Abstract
This article proposes functional median polish, an extension of univariate median polish, for one-way and two-way functional analysis of variance (ANOVA). The functional median polish estimates the functional grand effect and functional main factor effects based on functional medians in an additive functional ANOVA model assuming no interaction among factors. A functional rank test is used to assess whether the functional main factor effects are significant. The robustness of the functional median polish is demonstrated by comparing its performance with the traditional functional ANOVA fitted by means under different outlier models in simulation studies. The functional median polish is illustrated on various applications in climate science, including one-way and two-way ANOVA when functional data are either curves or images. Specifically, Canadian temperature data, U.S. precipitation observations and outputs of global and regional climate models are considered, which can facilitate the research on the close link between local climate and the occurrence or severity of some diseases and other threats to human health.
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Ammann, C. M., Washington, W. M., Meehl, G. A., Buja, L., and Teng, H. (2010), “Climate Engineering Through Artificial Enhancement of Natural Forcings: Magnitudes and Implied Consequences,” Journal of Geophysical Research, 115, D22109. doi:10.1029/2009JD012878.
Christensen, J., Carter, T., and Giorgi, F. (2002), “PRUDENCE Employs New Methods to Assess European Climate Change,” EOS, Transactions—American Geophysical Union, 83.
Collins, W. D. et al.(2006), “The Community Climate System Model Version 3 (CCSM3),” Journal of Climate, 19, 2122–2143. doi:10.1175/JCLI3761.1.
Cressie, N. A. C. (1993), Statistics for Spatial Data, New York: Wiley.
Emerson, J. D., and Hoaglin, D. C. (1983), “Analysis of Two-Way Tables by Medians,” in Understanding Robust and Exploratory Data Analysis, eds. D. C. Hoaglin, F. Mosteller, and J. W. Tukey, New York: Wiley, pp. 166–210.
Fink, A. M. (1988), “How to Polish off Median Polish,” SIAM Journal on Scientific and Statistical Computing, 9, 932–940.
Hoaglin, D. C., Mosteller, F., and Tukey, J. W. (eds.) (1983), Understanding Robust and Exploratory Data Analysis, New York: Wiley.
— (1985), Exploring Data Tables, Trends, and Shapes, New York: Wiley.
Intergovernmental Panel on Climate Change (IPCC) (2000), Special Report on Emission Scenarios. A special report of Working Group III of the Intergovernmental Panel on Climate Change, eds. N. Nakicenovic and R. Swart, 612, Cambridge: Cambridge Univ. Press.
Kaufman, C. G., and Sain, S. R. (2010), “Bayesian Functional ANOVA Modeling Using Gaussian Process Prior Distributions,” Bayesian Analysis, 5, 123–150.
López-Pintado, S., and Romo, J. (2009), “On the Concept of Depth for Functional Data,” Journal of the American Statistical Association, 104, 718–734.
López-Pintado, S., and Wei, Y. (2011), “Depth for Sparse Functional Data,” in Recent Advances in Functional Data Analysis and Related Topics, ed. F. Ferraty, Berlin: Springer, pp. 209–212.
Mosteller, F., and Tukey, J. W. (1977), Data Analysis and Regression, Cambridge: Addison-Wesley.
Ramsay, J. O., Hooker, G., and Graves, S. (2009), Functional Data Analysis With R and Matlab, New York: Springer.
Ramsay, J. O., and Silverman, B. W. (2005), Functional Data Analysis (2nd ed.), New York: Springer.
Sain, S. R., Nychka, D., and Mearns, L. (2011), “Functional ANOVA and Regional Climate Experiments: A Statistical Analysis of Dynamic Downscaling,” Environmetrics, 22, 700–711.
Sun, Y., and Genton, M. G. (2011), “Functional Boxplots,” Journal of Computational and Graphical Statistics, 20, 316–334.
— (2012), “Adjusted Functional Boxplots for Spatio-temporal Data Visualization and Outlier Detection,” Environmetrics, 23, 54–64.
Sun, Y., Genton, M. G., and Nychka, D. W. (2012), “Exact Fast Computation of Band Depth for Large Functional Datasets,” manuscript.
Tukey, J. W. (1970), Exploratory Data Analysis, Vol. 1, Reading: Addison-Wesley, Chapter 5 (Limited Preliminary Edition).
— (1977), Exploratory Data Analysis, Reading: Addison-Wesley.
Velleman, P. F., and Hoaglin, D. C. (1981), Applications, Basics, and Computing of Exploratory Data Analysis, N. Scituate: Duxbury Press.
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Sun, Y., Genton, M.G. Functional Median Polish. JABES 17, 354–376 (2012). https://doi.org/10.1007/s13253-012-0096-8
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DOI: https://doi.org/10.1007/s13253-012-0096-8