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Register a new userProspects of ratio and differential ({delta}) ratio based measurement-models: a case study for IRMS evaluation
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B. P. Datta
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The suitability of a mathematical-model Y = f({Xi}) in serving a purpose whatsoever (should be preset by the function f specific input-to-output variation-rates, i.e.) can be judged beforehand. We thus evaluate here the two apparently similar models: YA = fA(SRi,WRi) = (SRi/WRi) and: YD = fd(SRi,WRi) = ([SRi,WRi] - 1) = (YA - 1), with SRi and WRi representing certain measurable-variables (e.g. the sample S and the working-lab-reference W specific ith-isotopic-abundance-ratios, respectively, for a case as the isotope ratio mass spectrometry (IRMS)). The idea is to ascertain whether fD should represent a better model than fA, specifically, for the well-known IRMS evaluation. The study clarifies that fA and fD should really represent different model-families. For example, the possible variation, eA, of an absolute estimate as the yA (and/ or the risk of running a machine on the basis of the measurement-model fA) should be dictated by the possible Ri-measurement-variations (u_S and u_W) only: eA = (u_S + u_W); i.e., at worst: eA = 2ui. However, the variation, eD, of the corresponding differential (i.e. YD) estimate yd should largely be decided by SRi and WRi values: ed = 2(|m_i |x u_i) = (|m_i | x eA); with: mi = (SRi/[SRi - WRi]). Thus, any IRMS measurement (i.e. for which |SRi - WRi| is nearly zero is a requirement) should signify that |mi| tends to infinity. Clearly, yD should be less accurate than yA, and/ or even turn out to be highly erroneous (eD tends to infinity). Nevertheless, the evaluation as the absolute yA, and hence as the sample isotopic ratio Sri, is shown to be equivalent to our previously reported finding that the conversion of a D-estimate (here, yD) into Sri should help to improve the achievable output-accuracy and -comparability.
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In isotope ratio mass spectrometry (IRMS), any sample (S) measurement is performed as a relative-difference ((S/W)di) from a working-lab-reference (W), but the result is evaluated relative to a recommended-standard (D): (S/D)di. It is thus assumed that different source specific results ((S1/D)di, (S2/D)di) would represent their sources (S1, S2), and be accurately intercomparable. However, the assumption has never been checked. In this manuscript we carry out this task by considering a system as CO2+-IRMS. We present a model for a priori predicting output-uncertainty. Our study shows that scale-conversion, even with the aid of auxiliary-reference-standard(s) Ai(s), cannot make (S/D)di free from W; and the ((S/W)di,(A1/W)di,(A2/W)di) To (S/D)di conversion-formula normally used in the literature is invalid. Besides, the latter-relation has been worked out, which leads to e.g., fJ([(S/W)dJCO2pmp%],[(A1/W)dJCO2pmp%],[(A2/W)dJCO2pmp%]) = ((S/D)dJCO2pm4.5p%); whereas FJ([(S/W)dJCO2pmp%],[(A1/W)dJCO2pmp%]) = ((S/D)dJCO2pm1.2p%). That is, contrary to the general belief (Nature 1978, 271, 534), the scale-conversion by employing one than two Ai-standards should ensure (S/D)di to be more accurate. However, a more valuable finding is that the transformation of any d-estimate into its absolute value helps improve accuracy, or any reverse-process enhances uncertainty. Thus, equally accurate though the absolute-estimates of isotopic-CO2 and constituent-elemental-isotopic abundance-ratios could be, in contradistinction any differential-estimate is shown to be less accurate. Further, for S and D to be similar, any absolute estimate is shown to turn out nearly absolute accurate but any (S/D)d value as really absurd. That is, estimated source specific absolute values, rather than corresponding differential results, should really represent their sources, and/ or be closely intercomparable.
The evaluation of the error to be attributed to cut efficiencies is a common question in the practice of experimental particle physics. Specifically, the need to evaluate the efficiency of the cuts for background removal, when they are tested in a signal-free-background-only energy window, originates a statistical problem which finds its natural framework in the ample family of solutions for two classical, and closely related, questions, i.e. the determination of confidence intervals for the parameter of a binomial proportion and for the ratio of Poisson means. In this paper the problem is first addressed from the traditional perspective, and afterwards naturally evolved towards the introduction of non standard confidence intervals both for the binomial and Poisson cases; in particular, special emphasis is given to the intervals obtained through the application of the likelihood ratio ordering to the traditional Neyman prescription for the confidence limits determination. Due to their attractiveness in term of reduced length and of coverage properties, the new intervals are well suited as interesting alternative to the standard Clopper-Pearson PDG intervals.
A maximum likelihood method is used to deal with the combined estimation of multi-measurements of a branching ratio, where each result can be presented as an upper limit. The joint likelihood function is constructed using observed spectra of all measurements and the combined estimate of the branching ratio is obtained by maximizing the joint likelihood function. The Bayesian credible interval, or upper limit of the combined branching ratio, is given in cases both with and without inclusion of systematic error.
A method to include multiplicative systematic uncertainties into branching ratio limits was proposed by M. Convery. That solution used approximations which are not necessarily valid. This note provides a solution without approximations and compares the results.
Selection of the correct convergence angle is essential for achieving the highest resolution imaging in scanning transmission electron microscopy (STEM). Use of poor heuristics, such as Rayleighs quarter-phase rule, to assess probe quality and uncertainties in measurement of the aberration function result in incorrect selection of convergence angles and lower resolution. Here, we show that the Strehl ratio provides an accurate and efficient to calculate criteria for evaluating probe size for STEM. A convolutional neural network trained on the Strehl ratio is shown to outperform experienced microscopists at selecting a convergence angle from a single electron Ronchigram using simulated datasets. Generating tens of thousands of simulated Ronchigram examples, the network is trained to select convergence angles yielding probes on average 85% nearer to optimal size at millisecond speeds (0.02% human assessment time). Qualitative assessment on experimental Ronchigrams with intentionally introduced aberrations suggests that trends in the optimal convergence angle size are well modeled but high accuracy requires extensive training datasets. This near immediate assessment of Ronchigrams using the Strehl ratio and machine learning highlights a viable path toward rapid, automated alignment of aberration-corrected electron microscopes.
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