Shape outlier detection and visualization for functional data: the outliergram

Abstract

We propose a new method to visualize and detect shape outliers in samples of curves. In functional data analysis, we observe curves defined over a given real interval and shape outliers may be defined as those curves that exhibit a different shape from the rest of the sample. Whereas magnitude outliers, that is, curves that lie outside the range of the majority of the data, are in general easy to identify, shape outliers are often masked among the rest of the curves and thus difficult to detect. In this article, we exploit the relationship between two measures of depth for functional data to help to visualize curves in terms of shape and to develop an algorithm for shape outlier detection. We illustrate the use of the visualization tool, the outliergram, through several examples and analyze the performance of the algorithm on a simulation study. Finally, we apply our method to assess cluster quality in a real set of time course microarray data.

Functional outlier detection and taxonomy by sequential transformations

“…To tackle the shape outliers, some researchers proposed decomposing the overall functional depth (or outlyingness) into just magnitude and shape depth (or outlyingness) in order to capture the shape outliers more accurately. Examples include the outliergram (Arribas-Gil and Romo, 2014), the functional outlier map (Rousseeuw et al, 2018), the total variation depth (Huang and Sun 2016), and the magnitude-shape plot (Dai and Genton, 2018c). Researchers also defined depth notions Figure 1: Shape outliers can be changed into magnitude outliers through some type of transformation.…”

Testing Normality of Functional Time Series

“…Like most moment‐based statistics, the test statistic has breakdown value 0. Outliers for functional data are generally more difficult to detect than for scalar or vector observations (see Arribas‐Gil and Romo, and references therein), so robust approaches can be of value; the ideas of Brys et al could be potentially extended to the functional context or other robust tests developed. We note that it is always useful to complement significance tests by exploratory tools such as normal QQ plots of the scores or other tools described, for example, in Kosiorowski and Zawadzki .…”

Band Depths Based on Multiple Time Instances

“…If m = 1 and d = 1, then we recover the band introduced in [11] and given by (1), so that band(f 1 , . .…”