# Sensitivity analysis for metrology applications

### Description

Performs a sensitivity analysis for metrology applications

### Usage

1 | ```
sensitivityMet(model, x, y, nboot, method,conf)
``` |

### Arguments

`model` |
a function representing a measurement model with an explicit mathemtical expression |

`x` |
a data frame that contains the input sample. |

`y` |
a vector of model responses. |

`nboot` |
an integer that denotes the number of bootstrap replicates. |

`method` |
a method for the evaluation of the sensitivity. Two methods are currently supported by sensitivityMet : "SRRC" for the standardized Rank Regression Coefficient and "Sobol" for the Sobol indices. |

`conf` |
the confidence level of the bootstrap confidence intervals |

### Details

If method = "SRRC", the function uses src function from the package sensitivity, with the option rank=TRUE to compute SRRC sensitivity indices. This method needs the specification of both the input sample X and either the vector of model responses y either the measurement model as a R function in model.

If method = "Sobol", the function uses sobol2007 function from the package sensitivity. The input sample x is divided into two samples of equal dimensions. Xdesign is returned, containing the design for the computation of Sobol indices. The user should evaluate Xdesign with the computational code and provide the corresponding output values using tell function. Details on the computation of Sobol indices are given in Sobol et al. (2007).

Both methods are applicable whether the measurement model is an explicit function (defined in model as a R function) or an external code.

The argument nboot is required in order to evaluate a confidence interval with a specified confidence level for the sensitivity indices. Default is 0 : in this case no bootstrap replicates and no confidence intervals are computed.

The argument conf defines the confidence level for the bootstrap confidence intervals. Default is 0.95.

### Value

sensitivityMet returns a list with the following components :

`model` |
a function representing a measurement model with an explicit mathemtical expression |

`method` |
The method used for the evaluation of the sensitivity indices |

If method = "SRRC" :

`x` |
a data frame that contains the random sample of the input quantities. |

`y` |
a vector of model responses. |

`S1` |
a data frame wihch summarizes the first order sensitivity indices obtained with the SRRC. |

If method = "Sobol" :

`SI` |
an object of class sobol2007 for the computation of the Sobol sensitivity indices with the following list of components : |

`model` |
a function representing a measurement model with an explicit mathemtical expression |

`X1` |
the first random sample |

`X2` |
the second random sample |

`nboot` |
the number of bottstrap replicates |

`conf` |
the confidence level for bootstrap confidence intervals |

`X` |
a data.frame containing the design of experiment |

`call` |
the matched call of the function sobol2007 |

`y` |
the response used |

`V` |
the estimations of 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)

Alexandre Allard alexandre.allard@lne.fr

### References

I.M. Sobol, S. Tarantola, D. Gatelli, S.S. Kucherenko and W. Mauntz, 2007, Estimating the approximation errors when fixing unessential factors in global sensitivity analysis, Reliability Engineering and System Safety, 92, 957-960.

A. Saltelli, P. Annoni, I. Azzini, F. Campolongo, M. Ratto and S. Tarantola, 2010, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications 181, 259-270.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | ```
rm(list=ls())
##Simulate the input sample
M=10000
Xmass <- data.frame(X1 = rnorm(M, 100, 5e-5),
X2 = rnorm(M, 0.001234, 2e-5),
X3 = runif(M, 1100, 1300),
X4 = runif(M,7000000,9000000),
X5 = runif(M,7950000,8050000))#Data-frame
#Define the measurement model (GUM-S1, 9.3)
calibMass <-function(x){
return(((x[,1]+x[,2])*(1+(x[,3]-1200)*(1/x[,4]-1/x[,5]))-100)*1e3)
}
##### Use SRRC with a model function #####
#Apply sensitivityMet function to evaluate the associated SRRC indices
S_SRRC=sensitivityMet(model=calibMass,x=Xmass, nboot=100, method="SRRC", conf=0.95)
##Print the results
#First order indices
S_SRRC$S1
##### Use Sobol with a computational code #####
#Creation of the design for the computation of Sobol sensitivity indices
S_Sobol=sensitivityMet(model=NULL,x=Xmass,y=NULL, nboot=100, method="Sobol", conf=0.95)
#Obtain the design of experiment to submit to the code
XDesign=S_Sobol$SI$X
#Run the computational code with XDesign as a sample of the input quantities
#We use calibMass function (see GUM-S1) as an example
YDesign=calibMass(XDesign)
#Run the Sobol indices calculations with the outputs of the code
S_Sobol$SI=tell(x=S_Sobol$SI,y=YDesign)
##Print the results
#First order indices
S_Sobol$SI$S
#Total order indices
S_Sobol$SI$T
``` |