Analysis of magnetic field fluctuation thermometry using Bayesian inference

Wübbeler, Gerd; Schmähling, Franko; Beyer, J.; Engert, J.; Elster, Clemens

doi:10.1088/0957-0233/23/12/125004

Cited by 6 publications

(14 citation statements)

References 16 publications

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“…N. We assume that model (1) provides an adequate description of the S i . The product of independent Erlang distributions is assumed to model the observations, cf [3,8], and hence the likelihood is given by…”

Section: Problem Specification and Assumptionsmentioning

confidence: 99%

“…The terms w 2 i ( θ)/M cal , on the other hand, may be viewed as a type A evaluation of uncertainty as they are estimates for the variance of the according sampling distribution, obtained from the data. The fact that w 2 i ( θ)/M cal is a variance estimate follows directly from var(S cal,i ) = T 2 cal w 2 i (θ)/M cal (cf [3]). The matrix (5) thus combines uncertainties that have been obtained from type A and type B evaluation of uncertainty, as proposed by the GUM.…”

Section: Problem Specification and Assumptionsmentioning

confidence: 99%

“…Numerical calculations were carried out to validate the simplified analyses given in sections 3 and 4, and to explore the cases in which they can be safely applied. By a successful validation we mean providing similar results as the Bayesian treatment given in [3]. In the following, we will denote the Bayesian temperature estimate by TB and its associated uncertainty by u( TB ); TB and u( TB ) are determined as the mean and square root of the variance of the marginalized posterior distribution for the temperature, respectively.…”

Section: Validationmentioning

confidence: 99%

“…The calibration step is required since from the observed noise power spectrum only the product, aT , can be inferred in model (1) but not both a and T . In [3], a Bayesian analysis procedure for magnetic field fluctuation thermometry has been developed. The advantage of the Bayesian procedure is that the information gained in the calibration is fully taken into account in the measurement stage, thereby providing an estimate of the temperature with an uncertainty that accounts for all uncertainties in the calibration and measurement stage (see, e.g., [4,5] for further reading on Bayesian statistics).…”

Section: Introductionmentioning

confidence: 99%

“…However, the Bayesian procedure requires rather extensive numerical efforts in terms of Markov chain Monte Carlo sampling [6]. Some familiarity with such techniques is needed, or is at least advantageous, when utilizing corresponding software such as that provided in [3]. For these reasons, a simplified analysis yielding results that are similar to those obtained by the Bayesian proceeding would be welcome, in particular for the measurement stage.…”

Section: Introductionmentioning

confidence: 99%

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Simplified evaluation of magnetic field fluctuation thermometry

Wübbeler¹,

Elster²

2013

Meas. Sci. Technol.

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Recently, a Bayesian approach has been proposed for evaluating magnetic field fluctuation thermometry measurements. The approach provides a coherent use of calibration results in the inference of the temperature from subsequent magnetic field fluctuation thermometry measurements. It does, however, rely on an extensive numerical effort. In this paper, we develop simplified parametric and non-parametric analysis schemes. For all approaches we derive analytic expressions for the resulting temperature estimates and their uncertainties. We assess the new approaches and show in particular that an easy to apply non-parametric analysis yields results which are in good agreement with those obtained by the Bayesian inference.

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Section: Problem Specification and Assumptionsmentioning

confidence: 99%

Section: Problem Specification and Assumptionsmentioning

confidence: 99%

Section: Validationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simplified evaluation of magnetic field fluctuation thermometry

Wübbeler¹,

Elster²

2013

Meas. Sci. Technol.

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A tutorial on Bayesian Normal linear regression

et al. 2015

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Regression is a common task in metrology and often applied to calibrate instruments, evaluate inter-laboratory comparisons or determine fundamental constants, for example. Yet, a regression model cannot be uniquely formulated as a measurement function, and consequently the Guide to the Expression of Uncertainty in Measurement (GUM) and its supplements are not applicable directly. Bayesian inference, however, is well suited to regression tasks, and has the advantage of accounting for additional a priori information, which typically robustifies analyses. Furthermore, it is anticipated that future revisions of the GUM shall also embrace the Bayesian view.Guidance on Bayesian inference for regression tasks is largely lacking in metrology. For linear regression models with Gaussian measurement errors this tutorial gives explicit guidance. Divided into three steps, the tutorial first illustrates how a priori knowledge, which is available from previous experiments, can be translated into prior distributions from a specific class. These prior distributions have the advantage of yielding analytical, closed form results, thus avoiding the need to apply numerical methods such as Markov Chain Monte Carlo. Secondly, formulas for the posterior results are given, explained and illustrated, and software implementations are provided. In the third step, Bayesian tools are used to assess the assumptions behind the suggested approach.These three steps (prior elicitation, posterior calculation, and robustness to prior uncertainty and model adequacy) are critical to Bayesian inference. The general guidance given here for Normal linear regression tasks is accompanied by a simple, but real-world, metrological example. The calibration of a flow device serves as a running example and illustrates the three steps. It is shown that prior knowledge from previous calibrations of the same sonic nozzle enables robust predictions even for extrapolations.

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Reference measurements of SQUID-based magnetic-field fluctuation thermometers

Beyer¹,

Schmidt²,

Engert³

et al. 2013

Supercond. Sci. Technol.

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The magnetic-field fluctuation thermometer (MFFT) is a noise thermometer that has been developed for thermometry in sub-Kelvin refrigerators. Meaningful MFFT operation requires a prior measurement at one known temperature in the temperature range of operation. From this reference measurement a set of calibration parameters is determined that characterizes the thermometer. Aiming at improving the measurement quality of the MFFT and extending its acceptance, we have developed a setup and procedure for reference measurements at temperatures directly traceable to the PLTS-2000. The reference measurements allow the assessment of methods to determine the uncertainty of a MFFT measurement. Temperature measurements with relative uncertainties of <1% can be obtained within measurement times of ≈10 s.

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Analysis of magnetic field fluctuation thermometry using Bayesian inference

Cited by 6 publications

References 16 publications

Simplified evaluation of magnetic field fluctuation thermometry

Simplified evaluation of magnetic field fluctuation thermometry

A tutorial on Bayesian Normal linear regression

Reference measurements of SQUID-based magnetic-field fluctuation thermometers

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