![A depiction of compositional data and its representation in the probability simplez. All instances of compositional data can be represented as points in the probability simplex Δ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{3}$$\end{document}. In the example, points in pink represent possible foods in terms of their percentage of protein, carbohidrates and fat. Red meat foods are represented by red squares, the blue circles indicate vegetables and green triangles show some types of fish and seafood](https://www.researchgate.net/publication/371162668/figure/fig1/AS:11431281202357817@1698809761244/A-depiction-of-compositional-data-and-its-representation-in-the-probability-simplez-All_Q320.jpg)
![The k-neighborhood of a vector. LOF identifies the k-nearest neighbors for each vector based on the k-distance. Vector v has as its neighbors vectors w1,w2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{1}, w_{2}$$\end{document}, and w3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{3}$$\end{document}. The expected distance from v to its neighbors serves as the basis for the characterization of v. The neighbors of v have to be characterized in terms of their own neighbors. The characterization of v and those in its context (neighborhood) are to be compared to compute the local outlier factor of v](https://www.researchgate.net/publication/371162668/figure/fig2/AS:11431281202357818@1698809761377/The-k-neighborhood-of-a-vector-LOF-identifies-the-k-nearest-neighbors-for-each-vector_Q320.jpg)
![Precision, recall and accuracy of anomaly detection for the dataset of points defining an annulus in the probability simplex Δd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{d}$$\end{document}, for d=3,5,10,20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d = 3, 5, 10, 20$$\end{document}](https://www.researchgate.net/publication/371162668/figure/fig5/AS:11431281202357821@1698809761857/Precision-recall-and-accuracy-of-anomaly-detection-for-the-dataset-of-points-defining-an_Q320.jpg)
May 2023
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2 Citations
Information Geometry
An open problem in data science is that of anomaly detection. Anomalies are instances that do not maintain a certain property that is present in the remaining observations in a dataset. Several anomaly detection algorithms exist, since the process itself is ill-posed mainly because the criteria that separates common or expected vectors from anomalies are not unique. In the most extreme case, data is not labelled and the algorithm has to identify the vectors that are anomalous, or assign a degree of anomaly to each vector. The majority of anomaly detection algorithms do not make any assumptions about the properties of the feature space in which observations are embedded, which may affect the results when those spaces present certain properties. For instance, compositional data such as normalized histograms, that can be embedded in a probability simplex, constitute a particularly relevant case. In this contribution, we address the problem of detecting anomalies in the probability simplex, relying on concepts from Information Geometry, mainly by focusing our efforts in the distance functions commonly applied in that context. We report the results of a series of experiments and conclude that when a specific distance-based anomaly detection algorithm relies on Information Geometry-related distance functions instead of the Euclidean distance, the performance is significantly improved.