Propagation of Uncertainty - Non-linear Combinations

Non-linear Combinations

When f is a set of non-linear combination of the variables x, it must usually be linearized by approximation to a first-order Taylor series expansion, though in some cases, exact formulas can be derived that do not depend on the expansion as is the case for the exact variance of products. The Taylor expansion would be:

where denotes the partial derivative of f_k with respect to the i-th variable. Or in matrix notation,

where J is the Jacobian matrix. Since f0_k is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, A_ik and A_jk by the partial derivatives, and . In matrix notation,

.

That is, the Jacobian of the function is used to transform the rows and columns of the covariance of the argument.

Nonetheless, the most common formula among engineers and experimental scientist to calculate error propagation for independent variables is the one proposed by the NIST:

It is important to note that this formula is based on the linear characteristics of the gradient of and therefore it is a good estimation for the standard deviation of as long as are small compared to the partial derivatives.