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Wikipedia / Propagation of uncertainty

In statistics, propagation of uncertainty is the effect of variables' uncertainties 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 due to the combination of variables in the function.

The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error . Uncertainties can also be defined by the relative error , which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, , which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval . 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 from the central value , which means that the region 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.

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. For very large datasets or complex functions, the calculation of the error propagation may be very expensive so that a surrogate model or a parallel computing strategy may be necessary.

In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.

Linear combinations

be a set of m

Also let the variance–covariance matrix of is the outer product.

of f

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x 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 is a diagonal matrix 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

can be expressed in terms of the correlation coefficient, so that an alternative expression for the variance of f

In the case that the variables in x

, the result is the standard error of the mean

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 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. Or in matrix notation 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, Aki and Akj. In matrix notation,

.

Simplification

Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:, and so forth.

are small enough. Specifically, the linear approximation.

Example

Any non-linear differentiable function If we take the variance on both sides and use the formula for the variance of a linear combination.

In the particular case that .

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(1+x) increases as x increases, since the expansion to x is a good approximation only when x is near zero.

For highly non-linear functions, there exist five categories of probabilistic approaches for uncertainty propagation; see Uncertainty quantification for details.

Reciprocal and shifted reciprocal

follows a standard normal distribution, the resulting distribution is a reciprocal standard normal distribution, and there is no definable variance.

following a general normal distribution, then mean and variance statistics do exist in a principal value is real-valued.

Ratios

Ratios are also problematic; normal approximations exist under certain conditions.

Example formulae

covariance

are expectation values is the value of the function calculated at those values.

we also have Goodman's expression

Effect of correlation on differences

If A and B are uncorrelated, their difference A − B) will decrease the variance of the difference, converging to zero variance for perfectly correlated variables with the same variance) will further increase the variance of the difference, compared to the uncorrelated case.

For example, the self-subtraction f = A − A only if the variate is perfectly autocorrelated). If A And if A for f = aA − aA in the table above).

Example calculations

Inverse tangent function

We can calculate the uncertainty propagation for the inverse tangent function as an example of using partial derivatives to propagate error.

is the absolute uncertainty on our measurement of . The derivative of with respect to

is the absolute propagated uncertainty.

Resistance measurement

A practical application is an experiment in which one measures current, , and voltage, , on a resistor in order to determine the resistance, , using Ohm's law, .

Given the measured variables with uncertainties, and , and neglecting their possible correlation, the uncertainty in the computed quantity, , is:

See also

References

Further reading