Non Linear Error Propagation
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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 propagation of error division are the values of experimental measurements they have uncertainties due to measurement
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limitations (e.g., instrument precision) which propagate to the combination of variables in the function. The uncertainty u error propagation physics 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
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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, error propagation square root 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 ± one standard deviation from the value, that is, there is approximately a 68% probability that the true value lies in the region x ± σ. 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.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Resistance measurement 5 See also 6 References 7 Further reading 8 External links Linear combinations[edit] Let { f k ( x 1 , x 2 , … , x n ) } {
of errors is essential to be able to judge the relevance of observed trends. Below, a brief definition of the main concepts and a discussion of generic ways to obtain error estimates is provided. [1] [2] Of course, any particular measuring device generally requires specific
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techniques. Contents 1 The measurement process 2 Calibration 3 Error estimate (experimental error known) 4
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Systematic and random errors 5 Error estimate (experimental error unknown) 6 Test of statistical validity of the model 7 Fluctuations and noise 8 error propagation calculus Non-Gaussian statistics 9 Integrated data analysis 10 Summary 11 References The measurement process The measuring device performs measurements on a physical system P. As a result, it produces estimates of a set of physical parameters {p}. One https://en.wikipedia.org/wiki/Propagation_of_uncertainty may think of p as loose numbers (e.g., a confinement time), data along a spatial chord at a single time (e.g., a Thomson scattering profile), data at a point in space with time resolution (e.g., magnetic field fluctuations from a Mirnov coil), or data having both time and space resolution (e.g., tomographic data from Soft X-Ray arrays). The actual measurement hardware does not deliver the parameters {p} directly, but produces a set of numbers {s}, http://wiki.fusenet.eu/wiki/Error_propagation usually expressed in Volts, Amperes, or pixels. Calibration The first task of the experimentalist is to translate the measured signals {s} into the corresponding physical parameters {p}. The second task is to provide error estimates (discussed below). Generally, the translation of {s} into {p} requires having a (basic) model for the experiment studied and its interaction with the measuring device. In the simplest cases, the relation between {s} and {p} is linear (e.g. conversion of the measured voltages from Mirnov coils to magnetic fields). Taking s and p to be vectors, such a conversion can be written as $ p = A \cdot(s - b), $ where A is a (possibly diagonal) calibration matrix and b a vector for offset correction. However, in fusion science it is more common that the conversion from s to p involves some (non-linear) numerical modelling of the physical (and measurement) system. In this case, rather than assuming a linear relation, one assumes a non-linear map Mp between s and p: p = Mp(s). The subscript p indicates that Mp may depend on p. In principle, determining p from s now requires an iterative numerical approach. The map Mp should be tested to check that it is not ill-conditioned (i.e. small variations in s produce large variations in p), since that would render the measurements useless; ill-conditioning leads t
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