Uncertainty estimation with a small number of measurements, part I: new insights on the<i>t</i>-interval method and its limitations

Huang, Hening

doi:10.1088/1361-6501/aa96c7

Cited by 7 publications

(9 citation statements)

References 17 publications

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“…Deviations increase to values of >150% as the number of stations decreases (<5) (Figures 3 and S9). The significant increase in the range of deviation could be explained by problems with the t ‐test and resampling techniques in the presence of small (<20) or very small sample sizes (<5), as well as an unknown standard deviation (Huang, 2017). An alternative approach to estimating small sample size uncertainty could be to use well‐constrained uncertainties from independent estimates of larger sample sizes and applying their constraints (Steele et al, 1993).…”

Section: Discussionmentioning

confidence: 99%

Using a Large‐n Seismic Array to Explore the Robustness of Spectral Estimations

Kemna

Castro

Harrington

et al. 2020

Geophysical Research Letters

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Spectral analysis is widely used to estimate and refine earthquake source parameters such as source radius, seismic moment, and stress drop. This study aims to quantify the precision of the single spectra and empirical Green's function spectral ratio approach using the Large-n Seismic Survey in Oklahoma (LASSO) array. The dense station coverage in an area of local saltwater disposal offers a unique opportunity to observe and quantify radiation pattern effects and subsequent precision of spectral estimates of small earthquakes (M < 3). The results suggest that the precision of source properties estimated from direct phase arrivals for arrays with less than 20 stations should be assumed to be not less than 30% and could be as high as 150% if less than five stations are used. Furthermore, we do not see clear evidence for, or against, a scaling of stress drop with magnitude of small earthquakes (M < 3) as observed by other studies. Plain Language Summary Seismologists use ground motion recordings of seismic waves (seismograms) to infer details about the earthquake rupture process. While large earthquakes often generate a physical imprint on the earth's surface through surface rupture, small earthquakes can often only be studied from seismograms. Nevertheless, small earthquakes are of particular interest to learning about the rupture process for many reasons. For example, they are much more numerous than larger magnitude earthquakes and might rupture by the same physical process(es). While seismic arrays are usually restricted to a few to tens of stations, here we use a very large seismic array with >1,800 temporary stations to study small earthquakes and how station resolution may bias source property estimates. Source properties include the physical size of the rupture surface and the corresponding slip on that surface, which relate to the amount of stress released by the earthquake. The large number of stations allows us to estimate the source properties in unique detail and test the variability in measurements using different numbers of stations to estimate the precision. We find that the estimation of source properties is highly biased when using a small number of stations (<20), which should be taken under consideration in future studies.

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Section: Discussionmentioning

confidence: 99%

Using a Large‐n Seismic Array to Explore the Robustness of Spectral Estimations

Kemna

Castro

Harrington

et al. 2020

Geophysical Research Letters

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“…This paper (Part II of a series of two papers) continues our discussion in Part I (Huang 2017) on uncertainty estimation with n observations from a normally distributed quantity. Again, the conventional approach to estimating the uncertainty of the sample mean (taken as the measured value) is (1) if the population standard deviation σ is known or n ⩾ 30, use the z-based uncertainty (e.g.…”

Section: Introductionmentioning

confidence: 68%

“…It is not valid for the t-based uncertainty because the t-based uncertainty is not a probabilistic error bound and is not even a realistic estimate of the probabilistic error bound for ultra-small samples. Recall that the z-interval actually belongs to the classical theory of errors as discussed in Part I (Huang 2017). The z-interval is best interpreted with the concept of errors, although it can also be interpreted with the theory of confidence intervals.…”

Section: Misuse Of the T-interval In Uncertainty Estimationmentioning

confidence: 99%

Uncertainty estimation with a small number of measurements, part II: a redefinition of uncertainty and an estimator method

Huang¹

2017

Meas. Sci. Technol.

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This paper is the second (Part II) in a series of two papers (Part I and Part II). Part I has quantitatively discussed the fundamental limitations of the t-interval method for uncertainty estimation with a small number of measurements. This paper (Part II) reveals that the t-interval is an ‘exact’ answer to a wrong question; it is actually misused in uncertainty estimation. This paper proposes a redefinition of uncertainty, based on the classical theory of errors and the theory of point estimation, and a modification of the conventional approach to estimating measurement uncertainty. It also presents an asymptotic procedure for estimating the z-interval. The proposed modification is to replace the t-based uncertainty with an uncertainty estimator (mean- or median-unbiased). The uncertainty estimator method is an approximate answer to the right question to uncertainty estimation. The modified approach provides realistic estimates of uncertainty, regardless of whether the population standard deviation is known or unknown, or if the sample size is small or large. As an application example of the modified approach, this paper presents a resolution to the Du–Yang paradox (i.e. Paradox 2), one of the three paradoxes caused by the misuse of the t-interval in uncertainty estimation.

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“…It is known that the t-interval does not perform well when N is small and the data are skewed (see Huang 2017;Meeden 1999). Therefore, the assumptions of the t-interval were reasonably met in the first scenario, where the underlying population was Normal, but less reasonably met in the second scenario, where the population was right-skewed and N was small.…”

Section: Simulation Resultsmentioning

confidence: 99%

The Importance of Discussing Assumptions when Teaching Bootstrapping

Totty¹,

Molyneux²,

Fuentes³

2021

Preprint

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Bootstrapping and other resampling methods are progressively appearing in the textbooks and curricula of courses that introduce undergraduate students to statistical methods. Though bootstrapping may have more relaxed assumptions than its traditional counterparts, it is in no way assumption-free. Students and instructors of these courses need to be well-informed about the assumptions of bootstrapping. This article details some of the assumptions and conditions that instructors should be aware of when teaching the simple bootstrap for uncertainty quantification and hypothesis testing. We emphasize the importance of these assumptions and conditions using simulations which investigate the performance of these methods when they are or are not reasonably met. We also discuss software options for introducing undergraduate students to these bootstrap methods, including a newly developed R package. Supplementary materials for the article are available online.

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Uncertainty estimation with a small number of measurements, part I: new insights on thet-interval method and its limitations

Cited by 7 publications

References 17 publications

Using a Large‐n Seismic Array to Explore the Robustness of Spectral Estimations

Using a Large‐n Seismic Array to Explore the Robustness of Spectral Estimations

Uncertainty estimation with a small number of measurements, part II: a redefinition of uncertainty and an estimator method

The Importance of Discussing Assumptions when Teaching Bootstrapping

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