Monte Carlo Estimation of Sobol' Indices (formulas of Martinez (2011) and Touati (2016))

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Description

soboltouati implements the Monte Carlo estimation of the Sobol' indices for both first-order and total indices using correlation coefficients-based formulas, at a total cost of (p + 2) * n model evaluations. These are called the Martinez estimators. It also computes their confidence intervals based on asymptotic properties of empirical correlation coefficients.

Usage

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soboltouati(model = NULL, X1, X2, conf = 0.95, ...)
## S3 method for class 'soboltouati'
tell(x, y = NULL, return.var = NULL, ...)
## S3 method for class 'soboltouati'
print(x, ...)
## S3 method for class 'soboltouati'
plot(x, ylim = c(0, 1), ...)

Arguments

model

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

X1

the first random sample.

X2

the second random sample.

conf

the confidence level for confidence intervals, or zero to avoid their computation if they are not needed.

x

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

y

a vector of model responses.

return.var

a vector of character strings giving further internal variables names to store in the output object x.

ylim

y-coordinate plotting limits.

...

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

Details

This estimator supports missing values (NA or NaN) which can occur during the simulation of the model on the design of experiments (due to code failure) even if Sobol' indices are no more rigorous variance-based sensitivity indices if missing values are present. In this case, a warning is displayed.

Value

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

call

the matched call.

X

a data.frame containing the design of experiments.

y

the response used

V

the estimations of normalized variances of the Conditional Expectations (VCE) with respect to each factor and also with respect to the complementary set of each factor ("all but Xi").

S

the estimations of the Sobol' first-order indices.

T

the estimations of the Sobol' total sensitivity indices.

Author(s)

Taieb Touati, Khalid Boumhaout

References

J-M. Martinez, 2011, Analyse de sensibilite globale par decomposition de la variance, Presentation in the meeting of GdR Ondes and GdR MASCOT-NUM, January, 13th, 2011, Institut Henri Poincare, Paris, France.

T. Touati, 2016, Confidence intervals for Sobol' indices. Submitted to the SAMO 2016 Conference, Reunion Island.

T. Touati, 2016, Proprietes asymptotiques du coefficient de correlation empirique, draft.

See Also

sobol, sobol2002, sobol2007, soboljansen, sobolEff, sobolmara, sobolmartinez

Examples

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# Test case : the non-monotonic Sobol g-function

# The method of sobol requires 2 samples
# There are 8 factors, all following the uniform distribution
# on [0,1]

library(boot)
n <- 1000
X1 <- data.frame(matrix(runif(8 * n), nrow = n))
X2 <- data.frame(matrix(runif(8 * n), nrow = n))

# sensitivity analysis

x <- soboltouati(model = sobol.fun, X1, X2)
print(x)
plot(x)

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