Interval predictor model: Difference between revisions

Revision as of 23:54, 29 December 2021

Revision as of 13:33, 8 August 2022

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Interval Predictor Models are sometimes referred to as a [[nonparametric regression]] technique, because a potentially infinite set of functions are contained by the IPM, and no specific distribution is implied for the regressed variables.

Multiple-input multiple-output IPMs for multi-point data commonly used to represent functions have been recently developed.<ref>{{cite journal |last1=Crespo |first1=Luis |last2=Kenny |first2=Sean |last3=Colbert |first3=Brendon |last4=Slagel |first4=Tanner |title=Interval Predictor Models for Robust System Identification |journal=IEEE CDC 2021 |date=December 2021}}</ref> These IPM prescribe the parameters of the model as a path-connected, semi-algebraic set using sliced-normal <ref>{{cite journal |last1=Crespo |first1=Luis |last2=Colbert |first2=Brendon |last3=Kenny |first3=Sean |last4=Giesy |first4=Daniel |title=On the quantification of aleatory and epistemic uncertainty using Sliced-Normal distributions |journal=Systems and controlControl lettersLetters |date=2019 |volume=34 |page=104560 |doi=10.1016/j.sysconle.2019.104560 |s2cid=209339118 |url=https://doi.org/10.1016/j.sysconle.2019.104560}}</ref> or sliced-exponential distributions.<ref>{{cite journal |last1=Crespo |first1=Luis |last2=Colbert |first2=Brendon |last3=Slager |first3=Tanner |last4=Kenny |first4=Sean |title=Robust Estimation of Sliced-Exponential Distributions |journal=IEEE CDC |date=December 2021}}</ref> A key advantage of this approach is its ability to characterize complex parameter dependencies to varying fidelity levels. This practice enables the analyst to adjust the desired level of conservatism in the prediction.

As a consequence of the theory of [[scenario optimization]], in many cases rigorous predictions can be made regarding the performance of the model at test time.<ref name="CampiCalafiore2009">{{cite journal|last1=Campi|first1=M.C.|last2=Calafiore|first2=G.|last3=Garatti|first3=S.|title=Interval predictor models: Identification and reliability|journal=Automatica|volume=45|issue=2|year=2009|pages=382–392|issn=000510980005-1098|doi=10.1016/j.automatica.2008.09.004}}</ref>

Hence an interval predictor model can be seen as a guaranteed bound on [[quantile regression]].

Interval predictor models can also be seen as a way to prescribe the [[Support_(mathematics)#Support_of_a_distribution|support]] of random predictor models, of which a [[Gaussian process]] is a specific case

.<ref name="CrespoKenny2018">{{cite journal|last1=Crespo|first1=Luis G.|last2=Kenny|first2=Sean P.|last3=Giesy|first3=Daniel P.|title=Staircase predictor models for reliability and risk analysis|journal=Structural Safety|volume=75|year=2018|pages=35–44|issn=016747300167-4730|doi=10.1016/j.strusafe.2018.05.002|s2cid=126167977 }}</ref>

== Convex interval predictor models ==

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The reliability of such an IPM is obtained by noting that for a convex IPM the number of support constraints is less than the dimensionality of the [[trainable parameter]]s, and hence the scenario approach can be applied.

Lacerda (2017) demonstrated that this approach can be extended to situations where the training data is interval valued rather than point valued.<ref name="LacerdaCrespo2017">{{cite book|last1=Lacerda|first1=Marcio J.|title=2017 American Control Conference (ACC)|last2=Crespo|first2=Luis G.|chapter=Interval predictor models for data with measurement uncertainty|year=2017|pages=1487–1492|doi=10.23919/ACC.2017.7963163|isbn=978-1-5090-5992-8|hdl=2060/20170005690|s2cid=3713493 }}</ref>

== Non-convex interval predictor models ==

In Campi (2015) a non-convex theory of scenario optimization was proposed.<ref name="CampiGaratti2015">{{cite book|last1=Campi|first1=Marco C.|title=2015 54th IEEE Conference on Decision and Control (CDC)|last2=Garatti|first2=Simone|last3=Ramponi|first3=Federico A.|chapter=Non-convex scenario optimization with application to system identification|year=2015|pages=4023–4028|doi=10.1109/CDC.2015.7402845|isbn=978-1-4799-7886-1|s2cid=127406 }}</ref>

This involves measuring the number of support constraints, <math>S</math>, for the Interval Predictor Model after training and hence making predictions about the reliability of the model.

This enables non-convex IPMs to be created, such as a single layer neural network.

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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/ }}</ref>

This is achieved by proposing generalizations to the max-error loss function given by

:<math>

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== Applications ==

Initially, [[scenario optimization]] was applied to robust control problems.<ref name="CampiGaratti2009">{{cite journal|last1=Campi|first1=Marco C.|last2=Garatti|first2=Simone|last3=Prandini|first3=Maria|author3-link= Maria Prandini |title=The scenario approach for systems and control design|journal=Annual Reviews in Control|volume=33|issue=2|year=2009|pages=149–157|issn=136757881367-5788|doi=10.1016/j.arcontrol.2009.07.001}}</ref>

Crespo (2015) and (2021) applied Interval Predictor Models to the design of space radiation shielding <ref name="CrespoKenny2016a">{{cite book|last1=Crespo|first1=Luis G.|title=18th AIAA Non-Deterministic Approaches Conference|last2=Kenny|first2=Sean P.|last3=Giesy|first3=Daniel P.|last4=Norman|first4=Ryan B.|last5=Blattnig|first5=Steve|chapter=Application of Interval Predictor Models to Space Radiation Shielding|year=2016|doi=10.2514/6.2016-0431|isbn=978-1-62410-397-1|hdl=2060/20160007750|s2cid=124192684 }}</ref> and to system identification.<ref>{{cite journal |last1=Crespo |first1=Luis |last2=Kenny |first2=Sean |last3=Colbert |first3=Brendon |last4=Slagel |first4=Tanner |title=Interval Predictor Models for Robust System Identification |journal=IEEE CDC 2021 |date=December 2021}}</ref>

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 }}</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=187770581877-7058|doi=10.1016/j.proeng.2017.09.292|doi-access=free}}</ref>

Garatti (2019) proved that Chebyshev layers (i.e., the minimax layers around functions fitted by linear <math>\ell_\infty</math>-regression) belong to a particular class of Interval Predictor Models, for which the reliability is invariant with respect to the distribution of the data.<ref name="GarCamCar2019">{{cite journal|last1=Garatti|first1=S.|last2=Campi|first2=M.C.|last3=Carè|first3=A.|title=On a class of Interval Predictor Models with universal reliability|journal=Automatica|volume=110|year=2019|page=108542|issn=000510980005-1098|doi=10.1016/j.automatica.2019.108542|hdl=11311/1121161 |s2cid=204188183 }}</ref>