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  • 1
    Publication Date: 2017-07-17
    Print ISSN: 0017-467X
    Electronic ISSN: 1745-6584
    Topics: Energy, Environment Protection, Nuclear Power Engineering , Geosciences
    Published by Wiley
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  • 2
    Publication Date: 2020-10-24
    Print ISSN: 0043-1397
    Electronic ISSN: 1944-7973
    Topics: Architecture, Civil Engineering, Surveying , Geography
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  • 3
    Publication Date: 2020-01-21
    Description: Model averaging makes it possible to use multiple models for one modelling task, like predicting a certain quantity of interest. Several Bayesian approaches exist that all yield a weighted average of predictive distributions. However, often, they are not properly applied which can lead to false conclusions. In this study, we focus on Bayesian Model Selection (BMS) and Averaging (BMA), Pseudo-BMS/BMA and Bayesian Stacking. We want to foster their proper use by, first, clarifying their theoretical background and, second, contrasting their behaviours in an applied groundwater modelling task. We show that only Bayesian Stacking has the goal of model averaging for improved predictions by model combination. The other approaches pursue the quest of finding a single best model as the ultimate goal, and use model averaging only as a preliminary stage to prevent rash model choice. Improved predictions are thereby not guaranteed. In accordance with so-called M -settings that clarify the alleged relations between models and truth, we elicit which method is most promising.
    Electronic ISSN: 2073-4441
    Topics: Energy, Environment Protection, Nuclear Power Engineering
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  • 4
    Publication Date: 2018-08-13
    Description: When constructing discrete (binned) distributions from samples of a data set, applications exist where it is desirable to assure that all bins of the sample distribution have nonzero probability. For example, if the sample distribution is part of a predictive model for which we require returning a response for the entire codomain, or if we use Kullback–Leibler divergence to measure the (dis-)agreement of the sample distribution and the original distribution of the variable, which, in the described case, is inconveniently infinite. Several sample-based distribution estimators exist which assure nonzero bin probability, such as adding one counter to each zero-probability bin of the sample histogram, adding a small probability to the sample pdf, smoothing methods such as Kernel-density smoothing, or Bayesian approaches based on the Dirichlet and Multinomial distribution. Here, we suggest and test an approach based on the Clopper–Pearson method, which makes use of the binominal distribution. Based on the sample distribution, confidence intervals for bin-occupation probability are calculated. The mean of each confidence interval is a strictly positive estimator of the true bin-occupation probability and is convergent with increasing sample size. For small samples, it converges towards a uniform distribution, i.e., the method effectively applies a maximum entropy approach. We apply this nonzero method and four alternative sample-based distribution estimators to a range of typical distributions (uniform, Dirac, normal, multimodal, and irregular) and measure the effect with Kullback–Leibler divergence. While the performance of each method strongly depends on the distribution type it is applied to, on average, and especially for small sample sizes, the nonzero, the simple “add one counter”, and the Bayesian Dirichlet-multinomial model show very similar behavior and perform best. We conclude that, when estimating distributions without an a priori idea of their shape, applying one of these methods is favorable.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 5
    Publication Date: 2016-11-17
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 6
    Publication Date: 2018-07-01
    Print ISSN: 0309-1708
    Electronic ISSN: 1872-9657
    Topics: Geography , Geosciences
    Published by Elsevier
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  • 7
    Publication Date: 2021-10-22
    Print ISSN: 0043-1397
    Electronic ISSN: 1944-7973
    Topics: Architecture, Civil Engineering, Surveying , Geography
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  • 8
    Publication Date: 2021-07-04
    Description: For simulating reactive transport on aquifer scale, various modeling approaches have been proposed. They vary considerably in their computational demands and in the amount of data needed for their calibration. Typically, the more complex a model is, the more data are required to sufficiently constrain its parameters. In this study, we assess a set of five models that simulate aerobic respiration and denitrification in a heterogeneous aquifer at quasi steady state. In a probabilistic framework, we test whether simplified approaches can be used as alternatives to the most detailed model. The simplifications are achieved by neglecting processes such as dispersion or biomass dynamics, or by replacing spatial discretization with travel‐time‐based coordinates. We use the model justifiability analysis proposed by Schöniger, Illman, et al. (2015, https://doi.org/10.1016/j.jhydrol.2015.07.047) to determine how similar the simplified models are to the reference model. This analysis rests on the principles of Bayesian model selection and performs a tradeoff between goodness‐of‐fit to reference data and model complexity, which is important for the reliability of predictions. Results show that, in principle, the simplified models are able to reproduce the predictions of the reference model in the considered scenario. Yet, it became evident that it can be challenging to define appropriate ranges for effective parameters of simplified models. This issue can lead to overly wide predictive distributions, which counteract the apparent simplicity of the models. We found that performing the justifiability analysis on the case of model simplification is an objective and comprehensive approach to assess the suitability of candidate models with different levels of detail.
    Description: Plain Language Summary: In groundwater, chemical substances like nitrate are transported and undergo chemical reactions. Understanding such reactive transport processes plays a key role in securing our water resources and drinking water. We use computer models for understanding such reactive transport processes and for simulating their future behavior. In such models, we make many scientific decisions on which processes should be included and in what degree of detail. Here, we face a trade‐off: Usually, a complex model with many mathematical terms resolves many details of the process. Yet, such complex models require lots of data for calibration and lots of time for the computer simulation. In contrast, a simple model with fewer details comes with less effort in both respects. However, it might neglect important parts of the process. For the example of nitrate decay, we use a probabilistic approach to find the best simplification for a comparatively detailed reference model. Our results show that, in certain cases, it is justified to employ a simpler model instead of a complex alternative without deteriorating modeling results. Alongside, we explain how difficult it can be to define realistic parameter ranges for simplified models.
    Description: Key Points: We compare a set of four simplified models against a reference model for reactive transport at quasi steady state on aquifer scale. A Bayesian model justifiability analysis helps identifying the most suitable model simplification strategy. The proposed analysis reveals the difficulty of reasonably constraining parameter priors for simplified models.
    Description: DFG http://dx.doi.org/10.13039/501100001659
    Keywords: 551.49 ; conceptual uncertainty ; reactive transport ; Bayesian model comparison ; model complexity
    Type: article
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  • 9
    Publication Date: 2022-08-09
    Description: Deterministic hydrological models with uncertain, but inferred‐to‐be‐time‐invariant parameters typically show time‐dependent model errors. Such errors can occur if a hydrological process is active in certain time periods in nature, but is not resolved by the model or by its input. Such missing processes could become visible during calibration as time‐dependent best‐fit values of model parameters. We propose a formal time‐windowed Bayesian analysis to diagnose this type of model error, formalizing the question “In which period of the calibration time‐series does the model statistically disqualify itself as quasi‐true?” Using Bayesian model evidence (BME) as model performance metric, we determine how much the data in time windows of the calibration time‐series support or refute the model. Then, we track BME over sliding time windows to obtain a dynamic, time‐windowed BME (tBME) and search for sudden decreases that indicate an onset of model error. tBME also allows us to perform a formal, sliding likelihood‐ratio test of the model against the data. Our proposed approach is designed to detect error occurrence on various temporal scales, which is especially useful in hydrological modeling. We illustrate this by applying our proposed method to soil moisture modeling. We test tBME as model error indicator on several synthetic and real‐world test cases that we designed to vary in error sources (structure and input) and error time scales. Results prove the successful detection errors in dynamic models. Moreover, the time sequence of posterior parameter distributions helps to investigate the reasons for model error and provide guidance for model improvement.
    Description: Key Points: We propose a data‐driven method for model‐structural error detection. Our method rests on a statistically rigorous Bayesian framework without prior assumptions about error sources or patterns. We confirm successful error detection on various temporal scales in synthetic test cases and present insights from a real‐world case study.
    Description: German Research Foundation (DFG)
    Description: Cluster of Excellence
    Description: University of Stuttgart
    Description: https://doi.org/10.18419/darus-1836
    Keywords: ddc:550.285
    Language: English
    Type: doc-type:article
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