68:
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:
331:
960:
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
67:
Typically the interval predictor model is created by specifying a parametric function, which is usually chosen to be the product of a parameter vector and a basis. 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
543:, 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
39:
Multiple-input multiple-output IPMs for multi-point data commonly used to represent functions have been recently developed. These IPM prescribe the parameters of the model as a path-connected, semi-algebraic set using sliced-normal or sliced-exponential distributions. A key advantage of this
74:
731:
1079:
857:
955:
326:{\displaystyle \operatorname {arg\,min} _{p}\left\{\mathbb {E} _{x}({\bar {y}}_{p}(x)-{\underline {y}}_{p}(x)):{\bar {y}}_{p}(x^{(i)})>y^{(i)}>{\underline {y}}_{p}(x^{(i)}),i=1,\ldots ,N\right\}}
40:
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.
569:
443:
486:
1134:
47:, 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
1673:
Patelli, Edoardo; Broggi, Matteo; Tolo, Silvia; Sadeghi, Jonathan (2017). "Cossan
Software: A Multidisciplinary and Collaborative Software for Uncertainty Quantification".
563:
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
400:
367:
1580:
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P.; Norman, Ryan B.; Blattnig, Steve (2016). "Application of
Interval Predictor Models to Space Radiation Shielding".
967:
739:
561:
541:
506:
1136:-regression) belong to a particular class of Interval Predictor Models, for which the reliability is invariant with respect to the distribution of the data.
36:
technique, because a potentially infinite set of functions are contained by the IPM, and no specific distribution is implied for the regressed variables.
508:. 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
32:, where usually one wishes to estimate point values or an entire probability distribution. Interval Predictor Models are sometimes referred to as a
862:
515:
Lacerda (2017) demonstrated that this approach can be extended to situations where the training data is interval valued rather than point valued.
1106:
analysis problem. Brandt (2017) applies interval predictor models to fatigue damage estimation of offshore wind turbines jacket substructures.
1690:
1646:
1605:
1470:
1453:
Campi, Marco C.; Garatti, Simone; Ramponi, Federico A. (2015). "Non-convex scenario optimization with application to system identification".
1426:
1258:
1178:
1629:
Crespo, Luis G.; Kenny, Sean P.; Colbert, Brendon K.; Slagel, Tanner (2021). "Interval
Predictor Models for Robust System Identification".
1161:
Crespo, Luis G.; Kenny, Sean P.; Colbert, Brendon K.; Slagel, Tanner (2021). "Interval
Predictor Models for Robust System Identification".
1675:
Proceedings of the 2nd
International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2017)
1241:
Crespo, Luis G.; Colbert, Brendon K.; Slager, Tanner; Kenny, Sean P. (2021). "Robust
Estimation of Sliced-Exponential Distributions".
1099:
Crespo (2015) and (2021) applied
Interval Predictor Models to the design of space radiation shielding and to system identification.
523:
In Campi (2015) a non-convex theory of scenario optimization was proposed. This involves measuring the number of support constraints,
28:) is an approach to regression where bounds on the function to be approximated are obtained. This differs from other techniques in
1706:
Faes, Matthias; Sadeghi, Jonathan; Broggi, Matteo; De
Angelis, Marco; Patelli, Edoardo; Beer, Michael; Moens, David (2019).
1366:
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P. (2016). "Interval
Predictor Models With a Linear Parameter Dependency".
1323:
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P. (2018). "Staircase predictor models for reliability and risk analysis".
1853:
1753:
Brandt, Sebastian; Broggi, Matteo; Hafele, Jan; Guillermo
Gebhardt, Cristian; Rolfes, Raimund; Beer, Michael (2017).
726:{\displaystyle \operatorname {arg\,min} _{p}\left\{h:|{\hat {y}}_{p}(x^{(i)})-y^{(i)}|<h,i=1,\ldots ,N\right\},}
1794:
Garatti, S.; Campi, M.C.; Carè, A. (2019). "On a class of Interval Predictor Models with universal reliability".
1401:
Lacerda, Marcio J.; Crespo, Luis G. (2017). "Interval predictor models for data with measurement uncertainty".
33:
1285:
Campi, M.C.; Calafiore, G.; Garatti, S. (2009). "Interval predictor models: Identification and reliability".
405:
448:
1103:
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Garatti (2019) proved that Chebyshev layers (i.e., the minimax layers around functions fitted by linear
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In Patelli (2017), Faes (2019), and Crespo (2018), Interval Predictor models were applied to the
1712:
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering
1204:"On the quantification of aleatory and epistemic uncertainty using Sliced-Normal distributions"
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1074:{\displaystyle {\mathcal {L}}_{\text{max-error}}=\max _{i}|y^{(i)}-{\hat {y}}_{p}(x^{(i)})|,}
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852:{\displaystyle {\hat {y}}_{p}(x)=({\overline {y}}_{p}(x)+{\underline {y}}_{p}(x))\times 1/2}
56:
29:
1755:"Meta-models for fatigue damage estimation of offshore wind turbines jacket substructures"
1542:
546:
526:
491:
1708:"On the robust estimation of small failure probabilities for strong non-linear models"
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which is equivalent to solving the optimisation program proposed by Campi (2015).
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1754:
1511:
1203:
957:. This results in an IPM which makes predictions with homoscedastic uncertainty.
1682:
1410:
1825:
1780:
1731:
1566:
1462:
1387:
1344:
1306:
1597:
1496:"Efficient Training of Interval Neural Networks for Imprecise Training Data"
1418:
950:{\displaystyle h=({\overline {y}}_{p}(x)-{\underline {y}}_{p}(x))\times 1/2}
1519:
1495:
1144:
OpenCOSSAN provides a Matlab implementation of the work of Crespo (2015).
1816:
1589:
51:. Interval predictor models can also be seen as a way to prescribe the
1723:
1379:
1202:
Crespo, Luis; Colbert, Brendon; Kenny, Sean; Giesy, Daniel (2019).
1494:
Sadeghi, Jonathan C.; De Angelis, Marco; Patelli, Edoardo (2019).
1368:
Journal of Verification, Validation and Uncertainty Quantification
1545:(2009). "The scenario approach for systems and control design".
974:
1631:
2021 60th IEEE Conference on Decision and Control (CDC)
1455:
2015 54th IEEE Conference on Decision and Control (CDC)
1243:
2021 60th IEEE Conference on Decision and Control (CDC)
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2021 60th IEEE Conference on Decision and Control (CDC)
1115:
970:
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742:
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512:, and hence the scenario approach can be applied.
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361:
325:
1582:18th AIAA Non-Deterministic Approaches Conference
1318:
1316:
989:
736:where the interval predictor model center line
8:
488:are parameterised by the parameter vector
402:, and the Interval Predictor Model bounds
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1096:was applied to robust control problems.
1153:
438:{\displaystyle {\underline {y}}_{p}(x)}
55:of random predictor models, of which a
1403:2017 American Control Conference (ACC)
481:{\displaystyle {\overline {y}}_{p}(x)}
336:where the training data examples are
7:
519:Non-convex interval predictor models
1541:Campi, Marco C.; Garatti, Simone;
1121:
591:
588:
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43:As a consequence of the theory of
14:
1808:10.1016/j.automatica.2019.108542
1299:10.1016/j.automatica.2008.09.004
63:Convex interval predictor models
1559:10.1016/j.arcontrol.2009.07.001
1129:{\displaystyle \ell _{\infty }}
1337:10.1016/j.strusafe.2018.05.002
1220:10.1016/j.sysconle.2019.104560
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1:
1639:10.1109/CDC45484.2021.9683582
1251:10.1109/CDC45484.2021.9683584
1171:10.1109/CDC45484.2021.9683582
1772:10.1016/j.proeng.2017.09.292
1512:10.1016/j.neunet.2019.07.005
881:
783:
458:
1208:Systems and Control Letters
1870:
1683:10.7712/120217.5364.16982
1547:Annual Reviews in Control
1411:10.23919/ACC.2017.7963163
1463:10.1109/CDC.2015.7402845
1140:Software implementations
34:nonparametric regression
22:interval predictor model
395:{\displaystyle x^{(i)}}
362:{\displaystyle y^{(i)}}
1457:. pp. 4023–4028.
1405:. pp. 1487–1492.
1245:. pp. 6742–6748.
1130:
1104:structural reliability
1075:
951:
859:, and the model width
853:
727:
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59:is a specific case .
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1094:scenario optimization
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45:scenario optimization
1759:Procedia Engineering
1677:. pp. 212–224.
1633:. pp. 872–879.
1165:. pp. 872–879.
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968:
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510:trainable parameters
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1854:Regression analysis
1590:10.2514/6.2016-0431
49:quantile regression
18:regression analysis
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1724:10.1115/1.4044044
1692:978-618-82844-4-9
1648:978-1-6654-3659-5
1607:978-1-62410-397-1
1472:978-1-4799-7886-1
1428:978-1-5090-5992-8
1380:10.1115/1.4032070
1325:Structural Safety
1260:978-1-6654-3659-5
1180:978-1-6654-3659-5
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556:{\displaystyle S}
536:{\displaystyle S}
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30:machine learning
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1553:(2): 149–157.
1533:
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1312:
1293:(2): 382–392.
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1813:
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1765:: 1158–1163.
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1374:(2): 021007.
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1796:Automatica
1287:Automatica
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1148:References
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1826:0005-1098
1781:1877-7058
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1732:2332-9017
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