6022
J. Phys. Chem. A 2010, 114, 6022–6032
Global Sensitivity Analysis for Systems with Independent and/or Correlated Inputs Genyuan Li and Herschel Rabitz* Department of Chemistry, Princeton UniVersity, Princeton, New Jersey 08544
Paul E. Yelvington,† Oluwayemisi O. Oluwole, Fred Bacon, and Charles E. Kolb Aerodyne Research, Inc., 45 Manning Road, Billerica, Massachusetts 01821
Jacqueline Schoendorf SPARTA, Inc., 39 Simon Street, Suite 15, Nashua, New Hampshire 03060 ReceiVed: October 9, 2009; ReVised Manuscript ReceiVed: February 24, 2010
The objective of a global sensitivity analysis is to rank the importance of the system inputs considering their uncertainty and the influence they have upon the uncertainty of the system output, typically over a large region of input space. This paper introduces a new unified framework of global sensitivity analysis for systems whose input probability distributions are independent and/or correlated. The new treatment is based on covariance decomposition of the unconditional variance of the output. The treatment can be applied to mathematical models, as well as to measured laboratory and field data. When the input probability distribution is correlated, three sensitivity indices give a full description, respectively, of the total, structural (reflecting the system structure) and correlative (reflecting the correlated input probability distribution) contributions for an input or a subset of inputs. The magnitudes of all three indices need to be considered in order to quantitatively determine the relative importance of the inputs acting either independently or collectively. For independent inputs, these indices reduce to a single index consistent with previous variance-based methods. The estimation of the sensitivity indices is based on a meta-modeling approach, specifically on the random sampling-high dimensional model representation (RS-HDMR). This approach is especially useful for the treatment of laboratory and field data where the input sampling is often uncontrolled. 1. Introduction Suppose that an input-output system structure is described by a deterministic relation
y ) f(x) ) f(x1, x2, ..., xn)
Y to the inputs X in terms of a reduction in the variance of Y
Si ) Vi /V(Y) ) Var[E(Y|Xi)]/Var(Y)
(2)
(1)
where xi denotes the ith input and y is the output. We will use upper-case letters, i.e., Xi, Y, when referring to the generic aspects of variables. Lower-case letters, i.e., xi, y, represent their observed values. Boldface, as X or x, is used to designate vectors. Sensitivity analysis is concerned with understanding how the system input variations influence the changes of the output. This is often motivated by the fact that there is uncertainty about the true values of the inputs used in a particular application. Thus, in sensitivity analysis, the Xi’s are formally treated as random variables with specified distributions, and, consequently, Y is also a random variable with a probability distribution. The characterization of the empirical output distribution, given the input probability distribution, is the goal of uncertainty analysis. The assessment of the relative importance of the inputs in the above relation is the objective of sensitivity analysis.1-3 Variance-based methods are commonly used4-7 in global sensitivity analysis for quantifying the sensitivity of the output * Corresponding author. Tel: +1-609-258-3917. Fax: +1-609-258-0967. E-mail:
[email protected]. † Current address: Mainstream Engineering Corporation, 200 Yellow Place, Rockledge, FL 32955.
Sij ) Vij /V(Y) ) (Var[E(Y|Xi, Xj)] - Vi - Vj)/Var(Y) ···
(3) where E( · ) and Var( · ) represent the expected value and variance; Si and Sij are referred to as the main and first-order interaction effects for Xi and Xi, Xj, respectively. These measures reflect the reduced portions of the output uncertainty caused by the inputs and their interactions when the true values of a subset of inputs Xp (where p is a subset of {1, 2, · · · , n}) are known. Moreover, the total effect of Xi is defined as
STi ) VTi /V(Y) ) (Var(Y) - Var[E(Y|X-i)])/Var(Y)
(4) where X-i indicates all inputs except Xi. STi is the ratio of the remaining uncertainty of the output to the unconditional output uncertainty V(Y) when the true values of all inputs except Xi are known. The variance-based methods are closely related to the decomposition of f(x) itself:5
10.1021/jp9096919 2010 American Chemical Society Published on Web 04/26/2010
Sensitivity Analysis of Independent/Correlated Inputs
J. Phys. Chem. A, Vol. 114, No. 19, 2010 6023
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