sobolmartinez {sensitivity}

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

Description

sobolmartinez 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) \times n model evaluations. These are called the Martinez estimators.

Usage

sobolmartinez(model = NULL, X1, X2, nboot = 0, conf = 0.95, ...)
## S3 method for class 'sobolmartinez'
tell(x, y = NULL, return.var = NULL, ...)
## S3 method for class 'sobolmartinez'
print(x, ...)
## S3 method for class 'sobolmartinez'
plot(x, ylim = c(0, 1), y_col = NULL, y_dim3 = NULL, ...)
## S3 method for class 'sobolmartinez'
ggplot(data, mapping = aes(), ylim = c(0, 1), y_col = NULL,
                 y_dim3 = NULL, ..., environment = parent.frame())

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.

nboot

the number of bootstrap replicates, or zero to use theoretical formulas based on confidence interfaces of correlation coefficient (Martinez, 2011).

conf

the confidence level for bootstrap confidence intervals.

x

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

data

a list of class "sobolmartinez" 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.

y_col

an integer defining the index of the column of x$y to be used for plotting the corresponding sensitivity indices (only applies if x$y is a matrix or an array). If set to NULL (as per default) and x$y is a matrix or an array, the first column (respectively the first element in the second dimension) of x$y is used (i.e. y_col = 1).

y_dim3

an integer defining the index in the third dimension of x$y to be used for plotting the corresponding sensitivity indices (only applies if x$y is an array). If set to NULL (as per default) and x$y is a three-dimensional array, the first element in the third dimension of x$y is used (i.e. y_dim3 = 1).

mapping

Default list of aesthetic mappings to use for plot. If not specified, must be supplied in each layer added to the plot.

environment

[Deprecated] Used prior to tidy evaluation.

...

for sobolmartinez: 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.

This version of sobolmartinez also supports matrices and three-dimensional arrays as output of model. Bootstrapping (including bootstrap confidence intervals) is also supported for matrix or array output. However, theoretical confidence intervals (for nboot = 0) are only supported for vector output. If the model output is a matrix or an array, V, S and T are matrices or arrays as well (depending on the type of y and the value of nboot).

The bootstrap outputs V.boot, S.boot and T.boot can only be returned if the model output is a vector (using argument return.var). For matrix or array output, these objects can't be returned.

Value

sobolmartinez returns a list of class "sobolmartinez", 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

either a vector, a matrix or a three-dimensional array of model responses (depends on the output of model).

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 X_i").

S

the estimations of the Sobol' first-order indices.

T

the estimations of the Sobol' total sensitivity indices.

Users can ask more ouput variables with the argument return.var (for example, bootstrap outputs V.boot, S.boot and T.boot).

Author(s)

Bertrand Iooss, with contributions from Frank Weber (2016)

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.

M. Baudin, K. Boumhaout, T. Delage, B. Iooss and J-M. Martinez, 2016, Numerical stability of Sobol' indices estimation formula, Proceedings of the SAMO 2016 Conference, Reunion Island, France, December 2016

See Also

sobol, sobol2002, sobolSalt, sobol2007, soboljansen, soboltouati, sobolMultOut

Examples

# 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 <- sobolmartinez(model = sobol.fun, X1, X2, nboot = 0)
print(x)
plot(x)

library(ggplot2)
ggplot(x)


# Only for demonstration purposes: a model function returning a matrix
sobol.fun_matrix <- function(X){
  res_vector <- sobol.fun(X)
  cbind(res_vector, 2 * res_vector)
}
x_matrix <- sobolmartinez(model = sobol.fun_matrix, X1, X2)
plot(x_matrix, y_col = 2)
title(main = "y_col = 2")

# Also only for demonstration purposes: a model function returning a
# three-dimensional array
sobol.fun_array <- function(X){
  res_vector <- sobol.fun(X)
  res_matrix <- cbind(res_vector, 2 * res_vector)
  array(data = c(res_matrix, 5 * res_matrix), 
        dim = c(length(res_vector), 2, 2))
}
x_array <- sobolmartinez(model = sobol.fun_array, X1, X2)
plot(x_array, y_col = 2, y_dim3 = 2)
title(main = "y_col = 2, y_dim3 = 2")