This paper presents two approaches to quantifying and visualizing variation in datasets of trees. The first approach localizes subtrees in which significant population differences are found through hypothesis testing and sparse classifiers on subtree features. The second approach visualizes the global metric structure of datasets through low-distortion embedding into hyperbolic planes in the style of multidimensional scaling. A case study is made on a dataset of airway trees in relation to Chronic Obstructive Pulmonary Disease.
* Low-distortion embeddings The standard technique for visualising population structure in high-dimensional or non-Euclidean datasets is to extract the pairwise distances between data points, and then use (MDS), which attempts to embed the points into a lower dimensional Euclidean space such that the given distances between the points are preserved. This is expressed mathematically as minimizing the sum of the differences between original and embedded pairwise distances @cite_7 . In a sequence of work @cite_41 @cite_5 @cite_31 Amenta, St. investigate visualization of sets of phylogenetic, or evolutionary, trees using multidimensional scaling. In this work, inter-tree distances are given by the Robinson-Foulds distance @cite_25 , which only measures topological differences in the trees. More recently, @cite_29 compare several non-linear versions of MDS on phylogenetic trees, and find that a metric that places less weight on large distances gives more meaningful visualizations. Chakerian and Holmes @cite_40 use MDS with the geodesic distance between trees @cite_22 . A different approach is that of @cite_38 , who visualize phylogenetic trees by projecting them onto a hypersphere; this approach does not consider branch lengths, only tree topology.
@cite_38 @cite_22 @cite_7 @cite_41 @cite_29 @cite_40 @cite_5 @cite_31 @cite_25
2020141201 2797520557 2169504875 1968399077
We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the "hyperbolic tree browser" demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space (H2).In the new approach, H-MDS maps proximity data directly into the H2. This removes the restriction to "quasihierarchical", graph-based data--a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general.We review important properties of the hyperbolic space and, in particular, the circular Poincare model of the H2. It enables effective human-computer interaction: by mouse dragging the "focus", the user can navigate in the data without loosing the context. In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce "ring"-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H2--creating a "space of movies" for interactive exploration. Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while 's recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable. We propose a novel projection based visualization method for high-dimensional datasets by combining concepts from MDS and the geometry of the hyperbolic spaces. Our approach Hyperbolic Multi-Dimensional Scaling (H-MDS) extends earlier work [7] using hyperbolic spaces for visualization of tree structures data ( "hyperbolic tree browser" ).By borrowing concepts from multi-dimensional scaling we map proximity data directly into the 2-dimensional hyperbolic space (H2). This removes the restriction to "quasihierarchical", graph-based data -- limiting previous work. Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data this type of data can be considered the most general.We used the circular Poincare model of the H2 which allows effective human-computer interaction: by moving the "focus" via mouse the user can navigate in the data without loosing the "context". In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.We present several high-dimensional mapping examples including synthetic and real world data and a successful application for unstructured text. By analyzing and integrating multiple film critiques from news:rec.art.movies.reviews and the internet movie database, each movie becomes placed within the H2. Here the idea is, that related films share more words in their reviews than unrelated. Their semantic proximity leads to a closer arrangement. The result is a kind of high-level content structured display allowing the user to explore the "space of movies". We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances d(u,v) between leaves. It is well known that these determine the tree and the d length of all edges. Here, we consider distortions @math of d such that, for all leaves u and v, it holds that @math if either d(u,v)< M + f 2 or @math , where d satisfies f ≤ d(e) ≤ g for all edges e. Given such distortions, we show how to reconstruct in polynomial time a forest T1, ... ,Tα such that the true tree T may be obtained from that forest by adding α-1 edges and α-1 ≤ 2-Ω(M g) n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight.