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Fitting a defect non-linear model with or without prior, distinguishing nuclear reaction products as an example
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Applied Nuclear Physics. (Nuclear Reactions)
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Applied Nuclear Physics. (Nuclear Reactions)ORCID iD: 0000-0002-7595-8024
2017 (English)In: Review of Scientific Instruments, ISSN 0034-6748, E-ISSN 1089-7623, Vol. 88, article id 115114Article in journal (Refereed) Published
Abstract [en]

Fitting parametrized functions to data is important for many researchers and scientists. If the model is non-linear and/or defect, it is not trivial to do correctly and to include an adequate uncertainty analysis. This work presents how the Levenberg-Marquardt algorithm for non-linear generalized least squares fitting can be used with a prior distribution for the parameters, and how it can be combined with Gaussian processes to treat model defects. An example, where three peaks in a histogram are to be distinguished, is carefully studied. In particular, the probability r1 for a nuclear reaction to end up in one out of two overlapping peaks is studied. Synthetic data is used to investigate effects of linearizations and other assumptions. For perfect Gaussian peaks, it is seen that the estimated parameters are distributed close to the truth with good covariance estimates. This assumes that the method is applied correctly; for example, prior knowledge should be implemented using a prior distribution, and not by assuming that some parameters are perfectly known (if they are not). It is also important to update the data covariance matrix using the fit if the uncertainties depend on the expected value of the data (e.g., for Poisson counting statistics or relative uncertainties). If a model defect is added to the peaks, such that their shape is unknown, a fit which assumes perfect Gaussian peaks becomes unable to reproduce the data, and the results for r1 become biased. It is, however, seen that it is possible to treat the model defect with a Gaussian process with a covariance function tailored for the situation, with hyper-parameters determined by leave-one-out cross validation. The resulting estimates for r1 are virtually unbiased, and the uncertainty estimates agree very well with the underlying uncertainty.

Place, publisher, year, edition, pages
2017. Vol. 88, article id 115114
Keyword [en]
Non-linear fitting, Model defects, Gaussian processes, Uncertainty analysis
National Category
Subatomic Physics
Identifiers
URN: urn:nbn:se:uu:diva-326313DOI: 10.1063/1.4993697ISI: 000416780600071OAI: oai:DiVA.org:uu-326313DiVA, id: diva2:1120225
Available from: 2017-07-05 Created: 2017-07-05 Last updated: 2018-04-16Bibliographically approved
In thesis
1. Approaching well-founded comprehensive nuclear data uncertainties: Fitting imperfect models to imperfect data
Open this publication in new window or tab >>Approaching well-founded comprehensive nuclear data uncertainties: Fitting imperfect models to imperfect data
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Nuclear physics has a wide range of applications; e.g., low-carbon energy production, medical treatments, and non-proliferation of nuclear weapons. Nuclear data (ND) constitute necessary input to computations needed within all these applications.

This thesis considers uncertainties in ND and their propagation to applications such as ma- terial damage in nuclear reactors. TENDL is today the most comprehensive library of evaluated ND (a combination of experimental ND and physical models), and it contains uncertainty estimates for all nuclides it contains; however, TENDL relies on an automatized process which, so far, includes a few practical remedies which are not statistically well-founded. A longterm goal of the thesis is to provide methods which make these comprehensive uncertainties well-founded. One of the main topics of the thesis is an automatic construction of experimental covariances; at first by attempting to complete the available uncertainty information using a set of simple rules. The thesis also investigates using the distribution of the data; this yields promising results, and the two approaches may be combined in future work.

In one of the papers underlying the thesis, there are also manual analyses of experiments, for the thermal cross sections of Ni-59 (important for material damage). Based on this, uncertainty components in the experiments are sampled, resulting in a distribution of thermal cross sections. After being combined with other types of ND in a novel way, the distribution is propagated both to an application, and to an evaluated ND file, now part of the ND library JEFF 3.3.

The thesis also compares a set of different techniques used to fit models in ND evaluation. For example, it is quantified how sensitive different techniques are to a model defect, i.e., the inability of the model to reproduce the truth underlying the data. All techniques are affected, but techniques fitting model parameters directly (such as the primary method used for TENDL) are more sensitive to model defects. There are also advantages with these methods, such as physical consistency and the possibility to build up a framework such as that of TENDL.

The treatment of these model defects is another main topic of the thesis. To this end, two ways of using Gaussian processes (GPs) are studied, applied to quite different situations. First, the addition of a GP to the model is used to enable the fitting of arbitrarily shaped peaks in a histogram of data. This is shown to give a substantial improvement compared to if the peaks are assumed to be Gaussian (when they are not), both using synthetic and authentic data.

The other approach uses GPs to fit smoothly energy-dependent model parameters in an ND evaluation context. Such an approach would be relatively easy to incorporate into the TENDL framework, and ensures a certain level of physical consistency. It is used on a TALYS-like model with synthetic data, and clearly outperforms fits without the energy-dependent model parameters, showing that the method can provide a viable route to improved ND evaluation. As a proof of concept, it is also used with authentic TALYS, and with authentic data.

To conclude, the thesis takes significant steps towards well-founded comprehensive ND un- certainties.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. p. 119
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1669
Keyword
Evaluated nuclear data, uncertainty propagation, uncertainty quantification, model defects, Gaussian processes, TALYS, TENDL, covariances.
National Category
Subatomic Physics
Research subject
Physics with specialization in Applied Nuclear Physics
Identifiers
urn:nbn:se:uu:diva-348553 (URN)978-91-513-0334-5 (ISBN)
Public defence
2018-06-08, Häggsalen, Ångströmslaboratoriet, Lägerhyddsv. 1, Uppsala, 09:15 (English)
Opponent
Supervisors
Available from: 2018-05-17 Created: 2018-04-16 Last updated: 2018-05-17

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Helgesson, PetterSjöstrand, Henrik

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