Editing Propagation of uncertainty (section)

==Linear combinations==
Let <math>\{f_k(x_1, x_2, \dots, x_n)\}</math> be a set of ''m'' functions, which are linear combinations of <math>n</math> variables <math>x_1, x_2, \dots, x_n</math> with combination coefficients <math>A_{k1}, A_{k2}, \dots,A_{kn}, (k = 1, \dots, m)</math>:
<math display="block">f_k = \sum_{i=1}^n A_{ki} x_i,</math>
or in matrix notation,
<math display="block">\mathbf{f} = \mathbf{A} \mathbf{x}.</math>

Also let the [[variance–covariance matrix]] of {{math|1=''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} be denoted by <math>\boldsymbol\Sigma^x</math> and let the mean value be denoted by <math>\boldsymbol{\mu}</math>:
<math display="block">\begin{align}
\boldsymbol\Sigma^x = \operatorname{E}[(\mathbf{x}-\boldsymbol\mu)\otimes (\mathbf{x}-\boldsymbol\mu)]
&=
\begin{pmatrix}
   \sigma^2_1 & \sigma_{12} & \sigma_{13} & \cdots \\
   \sigma_{21} & \sigma^2_2 & \sigma_{23} & \cdots\\
   \sigma_{31} & \sigma_{32} & \sigma^2_3 & \cdots \\
   \vdots & \vdots & \vdots & \ddots
\end{pmatrix} \\[1ex]
&=
\begin{pmatrix}
   {\Sigma}^x_{11} & {\Sigma}^x_{12} & {\Sigma}^x_{13} & \cdots \\
   {\Sigma}^x_{21} & {\Sigma}^x_{22} & {\Sigma}^x_{23} & \cdots \\
   {\Sigma}^x_{31} & {\Sigma}^x_{32} & {\Sigma}^x_{33} & \cdots \\
   \vdots & \vdots & \vdots & \ddots
\end{pmatrix}.
\end{align}
</math>
<math>\otimes</math> is the [[outer product]].

Then, the variance–covariance matrix <math>\boldsymbol\Sigma^f</math> of ''f'' is given by
<math display="block">\begin{align}
\boldsymbol\Sigma^f
&= \operatorname{E}\left[(\mathbf{f} - \operatorname{E}[\mathbf{f}]) \otimes (\mathbf{f} - \operatorname{E}[\mathbf{f}])\right]
= \operatorname{E}\left[\mathbf{A}(\mathbf{x}-\boldsymbol\mu) \otimes \mathbf{A}(\mathbf{x}-\boldsymbol\mu)\right] \\[1ex]
&= \mathbf{A} \operatorname{E}\left[(\mathbf{x}-\boldsymbol\mu) \otimes (\mathbf{x}-\boldsymbol\mu)\right] \mathbf{A}^\mathrm{T}
= \mathbf{A} \boldsymbol\Sigma^x \mathbf{A}^\mathrm{T}.
\end{align}</math>

In component notation, the equation
<math display="block">\boldsymbol\Sigma^f = \mathbf{A} \boldsymbol\Sigma^x \mathbf{A}^\mathrm{T}</math>
reads
<math display="block">\Sigma^f_{ij} = \sum_k^n \sum_l^n A_{ik} {\Sigma}^x_{kl} A_{jl}.</math>

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on ''x'' are uncorrelated, the general expression simplifies to
<math display="block">\Sigma^f_{ij} = \sum_k^n A_{ik} \Sigma^x_k A_{jk},</math>
where <math>\Sigma^x_k = \sigma^2_{x_k}</math> is the variance of ''k''-th element of the ''x'' vector.
Note that even though the errors on ''x'' may be uncorrelated, the errors on ''f'' are in general correlated; in other words, even if <math>\boldsymbol\Sigma^x</math> is a diagonal matrix, <math>\boldsymbol\Sigma^f</math> is in general a full matrix.

The general expressions for a scalar-valued function ''f'' are a little simpler (here '''a''' is a row vector):
<math display="block">f = \sum_i^n a_i x_i = \mathbf{a x},</math>
<math display="block">\sigma^2_f = \sum_i^n \sum_j^n a_i \Sigma^x_{ij} a_j = \mathbf{a} \boldsymbol\Sigma^x \mathbf{a}^\mathrm{T}.</math>

Each covariance term <math>\sigma_{ij}</math> can be expressed in terms of the [[Pearson product-moment correlation coefficient|correlation coefficient]] <math>\rho_{ij}</math> by <math>\sigma_{ij} = \rho_{ij} \sigma_i \sigma_j</math>, so that an alternative expression for the variance of ''f'' is
<math display="block">\sigma^2_f = \sum_i^n a_i^2 \sigma^2_i + \sum_i^n \sum_{j (j \ne i)}^n a_i a_j \rho_{ij} \sigma_i \sigma_j.</math>

In the case that the variables in ''x'' are uncorrelated, this simplifies further to
<math display="block">\sigma^2_f = \sum_i^n a_i^2 \sigma^2_i.</math>

In the simple case of identical coefficients and variances, we find
<math display="block">\sigma_f = \sqrt{n}\, |a| \sigma.</math>

For the arithmetic mean, <math>a=1/n</math>, the result is the [[standard error of the mean]]:
<math display="block">\sigma_f = \frac{\sigma} {\sqrt{n}}.</math>