# Sobol Method

struct Sobol <: GSAMethod
order::Vector{Int}
nboot::Int
conf_level::Float64
end

The Sobol object has as its fields the order of the indices to be estimated.

• order - the order of the indices to calculate. Defaults to [0,1], which means the Total and First order indices. Passing 2 enables calculation of the Second order indices as well.

For confidence interval calculation nboot should be specified for the number (>0) of bootstrap runs and conf_level for the confidence level, the default for which is 0.95.

## Sobol Method Details

Sobol is a variance-based method and it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. This helps to get not just the individual parameter's sensitivities but also gives a way to quantify the affect and sensitivity from the interaction between the parameters.

$$$Y = f_0+ \sum_{i=1}^d f_i(X_i)+ \sum_{i < j}^d f_{ij}(X_i,X_j) ... + f_{1,2...d}(X_1,X_2,..X_d)$$$$$$Var(Y) = \sum_{i=1}^d V_i + \sum_{i < j}^d V_{ij} + ... + V_{1,2...,d}$$$

The Sobol Indices are "order"ed, the first order indices given by $S_i = \frac{V_i}{Var(Y)}$ the contribution to the output variance of the main effect of $X_i$, therefore it measures the effect of varying $X_i$ alone, but averaged over variations in other input parameters. It is standardised by the total variance to provide a fractional contribution. Higher-order interaction indices $S_{i,j}, S_{i,j,k}$ and so on can be formed by dividing other terms in the variance decomposition by $Var(Y)$.

### API

function gsa(f, method::Sobol, A::AbstractMatrix{TA}, B::AbstractMatrix;
batch=false, Ei_estimator = :Jansen1999, distributed::Val{SHARED_ARRAY} = Val(false), kwargs...) where {TA, SHARED_ARRAY}


Ei_estimator can take :Homma1996, :Sobol2007 and :Jansen1999 for which Monte Carlo estimator is used for the Ei term. Defaults to :Jansen1999. Details for these can be found in the corresponding papers: - :Homma1996 - Homma, T. and Saltelli, A., 1996. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering & System Safety, 52(1), pp.1-17. - :Sobol2007 - I.M. Sobol, S. Tarantola, D. Gatelli, S.S. Kucherenko and W. Mauntz, 2007, Estimating the approx- imation 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. - :Jansen1999 - M.J.W. Jansen, 1999, Analysis of variance designs for model output, Computer Physics Communi- cation, 117, 35–43.

### Example

using GlobalSensitivity, QuasiMonteCarlo

function ishi(X)
A= 7
B= 0.1
sin(X) + A*sin(X)^2+ B*X^4 *sin(X)
end

n = 600000
lb = -ones(4)*π
ub = ones(4)*π
sampler = SobolSample()
A,B = QuasiMonteCarlo.generate_design_matrices(n,lb,ub,sampler)

res1 = gsa(ishi,Sobol(order=[0,1,2]),A,B)

function ishi_batch(X)
A= 7
B= 0.1
@. sin(X[1,:]) + A*sin(X[2,:])^2+ B*X[3,:]^4 *sin(X[1,:])
end

res2 = gsa(ishi_batch,Sobol(),A,B,batch=true)