Multivariate analysis of variance for functional data. (English) Zbl 1516.62307
Summary: Functional data are being observed frequently in many scientific fields, and therefore most of the standard statistical methods are being adapted for functional data. The multivariate analysis of variance problem for functional data is considered. It seems to be of practical interest similarly as the one-way analysis of variance for such data. For the MANOVA problem for multivariate functional data, we propose permutation tests based on a basis function representation and tests based on random projections. Their performance is examined in comprehensive simulation studies, which provide an idea of the size control and power of the tests and identify differences between them. The simulation experiments are based on artificial data and real labeled multivariate time series data found in the literature. The results suggest that the studied testing procedures can detect small differences between vectors of curves even with small sample sizes. Illustrative real data examples of the use of the proposed testing procedures in practice are also presented.
MSC:
| 62-XX | Statistics |
Keywords:
basis function representation; MANOVA; multivariate functional data; orthonormal basis; random projection; stochastic processReferences:
| [1] | M.C. Aguilera-Morillo, A.M. Aguilera, M. Escabias and M.J. Valderrama, Penalized spline approaches for functional logit regression, Test 22 (2013), pp. 251-277. doi: 10.1007/s11749-012-0307-1 · Zbl 1367.62183 · doi:10.1007/s11749-012-0307-1 |
| [2] | T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed., Wiley, London, 2003. · Zbl 1039.62044 |
| [3] | J. Antoch, L. Prchal, M.R. De Rosa, and P. Sarda, Electricity consumption prediction with functional linear regression using spline estimators, J. Appl. Stat. 37 (2010), pp. 2027-2041. doi: 10.1080/02664760903214395 · Zbl 1511.62405 |
| [4] | Y. Araki, S. Konishi, S. Kawano, and H. Matsui, Functional logistic discrimination via regularized basis expansions, Commun. Statist. Theory Methods 38 (2009), pp. 2944-2957. doi: 10.1080/03610920902947246 · Zbl 1175.62061 |
| [5] | K. Bache and M. Lichman, UCI Machine Learning Repository, University of California, School of Information and Computer Science, Irvine, CA (2013). Available at http://archive.ics.uci.edu/ml. |
| [6] | K. Benhenni, F. Ferraty, M. Rachdi, and P. Vieu, Local smoothing regression with functional data, Comput. Statist. 22 (2007), pp. 353-369. doi: 10.1007/s00180-007-0045-0 · Zbl 1194.62042 · doi:10.1007/s00180-007-0045-0 |
| [7] | Y. Benjamini and D. Yekutieli, The control of the false discovery rate in multiple testing under dependency, Ann. Statist. 29 (2001), pp. 1165-1188. doi: 10.1214/aos/1013699998 · Zbl 1041.62061 · doi:10.1214/aos/1013699998 |
| [8] | J.R. Berrendero, A. Justel, and M. Svarc, Principal components for multivariate functional data, Comput. Statist. Data Anal. 55 (2011), pp. 2619-2634. doi: 10.1016/j.csda.2011.03.011 · Zbl 1464.62025 · doi:10.1016/j.csda.2011.03.011 |
| [9] | G. Boente, M.S. Barrera, and D.E. Tyler, A characterization of elliptical distributions and some optimality properties of principal components for functional data, J. Multivariate Anal. 131 (2014), pp. 254-264. doi: 10.1016/j.jmva.2014.07.006 · Zbl 1298.62093 · doi:10.1016/j.jmva.2014.07.006 |
| [10] | J.A. Cuesta-Albertos, A. Cuevas, and R. Fraiman, On projection-based tests for spherical and compositional data, Stat. Comput. 19 (2009), pp. 367-380. doi: 10.1007/s11222-008-9098-3 · doi:10.1007/s11222-008-9098-3 |
| [11] | J.A. Cuesta-Albertos, T. del Barrio, R. Fraiman, and C. Matrán, The random projection method in goodness of fit for functional data, Comput. Statist. Data Anal. 51 (2007), pp. 4814-4831. doi: 10.1016/j.csda.2006.09.007 · Zbl 1162.62363 · doi:10.1016/j.csda.2006.09.007 |
| [12] | J.A. Cuesta-Albertos and M. Febrero-Bande, A simple multiway ANOVA for functional data, Test 19 (2010), pp. 537-557. doi: 10.1007/s11749-010-0185-3 · Zbl 1203.62122 · doi:10.1007/s11749-010-0185-3 |
| [13] | J.A. Cuesta-Albertos, R. Fraiman, and T. Ransford, A sharp form of the Cramér-Wold theorem, J. Theoret. Probab. 20 (2007), pp. 201-209. doi: 10.1007/s10959-007-0060-7 · Zbl 1124.60004 · doi:10.1007/s10959-007-0060-7 |
| [14] | A. Cuevas, M. Febrero, and R. Fraiman, An ANOVA test for functional data, Comput. Statist. Data Anal. 47 (2004), pp. 111-122. doi: 10.1016/j.csda.2003.10.021 · Zbl 1429.62726 · doi:10.1016/j.csda.2003.10.021 |
| [15] | M. Febrero-Bande and W. González-Manteiga, Generalized additive models for functional data, Test 22 (2013), pp. 278-292. doi: 10.1007/s11749-012-0308-0 · Zbl 1367.62116 · doi:10.1007/s11749-012-0308-0 |
| [16] | F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice, Springer, New York, 2006. · Zbl 1119.62046 |
| [17] | S. Fremdt, L. Horváth, P. Kokoszka, and J.G. Steinebach, Functional data analysis with increasing number of projections, J. Multivariate Anal. 124 (2014), pp. 313-332. doi: 10.1016/j.jmva.2013.11.009 · Zbl 1359.62197 · doi:10.1016/j.jmva.2013.11.009 |
| [18] | T. Górecki, M. Krzyśko, and Ł. Waszak, Functional discriminant coordinates, Commun. Statist. Theory Methods 43 (2014), pp. 1013-1025. doi: 10.1080/03610926.2013.828074 · Zbl 1462.62384 |
| [19] | T. Górecki and M. Łuczak, Multivariate time series classification with parametric derivative dynamic time warping, Expert Syst. Appl. 42 (2015), pp. 2305-2312. doi: 10.1016/j.eswa.2014.11.007 · doi:10.1016/j.eswa.2014.11.007 |
| [20] | T. Górecki and Ł. Smaga, A comparison of tests for the one-way ANOVA problem for functional data, Comput. Statist. 30 (2015), pp. 987-1010. doi: 10.1007/s00180-015-0555-0 · Zbl 1329.65028 · doi:10.1007/s00180-015-0555-0 |
| [21] | B. Gregorutti, B. Michel, and P. Saint-Pierre, Grouped variable importance with random forests and application to multiple functional data analysis, Comput. Statist. Data Anal. 90 (2015), pp. 15-35. doi: 10.1016/j.csda.2015.04.002 · Zbl 1468.62069 · doi:10.1016/j.csda.2015.04.002 |
| [22] | M. Guo, L. Zhou, J.Z. Huang, and W.K. Härdle, Functional data analysis of generalized regression quantiles, Stat. Comput. 25 (2015), pp. 189-202. doi: 10.1007/s11222-013-9425-1 · Zbl 1331.62031 · doi:10.1007/s11222-013-9425-1 |
| [23] | L. Horváth and P. Kokoszka, Inference for functional data with applications, Springer, New York, 2012. · Zbl 1279.62017 · doi:10.1007/978-1-4614-3655-3 |
| [24] | L. Horváth, P. Kokoszka, and J.G. Steinebach, On sequential detection of parameter changes in linear regression, Statist. Probab. Lett. 77 (2007), pp. 885-895. doi: 10.1016/j.spl.2006.12.014 · Zbl 1117.62079 · doi:10.1016/j.spl.2006.12.014 |
| [25] | L. Horváth and G. Rice, Testing equality of means when the observations are from functional time series, J. Time Series Anal. 36 (2015), pp. 84-108. doi: 10.1111/jtsa.12095 · Zbl 1308.62172 · doi:10.1111/jtsa.12095 |
| [26] | L. Horváth and G. Rice, An introduction to functional data analysis and a principal component approach for testing the equality of mean curves, Rev. Mat. Comput. 28 (2015), pp. 505-548. doi: 10.1007/s13163-015-0169-7 · Zbl 1347.60028 · doi:10.1007/s13163-015-0169-7 |
| [27] | J. Jacques and C. Preda, Model-based clustering for multivariate functional data, Comput. Statist. Data Anal. 71 (2014), pp. 92-106. doi: 10.1016/j.csda.2012.12.004 · Zbl 1471.62096 · doi:10.1016/j.csda.2012.12.004 |
| [28] | E. Keogh, Q. Zhu, B. Hu, Y. Hao, X. Xi, L. Wei and C.A. Ratanamahatana, The UCR Time Series Classification/Clustering Homepage (2011). Available at http://www.cs.ucr.edu/eamonn/time_series_data/. |
| [29] | I.K. Keser and I.D. Kocakoç, Smoothed functional canonical correlation analysis of humidity and temperature data, J. Appl. Stat. 42 (2015), pp. 2126-2140. doi: 10.1080/02664763.2015.1019842 · Zbl 1514.62665 |
| [30] | M. Krzyśko and Ł. Waszak, Canonical correlation analysis for functional data, Biomet. Lett. 50 (2013), pp. 95-105. · doi:10.2478/bile-2013-0020 |
| [31] | R. Leeb, F. Lee, C. Keinrath, R. Scherer, H. Bischof, and G. Pfurtscheller, Brain-computer communication: motivation, aim, and impact of exploring a virtual apartment, IEEE Trans. Neural Syst. Rehab. Eng. 15 (2007), pp. 473-482. doi: 10.1109/TNSRE.2007.906956 · doi:10.1109/TNSRE.2007.906956 |
| [32] | H. Lian, Empirical likelihood confidence intervals for nonparametric functional data analysis, J. Statist. Plan. Inference 142 (2012), pp. 1669-1677. doi: 10.1016/j.jspi.2012.02.008 · Zbl 1238.62057 · doi:10.1016/j.jspi.2012.02.008 |
| [33] | H. Lian, Some asymptotic properties for functional canonical correlation analysis, J. Statist. Plan. Inference 153 (2014), pp. 1-10. doi: 10.1016/j.jspi.2014.05.008 · Zbl 1365.62140 · doi:10.1016/j.jspi.2014.05.008 |
| [34] | P. Martínez-Camblor and N. Corral, Repeated measures analysis for functional data, Comput. Statist. Data Anal. 55 (2011), pp. 3244-3256. doi: 10.1016/j.csda.2011.06.007 · Zbl 1261.65013 · doi:10.1016/j.csda.2011.06.007 |
| [35] | E.L. Montoya and W. Meiring, On the relative efficiency of a monotone parameter curve estimator in a functional nonlinear model, Stat. Comput. 23 (2013), pp. 425-436. doi: 10.1007/s11222-012-9320-1 · Zbl 1322.62123 · doi:10.1007/s11222-012-9320-1 |
| [36] | R.A. Olshen, E.N. Biden, M.P. Wyatt, and D.H. Sutherland, Gait analysis and the bootstrap, Ann. Statist. 17 (1989), pp. 1419-1440. doi: 10.1214/aos/1176347372 · Zbl 0695.62244 · doi:10.1214/aos/1176347372 |
| [37] | R.T. Olszewski, Generalized feature extraction for structural pattern recognition in time-series data, Ph.D. diss., Carnegie Mellon University, Pittsburgh, 2001. |
| [38] | J.O. Ramsay, G. Hooker, and S. Graves, Functional Data Analysis with R and MATLAB, Springer, Berlin, 2009. · Zbl 1179.62006 · doi:10.1007/978-0-387-98185-7 |
| [39] | J.O. Ramsay and B.W. Silverman, Applied Functional Data Analysis. Methods and Case Studies, Springer, New York, 2002. · Zbl 1011.62002 · doi:10.1007/b98886 |
| [40] | J.O. Ramsay and B.W. Silverman, Functional Data Analysis, 2nd ed., Springer, New York, 2005. · Zbl 1079.62006 · doi:10.1007/b98888 |
| [41] | J.O. Ramsay, H. Wickham, S. Graves and G. Hooker, fda: Functional Data Analysis, R package version 2.4.3, (2014). Available at http://CRAN.R-project.org/package=fda. |
| [42] | R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, (2015). Available at http://www.R-project.org/. |
| [43] | A. Redd, A comment on the orthogonalization of B-spline basis functions and their derivatives, Stat. Comput. 22 (2012), pp. 251-257. doi: 10.1007/s11222-010-9221-0 · Zbl 1322.62034 · doi:10.1007/s11222-010-9221-0 |
| [44] | S. Robbiano, M. Saumard and M. Curé, Improving prediction performance of stellar parameters using functional models, J. Appl. Stat. 43 (2015), pp. 1465-1476. doi: 10.1080/02664763.2015.1106448 · Zbl 1514.62826 |
| [45] | J.J. Rodriguez, C.J. Alonso and J.A. Maestro, Support vector machines of intervalbased features for time series classification, Knowledge-Based Syst 18 (2005), pp. 171-178. doi: 10.1016/j.knosys.2004.10.007 · doi:10.1016/j.knosys.2004.10.007 |
| [46] | J.R. Schott, Some high-dimensional tests for a one-way MANOVA, J. Multivariate Anal. 98 (2007), pp. 1825-1839. doi: 10.1016/j.jmva.2006.11.007 · Zbl 1130.62058 · doi:10.1016/j.jmva.2006.11.007 |
| [47] | C. Sguera, P. Galeano, and R. Lillo, Spatial depth-based classification for functional data, Test 23 (2014), pp. 725-750. doi: 10.1007/s11749-014-0379-1 · Zbl 1312.62083 · doi:10.1007/s11749-014-0379-1 |
| [48] | G. Shmueli, To explain or to predict? Statist. Sci. 25 (2010), pp. 289-310. doi: 10.1214/10-STS330 · Zbl 1329.62045 · doi:10.1214/10-STS330 |
| [49] | S. Tokushige, H. Yadohisa and K. Inada, Crisp and fuzzy \(k\)-means clustering algorithms for multivariate functional data, Comput. Statist. 22 (2007), pp. 1-16. doi: 10.1007/s00180-006-0013-0 · Zbl 1196.62089 · doi:10.1007/s00180-006-0013-0 |
| [50] | J.T. Zhang, Analysis of Variance for Functional Data, Chapman & Hall, London, 2013. · Zbl 1281.62160 · doi:10.1201/b15005 |
| [51] | J.T. Zhang and X. Liang, One-way ANOVA for functional data via globalizing the pointwise F-test, Scand. J. Stat. 41 (2014), pp. 51-71. doi: 10.1111/sjos.12025 · Zbl 1349.62331 · doi:10.1111/sjos.12025 |
| [52] | J. Zhou, N.Y. Wang and N. Wang, Functional linear model with zero-value coefficient function at sub-regions, Statist. Sin. 23 (2013), pp. 25-50. · Zbl 1426.62214 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
