## Functional data analysis of amplitude and phase variation.(English)Zbl 1426.62146

Summary: The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite-dimensionality of function spaces, observation noise, and so on. However, there is another interesting phenomenon that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures, and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly used alignment procedure, based on minimizing the $$\mathbb{L}^{2}$$ norm between functions, does not provide satisfactory results. In this paper, we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in the following: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them.

### MSC:

 62G99 Nonparametric inference 62H25 Factor analysis and principal components; correspondence analysis 62J99 Linear inference, regression
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### References:

 [1] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression. Wiley, New York. · Zbl 0246.62038 [2] Boudaoud, S., Rix, H. and Meste, O. (2010). Core shape modelling of a set of curves. Comput. Statist. Data Anal.54 308-325. · Zbl 1464.62032 [3] Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis. Wiley, Chichester. · Zbl 0901.62072 [4] Gervini, D. and Gasser, T. (2004). Self-modelling warping functions. J. R. Stat. Soc. Ser. B. Stat. Methodol.66 959-971. · Zbl 1061.62052 [5] Grenander, U. (1993). General Pattern Theory. Oxford Univ. Press, New York. · Zbl 0827.68098 [6] Grenander, U. and Miller, M. I. (1998). Computational anatomy: An emerging discipline. Quart. Appl. Math.56 617-694. · Zbl 0952.92016 [7] Hadjipantelis, P. Z., Aston, J. A. D., Müller, H. G. and Evans, J. P. (2015). Unifying amplitude and phase analysis: A compositional data approach to functional multivariate mixed-effects modeling of Mandarin Chinese. J. Amer. Statist. Assoc.110 545-559. [8] Hadjipantelis, P. Z., Aston, J. A. D., Müller, H.-G. and Moriarty, J. (2014). Analysis of spike train data: A multivariate mixed effects model for phase and amplitude. Electron. J. Stat.8 1797-1807. · Zbl 1305.62329 [9] Jung, S., Dryden, I. L. and Marron, J. S. (2012). Analysis of principal nested spheres. Biometrika99 551-568. · Zbl 1437.62507 [10] Kneip, A. and Ramsay, J. O. (2008). Combining registration and fitting for functional models. J. Amer. Statist. Assoc.103 1155-1165. · Zbl 1205.62073 [11] Lavine, B. K. and Workman, J. J. (2013). Chemometrics. Anal. Chem.85 705-714. [12] Liu, X. and Müller, H.-G. (2004). Functional convex averaging and synchronization for time-warped random curves. J. Amer. Statist. Assoc.99 687-699. · Zbl 1117.62392 [13] Liu, X. and Yang, M. C. K. (2009). Simultaneous curve registration and clustering for functional data. Comput. Statist. Data Anal.53 1361-1376. · Zbl 1452.62993 [14] Lu, X. and Marron, J. S. (2013). Principal nested spheres for time warped functional data analysis. Preprint. Available at arXiv:1304.6789. [15] Marron, J. S. and Alonso, A. M. (2014). Overview of object oriented data analysis. Biom. J.56 732-753. · Zbl 1309.62008 [16] Marron, J. S., Ramsay, J. O., Sangalli, L. M. and Srivastava, A. (2014). Statistics of time warpings and phase variations. Electron. J. Stat.8 1697-1702. · Zbl 1305.62015 [17] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York. · Zbl 1079.62006 [18] Sakoe, H. and Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. Acoust. Speech Signal Process.26 43-49. · Zbl 0371.68035 [19] Sangalli, L. M., Secchi, P. and Vantini, S. (2014). Analysis of AneuRisk65 data: $$k$$-mean alignment. Electron. J. Stat.8 1891-1904. · Zbl 1305.62377 [20] Sangalli, L. M., Secchi, P., Vantini, S. and Veneziani, A. (2009). A case study in exploratory functional data analysis: Geometrical features of the internal carotid artery. J. Amer. Statist. Assoc.104 37-48. · Zbl 1388.62191 [21] Sangalli, L. M., Secchi, P., Vantini, S. and Vitelli, V. (2010). $$k$$-mean alignment for curve clustering. Comput. Statist. Data Anal.54 1219-1233. · Zbl 1464.62153 [22] Sotiras, A., Davatzikos, C. and Paragios, N. (2013). Deformable medical image registration: A survey. IEEE Trans. Med. Imag.32 1153-1190. [23] Srivastava, A., Jermyn, I. and Joshi, S. H. (2007). Riemannian analysis of probability density functions with applications in vision. In IEEE Conference on Computer Vision and Pattern Recognition, 2007. CVPR ’07 1-8. Minneapolis, MN, USA. [24] Srivastava, A., Wu, W., Kurtek, S., Klassen, E. and Marron, J. S. (2011a). Registration of functional data using Fisher-Rao metric. Preprint. Available at arXiv:1103.3817v2. [25] Srivastava, A., Klassen, E., Joshi, S. H. and Jermyn, I. H. (2011b). Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell.33 1415-1428. [26] Tang, R. and Müller, H.-G. (2008). Pairwise curve synchronization for functional data. Biometrika95 875-889. · Zbl 1437.62625 [27] Tang, R. and Müller, H.-G. (2009). Time-synchronized clustering of gene expression trajectories. Biostatistics10 32-45. · Zbl 1437.62626 [28] Tucker, J. D., Wu, W. and Srivastava, A. (2013). Generative models for functional data using phase and amplitude separation. Comput. Statist. Data Anal.61 50-66. · Zbl 1349.62253 [29] Tuddenham, R. D. and Snyder, M. M. (1954). Physical growth of California boys and girls from birth to eighteen years. University of California Publication in Child Development1 183-364. [30] Vantini, S. (2012). On the definition of phase and amplitude variability in functional data analysis. TEST21 676-696. · Zbl 1284.62316 [31] Veeraraghavan, A., Srivastava, A., Roy-Chowdhury, A. K. and Chellappa, R. (2009). Rate-invariant recognition of humans and their activities. IEEE Trans. Image Process.18 1326-1339. · Zbl 1371.94384 [32] Wang, H. and Marron, J. S. (2007). Object oriented data analysis: Sets of trees. Ann. Statist.35 1849-1873. · Zbl 1126.62002 [33] Younes, L., Michor, P. W., Shah, J. and Mumford, D. (2008). A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.19 25-57. · Zbl 1142.58013
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