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Using Bagidis in nonparametric functional data analysis: Predicting from curves with sharp local features

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  • Timmermans, Catherine
  • Delsol, Laurent
  • von Sachs, Rainer

Abstract

Our goal is to predict a scalar value or a group membership from the discretized observation of curves with sharp local features that might vary both vertically and horizontally. To this aim, we propose to combine the use of the nonparametric functional regression estimator developed by Ferraty and Vieu (2006) [18] with the Bagidis semimetric developed by Timmermans and von Sachs (submitted for publication) [36] with a view of efficiently measuring dissimilarities between curves with sharp patterns. This association is revealed as powerful. Under quite general conditions, we first obtain an asymptotic expansion for the small ball probability indicating that Bagidis induces a fractal topology on the functional space. We then provide the rate of convergence of the nonparametric regression estimator in this case, as a function of the parameters of the Bagidis semimetric. We propose to optimize those parameters using a cross-validation procedure, and show the optimality of the selected vector. This last result has a larger scope and concerns the optimization of any vector parameter characterizing a semimetric used in this context. The performances of our methodology are assessed on simulated and real data examples. Results are shown to be superior to those obtained using competing semimetrics as soon as the variations of the significant sharp patterns in the curves have a horizontal component.

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  • Timmermans, Catherine & Delsol, Laurent & von Sachs, Rainer, 2013. "Using Bagidis in nonparametric functional data analysis: Predicting from curves with sharp local features," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 421-444.
  • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:421-444
    DOI: 10.1016/j.jmva.2012.10.013
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    References listed on IDEAS

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    1. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
    2. Fryzlewicz, Piotr, 2007. "Unbalanced Haar technique for nonparametric function estimation," LSE Research Online Documents on Economics 25216, London School of Economics and Political Science, LSE Library.
    3. Delsol, Laurent & Ferraty, Frédéric & Vieu, Philippe, 2011. "Structural test in regression on functional variables," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 422-447, March.
    4. Fryzlewicz, Piotr, 2007. "Unbalanced Haar Technique for Nonparametric Function Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1318-1327, December.
    5. Ferraty, Frédéric & Vieu, Philippe, 2009. "Additive prediction and boosting for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1400-1413, February.
    6. Alejandro Quintela-Del-Río, 2008. "Hazard function given a functional variable: Non-parametric estimation under strong mixing conditions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(5), pages 413-430.
    7. Ashish Sood & Gareth M. James & Gerard J. Tellis, 2009. "Functional Regression: A New Model for Predicting Market Penetration of New Products," Marketing Science, INFORMS, vol. 28(1), pages 36-51, 01-02.
    8. Aneiros-Pérez, Germán & Vieu, Philippe, 2008. "Nonparametric time series prediction: A semi-functional partial linear modeling," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 834-857, May.
    9. Timmermans, Catherine & Fryzlewicz, Piotr, 2012. "Shah: Shape-Adaptive Haar Wavelet Transform For Images With Application To Classification," LIDAM Discussion Papers ISBA 2012015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Timmermans, Catherine & von Sachs, Rainer, 2010. "BAGIDIS, a new method for statistical analysis of differences between curves with sharp discontinuities," LIDAM Discussion Papers ISBA 2010030, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Marron, James Stephen & Härdle, Wolfgang, 1986. "Random approximations to some measures of accuracy in nonparametric curve estimation," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 91-113, October.
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    1. Timmermans, Catherine & von Sachs, Rainer, 2013. "BAGIDIS: Statistically investigating curves with sharp local patterns using a new functional measure of dissimilarity," LIDAM Discussion Papers ISBA 2013031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Helander, Sami & Laketa, Petra & Ilmonen, Pauliina & Nagy, Stanislav & Van Bever, Germain & Viitasaari, Lauri, 2022. "Integrated shape-sensitive functional metrics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).

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