Title:	Compute Global Sensitivity Analysis Indices Using Optimal Transport
Version:	1.1.0
Description:	Computing Global Sensitivity Indices from given data using Optimal Transport, as defined in Borgonovo et al (2024) <doi:10.1287/mnsc.2023.01796>. You provide an input sample, an output sample, decide the algorithm, and compute the indices.
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.2
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
Config/testthat/edition:	3
Imports:	boot, ggplot2, patchwork (≥ 1.2.0), Rcpp, RcppEigen (≥ 0.3.4.0.0), Rdpack (≥ 2.4), stats, transport (≥ 0.15.0)
URL:	https://github.com/pietrocipolla/gsaot, https://pietrocipolla.github.io/gsaot/
BugReports:	https://github.com/pietrocipolla/gsaot/issues
LinkingTo:	Rcpp, RcppEigen
VignetteBuilder:	knitr
RdMacros:	Rdpack
NeedsCompilation:	yes
Packaged:	2025-07-18 09:21:57 UTC; dagileonardo
Author:	Leonardo Chiani [aut, cre, cph], Emanuele Borgonovo [rev], Elmar Plischke [rev], Massimo Tavoni [rev]
Maintainer:	Leonardo Chiani <leonardo.chiani@polimi.it>
Repository:	CRAN
Date/Publication:	2025-07-18 09:40:01 UTC

Compute confidence intervals for sensitivity indices

Description

Computes confidence intervals for a gsaot_indices object using bootstrap results.

Usage

## S3 method for class 'gsaot_indices'
confint(object, parm = NULL, level = 0.95, type = "norm", ...)

Arguments

Value

A data frame with the following columns:

• input: Name of the input variable.

• component: The index component for Wasserstein-Bures.

• index: Estimated indices

• original: Original estimates.

• bias: Bootstrap bias estimate.

• low.ci: Lower bound of the confidence interval.

• high.ci: Upper bound of the confidence interval.

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))

x <- data.frame(x)
y <- y

res <- ot_indices_wb(x, y, 10, boot = TRUE, R = 100)
confint(res, parm = c(1,3), level = 0.9)

Entropic lower bounds for entropic optimal transport sensitivity indices

Description

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

Details

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

Value

A scalar representing the entropic lower bound.

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)

Multivariate Gaussian linear model evaluation

Description

Generates samples from a multivariate Gaussian distribution and evaluates a simple linear transformation model.

Usage

gaussian_fun(N)

Arguments

Details

Inputs x are sampled from:

\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma), \quad \boldsymbol{\mu} = [1, 1, 1], \quad \Sigma = \begin{bmatrix} 1 & 0.5 & 0.5 \\ 0.5 & 1 & 0.5 \\ 0.5 & 0.5 & 1 \end{bmatrix}

The output is given by:

\mathbf{Y} = A \mathbf{X}^{\top}, \quad A = \begin{bmatrix} 4 & -2 & 1 \\ 2 & 5 & -1 \end{bmatrix}

Value

A list with two elements:

• x: a numeric matrix of size N x 8 containing the input samples.

• y: a numeric vector of length N with the corresponding function outputs.

See Also

sobol_fun, ishi_homma_fun

Examples

result <- gaussian_fun(1000)
head(result$x)
head(result$y)

Irrelevance threshold for optimal transport sensitivity indices

Description

Calculate irrelevance threshold using dummy variable for Optimal Transport sensitivity indices

Usage

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

Arguments

Details

The function allows the computation of irrelevance threshold. The function samples from a distribution defined in dummy_optns (by default a standard normal), independent from the output y and then computes the indices using the algorithm specified in solver. Under the hood, lower_bound calls the other available functions in the package:

• ot_indices_1d() (for solver="1d")

• ot_indices_wb() (for solver="wasserstein-bures")

• ot_indices() (for solver %in% c("sinkhorn", "sinkhorn_log", "wasserstein")) The user can choose the distribution of the dummy variable using the argument dummy_optns. dummy_optns should be a named list with at least a term called "distr" defining the sampling function. The other terms in the list are used as arguments to the sampling function.

Value

An object of class gsaot_indices.

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

dummy_lb <- irrelevance_threshold(y, M)

# Custom sampling funtion and network simplex solver
dummy_optns <- list(distr = "rgamma", shape = 3)
dummy_lb_cust <- irrelevance_threshold(y, M,
                                      dummy_optns = dummy_optns,
                                      solver = "transport")

Ishigami-Homma function evaluation

Description

Evaluates the Ishigami-Homma function. Input samples are drawn from a uniform distribution over [-\pi, \pi]^3

Usage

ishi_homma_fun(N, A = 2, B = 1)

Arguments

Details

The Ishigami-Homma function is defined as:

Y = \sin(X_1) + A \cdot \sin^2(X_2) + B \cdot X_3^4 \cdot \sin(X_1)

where X_i \sim \mathcal{U}(-\pi, \pi).

Value

A list with two elements:

• x: a numeric matrix of size N x 8 containing the input samples.

• y: a numeric vector of length N with the corresponding function outputs.

See Also

sobol_fun, gaussian_fun

Examples

result <- ishi_homma_fun(1000)
head(result$x)
head(result$y)

Estimate optimal transport sensitivity indices for generic outputs

Description

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.

Usage

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"
)

Arguments

Details

Solvers

OT is a widely studied topic in Operational Research and Calculus. The reference for the OT solvers in this package is Peyré et al. (2019). The default solver is "sinkhorn", the Sinkhorn's solver introduced in Cuturi (2013). 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.

Solver options

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.

Value

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.

• W_boot: list of bootstrap objects, one for each input.

References

Cuturi M (2013). “Sinkhorn distances: Lightspeed computation of optimal transport.” Advances in neural information processing systems, 26.

Peyré G, Cuturi M, others (2019). “Computational optimal transport: With applications to data science.” Foundations and Trends® in Machine Learning, 11(5-6), 355–607.

See Also

ot_indices_1d(), ot_indices_wb()

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))

x <- data.frame(x)

M <- 25

# Calculate sensitivity indices
sensitivity_indices <- ot_indices(x, y, M)
sensitivity_indices

Estimate optimal transport indices on one dimensional outputs

Description

Estimate optimal transport indices on one dimensional outputs

Usage

ot_indices_1d(
  x,
  y,
  M,
  p = 2,
  boot = FALSE,
  R = NULL,
  parallel = "no",
  ncpus = 1,
  conf = 0.95,
  type = "norm"
)

Arguments

Value

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.

• W_boot: list of bootstrap objects, one for each input.

See Also

ot_indices(), ot_indices_wb()

Examples

x <- rnorm(1000)
y <- 10 * x
ot_indices_1d(data.frame(x), y, 10)

Estimate sensitivity maps using optimal transport indices

Description

Estimate sensitivity maps using optimal transport indices

Usage

ot_indices_smap(x, y, M)

Arguments

Value

A matrix where each column represents an input and each row represents an output. The values are indices between 0 and 1 computed using ot_indices_1d().

Examples

N <- 1000

x1 <- rnorm(N)
x2 <- rnorm(N)
x <- cbind(x1, x2)

y1 <- 10 * x1
y2 <- x1 + x2
y <- cbind(y1, y2)

ot_indices_smap(data.frame(x), y, 30)

Estimate Wasserstein-Bures approximation of the optimal transport solution

Description

Estimate Wasserstein-Bures approximation of the optimal transport solution

Usage

ot_indices_wb(
  x,
  y,
  M,
  boot = FALSE,
  R = NULL,
  parallel = "no",
  ncpus = 1,
  conf = 0.95,
  type = "norm"
)

Arguments

Value

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.

• W_boot: list of bootstrap objects, one for each input.

See Also

ot_indices(), ot_indices_1d()

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))

x <- data.frame(x)
y <- y

ot_indices_wb(x, y, 10)

Plot optimal transport sensitivity indices

Description

Plot Optimal Transport based sensitivity indices using ggplot2 package.

Usage

## S3 method for class 'gsaot_indices'
plot(x, ranking = NULL, wb_all = FALSE, threshold = NULL, ...)

Arguments

Value

A ggplot object that, if called, will print

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))

x <- data.frame(x)

M <- 25

# Calculate sensitivity indices
sensitivity_indices <- ot_indices_wb(x, y, M)
sensitivity_indices

plot(sensitivity_indices)

Compare sensitivity indices across methods

Description

This function takes a list of gsaot_indices objects and generates a bar plot comparing the sensitivity indices across different methods.

Usage

plot_comparison(x_list, wb_all = FALSE)

Arguments

Value

A ggplot object representing the bar plot of sensitivity indices grouped by input and colored by method.

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))

x <- data.frame(x)

M <- 25

# Calculate sensitivity indices
ind_wb <- ot_indices_wb(x, y, M)
ind_sink <- ot_indices(x, y, M)

plot_comparison(list(ind_wb, ind_sink))

Plot optimal transport local separations

Description

Plot Optimal Transport based local separations for each partition using ggplot2 package. If provided, it plots also the uncertainty estimates.

Usage

plot_separations(x, ranking = NULL, wb_all = FALSE, ...)

Arguments

Value

A patchwork object that, if called, will print.

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))

x <- data.frame(x)

M <- 25

# Get sensitivity indices
sensitivity_indices <- ot_indices(x, y, M)
plot_separations(sensitivity_indices)

Print optimal transport sensitivity indices information

Description

Print optimal transport sensitivity indices information

Usage

## S3 method for class 'gsaot_indices'
print(x, ...)

Arguments

Value

The information contained in argument x

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))

x <- data.frame(x)

M <- 25

# Calculate sensitivity indices
sensitivity_indices <- ot_indices(x, y, M)
print(sensitivity_indices)

Sobol G function evaluation

Description

This function evaluates the Sobol G function on a set of input samples generated via crude Monte Carlos. It returns both the sampled inputs and the corresponding function outputs.

Usage

sobol_fun(N, a = c(0, 1, 4.5, 9, 99, 99, 99, 99))

Arguments

Details

The Sobol G function is defined as:

Y = \prod_{j=1}^{8} \frac{|\ 4 X_j - 2\ | + a_j}{1 + a_j}

where X_j \sim \mathcal{U}(0, 1) independently.

Value

A list with two elements:

• x: a numeric matrix of size N x 8 containing the input samples.

• y: a numeric vector of length N with the corresponding function outputs.

See Also

ishi_homma_fun, gaussian_fun

Examples

result <- sobol_fun(1000)
head(result$x)
head(result$y)

Summary method for gsaot_indices objects

Description

Summary method for gsaot_indices objects

Usage

## S3 method for class 'gsaot_indices'
summary(object, digits = 3, ranking = NULL, ...)

Arguments

Value

(Invisibly) a named list containing the main elements summarised on screen.