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

### 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

1 2 3 4 5 6 7 |

### Arguments

`model` |
a function, or a model with a |

`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 |

`y` |
a vector of model responses. |

`return.var` |
a vector of character strings giving further
internal variables names to store in the output object |

`ylim` |
y-coordinate plotting limits. |

`...` |
any other arguments for |

### 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 |

`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 |

`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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
# 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)
``` |