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Thus, even when using arguably the simplest nonlinear function, the square of a random variable, the process of finding the mean and variance of the derived quantity is difficult, and for more complicated functions it is safe to say that this process is not practical for experimental data analysis.
Simulation-based methods: Monte Carlo simulations, importance sampling, adaptive sampling, etc. General surrogate-based methods: In a non-instrusive approach, a surrogate model is learnt in order to replace the experiment or the simulation with a cheap and fast approximation. Surrogate-based methods can also be employed in a fully Bayesian fashion.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
In physical experiments uncertainty analysis, or experimental uncertainty assessment, deals with assessing the uncertainty in a measurement.An experiment designed to determine an effect, demonstrate a law, or estimate the numerical value of a physical variable will be affected by errors due to instrumentation, methodology, presence of confounding effects and so on.
Variance-based methods [27] are a class of probabilistic approaches which quantify the input and output uncertainties as random variables, represented via their probability distributions, and decompose the output variance into parts attributable to input variables and combinations of variables. The sensitivity of the output to an input variable ...
Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions.
The Hartley function only depends on the number of elements in a set, and hence can be viewed as a function on natural numbers. Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that satisfies = + (additivity)
Consider a set of data points, (,), (,), …, (,), and a curve (model function) ^ = (,), that in addition to the variable also depends on parameters, = (,, …,), with . It is desired to find the vector of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares = = is minimized, where the residuals (in-sample prediction errors) r i are ...