# hsic.gamma: Hilber Schmidt Independence Criterion gamma test In kpcalg: Kernel PC Algorithm for Causal Structure Detection

## Description

Test to check the independence between two variables x and y using HSIC. The hsic.gamma() function, uses Hilbert-Schmidt independence criterion to test for independence between random variables.

## Usage

 `1` ```hsic.gamma(x, y, sig = 1, numCol = 100) ```

## Arguments

 `x` data of first sample `y` data of second sample `sig` Gaussian kernel width for HSIC tests. Default is 1 `numCol` maximum number of columns that we use for the incomplete Cholesky decomposition

## Details

Let x and y be two samples of length n. Gram matrices K and L are defined as: K_{i,j} =exp((x_i-x_j)^2/sig^2) and L_{i,j} =exp((y_i-y_j)^2/sig^2). H_{i,j} = delta_{i,j} - 1/n. Let A=HKH and B=HLH, then HSIC(x,y)=Tr(AB)/n^2. Gamma test compares HSIC(x,y) with the alpha quantile of the gamma distribution with mean and variance such as HSIC under independence hypothesis.

## Value

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

 `method` description of test `statistic` observed value of the test statistic `estimate` HSIC(x,y) `estimates` a vector: [HSIC(x,y), mean of HSIC(x,y), variance of HSIC(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 (petras.verbyla@mrc-bsu.cam.ac.uk) and Nina Ines Bertille Desgranges

## References

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

hsic.perm, hsic.clust, kernelCItest

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```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

 -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.