ot_indices calculates sensitivity indices using Optimal
Transport (OT) for a multivariate output sample y with respect to input
data x. Sensitivity indices measure the influence of inputs on outputs,
with values ranging between 0 and 1.
ot_indices(
x,
y,
M,
cost = "L2",
discrete_out = FALSE,
solver = "sinkhorn",
solver_optns = NULL,
scaling = TRUE,
boot = FALSE,
stratified_boot = TRUE,
R = NULL,
parallel = "no",
ncpus = 1,
conf = 0.95,
type = "norm"
)A gsaot_indices object containing:
method: a string that identifies the type of indices computed.
indices: a names array containing the sensitivity indices between 0 and 1
for each column in x, indicating the influence of each input variable on
the output variables.
bound: a double representing the upper bound of the separation measure or
an array representing the mean of the separation for each input according
to the bootstrap replicas.
x, y: input and output data provided as arguments of the function.
inner_statistic: a list of matrices containing the values of the inner
statistics for the partitions defined by partitions. If method = wasserstein-bures, each matrix has three rows containing the
Wasserstein-Bures indices, the Advective, and the Diffusive components.
partitions: a matrix containing the partitions built to calculate the
sensitivity indices. Each column contains the partition associated to the
same column in x. If boot = TRUE, the object contains also:
indices_ci: a data.frame with first column the input, second and third
columns the lower and upper bound of the confidence interval.
inner_statistic_ci: a list of matrices. Each element of the list contains
the lower and upper confidence bounds for the partition defined by the row.
bound_ci: a list containing the lower and upper bounds of the confidence
intervals of the separation measure bound.
type, conf: type of confidence interval and confidence level, provided
as arguments.
A matrix or data.frame containing the input(s) values. The values
can be numeric, factors, or strings. The type of data changes the
partitioning. If the values are continuous (double), the function
partitions the data into M sets. If the values are discrete (integers,
strings, factors), the number of partitioning sets is data-driven.
A matrix containing the output values. Each column represents a different output variable, and each row represents a different observation. Only numeric values are allowed.
A scalar representing the number of partitions for continuous inputs.
(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.
(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 for the Optimal Transport problem. Currently supported options are:
"sinkhorn" (default), the Sinkhorn's solver cuturi2013sinkhorngsaot.
"sinkhorn_log", the Sinkhorn's solver in log scale peyre2019computationalgsaot.
"transport", a solver of the non regularized OT problem using transport::transport().
(optional) A list containing the options for the Optimal Transport solver. See details for allowed options and default ones.
(default TRUE) Logical that sets whether or not to scale the
cost matrix.
(default FALSE) Logical that sets whether or not to perform
bootstrapping of the OT indices.
(default TRUE) Logical that sets the type of
resampling performed. With stratified_boot=FALSE, the function resamples
the dataset and then creates the partitions. Otherwise, first, it
creates the partitions and then it performs stratified bootstrapping with
strata being the partitions.
(default NULL) Positive integer, number of bootstrap replicas.
(default "no") The type of parallel operation to be used
(if any). If missing, the default is taken from the option boot.parallel
(and if that is not set, "no"). Only considered if boot = TRUE. For
more information, check the boot::boot() function.
(default 1) Positive integer: number of processes to be used
in parallel operation: typically one would chose this to the number of
available CPUs. Check the ncpus option in the boot::boot() function of
the boot package.
(default 0.95) Number between 0 and 1 representing the
confidence level. Only considered if boot = TRUE.
(default "norm") Method to compute the confidence interval.
Only considered if boot = TRUE. For more information, check the type
option of boot::boot.ci().
OT is a widely studied topic in Operational Research and Calculus. The
reference for the OT solvers in this package is
peyre2019computational;textualgsaot. The default solver is
"sinkhorn", the Sinkhorn's solver introduced in
cuturi2013sinkhorn;textualgsaot. It solves the
entropic-regularized version of the OT problem. The "sinkhorn_log" solves
the same OT problem but in log scale. It is more stable for low values of
the regularization parameter but slower to converge. The option
"transport" is used to choose a solver for the non-regularized OT
problem. Under the hood, the function calls transport::transport() from
package transport. This option does not define the solver per se, but the
solver should be defined with the argument solver_optns. See the next
section for more information.
The argument solver_optns should be empty (for default options) or a list
with all or some of the required solver parameters. All the parameters not
included in the list will be set to default values. The solvers
"sinkhorn" and "sinkhorn_log" have the same options:
numIterations (default 1e3): a positive integer defining the maximum number
of Sinkhorn's iterations allowed. If the solver does not converge in the
number of iterations set, the solver will throw an error.
epsilon (default 0.01): a positive real number defining the regularization
coefficient. If the value is too low, the solver may return NA.
maxErr (default 1e-9): a positive real number defining the
approximation error threshold between the marginal histogram of the
partition and the one computed by the solver. The solver may fail to
converge in numIterations if this value is too low.
The solver "transport" has the parameters:
method (default "networkflow): string defining the solver of the OT
problem.
control: a named list of parameters for the chosen method or the result
of a call to transport::trcontrol().
threads (default 1): an Integer specifying the number of threads used
in parallel computing.
For details regarding this solver, check the transport::transport() help
page.
ot_indices_1d(), ot_indices_wb()
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))
x <- data.frame(x)
M <- 25
# Calculate sensitivity indices
sensitivity_indices <- ot_indices(x, y, M)
sensitivity_indices
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