dhsic {dHSIC}

d-variable Hilbert Schmidt independence criterion - dHSIC

Description

The d-variable Hilbert Schmidt independence criterion (dHSIC) is a non-parametric measure of dependence between an arbitrary number of variables. In the large sample limit the value of dHSIC is 0 if the variables are jointly independent and positive if there is a dependence. It is therefore able to detect any type of dependence given a sufficient amount of data.

Usage

dhsic(X, Y, K, kernel = "gaussian", 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 specified the other inputs will have no effect on the computations.

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.

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 all the variables are jointly independent. This function computes an estimator of dHSIC, which converges to the actual dHSIC in the large sample limit. It is therefore possible to detect any type of dependence in the large sample limit.

If X is a list with d matrices, the function computes dHSIC for the corresponding d random vectors. If X is a matrix and matrix.input is "TRUE" the functions dHSIC for 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 computes the dHSIC (HSIC) for the corresponding two random vectors.

For more details see the references.

Value

A list containing the following components:

dHSIC

the value of the empirical estimator of dHSIC

time

numeric vector containing computation times. time[1] is time to compute Gram matrix and time[2] is time to compute dHSIC.

bandwidth

bandwidth used during computations. 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 perform hypothesis tests based on dHSIC use the function dhsic.test.

Examples

### Three different input methods
set.seed(0)
x <- matrix(rnorm(200),ncol=2)
y <- matrix(rbinom(100,30,0.1),ncol=1)
# compute dHSIC of x and y (x is taken as a single variable)
dhsic(list(x,y),kernel=c("gaussian","discrete"))$dHSIC
dhsic(x,y,kernel=c("gaussian","discrete"))$dHSIC
# compute dHSIC of x[,1], x[,2] and y
dhsic(cbind(x,y),kernel=c("gaussian","discrete"), matrix.input=TRUE)$dHSIC

### Using a user-defined kernel (here: sigmoid kernel)
set.seed(0)
x <- matrix(rnorm(500),ncol=1)
y <- x^2+0.02*matrix(rnorm(500),ncol=1)
sigmoid <- function(x_1,x_2){
  return(tanh(sum(x_1*x_2)))
}
dhsic(x,y,kernel="sigmoid")$dHSIC