In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function.
Contents
- Linear combinations
- Non linear combinations
- Simplification
- Example
- Caveats and warnings
- Reciprocal
- Shifted reciprocal
- Example formulas
- Inverse tangent function
- Resistance measurement
- References
The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. 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, the positive square root of variance, σ2. The value of a quantity and its error are then expressed as an interval x ± u. If the statistical probability distribution of the variable is known or can be assumed, 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.
If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.
Linear combinations
Let
and let the variance-covariance matrix on x be denoted by
Then, the variance-covariance matrix
or, in matrix notation:
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
where
The general expressions for a scalar-valued function, f, are a little simpler.
(where a is a row-vector).
Each covariance term,
In the case that the variables in x are uncorrelated this simplifies further to
In the simplest case of identical coefficients and variances, we find
Non-linear combinations
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 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
where J is the Jacobian matrix. 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, Aik and Ajk 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
Simplification
Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:
where
It is important to note that this formula is based on the linear characteristics of the gradient of
Example
Any non-linear differentiable function, f(a,b), of two variables, a and b, can be expanded as
hence:
In the particular case that
or
Caveats and warnings
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. For example, the bias on the error calculated for log x increases as x increases, since the expansion to 1 + x is a good approximation only when x is small.
For highly non-linear functions, there exist five categories of probabilistic approaches for uncertainty propagation; see Uncertainty Quantification#Methodologies for forward uncertainty propagation for details.
Reciprocal
In the special case of the inverse or reciprocal
Shifted reciprocal
The statistics, mean and variance, of the shifted reciprocal function
Example formulas
This table shows the variances of simple functions of the real variables
For uncorrelated variables (
In this case, expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation gives,
For the case
and therefore we have:
Inverse tangent function
We can calculate the uncertainty propagation for the inverse tangent function as an example of using partial derivatives to propagate error.
Define
where σx is the absolute uncertainty on our measurement of x. The derivative of f(x) with respect to x is
Therefore, our propagated uncertainty is
where σf is the absolute propagated uncertainty.
Resistance measurement
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.
Given the measured variables with uncertainties, I ± σI and V ± σV, and neglecting their possible correlation, the uncertainty in the computed quantity, σR is