{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T13:23:09Z","timestamp":1752672189992,"version":"3.37.3"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,11,1]],"date-time":"2022-11-01T00:00:00Z","timestamp":1667260800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,11,1]],"date-time":"2022-11-01T00:00:00Z","timestamp":1667260800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Physikalisch-Technische Bundesanstalt (PTB)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Process Lett"],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A Bayesian treatment of deep learning allows for the computation of uncertainties associated with the predictions of deep neural networks. We show how the concept of Errors-in-Variables can be used in Bayesian deep regression to also account for the uncertainty associated with the input of the employed neural network. The presented approach thereby exploits a relevant, but generally overlooked, source of uncertainty and yields a decomposition of the predictive uncertainty into an aleatoric and epistemic part that is more complete and, in many cases, more consistent from a statistical perspective. We discuss the approach along various simulated and real examples and observe that using an Errors-in-Variables model leads to an increase in the uncertainty while preserving the prediction performance of models without Errors-in-Variables. For examples with known regression function we observe that this ground truth is substantially better covered by the Errors-in-Variables model, indicating that the presented approach leads to a more reliable uncertainty estimation.<\/jats:p>","DOI":"10.1007\/s11063-022-11066-3","type":"journal-article","created":{"date-parts":[[2022,11,1]],"date-time":"2022-11-01T11:02:53Z","timestamp":1667300573000},"page":"4799-4818","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Aleatoric Uncertainty for Errors-in-Variables Models in Deep Regression"],"prefix":"10.1007","volume":"55","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5066-7661","authenticated-orcid":false,"given":"J.","family":"Martin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Elster","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,1]]},"reference":[{"key":"11066_CR1","doi-asserted-by":"crossref","unstructured":"Bassu D, Lo JT, Nave J (1999) Training recurrent neural networks with noisy input measurements. In: IJCNN\u201999. International joint conference on neural networks. Proceedings (Cat. No. 99CH36339), vol 1, pp 359\u2013363. IEEE","DOI":"10.1109\/IJCNN.1999.831519"},{"key":"11066_CR2","unstructured":"Blundell C, Cornebise J, Kavukcuoglu K, Wierstra D (2015) Weight uncertainty in neural network. In: International conference on machine learning, pp 1613\u20131622. PMLR"},{"key":"11066_CR3","unstructured":"Depeweg S, Hernandez-Lobato J-M, Doshi-Velez F, Udluft S (2018) Decomposition of uncertainty in bayesian deep learning for efficient and risk-sensitive learning. In: International conference on machine learning, pp 1184\u20131193. PMLR"},{"key":"11066_CR4","unstructured":"Duvenaud D, Maclaurin D, Adams R (2016) Early stopping as nonparametric variational inference. In: Artificial intelligence and statistics, pp 1070\u20131077. PMLR"},{"key":"11066_CR5","volume-title":"Measurement error models","author":"WA Fuller","year":"2009","unstructured":"Fuller WA (2009) Measurement error models, vol 305. John Wiley and Sons, New Jersey"},{"key":"11066_CR6","unstructured":"Gal Y, Ghahramani Z (2016) Dropout as a bayesian approximation: representing model uncertainty in deep learning. In: international conference on machine learning, pp 1050\u20131059. PMLR"},{"key":"11066_CR7","unstructured":"Gal Y, Hron J, Kendall A (2017) Concrete dropout. arXiv preprintarXiv:1705.07832"},{"key":"11066_CR8","unstructured":"Gillard J (2006) An historical overview of linear regression with errors in both variables. Math. School, Cardiff Univ., Wales, UK, Tech. Rep"},{"issue":"3","key":"11066_CR9","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1002\/rob.21918","volume":"37","author":"S Grigorescu","year":"2020","unstructured":"Grigorescu S, Trasnea B, Cocias T, Macesanu G (2020) A survey of deep learning techniques for autonomous driving. J Field Robot 37(3):362\u2013386","journal-title":"J Field Robot"},{"key":"11066_CR10","doi-asserted-by":"crossref","unstructured":"Gustafsson FK, Danelljan M, Schon TB (2020) Evaluating scalable bayesian deep learning methods for robust computer vision. In: Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition workshops, pp 318\u2013319","DOI":"10.1109\/CVPRW50498.2020.00167"},{"key":"11066_CR11","unstructured":"Hern\u00e1ndez-Lobato JM, Adams R (2015) Probabilistic backpropagation for scalable learning of bayesian neural networks. In: International conference on machine learning, pp 1861\u20131869. PMLR"},{"issue":"3","key":"11066_CR12","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ac0495","volume":"2","author":"L Hoffmann","year":"2021","unstructured":"Hoffmann L, Fortmeier I, Elster C (2021) Uncertainty quantification by ensemble learning for computational optical form measurements. Mach Learn Sci Technol 2(3):035030","journal-title":"Mach Learn Sci Technol"},{"key":"11066_CR13","doi-asserted-by":"crossref","unstructured":"Huang Y, Chen Y (2020) Survey of state-of-art autonomous driving technologies with deep learning. In: 2020 IEEE 20th international conference on software quality, reliability and security companion (QRS-C), pp 221\u2013228. IEEE","DOI":"10.1109\/QRS-C51114.2020.00045"},{"issue":"3","key":"11066_CR14","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/s10994-021-05946-3","volume":"110","author":"E H\u00fcllermeier","year":"2021","unstructured":"H\u00fcllermeier E, Waegeman W (2021) Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods. Mach Learn 110(3):457\u2013506","journal-title":"Mach Learn"},{"key":"11066_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-14596-5","volume-title":"Deep learning for NLP and speech recognition","author":"U Kamath","year":"2019","unstructured":"Kamath U, Liu J, Whitaker J (2019) Deep learning for NLP and speech recognition, vol 84. Springer, Berlin"},{"key":"11066_CR16","unstructured":"Kendall A, Gal Y (2017) What uncertainties do we need in bayesian deep learning for computer vision? arXiv preprintarXiv:1703.04977"},{"key":"11066_CR17","unstructured":"Kingma DP, Salimans T, Welling M (2015) Variational dropout and the local reparameterization trick. arXiv preprint arXiv:1506.02557"},{"key":"11066_CR18","unstructured":"Kingma DP, Welling M (2013) Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114"},{"key":"11066_CR19","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/j.media.2016.07.007","volume":"35","author":"T Kooi","year":"2017","unstructured":"Kooi T, Litjens G, Van Ginneken B, Gubern-M\u00e9rida A, S\u00e1nchez CI, Mann R, den Heeten A, Karssemeijer N (2017) Large scale deep learning for computer aided detection of mammographic lesions. Med Image Anal 35:303\u2013312","journal-title":"Med Image Anal"},{"key":"11066_CR20","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.ejmp.2019.05.008","volume":"62","author":"T Kretz","year":"2019","unstructured":"Kretz T, Anton M, Schaeffter T, Elster C (2019) Determination of contrast-detail curves in mammography image quality assessment by a parametric model observer. Phys Med 62:120\u2013128","journal-title":"Phys Med"},{"key":"11066_CR21","unstructured":"Lakshminarayanan B, Pritzel A, Blundell C (2016) Simple and scalable predictive uncertainty estimation using deep ensembles. arXiv preprint arXiv:1612.01474"},{"issue":"33","key":"11066_CR22","doi-asserted-by":"publisher","first-page":"747","DOI":"10.21105\/joss.00747","volume":"4","author":"A LeNail","year":"2019","unstructured":"LeNail A (2019) Nn-svg: publication-ready neural network architecture schematics. J Open Source Softw 4(33):747","journal-title":"J Open Source Softw"},{"key":"11066_CR23","doi-asserted-by":"publisher","unstructured":"Li Z, Li S, Bamasag OO, Alhothali A, Luo X (2022) Diversified regularization enhanced training for effective manipulator calibration. IEEE Trans Neural Netw Learn Syst 1\u201313. https:\/\/doi.org\/10.1109\/TNNLS.2022.3153039","DOI":"10.1109\/TNNLS.2022.3153039"},{"issue":"1","key":"11066_CR24","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1109\/JAS.2020.1003381","volume":"8","author":"Z Li","year":"2021","unstructured":"Li Z, Li S, Luo X (2021) An overview of calibration technology of industrial robots. IEEE\/CAA J Autom Sin 8(1):23\u201336","journal-title":"IEEE\/CAA J Autom Sin"},{"key":"11066_CR25","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.media.2017.07.005","volume":"42","author":"G Litjens","year":"2017","unstructured":"Litjens G, Kooi T, Bejnordi BE, Setio AAA, Ciompi F, Ghafoorian M, Van Der Laak JA, Van Ginneken B, S\u00e1nchez CI (2017) A survey on deep learning in medical image analysis. Med Image Anal 42:60\u201388","journal-title":"Med Image Anal"},{"issue":"2","key":"11066_CR26","doi-asserted-by":"publisher","first-page":"3153","DOI":"10.1109\/LRA.2020.2974682","volume":"5","author":"A Loquercio","year":"2020","unstructured":"Loquercio A, Segu M, Scaramuzza D (2020) A general framework for uncertainty estimation in deep learning. IEEE Robot Autom Lett 5(2):3153\u20133160","journal-title":"IEEE Robot Autom Lett"},{"issue":"11","key":"11066_CR27","doi-asserted-by":"publisher","first-page":"5931","DOI":"10.1109\/TII.2019.2909142","volume":"15","author":"H Lu","year":"2019","unstructured":"Lu H, Jin L, Luo X, Liao B, Guo D, Xiao L (2019) Rnn for solving perturbed time-varying underdetermined linear system with double bound limits on residual errors and state variables. IEEE Trans Industr Inf 15(11):5931\u20135942","journal-title":"IEEE Trans Industr Inf"},{"key":"11066_CR28","unstructured":"Maddox WJ, Izmailov P, Garipov T, Vetrov DP, Wilson AG (2019) A simple baseline for bayesian uncertainty in deep learning. In: Wallach H, Larochelle H, Beygelzimer A, d\u2019Alch\u00e9-Buc F, Fox E, Garnett R (eds) Advances in neural information processing systems, vol 32. Curran Associates, Inc. https:\/\/proceedings.neurips.cc\/paper\/2019\/file\/118921efba23fc329e6560b27861f0c2-Paper.pdf"},{"issue":"4","key":"11066_CR29","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6501\/ab094b","volume":"30","author":"J Martin","year":"2019","unstructured":"Martin J, Bartl G, Elster C (2019) Application of bayesian model averaging to the determination of thermal expansion of single-crystal silicon. Meas Sci Technol 30(4):045012","journal-title":"Meas Sci Technol"},{"key":"11066_CR30","doi-asserted-by":"crossref","unstructured":"McAllister R, Gal Y, Kendall A, Van Der\u00a0Wilk M, Shah A, Cipolla R, Weller A (2017) Concrete problems for autonomous vehicle safety: advantages of bayesian deep learning. In: International joint conferences on artificial intelligence, Inc","DOI":"10.24963\/ijcai.2017\/661"},{"issue":"2","key":"11066_CR31","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1109\/TNNLS.2020.2979670","volume":"32","author":"DW Otter","year":"2020","unstructured":"Otter DW, Medina JR, Kalita JK (2020) A survey of the usages of deep learning for natural language processing. IEEE Trans Neural Netw Learn Syst 32(2):604\u2013624","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"issue":"3","key":"11066_CR32","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/S0167-7152(96)00140-X","volume":"33","author":"RK Pace","year":"1997","unstructured":"Pace RK, Barry R (1997) Sparse spatial autoregressions. Stat Probab Lett 33(3):291\u2013297","journal-title":"Stat Probab Lett"},{"issue":"10","key":"11066_CR33","doi-asserted-by":"publisher","first-page":"10K102","DOI":"10.1063\/1.5039286","volume":"89","author":"A. Pavone, J. Svensson, A. Langenberg, N. Pablant, U. Hoefel, S. Kwak, R. Wolf, W. -X. Team","year":"2018","unstructured":"A. Pavone, J. Svensson, A. Langenberg, N. Pablant, U. Hoefel, S. Kwak, R. Wolf, W. -X. Team (2018) Bayesian uncertainty calculation in neural network inference of ion and electron temperature profiles at w7\u2013x. Rev Sci Instrum 89(10):10K102","journal-title":"Rev Sci Instrum"},{"issue":"16","key":"11066_CR34","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1080\/01691864.2017.1365009","volume":"31","author":"HA Pierson","year":"2017","unstructured":"Pierson HA, Gashler MS (2017) Deep learning in robotics: a review of recent research. Adv Robot 31(16):821\u2013835","journal-title":"Adv Robot"},{"key":"11066_CR35","volume-title":"The Bayesian choice: from decision-theoretic foundations to computational implementation","author":"CP Robert","year":"2007","unstructured":"Robert CP et al (2007) The Bayesian choice: from decision-theoretic foundations to computational implementation, vol 2. Springer, Berlin"},{"key":"11066_CR36","doi-asserted-by":"crossref","unstructured":"Schm\u00e4hling F, Martin J, Elster C (2021) A framework for benchmarking uncertainty in deep regression. arXiv preprint arXiv:2109.09048","DOI":"10.1007\/s10489-022-03908-3"},{"key":"11066_CR37","doi-asserted-by":"crossref","unstructured":"Seghouane A-K, Fleury G (2001) A cost function for learning feedforward neural networks subject to noisy inputs. In: Proceedings of the sixth international symposium on signal processing and its applications (Cat. No. 01EX467), vol\u00a02, pp 386\u2013389. IEEE","DOI":"10.1109\/ISSPA.2001.950161"},{"key":"11066_CR38","doi-asserted-by":"crossref","unstructured":"Sragner L, Horvath G (2003) Improved model order estimation for nonlinear dynamic systems. In: Second IEEE international workshop on intelligent data acquisition and advanced computing systems: technology and applications, 2003. Proceedings, pp 266\u2013271. IEEE","DOI":"10.1109\/IDAACS.2003.1249564"},{"issue":"4\u20135","key":"11066_CR39","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1177\/0278364918770733","volume":"37","author":"N S\u00fcnderhauf","year":"2018","unstructured":"S\u00fcnderhauf N, Brock O, Scheirer W, Hadsell R, Fox D, Leitner J, Upcroft B, Abbeel P, Burgard W, Milford M et al (2018) The limits and potentials of deep learning for robotics. Int J Robot Res 37(4\u20135):405\u2013420","journal-title":"Int J Robot Res"},{"key":"11066_CR40","first-page":"141","volume":"8","author":"J Van Gorp","year":"1998","unstructured":"Van Gorp J, Schoukens J, Pintelon R (1998) The errors-in-variables cost function for learning neural networks with noisy inputs. Intell Eng Artif Neural Netw 8:141\u2013146","journal-title":"Intell Eng Artif Neural Netw"},{"issue":"2","key":"11066_CR41","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1109\/72.839010","volume":"11","author":"J Van Gorp","year":"2000","unstructured":"Van Gorp J, Schoukens J, Pintelon R (2000) Learning neural networks with noisy inputs using the errors-in-variables approach. IEEE Trans Neural Netw 11(2):402\u2013414","journal-title":"IEEE Trans Neural Netw"},{"key":"11066_CR42","doi-asserted-by":"publisher","first-page":"7068349","DOI":"10.1155\/2018\/7068349","volume":"2018","author":"A Voulodimos","year":"2018","unstructured":"Voulodimos A, Doulamis N, Doulamis A, Protopapadakis E (2018) Deep learning for computer vision: a brief review. Comput Intell Neurosci 2018:7068349. https:\/\/doi.org\/10.1155\/2018\/7068349","journal-title":"Comput Intell Neurosci"},{"issue":"6","key":"11066_CR43","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1109\/72.809073","volume":"10","author":"W Wright","year":"1999","unstructured":"Wright W (1999) Bayesian approach to neural-network modeling with input uncertainty. IEEE Trans Neural Netw 10(6):1261\u20131270","journal-title":"IEEE Trans Neural Netw"},{"issue":"1","key":"11066_CR44","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1023\/A:1008111920791","volume":"26","author":"W Wright","year":"2000","unstructured":"Wright W, Ramage G, Cornford D, Nabney IT (2000) Neural network modelling with input uncertainty: theory and application. J VLSI Signal Process Syst Signal Image Video Technol 26(1):169\u2013188","journal-title":"J VLSI Signal Process Syst Signal Image Video Technol"},{"key":"11066_CR45","doi-asserted-by":"publisher","DOI":"10.1016\/j.energy.2020.119007","volume":"211","author":"G Xie","year":"2020","unstructured":"Xie G, Chen X, Weng Y (2020) Input modeling and uncertainty quantification for improving volatile residential load forecasting. Energy 211:119007","journal-title":"Energy"},{"issue":"24","key":"11066_CR46","doi-asserted-by":"publisher","first-page":"10486","DOI":"10.3390\/su122410486","volume":"12","author":"J Yuan","year":"2020","unstructured":"Yuan J, Zhu J, Nian V (2020) Neural network modeling based on the bayesian method for evaluating shipping mitigation measures. Sustainability 12(24):10486","journal-title":"Sustainability"},{"key":"11066_CR47","unstructured":"Zhang G, Sun S, Duvenaud D, Grosse R (2018) Noisy natural gradient as variational inference. In: International conference on machine learning, pp 5852\u20135861. PMLR"},{"issue":"3\u20134","key":"11066_CR48","doi-asserted-by":"publisher","first-page":"696","DOI":"10.1016\/j.jhydrol.2011.09.002","volume":"409","author":"X Zhang","year":"2011","unstructured":"Zhang X, Liang F, Yu B, Zong Z (2011) Explicitly integrating parameter, input, and structure uncertainties into bayesian neural networks for probabilistic hydrologic forecasting. J Hydrol 409(3\u20134):696\u2013709","journal-title":"J Hydrol"}],"container-title":["Neural Processing Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-022-11066-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11063-022-11066-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-022-11066-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T00:42:55Z","timestamp":1728261775000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11063-022-11066-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,1]]},"references-count":48,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["11066"],"URL":"https:\/\/doi.org\/10.1007\/s11063-022-11066-3","relation":{},"ISSN":["1370-4621","1573-773X"],"issn-type":[{"type":"print","value":"1370-4621"},{"type":"electronic","value":"1573-773X"}],"subject":[],"published":{"date-parts":[[2022,11,1]]},"assertion":[{"value":"16 October 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 November 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors of this work are not aware of any conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}