When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate due to the combination of variables in the function.
Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance.
However, the most general way of characterizing uncertainty is by specifying its probability distribution.
If the probability distribution of the variable is known or can be assumed, in theory it is possible to get any of its statistics.
In particular, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found.
For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases.
[1] In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the Monte Carlo method family.
[2] For very large datasets or complex functions, the calculation of the error propagation may be very expensive so that a surrogate model[3] or a parallel computing strategy[4][5][6] may be necessary.
In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures.
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
The general expressions for a scalar-valued function f are a little simpler (here a is a row vector):
In the simple case of identical coefficients and variances, we find
When f is a set of non-linear combination of the variables x, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables.
In a probabilistic approach, the function f must usually be linearised by approximation to a first-order Taylor series expansion, though in some cases, exact formulae can be derived that do not depend on the expansion as is the case for the exact variance of products.
denotes the partial derivative of fk with respect to the i-th variable, evaluated at the mean value of all components of vector x.
Since f0 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, Aki and Akj by the partial derivatives,
That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument.
Note this is equivalent to the matrix expression for the linear case with
Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:[9]
Error estimates for non-linear functions are biased on account of using a truncated series expansion.
The extent of this bias depends on the nature of the function.
[13] However, in the slightly more general case of a shifted reciprocal function
following a general normal distribution, then mean and variance statistics do exist in a principal value sense, if the difference between the pole
[14] Ratios are also problematic; normal approximations exist under certain conditions.
This table shows the variances and standard deviations of simple functions of the real variables
we also have Goodman's expression[7] for the exact variance: for the uncorrelated case it is
) will further increase the variance of the difference, compared to the uncorrelated case.
A practical application is an experiment in which one measures current, I, and voltage, V, on a resistor in order to determine the resistance, R, using Ohm's law, R = V / I.