Propagation of Uncertainty - Linear Combinations

Linear Combinations

Let be a set of m functions which are linear combinations of variables with combination coefficients .

or

and let the variance-covariance matrix on x be denoted by .

\Sigma^x = \begin{pmatrix} \sigma^2_1 & \text{cov}_{12} & \text{cov}_{13} & \cdots \\ \text{cov}_{12} & \sigma^2_2 & \text{cov}_{23} & \cdots\\ \text{cov}_{13} & \text{cov}_{23} & \sigma^2_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}

Then, the variance-covariance matrix, of f is given by

.

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are un-correlated the general expression simplifies to

Note that even though the errors on x may be un-correlated, their errors on f are always correlated.

The general expressions for a single function, f, are a little simpler.

Each covariance term, can be expressed in terms of the correlation coefficient by, so that an alternative expression for the variance of f is

In the case that the variables x are uncorrelated this simplifies further to

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