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In [[regression analysis]], an '''interval predictor model''' ('''IPM
This differs from other techniques in [[machine learning]], where usually one wishes to estimate point values or an entire probability distribution.
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.
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=00051098|doi=10.1016/j.automatica.2008.09.004}}</ref>. ▼
Multiple-input multiple-output IPMs for multi-point data commonly used to represent functions have been recently developed.<ref>{{cite book | doi=10.1109/CDC45484.2021.9683582 | chapter=Interval Predictor Models for Robust System Identification | title=2021 60th IEEE Conference on Decision and Control (CDC) | year=2021 | last1=Crespo | first1=Luis G. | last2=Kenny | first2=Sean P. | last3=Colbert | first3=Brendon K. | last4=Slagel | first4=Tanner | pages=872–879 | isbn=978-1-6654-3659-5 | s2cid=246479771 }}</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 Control Letters |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|url-access=subscription }}</ref> or sliced-exponential distributions.<ref>{{cite book | doi=10.1109/CDC45484.2021.9683584 | chapter=Robust Estimation of Sliced-Exponential Distributions<sup>⋆</sup> | title=2021 60th IEEE Conference on Decision and Control (CDC) | year=2021 | last1=Crespo | first1=Luis G. | last2=Colbert | first2=Brendon K. | last3=Slager | first3=Tanner | last4=Kenny | first4=Sean P. | pages=6742–6748 | isbn=978-1-6654-3659-5 | s2cid=246476974 }}</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
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=
== Convex interval predictor models ==
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Usually the basis is made up of polynomial features or a radial basis is sometimes used.
Then a convex set is assigned to the parameter vector, and the size of the convex set is minimized such that every possible data point can be predicted by one possible value of the parameters.
Ellipsoidal parameters sets were used by Campi (2009), which yield a convex optimization program to train the IPM
Crespo (2016) proposed the use of a hyperrectangular parameter set, which results in a convenient, linear form for the bounds of the IPM
Hence the IPM can be trained with a linear optimization program:
:<math>
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</math>
where the training data examples are <math> y^{(i)}</math> and <math> x^{(i)}</math>, and the Interval Predictor Model bounds <math> \underline{y}_p(x)</math> and <math>\overline{y}_p(x) </math> are parameterised by the parameter vector <math> p </math>.
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
Lacerda (2017) demonstrated that this approach can be extended to situations where the training data is interval valued rather than point valued
== Non-convex interval predictor models ==
In Campi (2015) a non-convex theory of scenario optimization was proposed
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.
Campi (2015) demonstrates that an algorithm where the scenario optimization program is only solved <math>S</math> times which can determine the reliability of the model at test time without a prior evaluation on a validation set
This is achieved by solving the optimisation program
:<math>
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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
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=
Crespo (2015) and (2021) applied Interval Predictor Models to the design of space radiation shielding
In Patelli (2017), Faes (2019), and Crespo (2018), Interval Predictor models were applied to the [[structural reliability]] analysis problem.<ref name="PatelliBroggi2017">{{cite
<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
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=
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=0005-1098|doi=10.1016/j.automatica.2019.108542|hdl=11311/1121161 |s2cid=204188183 |hdl-access=free}}</ref>
== Software implementations ==
== References ==
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