Propagation of uncertainty
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 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 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.
Let be a set of m functions which are linear combinations of variables with combination coefficients :
Also let the variance-covariance matrix of x = (x1, ..., xn) be denoted by .
Then, the variance-covariance matrix of f is given by
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 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 is a diagonal matrix, is in general a full matrix.
The general expressions for a scalar-valued function, f, are a little simpler:
(where a is a row vector).
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 in x are uncorrelated this simplifies further to
In the simplest 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 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 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 by the partial derivatives, and . In matrix notation,
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:
where represents the standard deviation of the function , represents the standard deviation of , represents the standard deviation of , and so forth.
It is important to note that this formula is based on the linear characteristics of the gradient of and therefore it is a good estimation for the standard deviation of as long as are small enough. Specifically, the linear approximation of has to be close to inside a neighborhood of radius .
Any non-linear differentiable function, , of two variables, and , can be expanded as
where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and is the covariance between and .
In the particular case that , . Then
where is the correlation between and .
When the variables and are uncorrelated, . Then
Caveats and warningsEdit
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#Methodologies for forward uncertainty propagation for details.
In the special case of the inverse or reciprocal , where , the distribution is a reciprocal normal distribution, and there is no definable variance. For such inverse distributions and for ratio distributions, there can be defined probabilities for intervals, which can be computed either by Monte Carlo simulation or, in some cases, by using the Geary–Hinkley transformation.
The statistics, mean and variance, of the shifted reciprocal function for , however, exist in a principal value sense if the difference between the shift or pole and the mean is real. The mean of this transformed random variable is then indeed the scaled Dawson's function . In contrast, if the shift is purely complex, the mean exists and is a scaled Faddeeva function, whose exact expression depends on the sign of the imaginary part, . In both cases, the variance is a simple function of the mean. Therefore, the variance has to be considered in a principal value sense if is real, while it exists if the imaginary part of is non-zero. Note that these means and variances are exact, as they do not recur to linearisation of the ratio. The exact covariance of two ratios with a pair of different poles and is similarly available. The case of the inverse of a complex normal variable , shifted or not, exhibits different characteristics.
This table shows the variances of simple functions of the real variables , with standard deviations covariance and exactly known real-valued constants (i.e., ).
For uncorrelated variables ( ) the covariance terms are also zero, as .
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 we also have Goodman's expression for the exact variance: for the uncorrelated case it is
and therefore we have:
Inverse tangent functionEdit
We can calculate the uncertainty propagation for the inverse tangent function as an example of using partial derivatives to propagate error.
where 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 is the absolute propagated uncertainty.
Given the measured variables with uncertainties, I ± σI and V ± σV, and neglecting their possible correlation, the uncertainty in the computed quantity, σR, is:
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