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Propagation of Uncertainty from Random Error00:59

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
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The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...
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Scientists always try their best to record measurements with the utmost accuracy and precision. However, sometimes errors do occur. These errors can be random or systematic. Random errors are observed due to the inconsistency or fluctuation in the measurement process, or variations in the quantity itself that is being measured. Such errors fluctuate from being greater than or less than the true value in repeated measurements. Consider a scientist measuring the length of an earthworm using a...
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Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...
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The Wald-Wolfowitz runs test, commonly referred to as the runs test, is a nonparametric test used to assess the randomness of ordered data. The test evaluates the number of runs, which are consecutive sequences of similar elements within the data. If the number of runs is significantly higher or lower than expected, the data is considered non-random, indicating a detectable pattern or structure.
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Estimating the Complete Basis Set Extrapolation Error through Random Walks.

Jakub Lang1, Michał Przybytek1, Michał Lesiuk1

  • 1University of Warsaw, Faculty of Chemistry, Pasteura 1, 02-093 Warsaw, Poland.

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Summary
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We developed a new method to estimate uncertainty in complete basis set extrapolation. This approach uses random walks to simulate outcomes, providing reliable and conservative error bounds for scientific results.

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Area of Science:

  • Computational chemistry
  • Quantum chemistry

Background:

  • Extrapolation to the complete basis set (CBS) limit is crucial for accurate quantum chemical calculations.
  • Estimating the uncertainty associated with CBS extrapolation is challenging but essential for reliable predictions.

Purpose of the Study:

  • To propose a novel, parameter-free method for quantifying the uncertainty in results obtained via CBS extrapolation.
  • To provide a statistically rigorous framework for predicting error bounds at a desired confidence level.

Main Methods:

  • The method employs an ensemble of random walks to simulate all potential extrapolation outcomes.
  • Statistical analysis of the ensemble results enables uncertainty prediction.
  • The approach is designed to be compatible with any existing CBS extrapolation scheme.

Main Results:

  • Numerical trials demonstrate the reliability and tightness of the predicted error bounds.
  • The method provides conservative estimates, ensuring a high level of confidence in the uncertainty quantification.
  • The approach was validated by comparison with reliable reference data.

Conclusions:

  • The proposed random walk ensemble method offers a robust and versatile tool for uncertainty estimation in CBS extrapolation.
  • This technique enhances the reliability of computational chemistry results by providing accurate and conservative error bounds.
  • The parameter-free nature and compatibility with various schemes make it broadly applicable in scientific research.