A Geometric Approach to Visualization of Variability in Functional Data

Abstract

We propose a new method for the construction and visualization of boxplot-type displays for functional data. We use a recent functional data analysis framework, based on a representation of functions called square-root slope functions, to decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of functional data is a very complex concept. Thus, we propose to identify outliers based on any of the three main components after decomposition. We provide a variety of visualization tools for the proposed boxplot-type displays including surface plots. We evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface temperature functions, electrocardiogram functions and growth curves.Comment: Journal of the American Statistical Association, 201

Functional outlier detection and taxonomy by sequential transformations

“…1982 was also outlying due to the rapid increase from October to December. Figure 6 shows our result, as well as the outliers detected by the other two methods, the original functional boxplot based on the modified band depth (Sun and Genton, 2011) and the phase-amplitude decomposition (Xie et al, 2017). According to a National Climatic Data Center report, two of the strongest El Niño events happened during 1982-1983 and because we used the L ∞ depth to construct the functional boxplot, and this depth notion puts more weight on extremal events, which matches well with El Niño studies.…”

A geometric approach for computing tolerance bounds for elastic functional data

“…There are many ways to estimate quantiles for functional data, with the cross-sectional (pointwise) approach being most popular [18]. We propose to use the geometric method of Xie et al [35], which relies on the Riemannian geometry of the amplitude and phase spaces. In that paper, the authors only compute quartiles, but their method can be easily extended to calculate general quantiles.…”

“…Xie et al [35] provide a detailed commentary of this in their paper and propose a surface plot using the proper metrics to display the quantiles. We thus use the same approach here, and present such surface plots for the amplitude and phase components in Figure 6 case, it is more effective to display the difference between the median/bounds and the identity element γ id .…”

Trajectory functional boxplots

“…When the underlying dataset is possibly contaminated, the detection of outliers becomes an important step of exploratory data analysis. For functional data, the existing outlier detection rules consist of three different subtypes: discarding a prefixed proportion of data with respect to the depth values (Fraiman & Muniz, ), using graphical tools based on the raw curves (Hyndman & Shang, ; Sun & Genton, ; Xie, Kurtek, Bharath, & Sun, ), and approximating the distribution of the depth (or its transformation) values (Dai & Genton, , Rousseeuw et al, ). We use two of them that belong to the last two categories, respectively.…”