# SobolIndices-class: Estimating Sobol Indices Using Sensitivity Analysis In SobolIndices: Computing the Sobol Indices

## Description

Sobol  proposed a definition called Sobol Indices for estimating the importance of single variable or multiple variales' interaction. We have derived the formulas for Sobol Indices by using sensitivity analysis under GLM of three link functions in `SobolIndices` class, and enhanced the computation by automating the whole procedure.

## Usage

 ```1 2``` ```SobolIndices(xdata, varinput=1, beta=0, link=c("identity","log","logit")) summary(object, ...) ```

## Arguments

 `xdata` A data set of class 'matrix' or 'data.frame' which only includes the variables or features. `varinput` A vector; the indices of the variables which are of interest for computing their interaction (usually high order) sobol indices main effect. `beta` A vector; the coefficients of the variables estimated by the regression model. `link` A character; the link function used under the GLM model. `object` An object of the `SobolIndices` class. `...` Other arguments that could be added.

## Details

The proposed algorithm for computing the Sobol Indices is to use a simple strategy under the GLM model with independent or multivariate normal inputs. We derive the conditional expectations of the response with respect to the input subsets, and then estimate the Sobol' sensitivity indices directly using closed formulas or approximately numerically using empirical variance estimates for a large number of GLMs. The results can enable us to perform ANOVA-type variance decomposition analysis on data with multicollinearity issue, not only under Gaussian regression but also under other types of GLMs such as Poisson and logistic regression. The resulting sobol indices for the variables interaction (or single variable) of interest are stored in the `sobol.indices` slot.

## Value

The `SobolIndices` function computes the sobol indices for variables of interest, constructs and returns an object of the `SobolIndices` class.

## Objects from the Class

Objects should be created using the `SobolIndices` constructor.

## Slots

`xdata`:

A data set of class 'matrix' or 'data.frame' which only includes the variables or features.

`varinput`

A vector which include the indices of the variables which are of interest for computing their interaction (usually high order) sobol indices main effect.

`beta`:

A vector which are the coefficients of the variables in a regression model.

`link`:

A character which is the link function used under the GLM model.

`sobol.indices`:

A numeric number which is the sobol indices of variable(s) of interest.

## Methods

summary

`signature(object = "SobolIndices")`: ...

## Author(s)

Min Wang <wang.1807@mbi.osu.edu>

## References

 Sobol, I. M. (1990). On sensitivity estimation for nonlinear mathematical models, Matematicheskoe Modelirovanie, 2, 112-118.  Lu, R., Rempala, G. and Wang, M. (2016). Sensitivity Analysis of Generalized Linear models, submitted.

`identitySIfunction`, `logSIfunction` and `logitSIfunction` to get a complete list of the functions under different link functions to compute the sobol indices.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```showClass("SobolIndices") # simulate xdata and beta xdata <- matrix(rnorm(20*5, 1), ncol=5) beta <- runif(6, min=-1, max=1) # variables 1 and 2 interaction is of interest varinput <- c(1,2) # link function is identity link (gaussian, possion, etc.) link <- "identity" # apply the proposed method si <- SobolIndices(xdata, varinput=varinput, beta, link="identity") # Review the results summary(si) ```