sensitivity (version 1.14.0)

sobolroauc: Sobol' Indices estimation under inequality constraints

Description

sobolroauc deals with the estimation of Sobol' sensitivity indices when there exists one or multiple sets of constrained factors. Constraints within a set are expressed as inequality constraints (simplex constraint). This function generalizes the procedure of Tissot and Prieur (2015) to estimate either all first-order indices or all closed second-order indices at a total cost of $2*N$ model evaluations. For closed second-order indices $N=q^2$ where $q >= d-1$ is a prime number denoting the number of levels of the orthogonal array, and where $d$ indicates the number of independent factors or sets of factors.

Usage

sobolroauc(model = NULL, factors, constraints = NULL, N, p = 1, order,  tail = TRUE, conf = 0.95, nboot = 0, ...)
"tell"(x, y = NULL, ...)
"print"(x, ...)
"plot"(x, ylim = c(0,1), ...)

Arguments

model

a function, or a model with a predict method, defining the model to analyze.

factors

an integer giving the number of factors, or a vector of character strings giving their names.

constraints

a list giving the sets of constrained factors (see "Details").

N

an integer giving the size of each replicated design (for a total of $2*N$ model evaluations).

p

an integer giving the number of model outputs.

order

an integer giving the order of the indices (1 or 2).

tail

a boolean specifying the method used to choose the number of levels of the orthogonal array (see "Warning messages").

conf

the confidence level for confidence intervals.

nboot

the number of bootstrap replicates.

x

a list of class "sobolroauc" storing the state of the sensitivity study (parameters, data, estimates).

y

a vector of model responses.

ylim

y-coordinate plotting limits.

...

any other arguments for model which are passed unchanged each time it is called.

Value

sobolroauc returns a list of class "sobolroauc", containing all the input arguments detailed before, plus the following components:

Warning messages

Details

constraints list the sets of factors depending on each other through inequality constraints (see "Examples"). A same factor is not allowed to appear in multiple sets. Factors not appearing in constraints are assumed to be independent and follow each a uniform distribution on [0,1]. One Sobol' index is estimated for each independent factor or set of factors.

Missing values (i.e NA values) in the model responses are automatically handled by the function.

This function also supports multidimensional outputs (matrices in y or as output of model). In this case, aggregated Sobol' indices are returned (see sobolMultOut).

References

L. Devroye, 1986, Non-Uniform Random Variate Generation. Springer-Verlag.

J. Jacques, C. Lavergne and N. Devictor, 2006, Sensitivity Analysis in presence of model uncertainty and correlated inputs. Reliability Engineering & System Safety, 91:1126-1134.

L. Gilquin, C. Prieur and E. Arnaud, 2015, Replication procedure for grouped Sobol' indices estimation in dependent uncertainty spaces, Information and Inference, 4:354-379.

J.Y. Tissot and C. Prieur, 2015, Estimating Sobol's indices combining Monte Carlo integration and Latin hypercube sampling, J. Statist. Comput. Simulation, 85:1358-1381.

See Also

sobolroalhs, sobolmara

Examples

library(boot)
library(numbers)

# Test case: the non-monotonic Sobol g-function
# (there are 8 factors, all following the uniform distribution on [0,1])

# Suppose we have the inequality constraints: X1 <= X3 and X4 <= X6.

# first-order sensitivity indices
x <- sobolroauc(model = sobol.fun, factors = 8, constraints = list(c(1,3),c(4,6)), 
                N = 1000, order = 1, nboot=100)
print(x)
plot(x)

# closed second-order sensitivity indices
x <- sobolroauc(model = sobol.fun, factors = 8, constraints = list(c(1,3),c(4,6)), 
                N = 1000, order = 2, nboot=100)
print(x)
plot(x)