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. 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 { f k ( x 1 , x 2 , … , x n ) } {displaystyle {f_{k}(x_{1},x_{2},dots ,x_{n})}} be a set of m functions which are linear combinations of n {displaystyle n} variables x 1 , x 2 , … , x n {displaystyle x_{1},x_{2},dots ,x_{n}} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 , … , m ) {displaystyle A_{k1},A_{k2},dots ,A_{kn},(k=1,dots ,m)} : Also let the variance-covariance matrix of x = (x1, ..., xn) be denoted by Σ x {displaystyle Sigma ^{x},} . Then, the variance-covariance matrix Σ f {displaystyle Sigma ^{f},} of f is given by