Abstract
In this paper, we propose a robust estimator for the location function from multi-dimensional functional data. The proposed estimators are based on the deep neural networks with ReLU activation function. At the meanwhile, the estimators are less susceptible to outlying observations and model-misspecification. For any multi-dimensional functional data, we provide the uniform convergence rates for the proposed robust deep neural networks estimators. Simulation studies illustrate the competitive performance of the robust deep neural network estimators on regular data and their superior performance on data that contain anomalies. The proposed method is also applied to analyze 2D and 3D images of patients with Alzheimer’s disease obtained from the Alzheimer Disease Neuroimaging Initiative database.
Funding Statement
Wang’s and Cao’s research was partially supported by NSF award DMS 1736470. Cao’s research was also partially supported by the Simons Foundation under Grant #849413.
Acknowledgments
We thank an Associate Editor and two referees for their constructive comments which have helped to improve the presentation of the paper.
Data used in preparation of this article were obtained from the Alzheimers Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.
Citation
Shuoyang Wang. Guanqun Cao. "Robust deep neural network estimation for multi-dimensional functional data." Electron. J. Statist. 16 (2) 6461 - 6488, 2022. https://doi.org/10.1214/22-EJS2093
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