Propagation of uncertainty
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. Mainly, the variables are measured in an experiment, and have uncertainties due to measurement limitations (e.g. instrument precision) which propagate to the result.
The uncertainty is usually defined by the absolute error — a variable that is probable to get the values x±Δx is said to have an uncertainty (or margin of error) of Δx. In other words, for a measured value x, it is probable that the true value lies in the interval [x−Δx, x+Δx]. Uncertainties can also be defined by the relative error Δx/x, which is usually written as a percentage. In many cases it is assumed that the difference between a measured value and the true value is normally distributed, with the mean being the true value and the standard deviation used as the uncertainty of the measurement (although this means the true value will only sit within the error bar 68% of the time).
This article explains how to calculate the uncertainty of a function if the variables' uncertainties are known.
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General formula
Let f(x1,x2,...,xn) be a function which depends on n variables x1,x2,...,xn. The uncertainty of each variable is given by Δxj:
If the variables are uncorrelated, we can calculate the uncertainty Δf of f that results from the uncertainties of the variables:
where designates the partial derivative of f for the j-th variable.
If the variables are correlated, the covariance between variable pairs, Ci,k := cov(xi,xk), enters the formula with a double sum over all pairs (i,k):
where
After calculating Δf, we can say that the value of the function with its uncertainty is:
In terms of variances, .
This comes from calculating the variance of a first-order Taylor series expansion of f.
In some examples, exact formulas can be derived that do not depend on the Taylor expansion [1].
Caveats and warnings
The general formulas given here rely on the convergence of the Taylor series of f. In some cases this cannot be assumed and the formulas will be grossly in error. In particular, the formula in the case of the ratio of two variables A / B (a ratio distribution) will be undefined in the case where the probability of | B | − > 0 is finite. In the special case of the inverse 1 / B where B = N(0,1), the distribution is a Cauchy distribution and there is no definable variance. In such cases, there can be defined probabilities for intervals which can be defined either by Monte Carlo simulation, or, in some cases, by using the Geary-Hinkel transformation [2].
Example formulas
This table shows the uncertainty of simple functions, resulting from uncorrelated, normally-distributed real variables with uncertainties
, and precisely-known real-valued constants
and
.
(Note that some references use to mean the variance of
. In this table,
is the variance of
.)
-
Function Uncertainty
Partial derivatives
Given
-
Absolute Error Variance [3]
Example calculation: 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
- f(θ) = arctanθ,
where σθ is the absolute uncertainty on our measurement of θ.
The partial derivative of f(θ) with respect to θ is
.
Therefore, our propagated uncertainty is
,
where σf is the absolute propagated uncertainty.
Example application: 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±ΔI and V±ΔV, the uncertainty in the computed quantity, ΔR is
Thus, in this simple case, the relative error ΔR/R is simply the square root of the sum of the squares of the two relative errors of the measured variables.
Notes
- ^ (1960). "On the Exact Variance of Products". Journal of the American Statistical Association 55 (292): 708-713.
- ^ Jack Hayya, and (July 1975). "A Note on the Ratio of Two Normally Distributed Variables". 21 (11): 1338-1341.
- ^ [Lindberg] (2000-07-01). Uncertainties and Error Propagation (eng). Uncertainties, Graphing, and the Vernier Caliper 1. Rochester Institute of Technology. Archived from the original on 2004-11-12. Retrieved on 2007-04-20. “The guiding principle in all cases is to consider the most pessimistic situation.”
External links
- Uncertainties and Error Propagation, Appendix V from the Mechanics Lab Manual, Case Western Reserve University.
- A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and monte carlo simulations instead of simple significance arithmetic.
- Uncertainty calculator can be used to calculate propagated error