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The Mutual Nearest Neighbor Method in Functional Nonparametric Regression

Received: 18 July 2018     Published: 19 July 2018
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Abstract

In recent decades, functional data have become a commonly encountered type of data. Its ideal units of observation are functions defined on some continuous domain and the observed data are sampled on a discrete grid. An important problem in functional data analysis is how to fit regression models with scalar responses and functional predictors (scalar-on-function regression). This paper focuses on the nonparametric approaches to this problem. First there is a review of the classical k-nearest neighbors (kNN) method for functional regression. Then the mutual nearest neighbors (MNN) method, which is a variant of kNN method, is applied to functional regression. Compared with the classical kNN approach, the MNN method takes use of the concept of mutual nearest neighbors to construct regression model and the pseudo nearest neighbors will not be taken into account during the prediction process. In addition, any nonparametric method in the functional data cases is affected by the curse of infinite dimensionality. To prevent this curse, it is legitimate to measure the proximity between two curves via a semi-metric. The effectiveness of MNN method is illustrated by comparing the predictive power of MNN method with kNN method first on the simulated datasets and then on a real chemometrical example. The comparative experimental analyses show that MNN method preserves the main merits inherent in kNN method and achieves better performances with proper proximity measures.

Published in Science Journal of Applied Mathematics and Statistics (Volume 6, Issue 3)
DOI 10.11648/j.sjams.20180603.13
Page(s) 81-89
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Functional Data, Nonparametric Estimation, Mutual Nearest Neighbors Estimator, Semi-Metric

References
[1] Ramsay J O, Silverman B W. Functional Data Analysis [M]. Springer New York, 1997.
[2] Ferraty F, Vieu P. Nonparametric functional data analysis: theory and practice [M]. Springer New York, 2006.
[3] Goia A, Vieu P. An introduction to recent advances in high/infinite dimensional statistics [J]. Journal of Multivariate Analysis, 2016, 146 (2):1-6.
[4] Wang J L, Chiou J M, Mueller H G. Review of Functional Data Analysis [J]. Statistics, 2015.
[5] Morris J S. Functional regression [J]. Annual Review of Statistics and Its Application, 2015, 2: 321-359.
[6] Reiss P T, Goldsmith J, Shang H L, et al. Methods for Scalar‐on-Function Regression [J]. International Statistical Review, 2017, 85 (2): 228-249.
[7] Royall R M. A class of non-parametric estimates of a smooth regression function [D]. Department of Statistics, Stanford University, 1966.
[8] Stone C J. Consistent nonparametric regression [J]. The annals of statistics, 1977: 595-620.
[9] Györfi L, Kohler M, Krzyzak A, et al. A distribution-free theory of nonparametric regression [M]. Springer Science & Business Media, 2006.
[10] Laloë T. A k-nearest neighbor approach for functional regression [J]. Statistics & probability letters, 2008, 78 (10): 1189-1193.
[11] Burba F, Ferraty F, Vieu P. k-Nearest Neighbour method in functional nonparametric regression [J]. Journal of Nonparametric Statistics, 2009, 21 (4): 453-469.
[12] Gowda K C, Krishna G. Agglomerative clustering using the concept of mutual nearest neighbourhood [J]. Pattern recognition, 1978, 10 (2): 105-112.
[13] Liu H, Zhang S, Zhao J, et al. A new classification algorithm using mutual nearest neighbors [C]. Grid and Cooperative Computing (GCC), 2010 9th International Conference on. IEEE, 2010: 52-57.
[14] Guyader A, Hengartner N. On the mutual nearest neighbors estimate in regression [J]. The Journal of Machine Learning Research, 2013, 14 (1): 2361-2376.
[15] Geenens G. Curse of dimensionality and related issues in nonparametric functional regression [J]. Statistics Surveys, 2011, 5: 30-43.
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  • APA Style

    Xingyu Chen, Dirong Chen. (2018). The Mutual Nearest Neighbor Method in Functional Nonparametric Regression. Science Journal of Applied Mathematics and Statistics, 6(3), 81-89. https://doi.org/10.11648/j.sjams.20180603.13

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    ACS Style

    Xingyu Chen; Dirong Chen. The Mutual Nearest Neighbor Method in Functional Nonparametric Regression. Sci. J. Appl. Math. Stat. 2018, 6(3), 81-89. doi: 10.11648/j.sjams.20180603.13

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    AMA Style

    Xingyu Chen, Dirong Chen. The Mutual Nearest Neighbor Method in Functional Nonparametric Regression. Sci J Appl Math Stat. 2018;6(3):81-89. doi: 10.11648/j.sjams.20180603.13

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  • @article{10.11648/j.sjams.20180603.13,
      author = {Xingyu Chen and Dirong Chen},
      title = {The Mutual Nearest Neighbor Method in Functional Nonparametric Regression},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {6},
      number = {3},
      pages = {81-89},
      doi = {10.11648/j.sjams.20180603.13},
      url = {https://doi.org/10.11648/j.sjams.20180603.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20180603.13},
      abstract = {In recent decades, functional data have become a commonly encountered type of data. Its ideal units of observation are functions defined on some continuous domain and the observed data are sampled on a discrete grid. An important problem in functional data analysis is how to fit regression models with scalar responses and functional predictors (scalar-on-function regression). This paper focuses on the nonparametric approaches to this problem. First there is a review of the classical k-nearest neighbors (kNN) method for functional regression. Then the mutual nearest neighbors (MNN) method, which is a variant of kNN method, is applied to functional regression. Compared with the classical kNN approach, the MNN method takes use of the concept of mutual nearest neighbors to construct regression model and the pseudo nearest neighbors will not be taken into account during the prediction process. In addition, any nonparametric method in the functional data cases is affected by the curse of infinite dimensionality. To prevent this curse, it is legitimate to measure the proximity between two curves via a semi-metric. The effectiveness of MNN method is illustrated by comparing the predictive power of MNN method with kNN method first on the simulated datasets and then on a real chemometrical example. The comparative experimental analyses show that MNN method preserves the main merits inherent in kNN method and achieves better performances with proper proximity measures.},
     year = {2018}
    }
    

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  • TY  - JOUR
    T1  - The Mutual Nearest Neighbor Method in Functional Nonparametric Regression
    AU  - Xingyu Chen
    AU  - Dirong Chen
    Y1  - 2018/07/19
    PY  - 2018
    N1  - https://doi.org/10.11648/j.sjams.20180603.13
    DO  - 10.11648/j.sjams.20180603.13
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
    SP  - 81
    EP  - 89
    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20180603.13
    AB  - In recent decades, functional data have become a commonly encountered type of data. Its ideal units of observation are functions defined on some continuous domain and the observed data are sampled on a discrete grid. An important problem in functional data analysis is how to fit regression models with scalar responses and functional predictors (scalar-on-function regression). This paper focuses on the nonparametric approaches to this problem. First there is a review of the classical k-nearest neighbors (kNN) method for functional regression. Then the mutual nearest neighbors (MNN) method, which is a variant of kNN method, is applied to functional regression. Compared with the classical kNN approach, the MNN method takes use of the concept of mutual nearest neighbors to construct regression model and the pseudo nearest neighbors will not be taken into account during the prediction process. In addition, any nonparametric method in the functional data cases is affected by the curse of infinite dimensionality. To prevent this curse, it is legitimate to measure the proximity between two curves via a semi-metric. The effectiveness of MNN method is illustrated by comparing the predictive power of MNN method with kNN method first on the simulated datasets and then on a real chemometrical example. The comparative experimental analyses show that MNN method preserves the main merits inherent in kNN method and achieves better performances with proper proximity measures.
    VL  - 6
    IS  - 3
    ER  - 

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Author Information
  • School of Mathematics and System Science, Beihang University, Beijing, P. R. China

  • School of Mathematics and Computer Sciences, Wuhan Textile University, Wuhan, P. R. China

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