  Uncertainties exist in both physics-based and data-driven models.
Variance-based sensitivity analysis characterizes how the variance of a model
output is propagated from the model inputs. The Sobol index is one of the most
widely used sensitivity indices for models with independent inputs. For models
with dependent inputs, different approaches have been explored to obtain
sensitivity indices in the literature. Typical approaches are based on
procedures of transforming the dependent inputs into independent inputs.
However, such transformation requires additional information about the inputs,
such as the dependency structure or the conditional probability density
functions. In this paper, data-driven sensitivity indices are proposed for
models with dependent inputs. We first construct ordered partitions of linearly
independent polynomials of the inputs. The modified Gram-Schmidt algorithm is
then applied to the ordered partitions to generate orthogonal polynomials with
respect to the empirical measure based on observed data of model inputs and
outputs. Using the polynomial chaos expansion with the orthogonal polynomials,
we obtain the proposed data-driven sensitivity indices. The sensitivity indices
provide intuitive interpretations of how the dependent inputs affect the
variance of the output without a priori knowledge on the dependence structure
of the inputs. Three numerical examples are used to validate the proposed
approach.
