Taylor-series and Monte-Carlo-method uncertainty estimation of the width of a probability distribution based on varying bias and random error

Abstract

Uncertainties are typically assumed to be constant or a linear function of the measured value; however, this is generally not true. Particle image velocimetry (PIV) is one example of a measurement technique that has highly nonlinear, time varying local uncertainties. Traditional uncertainty methods are not adequate for the estimation of the uncertainty of measurement statistics (mean and variance) in the presence of nonlinear, time varying errors. Propagation of instantaneous uncertainty estimates into measured statistics is performed allowing accurate uncertainty quantification of time-mean and statistics of measurements such as PIV. It is shown that random errors will always elevate the measured variance, and thus turbulent statistics such as . Within this paper, nonlinear, time varying errors are propagated from instantaneous measurements into the measured mean and variance using the Taylor-series method. With these results and knowledge of the systematic and random uncertainty of each measurement, the uncertainty of the time-mean, the variance and covariance can be found. Applicability of the Taylor-series uncertainty equations to time varying systematic and random errors and asymmetric error distributions are demonstrated with Monte-Carlo simulations. The Taylor-series uncertainty estimates are always accurate for uncertainties on the mean quantity. The Taylor-series variance uncertainty is similar to the Monte-Carlo results for cases in which asymmetric random errors exist or the magnitude of the instantaneous variations in the random and systematic errors is near the ‘true’ variance. However, the Taylor-series method overpredicts the uncertainty in the variance as the instantaneous variations of systematic errors are large or are on the same order of magnitude as the ‘true’ variance.

Estimating model- and sampling-related uncertainty in large-area growth predictions

“…However, first-order approximations imply an additional source of error and provide estimates of the statistical moments, not a distribution (Dettinger and Wilson, 1981). In the presence of a complex system such as nonlinear and time-varying errors, traditional uncertainty methods based on algebra are not adequate for the corrected estimation (Wilson and Smith, 2013). As an alternative to approximations, Breidenbach et al (2014) and McRoberts and Westfall (2014) proposed the use of a bootstrap variance estimator.…”

WITHDRAWN: A software tool for calculating the uncertainty of diagnostic accuracy measures

“…The standard combined uncertainty u c of each diagnostic accuracy measure is calculated by applying the rules of uncertainty propagation from the input values to the calculated diagnostic accuracy measure, according to GUM, with a first order Taylor series approximation to uncertainty propagation (16).…”

Uncertainty propagation analysis by an extended sparse grid technique

“…A jþn yðX jþn Þ, (12) where yðX Þ is the integrand with integration interval ½a,b, wðX Þ is the weight function, and X i and A i denote the integration nodes and weights, respectively.…”

“…Some existing methods, namely, Monte Carlo simulation (MCS) [11][12][13], Taylor expansion-based method [14,15], reliability-based method [16][17][18], polynomial chaos expansion (PCE) [19][20][21] and numerical integration-based method [1,22,23] have already contributed to UP. The MCS method was first used in simulating a neutron chain reaction.…”