Robust estimation and classification for functional data via projection-based depth notions. (English) Zbl 1195.62032
Different notions of data depth are considered for functional data, such as the functional integrated depth, the \(h\)-mode depth and different versions of the depth calculation procedures based on random projections. Applications to supervised classification and robust estimation are discussed. Results of simulations and applications to real data sets are presented.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 62G05 | Nonparametric estimation |
| 62G35 | Nonparametric robustness |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
References:
| [1] | Biau G, Bunea F, Wegkamp M (2005) Functional classification in Hilbert spaces. IEEE Trans Inf Theory 51:2163–2172 · Zbl 1285.94015 · doi:10.1109/TIT.2005.847705 |
| [2] | Cérou F, Guyader A (2005) Nearest neighbor classification in infinite dimension. Preprint · Zbl 1187.62115 |
| [3] | Cuesta-Albertos JA, Fraiman R, Ransford T (2006a) A sharp form of the Cramer–Wold theorem. J Theor Probab (in press) · Zbl 1124.60004 |
| [4] | Cuesta-Albertos JA, Fraiman R, Ransford T (2006b) Random projections and goodness-of-fit tests in infinite-dimensional spaces. Boletim da Sociedade Brasileira de Matematica 37:1–25 · Zbl 1098.14048 · doi:10.1007/s00574-006-0001-6 |
| [5] | Cuevas A, Febrero M, Fraiman R (2006) On the use of the bootstrap for estimating functions with functional data. Comput Stat Data Anal 51:1063–1074 · Zbl 1157.62390 · doi:10.1016/j.csda.2005.10.012 |
| [6] | Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, Heidelberg · Zbl 0853.68150 |
| [7] | Donoho DL (1982) Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statist., Harvard University |
| [8] | Ferraty F, Vieu P (2003) Curves discrimination: a nonparametric functional approach. Comput Stat Data Anal 44:161–173 · Zbl 1429.62241 · doi:10.1016/S0167-9473(03)00032-X |
| [9] | Ferraty F, Vieu P (2006) Nonparametric modelling for functional data. Springer, Heidelberg · Zbl 1119.62046 |
| [10] | Fishman GS (1996) Monte Carlo: concepts, algorithms and applications. Springer, Heidelberg · Zbl 0859.65001 |
| [11] | Fraiman R, Meloche J (1999) Multivariate L-estimation. Test 8:255–317 · Zbl 0942.62062 · doi:10.1007/BF02595872 |
| [12] | Fraiman R, Muniz G (2001) Trimmed means for functional data. Test 10:419–440 · Zbl 1016.62026 · doi:10.1007/BF02595706 |
| [13] | Ghosh AK, Chaudhuri P (2005) On data depth and distribution-free discriminant analysis using separating surfaces. Bernoulli 11:1–27 · Zbl 1059.62064 · doi:10.3150/bj/1110228239 |
| [14] | Hastie T, Buja A, Tibshirani R (1995) Penalized discriminant analysis. Ann Stat 23:73–102 · Zbl 0821.62031 · doi:10.1214/aos/1176324456 |
| [15] | Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Data mining, inference, and prediction. Springer, New York · Zbl 0973.62007 |
| [16] | James GM, Hastie TJ (2001) Functional linear discriminant analysis for irregularly sampled curves. J R Stat Soc B 63:533–550 · Zbl 0989.62036 · doi:10.1111/1467-9868.00297 |
| [17] | Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18:405–414 · Zbl 0701.62063 · doi:10.1214/aos/1176347507 |
| [18] | Liu RY, Parelius JM, Singh K (1999) Multivariate analysis by data depth: descriptive statistics, graphics and inference. With discussion and a reply by the authors. Ann Stat 27:783–858 · Zbl 0984.62037 |
| [19] | Ramsay JO, Silverman BW (2002) Applied functional data analysis. Methods and case studies. Springer, New York · Zbl 1011.62002 |
| [20] | Ramsay JO, Silverman BW (2005a) Applied functional data analysis. Springer, Heidelberg |
| [21] | Ramsay JO, Silverman BW (2005b) Functional data analysis, 2nd edn. Springer, Heidelberg |
| [22] | Rousseeuw PJ, Hubert M (1999) Regression depth (with discussion). J Am Stat Assoc 94:388–433 · Zbl 1007.62060 · doi:10.1080/01621459.1999.10474129 |
| [23] | Stone CJ (1977) Consistent nonparametric regression. With discussion and a reply by the author. Ann Stat 5:595–645 · Zbl 0366.62051 |
| [24] | Tukey JW (1975) Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, pp 523–531. Canad. Math. Congress, Montreal |
| [25] | Zuo Y (2003) Projection based depth functions and associated medians. Ann Stat 31:1460–1490 · Zbl 1046.62056 · doi:10.1214/aos/1065705115 |
| [26] | Zuo Y, Serfling R (2000) General notions of statistical depth function. Ann Stat 28:461–482 · Zbl 1106.62334 · doi:10.1214/aos/1016218226 |
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.
