dhsic.test {dHSIC}

Independence test based on dHSIC

Description

Hypothesis test for finding statistically significant evidence of dependence between several variables. Uses the d-variable Hilbert Schmidt independence criterion (dHSIC) as measure of dependence. Several types of hypothesis tests are included. The null hypothesis (H_0) is that all variables are jointly independent.

Usage

dhsic.test(X, Y, K, alpha = 0.05, method = "permutation",
            kernel = "gaussian", B = 1000, pairwise = FALSE,
            bandwidth = 1, matrix.input = FALSE)

Arguments

X

either a list of at least two numeric matrices or a single numeric matrix. The rows of a matrix correspond to the observations of a variable. It is always required that there are an equal number of observations for all variables (i.e. all matrices have to have the same number of rows). If X is a single numeric matrix than one has to specify the second variable as Y or set matrix.input to "TRUE". See below for more details.

Y

a numeric matrix if X is also a numeric matrix and omitted if X is a list.

K

a list of the gram matrices corresponding to each variable. If K the following inputs X, Y, kernel, pairwise, bandwidth and matrix.input will be ignored.

alpha

a numeric value in (0,1) specifying the confidence level of the hypothesis test.

method

a character string specifying the type of hypothesis test used. The available options are: "gamma" (gamma approximation based test), "permutation" (permutation test (slow)), "bootstrap" (bootstrap test (slow)) and "eigenvalue" (eigenvalue based test).

kernel

a vector of character strings specifying the kernels for each variable. There exist two pre-defined kernels: "gaussian" (Gaussian kernel with median heuristic as bandwidth) and "discrete" (discrete kernel). User defined kernels can also be used by passing the function name as a string, which will then be matched using match.fun. If the length of kernel is smaller than the number of variables the kernel specified in kernel[1] will be used for all variables.

B

an integer value specifying the number of Monte-Carlo iterations made in the permutation and bootstrap test. Only relevant if method is set to "permutation" or to "bootstrap".

pairwise

a logical value indicating whether one should use HSIC with pairwise comparisons instead of dHSIC. Can only be true if there are more than two variables.

bandwidth

a numeric value specifying the size of the bandwidth used for the Gaussian kernel. Only used if kernel="gaussian.fixed".

matrix.input

a boolean. If matrix.input is "TRUE" the input X is assumed to be a matrix in which the columns correspond to the variables.

Details

The d-variable Hilbert Schmidt independence criterion is a direct extension of the standard Hilbert Schmidt independence criterion (HSIC) from two variables to an arbitrary number of variables. It is 0 if and only if the variables are jointly independent.

4 different statistical hypothesis tests are implemented all with null hypothesis (H_0: X[[1]],...,X[[d]] are jointly independent) and alternative hypothesis (H_A: X[[1]],...,X[[d]] are not jointly independent): 1. Permutation test for dHSIC: exact level, slow 2. Bootstrap test for dHSIC: pointwise asymptotic level and pointwise consistent, slow 3. Gamma approximation based test for dHSIC: only approximate, fast 4. Eigenvalue based test for dHSIC: pointwise asymptotic level and pointwise consistent, medium

The null hypothesis is rejected if statistic is strictly greater than crit.value.

If X is a list with d matrices, the function tests for joint independence of the corresponding d random vectors. If X is a matrix and matrix.input is "TRUE" the functions tests the independence between the columns of X. If X is a matrix and matrix.input is "FALSE" then Y needs to be a matrix, too; in this case, the function tests the (pairwise) independence between the corresponding two random vectors.

For more details see the references.

Value

A list containing the following components:

statistic

the value of the test statistic

crit.value

critical value of the hypothesis test. The null hypothesis (H_0: joint independence) is rejected if statistic is greater than crit.value.

p.value

p-value of the hypothesis test, i.e. the probability that a random version of the test statistic is greater than statistic under the calculated null hypothesis (H_0: joint independence) based on the data.

time

numeric vector containing computation times. time[1] is time to compute Gram matrix, time[2] is time to compute dHSIC and time[3] is the time to compute crit.value and p.value.

bandwidth

bandwidth used during the computation. Only relevant if Gaussian kernel was used.

Author(s)

Niklas Pfister and Jonas Peters

References

Gretton, A., K. Fukumizu, C. H. Teo, L. Song, B. Sch\"olkopf and A. J. Smola (2007). A kernel statistical test of independence. In Advances in Neural Information Processing Systems (pp. 585-592).

Pfister, N., P. B\"uhlmann, B. Sch\"olkopf and J. Peters (2017). Kernel-based Tests for Joint Independence. To appear in the Journal of the Royal Statistical Society, Series B.

See Also

In order to only compute the test statistic without p-values, use the function dhsic.

Examples

### pairwise independent but not jointly independent (pairwise HSIC vs dHSIC)
set.seed(0)
x <- matrix(rbinom(100,1,0.5),ncol=1)
y <- matrix(rbinom(100,1,0.5),ncol=1)
z <- matrix(as.numeric((x+y)==1)+rnorm(100),ncol=1)
X <- list(x,y,z)

dhsic.test(X, method="permutation",
           kernel=c("discrete", "discrete", "gaussian"),
           pairwise=TRUE, B=1000)$p.value
dhsic.test(X, method="permutation",
           kernel=c("discrete", "discrete", "gaussian"),
           pairwise=FALSE, B=1000)$p.value