For most applications of uncertainty analysis, it is very important to obtain not just the response surface for the outputs, but also the variance contribution from each uncertain input parameter. Such information can be used as screening criteria, as weighting factors to guide the refinement of a model, or as the basis for robust-design decisions.

From the polynomial chaos expansion of the model response variables, we have

(20–20)

the variance contribution from the uncertain parameters can be evaluated as

(20–21)

Since the polynomials are all orthogonal, the expected value of cross products becomes zero. For all of the special polynomials listed in Table 20.1: Summary of General Orthogonal Expansions , we also have

(20–22)

Therefore the variance can be simplified as

(20–23)

This simple relationship between the polynomial chaos expansion coefficients and the variance contribution from all of the uncertain parameters suggests that the contribution of each input parameter can be determined from the relevant terms in the variance calculation. Then all of the terms involving only account for the variance contribution of the input parameter , while all of the terms involving only account for the variance contribution of . For the cross terms involving both and , the variance contribution to the response variable is apportioned between and , respectively.

From the variance analysis, the relative importance of uncertain input parameters can be determined simply by their contribution to the variance of the model response variable. Therefore, for large and complicated physical models, the user can readily identify the primary sources of both simulation and approximation error. This information can be used to focus resources on reducing the distribution of the output predictions by narrowing the spread of certain inputs. In addition, as we iteratively improve the error tolerances through additional model runs to extend the polynomial expansions, the variance contributions of the parameters can be used to determine which parameters need to be refined and which do not. This approach will assure the most effective use of resources in solving a particular problem.