      dCov 
0.01541517 

[1] -0.03614353

	HSIC test of independence

data:  Gamma approximation
HSIC = 0.01221, p-value < 2.2e-16
sample estimates:
     HSIC 
0.0122098 


	HSIC test of independence

data:  Permutation approximation
HSIC = 0.012203, p-value = 0.009901
sample estimates:
      HSIC 
0.01220283 


	dCov test of independence

data:  index 1, Gamma approximation
nV^2 = 762.29, p-value < 2.2e-16
sample estimates:
    dCov 
1.594046 


	Specify the number of replicates R (R > 0) to perform the test of
	independence

data:  index 1, replicates 0
nV^2 = 762.43, p-value = NA
sample estimates:
   dCov 
1.59419 

kpcalg documentation built on May 2, 2019, 12:39 p.m.