gsaot

The package gsaot provides a set of tools to compute and plot Optimal Transport (OT) based sensitivity indices. The core functions of the package are:

• ot_indices(): compute OT indices for multivariate outputs using different solvers for OT (network simplex, Sinkhorn, and so on).

• ot_indices_wb(): compute OT indices for univariate or multivariate outputs using the Wasserstein-Bures semi-metric.

• ot_indices_1d(): compute OT indices for univariate outputs using OT solution in one dimension.

The package also provides functions to plot the resulting indices and the separation measures.

Installation

install.packages("gsaot")

You can install the development version of gsaot from GitHub with:

# install.packages("remotes")
remotes::install_github("pietrocipolla/gsaot")

:exclamation: :exclamation: Installation note

The sinkhorn and sinkhorn_log solvers in gsaot greatly benefit from optimization in compilation. To add this option (before package installation), edit your .R/Makevars file with the desired flags. Even though different compilers have different options, a common flag to enable a safe level of optimization is

CXXFLAGS+=-O2

More detailed information on how to customize the R packages compilation can be found in the R guide.

Example

We can use a gaussian toy model with three outputs as an example:

library(gsaot)

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)

After having defined the number of partitions, we compute the sensitivity indices using different solvers. First, Sinkhorn solver and default parameters:

M <- 25

sensitivity_indices <- ot_indices(x, y, M)
sensitivity_indices
#> Method: sinkhorn 
#> 
#> Solver Options:
#> $numIterations
#> [1] 1000
#> 
#> $epsilon
#> [1] 0.01
#> 
#> $maxErr
#> [1] 1e-09
#> 
#> 
#> Indices:
#>        X1        X2        X3 
#> 0.5856179 0.6321027 0.2833714 
#> 
#> Upper bound: 93.27558

Second, Network Simplex solver:

sensitivity_indices <- ot_indices(x, y, M, solver = "transport")
sensitivity_indices
#> Method: transport 
#> 
#> Solver Options:
#> $fullreturn
#> [1] TRUE
#> 
#> 
#> Indices:
#>        X1        X2        X3 
#> 0.4867229 0.5197051 0.1618929 
#> 
#> Upper bound: 93.27558

Third, Wasserstein-Bures solver, with bootstrap:

sensitivity_indices <- ot_indices_wb(x, y, M, boot = TRUE, R = 100)
sensitivity_indices
#> Method: wasserstein-bures 
#> 
#> Indices:
#>         X1         X2         X3 
#> 0.45682255 0.48218997 0.09602267 
#> 
#> Advective component:
#>         X1         X2         X3 
#> 0.27698647 0.30531330 0.09049933 
#> 
#> Diffusive component:
#>          X1          X2          X3 
#> 0.179836078 0.176876664 0.005523346 
#> 
#> Type of confidence interval: norm 
#> Number of replicates: 100 
#> Confidence level: 0.95 
#> Bootstrap statistics:
#>   input component   original        bias       low.ci    high.ci
#> 1    X1        WB 0.46632977 0.009507222 0.4386714616 0.47497364
#> 2    X2        WB 0.49360786 0.011417898 0.4647262667 0.49965367
#> 3    X3        WB 0.11565245 0.019629776 0.0772420360 0.11480331
#> 4    X1 Advective 0.28174446 0.004757991 0.2651150430 0.28885790
#> 5    X2 Advective 0.31059858 0.005285280 0.2941242206 0.31650239
#> 6    X3 Advective 0.10105252 0.010553192 0.0744003614 0.10659829
#> 7    X1 Diffusive 0.18458531 0.004749231 0.1717065389 0.18796562
#> 8    X2 Diffusive 0.18300928 0.006132618 0.1690000331 0.18475330
#> 9    X3 Diffusive 0.01459993 0.009076584 0.0009580535 0.01008864
#> 
#> Upper bound: 93.16529

Fourth, we can use the package to compute the sensitivity map on the output:

sensitivity_indices <- ot_indices_smap(x, y, M)
sensitivity_indices
#>             X1        X2        X3
#> [1,] 0.5897603 0.0426077 0.1493735
#> [2,] 0.2822771 0.7039011 0.1233232