Estimating All Sobol Indices of Certain Order Using Sensitivity Analysis

Description

Sobol [1] 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 SobolIndicesAll class, and compute the sobol indices of all possible variables interactions of given order by using this algorithm.

Usage

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SobolIndicesAll(xdata, orderinput=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.

orderinput

A integer; the order of the interaction of the variables which are of interest for computing all possible interactions' 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 SobolIndicesAll 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 all the variables interaction (of order orderinput) of interest are stored in the sobol.indices.all slot.

Value

The SobolIndicesAll function computes all the sobol indices for variables interactions of order orderinput, constructs and returns an object of the SobolIndicesAll class.

Objects from the Class

Objects should be created using the SobolIndicesAll constructor.

Slots

xdata:

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

orderinput

A integer which is the order of the interaction of the variables of interest for computing all possible interactions' 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.all:

A list or a numeric object which stores sobol indices of variables interactions of order orderinput.

Methods

summary

signature(object = "SobolIndicesAll"): ...

Author(s)

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

References

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

See Also

identitySIfunction, logSIfunction and logitSIfunction to get a complete list of the functions under different link functions to compute the sobol indices.

Examples

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showClass("SobolIndicesAll")
# simulate xdata and beta
xdata <- matrix(rnorm(20*5, 1), ncol=5)
beta <- runif(6, min=-1, max=1)
# all paired variables interactions are of interest
orderinput <- 2
# link function is logit link (binomial, etc.)
link <- "logit"
# apply the proposed method
siall <- SobolIndicesAll(xdata, orderinput=orderinput, beta, link="logit")
# Review the results
summary(siall)