主元分析中的平滑性

doi:10.3969/j.iss.0372-2012-2014.03.019

电子学报 ›› 2014, Vol. 42 ›› Issue (3): 547-555.DOI: 10.3969/j.iss.0372-2012-2014.03.019

主元分析中的平滑性

向馗¹, 周申培¹, 李炳南²

1. 武汉理工大学自动化学院, 湖北武汉 430070;
2. 合肥工业大学医学工程学院, 安徽合肥 230009

收稿日期:2013-05-23 修回日期:2013-09-28 出版日期:2014-03-25
- 通讯作者:
- 李炳南
- 作者简介:
- 向馗男，1976年10月出生于湖北省秭归县.现为武汉理工大学自动化学院副教授，从事生理信号处理和人机协作方面的研究工作.E-mail：xkarcher@126.com
- 基金资助:
- 国家自然科学基金 (No.61101022）; 湖北省自然科学基金 (No.2012FFB05004）; 武汉理工大学自主创新研究基金 (No.2012-Ⅱ-017，No.2013-IV-063）

Smoothness in Principal Component Analysis：A Survey

XIANG Kui¹, ZHOU Shen-pei¹, LI Bing-nan²

1. School of Automation, Wuhan University of Technology, Wuhan, Hubei 430070, China;
2. School of Medical Engineering, Hefei University of Technology, Hefei, Anhui 230009, China

Received:2013-05-23 Revised:2013-09-28 Online:2014-03-25 Published:2014-03-25
- Supported by:
- National Natural Science Foundation of China (No.61101022); Natural Science Foundation of Hubei Province, China (No.2012FFB05004); Independent Innovation Research Fund of Wuhan University of Technology (No.2012-Ⅱ-017, No.2013-IV-063)

摘要/Abstract

摘要： 某些样本观测形如时间序列或离散信号，其本质为平滑曲线（即函数型数据），代表一个潜在的连续过程.在主元分析中引入平滑性，可更加全面地刻画样本观测中包含的连续动态特性.本文介绍了从离散样本过渡到连续曲线的平滑处理方法，陈述了线性平滑主元的基本框架——基函数空间下的多元统计.平滑曲线兼具幅度变异和相位变异，可通过配准分离两种变异.据此重点讨论了非线性平滑主元分析：既可采用混合数据形式，一并考察两种变异性；也可借助微分流形，在非欧氏空间描述相位变异.基于开源的步态数据集，给出了3组分析结果：未经配准的平滑主元分析；配准后的幅度变异分析和相位变异分析.最后，综述了平滑主元在生物信号处理中的典型应用.

关键词: 主元分析, 平滑, 函数型数据, 相位变异

Abstract: Some of the sample observations,which seem like time series or discrete signals,are in fact smooth curves (functional data) corresponding to a latent continuous process.The smooth principal component analysis (PCA) focusing on functional data variation can fully characterize the dynamic features hidden in observations.The approaches smoothing discrete samples to continuous curves were introduced.The linear framework of smooth PCA was described as multivariate statistics in basis function spaces.The amplitude variation and phase variation embedded in smooth curves needed registration operations to separate themselves.The nonlinear framework of smooth PCA was discussed in two aspects:depicting two types of variation together with mixed data;depicting phase variation separately with differential manifolds in non-Euclidean space.Three groups of smooth PCA results were presented,which are raw gait data without registration,gait amplitude variation with registration and phase variation.Finally,the applications of smooth PCA in bio-signal processing were reviewed.

Key words: principal component analysis, smooth, functional data, phase variation

中图分类号:

TP391

向馗, 周申培, 李炳南. 主元分析中的平滑性[J]. 电子学报, 2014, 42(3): 547-555.

XIANG Kui, ZHOU Shen-pei, LI Bing-nan . Smoothness in Principal Component Analysis：A Survey[J]. Acta Electronica Sinica, 2014, 42(3): 547-555.

参考文献

[1] I T Jolliffe.Principal Component Analysis (2nd edition)[M].New York:Springer-Verlag, 2002.
[2] J Shlens.A Tutorial on Principal Component Analysis[R].USA:Systems Neurobiology Laboratory, University of California at San Diego, 2005.
[3] 向馗, 李炳南.主元分析中的稀疏性[J].电子学报, 2012, 40(12):2525-2532. K Xiang, B Li.Sparsity in principal component analysis:A survey[J].Acta Electronica Sinica, 2012, 40(12):2525-2532.(in Chinese)
[4] 米子川, 赵丽琴.函数型数据分析的研究进展和技术框架[J].统计与信息论坛, 2012, 27(6):13-20. Z Mi, LZhao.The research development and technical framework of function data analysis[J].Statistics & Information Forum, 2012, 27(6):13-20.(in Chinese)
[5] H G Müller.Functional data analysis[A].M Lovric.International Encyclopedia of Statistical Science[C].Heidelberg:Springer Science Business Media, 2010.
[6] J O Ramsay.When the data are functions[J].Psychometrika, 1982, 47(4):379-396.
[7] J O Ramsay, B W Silverman.Functional Data Analysis (2nd Edition)[M].New York:Springer-Vetlag, 2005.
[8] J O Ramsay.Functional Data Analysis[EB/OL].http://www.psych.mcgill.ca/misc/fda/index.html, 2013-5-30.
[9] H G Müller.PACE:Principal Component by Conditional Expectation[EB/OL].http://anson.ucdavis.edu/～ntyang/PACE/, 2012-12-17.
[10] T Hastie, R Tibshirani, J Friedman.The Elements of Statistical Learning Data Mining, Inference, and Prediction (2nd edition)[M].New York:Springer-Verlag, 2002.
[11] J O Ramsay.Estimating smooth monotone functions[J].Journal of the Royal Statistical Society:Series B (Statistical Methodology), 1998, 60(2):365-375.
[12] D Pigoli, L M Sangalli.Wavelets in functional data analysis:Estimation of multidimensional curves and their derivatives[J].Computational Statistics & Data Analysis, 2012, 56(6):1482-1498.
[13] J Dauxois, A Pousse, Y Romain.Asymptotic theory for the principal component analysis of a vector random function:Some applications to statistical inference[J].Journal of Multivariate Analysis, 1982, 12(1):136-154.
[14] J A Rice, BW Silverman.Estimating the mean and covariance structure nonparametrically when the data are curves[J].Journal of the Royal Statistical Society.Series B (Methodological), 1991, 53(1):233-243.
[15] H L Shang.Visualizing and Forecasting Functional Time Series[D].Melbourne:Monash University, 2010.
[16] B W Silverman.Smoothed functional principal components analysis by choice of norm[J].The Annals of Statistics, 1996, 24(1):1-24.
[17] A M Aguilera, M C Aguilera-Morillo.Penalized PCA approaches for B-spline expansions of smooth functional data[J].Applied Mathematics and Computation, 2013, 219(14):7805-7819.
[18] F A Ocaña, A M Aguilera, M Escabias.Computational considerations in functional principal component analysis[J].Computational Statistics, 2007, 22(3):449-465.
[19] X Qi, H Zhao.Some theoretical properties of Silverman's method for smoothed functional principal component analysis[J].Journal of Multivariate Analysis, 2011, 102(4):741-767.
[20] Y Li, N Wang, R J Carroll.Selecting the number of principal components in functional data[J].Journal of the American Statistical Association, 2013, 108(504):1284-1294.
[21] P Sawant, N Billor, H Shin.Functional outlier detection with robust functional principal component analysis[J].Computational Statistics, 2012, 27(1):83-102.
[22] J Z Huang, H Shen, A Buja.Functional principal components analysis via penalized rank one approximation[J].Electronic Journal of Statistics, 2008, 2:678-695
[23] J O Ramsay, B W Silverman.Applied Functional Data Analysis:Methods and Case Studies[M].New York:Springer, 2002.
[24] R A Olshen, E N Biden, M P Wyatt, et al.Gait analysis and the bootstrap[J].Annals of Statistics, 1989, 17(4):1419-1440.
[25] J O Ramsay, X Li.Curve registration[J].Journal of the Royal Statistical Society:Series B (Statistical Methodology), 1998, 60(2):351-363.
[26] J Bigot.Landmark-based registration of curves via the continuous wavelet transform[J].Journal of Computational and Graphical Statistics, 2006, 15(3):542-564.
[27] L Slaets, G Claeskens, M Hubert.Phase and amplitude-based clustering for functional data[J].Computational Statistics & Data Analysis, 2012, 56(7):2360-2374.
[28] A Kneip, J O Ramsay.Combining registration and fitting for functional models[J].Journal of the American Statistical Association, 2008, 103(483):1155-1165.
[29] D Chen, H G Müller.Nonlinear manifold representations for functional data[J].Annals of Statistics, 2012, 40(1):1-29.
[30] J D Tucker, W Wu, A Srivastava.Generative models for functional data using phase and amplitude separation[J].Computational Statistics & Data Analysis, 2013, 61(1):50-66.
[31] X Lu, J S Marron.Principal nested spheres for time warped functional data analysis[J/OL].Math ST arXiv:1304.6789, April, 2013.
[32] S Ullah, C F Finch.Applications of functional data analysis:A systematic review[J].BMC Medical Research Methodology, 2013, 13(1):1-12.
[33] S Kurtek, W Wu, G E Christensen, et al.Segmentation, alignment and statistical analysis of biosignals with application to disease classification[J].Journal of Applied Statistics, 2013, 40(6):1270-1288.
[34] N Coffey, A J Harrison, O A Donoghue, et al.Common functional principal components analysis:A new approach to analyzing human movement data[J].Human Movement Science, 2011, 30(6):1144-1166.
[35] W Ryan, A Harrison, K Hayes.Functional data analysis of knee joint kinematics in the vertical jump[J].Sports Biomechanics, 2006, 5(1):121-138.
[36] I Epifanio, C Avila, A Page, et al.Analysis of multiple waveforms by means of functional principal component analysis:normal versus pathological patterns in sit-to-stand movement[J].Medical & Biological Engineering & Computing, 2008, 46(6):551-561.
[37] A A Samadani, A Ghodsi, D Kulic.Discriminative functional analysis of human movements[J].Pattern Recognition Letters, 2013, 34(15):1829-1839.
[38] 林辉杰, 严波涛, 许崇高, 等.基于函数型数据分析技术的运动协调量化方法应用研究[J].体育科学, 2012, 32(9):81-87. H Lin, B Yan, C Xu, et al.Applied research on quantitative method of motor coordination basing on functional data analysis technique[J].China Sport Science, 2012, 32(9):81-87.(in Chinese)
[39] T S Tian.Functional data analysis in brain imaging studies[J/OL].Frontiers in Psychology, 1:35, Oct 8, 2010.
[40] R Viviani, G Grön, M Spitzer.Functional principal component analysis of fMRI data[J].Human Brain Mapping, 2005, 24(2):109-129.
[41] E O'Connor, N Fieller, A Holmes, et al.Functional principal component analyses of biomedical images as outcome measures[J].Journal of the Royal Statistical Society:Series C (Applied Statistics), 2010, 59(1):57-76.
[42] C R Jiang, J A D Aston, J L Wang.Smoothing dynamic positron emission tomography time courses using functional principal components[J].NeuroImage, 2009, 47(1):184-193.
[43] D Ormoneit, M J Black, T Hastie, et al.Representing cyclic human motion using functional analysis[J].Image and Vision Computing, 2005, 23(14):1264-1276.

主元分析中的平滑性

Smoothness in Principal Component Analysis：A Survey

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