Conformal prediction bands for multivariate functional data. (English) Zbl 1520.62424
Summary: Motivated by the pressing request of methods able to create prediction sets in a general regression framework for a multivariate functional response, we propose a set of conformal predictors that produce finite-sample either valid or exact multivariate simultaneous prediction bands under the mild assumption of exchangeable regression pairs. The fact that the prediction bands can be built around any regression estimator and that can be easily found in closed form yields a very widely usable method, which is fairly straightforward to implement. In addition, we first introduce and then describe a specific conformal predictor that guarantees an asymptotic result in terms of efficiency and inducing prediction bands able to modulate their width based on the local behavior and magnitude of the functional data. The method is investigated and analyzed through a simulation study and a real-world application in the field of urban mobility.
MSC:
| 62R10 | Functional data analysis |
| 62G08 | Nonparametric regression and quantile regression |
| 62G15 | Nonparametric tolerance and confidence regions |
| 62H12 | Estimation in multivariate analysis |
Keywords:
conformal prediction; distribution-free prediction set; exact prediction set; finite-sample prediction set; functional data; prediction bandReferences:
| [1] | Aneiros, G.; Cao, R.; Fraiman, R.; Genest, C.; Vieu, P., Recent advances in functional data analysis and high-dimensional statistics, J. Multivariate Anal., 170, 3-9 (2019) · Zbl 1415.62043 |
| [2] | Antoniadis, A.; Brossat, X.; Cugliari, J.; Poggi, J.-M., A prediction interval for a function-valued forecast model: Application to load forecasting, Int. J. Forecast., 32, 3, 939-947 (2016) |
| [3] | Balasubramanian, V.; Ho, S.-S.; Vovk, V., Conformal Prediction for Reliable Machine Learning: Theory, Adaptations and Applications (2014), Morgan Kaufmann: Morgan Kaufmann Boston · Zbl 1290.68003 |
| [4] | Cao, G.; Yang, L.; Todem, D., Simultaneous inference for the mean function based on dense functional data, J. Nonparametr. Stat., 24, 2, 359-377 (2012) · Zbl 1241.62119 |
| [5] | Choi, H.; Reimherr, M., A geometric approach to confidence regions and bands for functional parameters, J. R. Stat. Soc. Ser. B Stat. Methodol., 80, 1, 239-260 (2018) · Zbl 1381.62143 |
| [6] | Degras, D. A., Simultaneous confidence bands for nonparametric regression with functional data, Statist. Sinica, 21, 4, 1735-1765 (2011) · Zbl 1225.62052 |
| [7] | Degras, D., Simultaneous confidence bands for the mean of functional data, Wiley Interdiscip. Rev. Comput. Stat., 9, 3, Article e1397 pp. (2017) · Zbl 07914924 |
| [8] | Diquigiovanni, J.; Fontana, M.; Vantini, S., The importance of being a band: Finite-sample exact distribution-free prediction sets for functional data (2021), arXiv:2102.06746 |
| [9] | Ferraty, F.; Vieu, P., Additive prediction and boosting for functional data, Comput. Statist. Data Anal., 53, 4, 1400-1413 (2009) · Zbl 1452.62989 |
| [10] | Gijbels, I.; Nagy, S., On a general definition of depth for functional data, Statist. Sci., 32, 4, 630-639 (2017) · Zbl 1381.62098 |
| [11] | Goia, A.; Vieu, P., An introduction to recent advances in high/infinite dimensional statistics, J. Multivariate Anal., 146, 1-6 (2016) · Zbl 1384.00073 |
| [12] | Hastie, T.; Tibshirani, R.; Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2009), Springer Science & Business Media · Zbl 1273.62005 |
| [13] | Hyndman, R. J.; Shahid Ullah, M., Robust forecasting of mortality and fertility rates: A functional data approach, Comput. Statist. Data Anal., 51, 10, 4942-4956 (2007) · Zbl 1162.62434 |
| [14] | Inselberg, A., The plane with parallel coordinates, Vis. Comput., 1, 2, 69-91 (1985) · Zbl 0617.52005 |
| [15] | Lei, J.; G’Sell, M.; Rinaldo, A.; Tibshirani, R. J.; Wasserman, L., Distribution-free predictive inference for regression, J. Amer. Statist. Assoc., 113, 523, 1094-1111 (2018) · Zbl 1402.62155 |
| [16] | Lei, J.; Robins, J.; Wasserman, L., Distribution-free prediction sets, J. Amer. Statist. Assoc., 108, 501, 278-287 (2013) · Zbl 06158342 |
| [17] | Liebl, D.; Reimherr, M., Fast and fair simultaneous confidence bands for functional parameters (2019), arXiv:1910.00131 |
| [18] | Lopez-Pintado, S.; Qian, K., A depth-based global envelope test for comparing two groups of functions with applications to biomedical data, Stat. Med., 40, 7, 1639-1652 (2021) |
| [19] | López-Pintado, S.; Romo, J., On the concept of depth for functional data, J. Amer. Statist. Assoc., 104, 486, 718-734 (2009) · Zbl 1388.62139 |
| [20] | Mosler, K., Depth statistics, (Robustness and Complex Data Structures (2013), Springer: Springer New York, NY), 17-34 |
| [21] | Myllymäki, M.; Mrkvika, T.s.; Grabarnik, P.; Seijo, H.; Hahn, U., Global envelope tests for spatial processes, J. R. Stat. Soc. Ser. B Stat. Methodol., 79, 2, 381-404 (2017) · Zbl 1414.62404 |
| [22] | Nagy, S., Statistical Depth for Functional Data (2016), KU Leuven and Charles University: KU Leuven and Charles University xxvi+216 pp, (Ph.D. thesis) |
| [23] | Nagy, S.; Gijbels, I.; Hlubinka, D., Depth-based recognition of shape outlying functions, J. Comput. Graph. Statist., 26, 4, 883-893 (2017) |
| [24] | Narisetty, N. N.; Nair, V. N., Extremal depth for functional data and applications, J. Amer. Statist. Assoc., 111, 516, 1705-1714 (2016) |
| [25] | Papadopoulos, H.; Proedrou, K.; Vovk, V.; Gammerman, A., Inductive confidence machines for regression, (European Conference on Machine Learning (2002), Springer), 345-356 · Zbl 1014.68514 |
| [26] | R Core Team, H., R: A language and environment for statistical computing (2020), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria |
| [27] | Ramsay, J. O., When the data are functions, Psychometrika, 47, 4, 379-396 (1982) · Zbl 0512.62004 |
| [28] | Ramsay, J. O.; Silverman, B. W., (Functional Data Analysis. Functional Data Analysis, Springer series in statistics (2005), Springer: Springer New York, NY) · Zbl 1079.62006 |
| [29] | Rao, P., Some notes on misspecification in multiple regressions, Amer. Statist., 25, 5, 37-39 (1971) |
| [30] | Sun, Y.; Genton, M. G., Functional boxplots, J. Comput. Graph. Statist., 20, 2, 316-334 (2011) |
| [31] | Telschow, F. J.; Schwartzman, A., Simultaneous confidence bands for functional data using the Gaussian kinematic formula, J. Statist. Plann. Inference, 216, 70-94 (2022) · Zbl 1477.62398 |
| [32] | Torti, A.; Pini, A.; Vantini, S., Modelling time-varying mobility flows using function-on-function regression: Analysis of a bike sharing system in the city of milan, J. R. Stat. Soc. Ser. C. Appl. Stat., 70, 1, 226-247 (2021) |
| [33] | Vapnik, V., Principles of risk minimization for learning theory, (Advances in Neural Information Processing Systems (1992), Denver: Denver CO), 831-838 |
| [34] | Vovk, V.; Gammerman, A.; Shafer, G., Algorithmic Learning in a Random World (2005), Springer Science & Business Media: Springer Science & Business Media New York, NY · Zbl 1105.68052 |
| [35] | Wooldridge, J. M., A simple specification test for the predictive ability of transformation models, Rev. Econ. Stat., 76, 1, 59-65 (1994) |
| [36] | Zeni, G.; Fontana, M.; Vantini, S., Conformal prediction: a unified review of theory and new challenges (2020), arXiv:2005.07972 |
| [37] | Zuo, Y., Projection-based depth functions and associated medians, Ann. Statist., 31, 5, 1460-1490 (2003) · Zbl 1046.62056 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
