A Measure of Directional Outlyingness With Applications to Image Data and Video

Abstract

Functional data covers a wide range of data types. They all have in common that the observed objects are functions of of a univariate argument (e.g. time or wavelength) or a multivariate argument (say, a spatial position). These functions take on values which can in turn be univariate (such as the absorbance level) or multivariate (such as the red/green/blue color levels of an image). In practice it is important to be able to detect outliers in such data. For this purpose we introduce a new measure of outlyingness that we compute at each gridpoint of the functions' domain. The proposed directional outlyingness (DO) measure accounts for skewness in the data and only requires O(n) computation time per direction. We derive the influence function of the DO and compute a cutoff for outlier detection. The resulting heatmap and functional outlier map reflect local and global outlyingness of a function. To illustrate the performance of the method on real data it is applied to spectra, MRI images, and video surveillance 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.…”

Blind source separation of tensor-valued time series

“…The fourth series on the left corresponds to a moment when the man has reached the edge of the screen and starts to slowly turn around. Note that contrary to the outlier detection method in Rousseeuw et al (2016) which flagged all frames after the change point as outlying, TSOBI produced signals that identify the frames during which big changes occur. Interestingly, the first two source series on the right also capture some systematic component in the video although exactly which is not evident from the video.…”

“…edu/~jiaxu/projects/gosus/supplement that has been used for background subtraction in Toyama et al (1999); Li et al (2004). The same video was also used for outlier detection in Rousseeuw et al (2016) and they have the preprocessed video available at https://wis.kuleuven.be/stat/robust/software. The video data tensor consists of a total of T = 633 color frames of size 128×160 depicting a beach, sea and a tree, with a man entering the scene from left and passing the tree during frames 480-500 and staying in the picture for the rest of the clip.…”

“…As an illustration Figure 5 shows frames 1 and 520 representing the typical view of the video without and with the man. Our objective with the video is similar as in Rousseeuw et al (2016), trying to detect the time window during which the man is in the scene. Assuming that the model (9) holds in this case, the relative staticity of the video before the man's entrance makes it reasonable to assume that the change point is also visible in the extracted source signals in Z t .…”

Outlier Detection using Graphical and Nongraphical Functional Methods in Hydrology

“…Outliergram was introduced by [11] for detecting shape outliers, [12] presented functional outliers taxonomy by proposing methods of visualization for detecting outliers whereas [13] introduced plots for displaying outlier curves having combined magnitudeshape features. Several other methods have also been recently developed by [14][15][16][17] for data visualization and outlier detection in functional framework. The study conducted by [2] in hydrology has been employed for classification of hydrograph by [18] and for the purpose of streamflow forecasting by [19].…”