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where the interval predictor model center line <math> \hat{y}_p(x) = (\overline{y}_p(x) + \underline{y}_p(x)) \times 1/2</math>, and the model width <math> h = (\overline{y}_p(x) - \underline{y}_p(x)) \times 1/2 </math>. This results in an IPM which makes predictions with homoscedastic uncertainty.
Sadeghi (2019) demonstrates that the non-convex scenario approach from Campi (2015) can be extended to train deeper neural networks which predict intervals with hetreoscedastic uncertainty on datasets with imprecision.<ref name="Sadeghi2019">{{cite journal|last1=Sadeghi|first1=Jonathan C.|last2=De Angelis|first2=Marco|last3=Patelli|first3=Edoardo|title=Efficient Training of Interval Neural Networks for Imprecise Training Data|year=2019|journal=Neural Networks|volume=118|pages=338–351|doi=10.1016/j.neunet.2019.07.005|pmid=31369950|s2cid=199383010 |url=https://strathprints.strath.ac.uk/71230/ |url-access=subscription}}</ref>
This is achieved by proposing generalizations to the max-error loss function given by
:<math>
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In Patelli (2017), Faes (2019), and Crespo (2018), Interval Predictor models were applied to the [[structural reliability]] analysis problem.<ref name="PatelliBroggi2017">{{cite book|last1=Patelli|first1=Edoardo|title=Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2017)|last2=Broggi|first2=Matteo|last3=Tolo|first3=Silvia|last4=Sadeghi|first4=Jonathan|year=2017|pages=212–224|doi=10.7712/120217.5364.16982|chapter=Cossan Software: A Multidisciplinary and Collaborative Software for Uncertainty Quantification|isbn=978-618-82844-4-9}}</ref>
<ref name="CrespoKenny2018"/>
<ref name="FaesSadeghi2019">{{cite journal|last1=Faes|first1=Matthias|last2=Sadeghi|first2=Jonathan|last3=Broggi|first3=Matteo|last4=De Angelis|first4=Marco|last5=Patelli|first5=Edoardo|last6=Beer|first6=Michael|last7=Moens|first7=David|title=On the robust estimation of small failure probabilities for strong non-linear models|journal=ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering|volume=5|issue=4|year=2019|issn=2332-9017|doi=10.1115/1.4044044|s2cid=197472507 |url=https://lirias.kuleuven.be/handle/123456789/633621 |url-access=subscription}}</ref>
Brandt (2017) applies interval predictor models to fatigue damage estimation of offshore wind turbines jacket substructures.<ref name="BrandtBroggi2017">{{cite journal|last1=Brandt|first1=Sebastian|last2=Broggi|first2=Matteo|last3=Hafele|first3=Jan|last4=Guillermo Gebhardt|first4=Cristian|last5=Rolfes|first5=Raimund|last6=Beer|first6=Michael|title=Meta-models for fatigue damage estimation of offshore wind turbines jacket substructures|journal=Procedia Engineering|volume=199|year=2017|pages=1158–1163|issn=1877-7058|doi=10.1016/j.proeng.2017.09.292|doi-access=free}}</ref>
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