Importance measure of correlated variables in polynomial output

With the case of the quadratic polynomial outputs without cross-term, the correlated and uncorrected contributions by correlated input variables to variance of output response are derived analytically through the properties of the multi-dimensional correlated normal distribution and conditional distribution. The results of examples demonstrate that the derived analytical expressions are correct. The derived analytical expressions can be used directly in recognition of the contribution by input variables in quadratic or one-order polynomial output without cross terms and can be compared for other new algorithms. This method can also be extended to higher order polynomial with cross terms, to solve the recognition of contribution by input variables in more complicated outputs.