Entropic lower bounds for entropic Optimal Transport sensitivity indices

Calculate entropic lower bounds for entropic Optimal Transport sensitivity indices

Usage

entropic_bound(
  y,
  M,
  cost = "L2",
  discrete_out = FALSE,
  solver = "sinkhorn",
  solver_optns = NULL,
  scaling = TRUE
)

Arguments

y

An array or a matrix containing the output values.

M

A scalar representing the number of partitions for continuous inputs.

cost

(default "L2") A string or function defining the cost function of the Optimal Transport problem. It should be "L2" or a function taking as input y and returning a cost matrix. If cost="L2", ot_indices uses the squared Euclidean metric.

discrete_out

(default FALSE) Logical, by default the output sample in y are equally weighted. If discrete_out=TRUE, the function tries to create an histogram of the realizations and to use the histogram as weights. It works if the output is discrete or mixed and the number of realizations is large. The advantage of this option is to reduce the dimension of the cost matrix.

solver

Solver for the Optimal Transport problem. Currently supported options are:

• "sinkhorn" (default), the Sinkhorn's solver (Cuturi 2013) .

• "sinkhorn_log", the Sinkhorn's solver in log scale (Peyré et al. 2019) .

solver_optns

(optional) A list containing the options for the Optimal Transport solver. See details for allowed options and default ones.

scaling

(default TRUE) Logical that sets whether or not to scale the cost matrix.

Value

A scalar representing the entropic lower bound.

Details

The function allows the computation of the entropic lower bounds. solver should be either "sinkhorn" or "sinkhorn_log".

Examples

N <- 1000

mx <- c(1, 1, 1)
Sigmax <- matrix(data = c(1, 0.5, 0.5, 0.5, 1, 0.5, 0.5, 0.5, 1), nrow = 3)

x1 <- rnorm(N)
x2 <- rnorm(N)
x3 <- rnorm(N)

x <- cbind(x1, x2, x3)
x <- mx + x %*% chol(Sigmax)

A <- matrix(data = c(4, -2, 1, 2, 5, -1), nrow = 2, byrow = TRUE)
y <- t(A %*% t(x))

M <- 25

sink_lb <- entropic_bound(y, M)